Descriptive Theories of Bargaining: An Experimental Analysis of Two- and Three-Person Characteristic Function Bargaining

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continuation on peg- 169

Lectu re Notes in Economics and Mathematical Systems

Managing Editors: M. Beckmann and W. Krelle

341

Gerald R. Uhlich

Descriptive Theories of Bargaining An Experimental Analysis of Two- and Three-Person Characteristic Function Bargaining

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

editorial Board

H.Albach M.Beckmann (Managing Editor) p. Ohrymes G. Fandel G. Feichtinger J. Green W. HildenbrandW. Krelle (Managing Editor) H. P. Kunzi K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten

Managing Editors

Prof. Dr. M. Beckmann Brown University Providence, RI 02912, USA

Prof. Dr. W. Krelle Institut fur Gesellschafts- und Wirtschaftswissenschaften der Universitat Bonn Adenauerallee 24-42, 0-5300 Bonn, FRG

Author

Gerald R. Uhlich Universitiit Bonn Institut fUr Gesellschafts- und Wirtschaftswissenschaften Adenauerallee 24-42, 0-5300 Bonn 1, FRG

ISBN-13: 978-3-540-52483-0 001: 10.1007/978-3-642-45672-5

e-ISBN-13: 978-3-642-45672-5

This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

e Springer-Verlag Berlin Heidelberg 1990

To Kerstin and my parents

ACKNOWLEDGEMENTS

The author expresses his appreciation to Reinhard Selten, who first stimulated my

interest in experimental economics, for his helpful comments and suggestions. I would like to

thank Abdolkarim Sadrieh, who did a lot of the programming for the two-person bargaining

experiments, and Anja Rosenbaum for typing parts of the manuscript. Without mentioning

names, I want to thank those colleagues, who supported me during the experimental sessions,

and all researchers who made unpublished data available to me. Sincere appreciation is also

extended to my wife Kerstin and to my parents, who enabled me to study economics.

1.

2.

2.1

2.2

3.

3.1

3.1.1

3.1.2

3.1.3

3.2

3.2.1

3.2.2

3.3

3.3.1

3.3.2

3.4

3.4.1

3.4.2

3.4.3

3.5

CONTENTS

Introduction

Notations and Definitions of Characteristic Function Games

Measurement of Predictive Success

Prominence Level

Two-Person Bargaining Games

Programs for Computer-Controlled Two-Person Bargaining Games

System Requirements

The Two-Person Bargaining Master-Program

The Two-Person Bargaining Terminal-Program

Experimental Design

Experimental Procedure

The Data Base

The Negotiation Agreement Area

Power, Justice Norms and Aspirations

A Descriptive Theory

Evaluation of Two Pilot Experiments

General Results

Comparison of Different Point-Solution Concepts

Comparison of Different Area Theories i Summary on Two-Person Games

1

4

9

11

15

15

15

16

11

18

18

19

22

23

25

3(

30

33

38

41

VIII

4. Three-Person Bargaining Games 43

4.1 Programs for Computer-Controlled Three-Person Bargaining Games 44

4.1.1 System Requirements 47

4.1.2 The Three-Person Bargaining Master-Program 48

4.1.3 The Three-Person Bargaining Terminal-Program 49

4.2 Experimental Design 52

4.2.1 Experimental Procedure 52

4.2.2 The Data Base 52

4.3 Theories of Coalition Formation 58

4.3.1 The Core. 58

4.3.2 Bargaining Set 59

4.3.3 Equal Excess Theory 62

4.3.4 Equal Division Payoff Bounds 64

4.3.5 Proportional Division Payoff Bounds 68

4.4 Experimental Results 85

4.4.1 Overall Comparisons 85

4.4.2 Games with Zero and Positive Payoffs to the One-Person Coalitions 94

4.4.3 Additional Hypotheses 98

4.4.4 The Relevance of the Core in Games with a Thick Core .107

4.4.5 Experience of Negotiators in Bargaining Games .113

4.5 Summary on Three-Person Games. .115

5. S~ary and Conclusion .119

Appendix .

A. Introduction to the Rules and the Experimental Apparatus of a

Two-Person Bargaining Experiment .

B.

c.

Introduction to the Rules and the Experimental Apparatus of a

Three-Person Bargaining Experiment

Listing of All Results

References

IX

.122

.122

.128

.139

.162

1. INTRODUCTION

The aim of this book is the presentation of two new descriptive theories for experimental

bargaining games and a comparison with other descriptive and normative theories. To obtain

data it was necessary to develop two sets of computer programs for computer controlled ex

periments. Moreover, data obtained by other researchers, which are available to us will be

included in this study.

The use of laboratory experiments in economics was introduced by THURSTONE [1931]

in the field of utility theory. CHAMBERLIN [1948] was the first person to establish an expe

rimental market for the purpose of testing a theory. The first experiment on characteristic

function games was done by KALISH, MILNOR, NASH, and NERING [1954]. Today the use

of experiments in controlled laboratory settings has become widespread. Earlier, economists

went into the field to observe phenomena as the behavior of individuals, corporations and

nations in action, then they formulated theories to explain what they saw. But unlike natural

scientists, economists have not been able to test their theories under controlled conditions.

Now experimental economists are able to replicate their results. Replication is very proble

matic for field studies, because rarely the same conditions can be established again. Moreover,

experimenters are able to test theories for situations described by simplified models which are

not observable in the real world.

Since some years it is convenient to use computers in experimental economics. There are

many advantages of using computers, such as bookkeeping, automatic data recording, check

ing subjects' behavior for procedural errors, and reduction of time necessary to run an experi

ment. More importantly, the computer insures experimental constancy in presentation across

conditions that might be very different psychologically. A human experimenter might un

consciously influence behavior towards his own hypotheses. In the light of these arguments

KAHAN and HELL WIG [1971] developed a set of comput.er controlled bargaining games

2 1. Introduction

written for the PDP-8 computer. In 1985 SELTEN founded the Bonn Laboratory of Experi

mental Economics, equipped with 17 personal computers connected through a local area net

work. All computers are placed in separate cubicles in order to secure anonymity of the sub

jects. With this environment a number of different experiments have been done up to now.

The present book will report our research on experimental bargaining games in characte

ristic function form. Our main interest is in three-person bargaining. The reevaluation of

data obtained by different researchers and the evaluation of our own data lead to the develop

ment of a new descriptive theory for experimental three-person games in characteristic func

tion form called the PROPORTIONAL DIVISION PAYOFF BOUNDS. It will be shown

that this theory is significantly more successful in the prediction of the results than other

descriptive and normative theories.

Some of the theoretical questions posed by the body of our data already arise in two-per

son games, but the literature on two-person games does not offer a well supported descriptive

theory. Therefore it was necessary to conduct an experiment on two-person games, which has

lead to the formulation of a new descriptive theory for experimental two-person games in

characteristic function form called the NEGOTIATION AGREEMENT AREA. This theory

seems to be more successful in prediction than other theories, even if more replicatjons would

be required for a stronger statistical analysis. Nevertheless, the results are suggestive. The

two-person experiments improve our understanding of subjects' behavior in three-person

games.

The structure of the book is as follows: after introducing some notations and definitions

we shall present our measures of predictive success for the comparisons of different theories.

The last section of chapter 2 is concerned with the fact that subjects prefer round numbers in

their decisions, therefore a method to calculate prominence levels in data sets will be

described. Chapter 3 reports our pilot study on two-person games. First a detailed descrip

tion of the experimental setup and the computer programs will be given, then we shall intro-

3

duce the NEGOTIATION AGREEMENT AREA, which will be compared with other point

and area solution concepts. The next chapter is concerned with three-person bargaining

games. Section 4.1 describes the hard- and software which was used to obtain our data. After

a detailed description of our experimental setup, a short description of each data set in our

data base will be given. Our new descriptive theory, the PROPORTIONAL DIVISION PAY

OFF BOUNDS will be introduced in section 4.3. Moreover, other descriptive and normative

theories, to be compared with our theory will be discussed. Section 4.4 is concerned with the

evaluation and reevaluation of 3088 plays of different games contained in 49 data sets, which

partially are played under different conditions. We start with comparisons of the different

theories over all data sets and continue with a separate analysis of games with and without

positive payoffs to the one-person coalitions. Section 4.4.4 is restricted to games with a non

empty core in order to get an impression of the relevance of the CORE-eoncept. The last

section analyzes the effects on the results of games played by experienced and unexperienced

subiects. Annendix A. and B. contains a translation of the instructions for two- and three-

2. NOTATIONS AND DEFINITIONS OF CHARACTERISTIC FUNCTION GAMES

Our particular interest is in cooperative n-person games ir. characteristic function form

with sidepayments. Before turning to the theories of coalition formation, we need to define the

n-person games these theories address.

An autonomous decision making unit with a unitary interest motivating its decisions is

called a player. A cooperative game describes the possibilities for binding agreements among

players, without a detailed specification of strategic possibilities. In our context binding

agreements are formed by coalitions. The set of players in a game will be denoted by

N = {I, ... ,n}. A coalition is a non-empty subset C of the set of all players N. To form a coali

tion C it is required that an agreement takes place involving approval by every player in C.

This agreement binds the players to each other and reconstitutes the separate individuals as a

coordinated whole. Agreements between any member of C and any member not in C then are

not any more permitted.

A simplified notation will be used for a specific coalition : coalition i stands for coalition

{i} and ij for {i,j}. The grand coalition of all n players will be referred as coalition N. The

number of elements in the set C is denoted by 1 C I. Coalitions with 1 C 1 = I are called one

person coalitions and those with 1 C 1 > I are referred to as genuine coalitions.

A characteristic function v assigns a real number v(C) to every element C of a set P of

non-empty subsets of the player set N. P contains at least all one-element subsets of N. The

non-empty subsets of N are called coalitions, and those in P are called permissible coalitions.

The set of all permissible genuine coalitions is denoted by Q. In the case of a three-person

game Q may contain the coalitions {I23}, {I2}, {I3}, and {23}.

Now a characteristic function game can be described by a triple r = (N,Q,v) where N is

the player set, Q is the set of permissible genuine coalition}! tnd v is a characteristic function

5

defined on the set P of permissible coalitions, which contains the elements of Q and all one

person coalitions.

A coalition structure of a game describes how the players divide themselves into mutu

ally exclusive coalitions. Any proposed or actual partition of the players can be described by

a set

(2.1)

of the genuine coalitions that formed. The set It is a partition of a subset of N, where for all

elements Cj E It :

Cj E Q, j= 1, .. ,m

for all it j, i,j = 1, ... ,m (2.2)

The players not in one of the Cj form one-person coalitions.

In the end-state of the game each player receives a payoff. It will be assumed, that the

payoffs are in money. Many normative theories like the BARGAINING SET depend only on

ordinal utility comparisons. As far as these theories are concerned, money can be identified

with utility in characteristic function games with sidepayments (see AUMANN [19671, KA

HAN and RAPOPORT [19841, pp. 21-23). Behavioral theories are formulated directly in

terms of money. The issue of utility measurement does not arise there either. The payoff of

player i will be denoted by Xi. The vector

(2.3)

is the collection of payoffs to all players.

The end result of a play is described by a conftguraton:

(2.4)

which shows the coalition structure and the payoff vector reached by the players. In a confi-

6 2. Notations and Definitions

guration the payoffs Xi are subject to the following restrictions:

Xi = v(i) if i t Cj , for j = 1, ... ,m

Xi ~ v(i)

LXi = v(Cj), ieCj

if i e Cj , for j = 1, ... ,m

for j = 1, ... ,m

A characteristic function v is called admissible, if we have

v(C)~ L v(i) ieC

(2.5)

(2.6)

for every permissible genuine coalition CeQ. A genuine coalition CeQ is called profitable, if

v(C» L v(i) ieC

and is called attractive, if

v(C) > V(Ci) + v(Cj) for all Cj, Cj C C such that

Cj n Cj = 0 and Cj U Cj = C.

(2.7)

(2.8)

There are several implicit assumptions incorporated in the definition of characteristic

functions (KAHAN and RAPOPORT [1984], pp. 26-27):

a) The value of any coalition is in money, and the players prefer more money to less.

b) A coalition C forms by making a binding agreement on the way its value v(C) is to

be distributed among its members. Any distribution of v(C) is permitted, given the

unanimous consent of all members of C.

c) The amount v(C) does not in any way depend on how the set N-C might partition

itself into coalitions. It is not possible to give parts of the amount v(C) to a member

of N-C, and no member of C can receive payment from N-C, within the episode

defined as the present game.

d) The characteristic function v is known to all players. ,Any agreement concerning the

7

formation of a coalition and the disbursement of its value is known to all n players as

soon as it is made. The termination of negotiations with respect to a proposed agree

ment is also publicly known.

e) Only the characteristic function influences player affinities for each other.

For experimental purposes some of these implicit assumptions have to be discussed in more

detail.

Most experimenters do not allow an infinitely fine division of v(C). There is a smallest

money unit 1 that cannot be subdivided, hence the range of possible outcomes is not a conti

nuum, but rather a finite set of configurations. A pair (r,1), where r = (N,Q,v) is a charac

teristic function game and 1 > 0 is a smallest money unit, is called a grid game if the follow

ing condition is satisfied for r [SELTEN 1987]. The values v(C) of all permissible coalitions

are integer multiples of 1- A grid configuration a = (xt, ... ,xn; Ct, ... ,Cm) for a grid game is a

configuration for r with the property that the payoffs Xt, ... ,Xn are integer multiples of the

smallest money unit 1.

Assumption e) is very problematic, because in real game situations there may exist many

additional influences on the players affinities for each other. This will be discussed in detail in

section 4.1.

In other definitions of characteristic functions, conceptually every non-ilmpty coalition

can form. Characteristic functions then may be constructed so as to make the formation of

certain coalitions so unrewarding to their members as to be practically infeasible. Unless

otherwise specified such coalitions in our context are permissible.

Sometimes a property of characteristic function games called superadditivity, which is

defined as

V(Ci U Cj) ~ V(Ci) + v(Cj)

such that Ci n Cj = 0,

for all Ci, Cj ~ N

i, j = 1, ... ,m (2.9)


will be used. While this property was part of the original definition of a game in characteris

tic function form [v. NEUMANN and MORGENSTERN 1947] and is used by some authors

without comment, we shall mention this property only if required.

For experimental purposes only games with at least one profitable coalition in Q are of

interest, such games will be called essential.

Two games r = (N,Q,v) and r' = (N,Q,v') are called strategicaUy equivalent, if there

exists a A > 0 and a vector a = (al, ... ,an) such that

v'(C) = A v(C) + Lai

ieC for all C £ N (2.10)

For every game r = (N,Q,v) with v(N) > 0 we define a one-normalized game r 1 = (N,Q,Vl)

with

Vl(C) = ~f~l for every permissible coalition C and a zero-normalized game ro = (N,Q,vo) with

vo(C) = v(C) - L v(i) ieC

(2.11)

(2.12)

for every permissible coalition C. A game r = (N,Q,v) is called zero-normalized if v(i) = 0

holds for i = 1, ... ,n.

Cooperative solution theories are usually based on the implicit or explicit assumption

that the behavior of players is invariant with respect to strategic equivalence. Already the

first experiments on characteristic function games by KALISH, MILNOR, NASH and NE

RING [1954] supplied evidence against this hypothesis. Therefore it is not sufficient to per

form experiments on zero-normalized or one-normalized games in order to obtain data for

the comparison of descriptive theories. Especially it seems to matter whether a game is zero

normalized or not.

2.1 Measurement of Predictive Success 9

When talking about zero-normalized games, the following notation will be used for

three-person games :

g = v(123)

a = v(12)

b=v(13)

c = v(23)

2.1 MEASUREMENT OF PREDICTIVE SUCCESS

(2.13)

For the comparison of different theories two measures will be applied: one for point pre

diction theories and one for area theories.

The first measure, used for the comparison of area theories, called success measure was

introduced by SELTEN and KRISCHKER [1983]. If the success of area theories has to be

compared, it is not sufficient to check which theory yields more correct predictions, since the

predicted area of one theory may be larger than the other. Therefore the size of the predicted

area has to be taken into account. In his recent paper SELTEN [1989] introduced an axioma

tic characterization of the success measure, which will not be discussed here.

Let K be the number of possible coalition structures for a given characteristic function,

and let N(Ct, ... ,Cm) be the number of configurations within the coalition structure (C1, ... ,Cm),

then for every grid configuration a = (xl, ... ,Xni Ct, ... ,Cm) the weight A( a) of a is defined as

1 A(a)= K.N(C1, ... ,Cm) •

Let Z be a set of predicted configurations, then the area A(Z) is defined as follows:

A(Z)= L A(a) aeZ

(2.14)

(2.15)


A(Z) is the measure of the size of the predicted range of outcomes relative to the size of all

possible outcomes. The measures gives equal weight to all possible coalition structures.

Suppose a data set consists of k plays of different games v and s is the number of correct

predictions by a specific theory, then the hit rate is defined as

s R =][. (2.16)

Let Aj{Zj) be the size of the predicted range of play j of any game v in the data set, then the

average area is defined as

k

A = ~ L,Aj(Zj) j=1

The success measure is the difference between the hit rate and the average area:

M=R-A

(2.17)

(2.18)

Since R is a number between 0 and 1, and A is in the same range if the predicted size is lower

or equal to the size of the set of all possible configurations, then the success measure M has a

range between -1 and 1. This is always satisfied within this framework.

The second measure, used for point solution concepts, was introduced by RAPOPORT

and KAHAN [1976] as the mean absolute deviation score which will be used in a normalized

form, in order to permit aggregation of data obtained with different characteristic functions.

While the success measure will be used for two- and three-person games, the mean absolute

deviation score will be used only for two-person games.

Let k be the number of plays of the data set. Different plays may be plays of different

games. Let fi be the theoretical payoff of player i in play j, and let xji be the actual payoff of

player i in play j. The mean absolute deviation score D for the data set is defined as follows:

with dj = I xjl-fll + ~~-.d I 2(v j (12)-v j (1 )-v j (2))

(2.19)

2.2 Prominence Level 11

If actual payoffs and solution payoffs are individually rational, i.e. if xji ~ vj (i) and

cd ~ veil holds for i=I,2 then this measure is between 0 and 1. In most cases this assumption

is satisfied.

While a theory with a mean absolute deviation score lower than the score of another

solution concept is more successful, the reverse is true for the success measure.

2.2 PROMINENCE LEVEL

The phenomenon that subjects prefer "round" numbers is known to every researcher in

experimental economics. The idea of prominence was first introduced by SCHELLING [1960].

Investigation of prominence in the decimal system suggest that numbers are perceived as

"round", if they are divisible without remainder by a prominence level .:l, which depends on

the context [ALBERS and ALBERS 1983, TIETZ 1984, SELTEN 1987].

A prominence level .:l must be an integer multiple of the smallest money unit 1 of the

form .:l=p.1O'fl1 with p.=1, 2, 2.5, 5 and 1]=0, 1, 2, .... The method used in this book for the

determination of the prominence level of a data set was developed by SELTEN [1987].

A prominence level is assigned to every integer multiple n1 of 1. The prominence level of

n1 is the greatest prominence level 6. such that n1 is divisible by 6. without remainder.

In a data set one must distinguish between the number of observations and the number of

values at a prominence level 6.. In the number of observations every observation of a number

with the prominence level 6. is counted as often as it occurs in the data set, but in the num

ber of values an observed number which occurs several times in the data set is counted only

once.


Table 2.1 shows an example for the computation of the prominence level of a data set.

The greatest value in the data of our two-person experiment in session 2 is 320. Therefore we

get a list of possible prominence levels beginning with 250 up to 1, which is shown in the first

column. The smallest money unit 'Y is 1. The next column m(d) shows the number of

values. For example for d = 100 we find the values 100 and 300, therefore m(IOO) = 2. We

observe 100 five times and 300 two times, which gives us the numbers of observations on the

prominence level 100, hence h(100) = 7. Such computations have to be done for all

prominence levels. Now the cumulative number of values M(d) and observations H(d) have

to be computed beginning with the highest d. Let M and H be the sum of values and

observations, respectively, then the cumulative distributions M(d)/M and H(d)/H can be

* calculated. The prominence level d is defined as the greatest maximizer of the surplus

* D(d)=H(d)/H-M(d)/M. In our example d is 5. A binomial test proposed by SELTEN

* [1987 pp. 89-91] yields a significance level of 0.001175, so we can trust d = 5. If the test is

* not significant on a certain level, one should not trust d . In the present study this

sometimes occurs in data sets with a small number of observations. In the light of

comparisons with other data involving coalition values of similar size, we found that in these

cases it is appropriate to use the next lower level, but this may not be true in general.

Wherever it is possible one should use not only the final results, but the complete negotiation

protocols to determine the prominence level. However, not all values can be used because we

only have I C 1-1 degrees of freedom in a proposed coalition. Therefore one payoff should be

left out in every proposal. The computations reported for two-person games are based on

values a proposer demands for himself. The computations for the three-person games are

based on the rule, that the smallest payoff in a proposal is left out or if there are two or more

smallest payoffs, one of the smallest ones is left out.

Tab

le 2

.1:

Det

erm

inat

ion

of th

e pr

omin

ence

leve

l in

the

tw

o-pe

rson

gam

es o

f se

ssio

n 2

Cum

ulat

ive

Cm

ulat

ive

Num

ber

of

num

ber

of

dist

ribu

tion

s

Prom

inen

ce

Val

ues

Obs

erva

tions

Val

ues

Obs

erva

tions

I(

Ll)

H

(Ll)

Surp

lus

leve

l Ll

m

(Ll)

h(L

l)

I(L

l)

H(L

l) I

H

D(L

l)

250.

00

1 12

1

12

0.00

60

0.02

29

0.01

70

200.

00

1 11

2

23

0.01

20

0.04

40

0.03

20

100.

00

2 7

4 30

0.

0240

0.

0574

0.

0334

50

.00

2 14

6

44

0.03

59

0.08

41

0.04

82

25.0

0 5

16

11

60

0.06

59

0.11

47

0.04

89

20.0

0 11

71

22

13

1 0.

1317

0.

2505

0.

1187

10

.00

9 47

31

17

8 0.

1856

0.

3403

0.

1547

5.

00

19

75

50

253

0.29

94

0.48

37

0.18

43

2.00

61

15

3 11

1 40

6 0.

6647

0.

7763

0.

1116

1.

00

56

117

167

523

1.00

00

1.00

00

0.00

00

"" ~ ""d

.... o ~. m

~

t"' [ ......

<:,.

)


The computation of a prominence level does not make any sense if the games played are

very different, because we cannot expect to get a unique prominence level if we have some

games with high coalition values and other games with small coalition values in the same

data set. Therefore prominence levels should not be determined for data sets with great diffe

rences between individual games with respect to the range of coalition values. However, it is

not clear to what extent such differences can be tolerated. The data sets included in this

study are quite homogeneous with respect to the range of coalition values.

3. TWO-PERSON BARGAINING GAMES

There are many theories of two-person bargaining games, but most of them do not seem

to have much relevance for the explanation of laboratory experiments. This may be due to

the fact, that most of the theories are normative rather than descriptive. Thousands of plays

of different three-person games in characteristic function form have been evaluated at the

Bonn Laboratory of Experimental Economics. Some of the theoretical questions posed by this

body of data already arise in two-person games. The experimental literature on two-person

games does not offer a well supported descriptive theory.

Since no experimental data were available, it was necessary to conduct an experiment.

The aim of this chapter is to compose reasonable theories which are based on equity and

parity norms. Moreover, the present chapter is concerned with the experimental testing of a

new descriptive approach to the bargaining problem.

3.1 PROGRAMS FOR COMPUTER-CONTROLLED TWO-PERSON BARGAINING

GAMES

"NEGOTIATIONS 2" is a set of programs designed for two-person games in characte

ristic function form. To run the two-person game two different programs are necessary: The

Master-Program and the Terminal-Program.

3.1.1 SYSTEM REQUIREMENTS

In the maximal configuration 16 subjects can participate simultaneously . All in all 17

IBM PC, XT or AT computers (or close compatibles) running DOS 3.1 or a later version are

16 3. Two-Person Bargaining Games

required. One PC is needed to run the Master-Program and one PC is needed for every sub

ject to run the Terminal-Program, thus in the minimal configuration for two participants

three Personal Computers are required. The programs support all known Graphic Adapters

including VGA. No commands specific to a Network Program are used, thus the set of pro

grams may run in all Network environments which allow to share a hard-disk or RAM-disk.

For use with the IBM PC Network Programs one PC must have 640 KB of memory and one

disk drive to run the Master-Program. A hard disk is desirable. To run the

Terminal-Program 256 KB of memory and one disk drive is required.

3.1.2 THE TWO-PERSON BARGAINING MASTER-PROGRAM

The main task of this program is to control the whole experiment. All data will be re

corded and stored to disk. The experimenter can observe every negotiation step on the screen

of the PC running the Master-Program.

Further the program supports an interactive development of the experimental design.

First the program asks for the games to be played, then the combination of the players has to

be set and wh.jch game in the data base they shall play. Moreover, it has to be decided which

player moves first.

Finally the experimenter is asked for a point to cash rate. The complete setup will be

stored in a file. At the beginning of a session the experimenter is asked for the name of the

setup file. Now he turns into a spectator until it becomes necessary to change the setup or the

session ends. If a printer is connected, then a list with the money payoffs for the subjects is

printed.

3.1 Programs for Computer-Controlled Two-Person Bargaining Games 17

3.1.3 THE TWO-PERSON BARGAINING TERMINAL-PROGRAM

This is the computer-program, running on all terminals, which the subjects use to trans

mit proposals to other subjects. Communication is restricted to a formalized interaction. The

players act in alternate order. A player in the decision mode has four options:

1) The player can propose a non-negative integer valued allocation of the coalition

value v(12), independent of any proposals made before.

2) The player can shift the initiative to the other player.

3) The player can accept an outstanding proposal. A proposal is called outstanding if it

has been made by the other player in his last decision mode.

4) The player can abort the negotiations.

There are no time restrictions or restrictions on the number of proposals. The game only

ends, if one player aborts the negotiations or a proposal is accepted. Figure 1 shows a

hardcopy from the display of the Terminal-Program. Due to the fact that the experiments

were conducted in Germany all instructions to the subjects are in German language.

TWO PERSON BARGAINING GAME - TERMINAL PROGRAM •

r-- AKTUELLER VORSCHLAG - -AUSZAHWNGEN- r-:~UST~=-1 vorrh1aq Ich Er/Sie ICH ER/SIE Entscheiden Nr. von erhalte erhilt

32 128 I SPIELER 11 3 ihm(r) 60 260 320 :

VERHAHDWNGSGESCHICHTE KOALITION I SPIEL NR : 11

ihm(r) 280 IIEINGEBEN IIABBRECHENI

1 40 I ANNEIIMEN II SCHIEBEN 2 mir 160 160

3 ih1D(r) 60 260

PqUp/PqDn : Blattern in Home/End : Anfanq/Ende t / l : Auf/Job in

} der Geschichte

"/~ : Links/Rechts Bewequnq <~ : Akzeptieren ESC : Zuriick oal : tOschen einer Ziffer Leertaste : tOschen der qanzen Zahl

Figure 3.1: Hardcopy from the display of the Terminal Program


3.2 EXPERIMENTAL DESIGN

The experiments reported below were designed as pilot studies. Game theorists have

developed many theories, which in the case of two-person games in characteristic function

form either do not restrict the imputation space or are equal to Schelling's "Split the

difference" [1960]. These solution concepts, in the following called equal surplus solution, are

good predictors, if the game is symmetric. However, the games for this study have been

selected in such a way, that there is a large variation in the "threat-point".

3.2.1 EXPERIMENTAL PROCEDURE

The subjects were 24 male and female undergraduate students of economics and law, who

never participated in two-person bargaining games before. The experiment was conducted in

two sessions at the Bonn Laboratory of Experimental Economics. The bargaining procedure

was computer controlled by the software package described above.

All participants were introduced to the experimental apparatus and the rules of the

games in a 30 minutes session immediately before the experiment started.

In each session two groups of 6 subjects played 5 different games in different dyads. The

games were played for money converted to cash at a fixed rate. The point to cash rate was

1:0.05 Deutsche Mark in session 1 and 1:0.025 Deutsche Mark in session 2. All dyads in one

subject group played the same games but in a randomized order. Therefore 60 results of 20

different games with four independent subject groups were obtained.


3.2.2 THE DATA BASE

The 10 games of session 1 were selected such that the relative distance between the one-per

son payoffs (v(1)-v(2»/v(12) varies from 0.0 to 0.9 (see table 3.1). If one looks at the

one-normalization of the games, it can be seen, that the equal surplus solution point is varied

systematically (see figure 3.2).

In session 2 the 10 games were selected such that the "threat-point" of the one

normalized games varies systematically in the range for superadditive games (see table 3.2

and figure 3.3). One play will be excluded from the analysis, because a subject declared that

he has made a clerical error in typing his demand (12 instead of 129).

Table 3.1: Games and Results of Session 1

lesults

No v(12) v{l) v(2) v(1)-v(2) 1 2 1 2 1 2 v(12)

1 160 40 40 0.0 80 80 80 80 80 80 2 200 40 20 0.1 133 67 113 67 109 81 3 180 81 45 0.2 105 75 95 85 125 55 4 180 74 20 0.3 158 22 123 57 74 20 5 160 84 20 0.4 105 55 112 48 102 58 6 170 115 30 0.5 120 50 129 41 128 42 7 170 122 20 0.6 137 33 136 34 122 20 8 150 110 5 0.7 110 5 117 33 105 45 9 120 96 0 0.8 101 19 96 0 107 13 10 110 99 0 0.9 104 6 102 8 109 1


Player 1

I-____________ ~_ Player 2

Figure 3.2: Graphical representation of the 10 'one-normalized' games contained in session 1. The intersection with the Pareto-Efficient line shows the equal surplus solution point.

Table 3.2: Games and Results of Session 2

lesults

No v(12) v(l) v(2) v(1)-v(2) 1 2 1 2 1 2 v(12)

1 320 64 32 0.1 170 150 190 130 160 160 2 320 96 64 0.1 200 120 187 133 208 112 3 320 128 96 0.1 128 96 182 138 179 141 4 320 160 128 0.1 175 145 170 150 182 138 5 320 128 32 0.3 172 148 error in data 210 110 6 320 160 64 0.3 185 135 160 64 207 113 7 320 192 96 0.3 207 113 209 111 192 96 8 320 192 32 0.5 192 32 270 50 250 70 9 320 224 64 0.5 224 64 250 70 250 70 10 320 256 32 0.7 272 48 256 32 275 45


Player 1

1--____________ -->0._ Player 2

Figure 3.3: Graphical representation of the 10 'one-normalized' games contained in session 2. The intersection with the Pareto-Efficient line shows the equal surplus solution point.


3.3 THE NEGOTIATION AGREEMENT AREA

The aim of this chapter is to introduce a new descriptive theory for two-person games in

characteristic function form. SELTEN [1983, 1987] has introduced the theory of EQUAL

DIVISION PAYOFF BOUNDS. This theory has been conceived for thr~person games with

zero-payoffs for the one-person coalitions only. It is not clear how the EQUAL DIVISION

PAYOFF BOUNDS should be generalized to thr~person games with non-zero payoffs to

one-person coalitions. One way of doing this is the computation of bounds for the strategi

cally equivalent zero-normalized game with a subsequent retransformation of the bounds.

Whenever we talk of EQUAL DIVISION PAYOFF BOUNDS for thr~person games with

non-zero payoffs to one-person coalitions we refer to this generalization based on the prin

ciple of strategic equivalence. However, it should be emphasized that this generalization is

not really adequate, since the behavior of the subjects is not invariant with respect to strate

gic equivalence. Nevertheless it has been shown by SELTEN and UHLICH [1988] that the

generalization based on strategic equivalence is more successful in prediction than various

versions of the BARGAINING SET [AUMANN and MASCHLER 1964; MASCHLER 1963,

1978]. Therefore the generalized version of the EQUAL DIVISION PAYOFF BOUNDS con

sidered here is not without interest, even if it cannot be considered to be an adequate genera

lization of the concept.

In the case of games with zero-payoffs to one-person coalitions the predictive power of

the EQUAL DIVISION PAYOFF BOUNDS can be improved by the order of strength hypo

thesis which maintains that within a coalition a stronger player does not receive less than a

weaker one [SELTEN 1972]. This is not true for games with non-zero payoffs for one-person

coalitions.

Since in games with non-zero payoffs for one-person coalitiOns, subjects do not behave

in the same way as in the stra.tegically equivalent zero-l!otmalized game, one has to look

3.3 The Negotiation Agreement Area 23

again at the basic assumptions of the theory of EQUAL DMSION PAYOFF BOUNDS.

Especially the "order of strength hypothesis" needs to be modified. There is only one plau

sible way to define an order of strength for zero-normalized games, but for games with

non-flero payoffs for one-person coalitions it is not obvious how an order of strength should

be defined.

H one looks at the negotiation protocols of three-person characteristic function games, it

can be seen that there are sequences of bilateral negotiations. Therefore a theory for three

person games has to include a concept for two-person negotiations.

3.3.1 POWER, JUSTICE NORMS AND ASPIRATIONS

There is some evidence to suggest that any theory that ignores either power or justice

norms is not likely to be very accurate [KOMORITA 1984]. Moreover, because there is

considerable evidence that power as well as justice norms have strong influence on a

bargainer's share of the reward, the theory should take account of both. Of course, there are

some additional influences on the result of a negotiation such as personal skills and

aspirations. The effect of personal skills has to be neglected because it is very problematic to

translate them into parametric values. This does no harm in the framework of this chapter,

because only games with non-verbal communication will be analyzed.

The first question to answer is, where does power come from. An answer is given by

FISHER and URY [1981]: "The better your BATNA (Best Alternative to a Negotiated

Agreement), the greater your power. People think of negotiating power as being determined

by resources like wealth, political connections, physical strength, friends, and military might.

In fact, the relative power of two parties depends primarily upon how attractive to each is the

option of not reaching agreement." Therefore in terms of the t;wb-person characteristic func

tion game player 1 is stronger than player 2, if and only if v(1»v(2), and player 1 and 2 are

24 3. TW<rPerson Bargaining Games

equally strong, if v(I)=v(2). Without loss of generality it will be assumed that we have

v(I)~v(2). Moreover, it will be assumed that v(12), v(I), v(2) ~ O.

While the question of power is easy to answer in the two-person case it becomes more

difficult to think about justice norms, because normally there is more than one justice norm,

which also may be called a fair solution. If two players have to distribute 100 units of money,

player l's alternative is to receive 40 units and player 2's alternative are 10 units, then player

1 may think a fair solution will be to divide the surplus (100 - 40 -10) equally in addition to

the alternative value, while player 2 proposes to divide the value of 100 equally. Both

distribution schemes can be derived by justice norms, and an astute negotiator will select

that principal of fairness that favors his side. Before answering the question how this non

unique concept of justice norms influences the outcome, the effects of aspiration levels have to

be discussed.

Unfortunately, with one exception aspiration levels cannot be observed directly. Only

the first proposals of both players in the negotiation can serve as aspiration levels and will be

called revealed aspirations A~ev. It is plausible that these revealed aspirations have an

influence on the outcome, because it seems to be impossible to raise a demand in a later

stage. Therefore the first demand has to be high, but not too high, because of the risk of a

break-<>ff of the negotiations. The maximal aspiration will be defined as the contribution of

player i, if he joins the coalition:

A~X = v(12)-v(j), i=I,2 and i;j. (3.1)

If player j is individuaUy rational, which means that he will join a coalition, if his payoff Xj is

greater than or equal to v(j), then i cannot receive more than v(12}-v(j). It is clear that the

first proposal will not be lower than an expected attainable aspiration level. The attainable

aspiration level A~tt will depend on justice norms, but for. the moment it is not yet defined.


3.3.2 A DESCRIPTIVE THEORY

In the sections above it has been explained that power, justice norms and aspiration

levels influence the outcome of negotiations. The descriptive theory introduced here proceeds

from the assumption that power and justice norms influence the aspiration levels and that

only the aspiration levels influence the outcome directly. Two questions need to be discussed:

1) How do aspiration levels depend on power and justice norms? 2) How do aspiration levels

influence the outcome of negotiations?

We shall distinguish four kinds of aspiration levels: maximal, minimal, attainable, and

revealed aspiration levels. The upper bounds for the aspiration levels of both players are the

maximal aspirations A ~ax as defined in the section above. These bounds depend on the power

of the players, therefore one has A ~x~A ~x , if v(1 )~v(2). The minimal aspiration level A ~in

is given by the assumption of individual rationality (A~in = v(i), i=1,2). At last the

attainable aspiration levels depend on justice norms, but because these are not unique the

worst case for each player will be assumed:

A~tt = max [v(1), vQ2)] and

A~tt = v(2) + (V(12)-;(1)-V(2))

In the case v(1) = v(2) both right hand sides are equal to v(12)/2.

(3.2)

It is plausible that the revealed aspiration level will be somewhere between the attain

able aspiration level and the maximal aspiration level. In the following the question will be

examined how the final agreement depends on revealed aspiration levels. Mostly, there will be

a conflict and the negotiators have to reduce their aspirations to achieve an agreement, but

no reliable theory on the dependencies of the concessions made by negotiators is known to the


author. In view of the fact that many field studies show that opponent's behavior does

influence the negotiator I and the finding of BARTOS, TIETZ and MCLEAN [1983] that those

who made a large first demand tended to make a large total concession, it will be assumed

that the relative concession of a negotiator is equal to the relative concession of the opponent.

If the process runs until the demands are compatible then the solution point Srev =(Xt,X2) is

given by

A rev Xl = v(12)· rev I rev and

Al +A2

A rev X2 = v(12). re v 2 rev (3.3)

Al +A2

This point solution concept predicts that the negotiators distribute the coalition value in

proportion to their revealed aspirations (figure 3.4).

Player 1

X,

(v(1},v(2))

/

/ /

I

/

/

/

v(12}

'-------=X2:--------:>....-- Player 2

Figure 3.4: Graphical representation of the solution point Srev

I See, for example, the HOPMANN /SMITH [1978] analysis of the Soviet-American test ban negotiations.


Though this concept will be used, it will not be expected that the underlying process is a

good description of observable data. The process must be thought of as an approximation,

because in laboratory experimentations there are tendencies of the subjects to propose round

numbers and payoffs are not infinitely divisible, since there is a smallest money unit which

cannot be further subdivided. Moreover there may exist sociological or psychological

differences between different subjects which lead to short-term deviations from the described

process rule but it will be assumed that on the average the negotiators behave according to

this rule.

For given revealed aspiration levels (3.3) is a point theory for the negotiation outcome.

However, it is the intention of this section to develop an area theory. Area theories have

important advantages over point theories. "An area theory is one that predicts a range of

outcomes. Other kinds of theories predict only average outcomes or are even less specific. The

advantage of area theories is that for every single play of the game one can check whether the

prediction was correct or false. This is a great heuristic advantage if one wants to improve

theories in the light of data. In every case in which a prediction fails one can ask oneself what

went wrong" [SELTEN 1987, p.43]. This is the reason for our aim to find a solution area

determined through lower bounds for the predicted payoffs.

If negotiators distribute the coalition value v(12) proportional to the revealed aspira

tions, then a lower bound Xl for the strong player 1 (v(l) > v(2» can be deduced from (3.3),

if the maximal aspirations become revealed aspirations:

A~ax Xl = v(12) • ---;;:;;':;:-"-----;;;=

A max Amax 1 + 2

for v(l) > v(2). (3.4)

Player 2 may think that a high demand is very risky because player l's BATNA is very

attractive and a break-{)ff of the negotiations must be taken into account. In order to avoid a

break-{)ff he could try to propose a fair solution and the attainable aspiration level A~tt be

comes the revealed aspiration level. A similar argument does' not hold for the strong player;

28 3. TW<rPerson Bargaining Games

"equality and justice are always sought by the weaker party, but those who have the upper

hand pay no attention to them" [ARISTOTLE, Pol. 6, I, 14, 1318 b5]. Following this a lower

bound for player 2 can be deduced from (3.3):

Aatt X2 = v(12)· max 2 att , for vel) > v(2).

Al + A2 (3.5)

H both players are equally strong (v(l) =v(2», then the obtained area, which will be

called NEGOTIATION AGREEMENT AREA for two-person games (NAA), shrinks to a

single point:

A max XI = v(12)· max I max and

Al + A2

A max X2 = v(12)· max 2 max , for vel) = v(2).

Al + A2 (3.6)

Only in this case the NAA equals the usual solution concepts which predict a "split the

difference" allocation, but it can be proved that the N AA always contains this solution point.

For the case of non-superadditive games it will be assumed that no coalition forms.

In order to test this theory in an experimental setup, one cannot ignore the phenomenon

that subjects prefer "round" numbers, therefore the bounds Xi are rounded to the next lower

number divisible by the prominence level 1:1. Whenever this yields an amount lower than or

equal to v(i), the final bound will be v(i)+'Y, because it is obvious that a player only joins a

coalition, if he receives at least one smallest money unit 'Y in addition to his alternative veil.

Hence, we have:

Ui = max [ v(i)+'Y, 1:1 int -¥-]. (3.7)

The symbol flint p." stands for the greatest integer not greater than p.. The NAA is the set of

all grid points (XI,x2) with xI+x2=v(12) and Xi~Ui for i=l,2.

3.3 The Negotiation Agreement Area

Player 1

v(l}

I

-+ /1

/1 . I

/NAA

i )" / / I . /

/ . x· / / . . /

/ . ./

I

- .~ /

/ I I

~.,..,-__ >--~ ___ = __ """_ Player 2

v(2}

29

Figure 3.5: Graphical representation of the NEGOTIATION AGREEMENT AREA for two-person games.


3.4 EVALUATION OF TWO PILOT EXPERIMENTS

Before going into details it must be said that the results only depend on two pilot

studies. Altogether only 59 observations from 20 different games with four independent

subject groups (one play is excluded due to an error) could be obtained. There have to be

more replications to get significant results, but some implications from this research are

worth talking about.

3.4.1 GENERAL RESULTS

Ten of the 59 plays ended with a break-{)ff of negotiation and will be excluded from

further analysis. On average a strong player received in the remaining 49 games a reward of

50.48% of the surplus v(12)-v(I)-v(2). This will be no surprise to the researchers who favor

the "split the difference" concept but only in 7 plays the subjects have divided the surplus

exactly equally. In three of these results the games played were symmetric (v(I)=v(2». If an

equal split was not possible due to the existence of the smallest money unit, the solution

point was rounded in favor of the strong player. In 22 cases a strong player earned more than

50% of the surplus and 20 cases less than 50%. If one takes a closer look at the two sessions;

it can be seen that a strong player only received a reward of 46.6% of the surplus in session 1,

while a strong player in session 2 earned 54.86% of the surplus. This difference seems to be

due to a "first move advantage". The strong player was first mover in ten of 26 plays

contained in session 1 (38.46%) and in 14 of 23 plays contained in session 2 (60.87%). Hence

on average over both sessions a first moving strong player received a share of 53.53% of the

surplus, while only 47.55% could be reached, if the weak player was the first mover. An

overview is given in table 3.3.

3.4 Evaluation of two Pilot Experiments

Table 3.3: First move advantage (Payoffs in percent of the surplus)

Strong player

First mover 53.53

Second mover 47.55

Total 50.48

31

Weak player Total

52.45 52.98

46.47 47.02

49.52 100

Moreover, a first moving player independently of his strength received 52.98% of the sur

plus. This very surprising result should be tested, but only four independent subject groups

are available and the first move advantage can be observed only in three groups, therefore a

reliable test on this small data base is impossible. Figures 3.6 and 3.7 show the distribution of

the surplus depending on the first moving player. For example, the bar in figure 6 labeled

with "50" means that 9 first moving strong players received a payoff of more than or equal

50% of the surplus but less than 60%. It can be seen that we have 23 first moving strong

players and 23 first moving weak players, but independently of the players' power 30 of the

first moving players earned more than or equal to 50% of the surplus and 16 less than 50%. If

the 46 plays are assumed to be independent from each other, then a Binomial-Test can be

used to test the null hypothesis that there is no difference in the probability for a first moving

player to earn more or less than 50% of the surplus against the alternative hypothesis that

the probability for a first moving player to earn more than 50% is greater than the

probability to get less than 50%. The null hypothesis has to be rejected (Significance 0.05).

Even if the independence assumption is dubious this suggests, that there is an advantage for

any player to have the first move. The main question to answer is, where does this come

from? There are no hints that this result depends on the structure of the games or on the

experimental setup. Further investigations are necessary. It seems to be obvious that theories

developed only on the basis of the characteristic function are very problematic. Rubinstein's

bargaining theory [1982] predicts a first move advantage depenc;ling on the discount rate, but .

only on the equilibrium path, on which the first offer is accepted. Even if our procedure does


not specify a discount rate, the theory can be applied, if one assumes that the discount rate is

part of the players' preferences. However, in the experiments the first offer was accepted only

in 4 out of 49 cases in which an agreement was reached. The average number of periods in a

play was 19.75.

• 18

• ... u e c • :J IT 6 • L

II.

• 4 _I. <J :J _1_1 .... 2 D • .D • a: 9

-20 -10 0 18 29 30 40 60 60 70 eo 90

Figure 3.6: Absolute profit frequencies of the first moving strong player in percents of the surplus. (Symmetric games are excluded)

3.4 Evaluation of two Pilot Experiments 33

• 8

• ... u C 6 • J IT • L 4

'" • 4J J 2 ... 0 • 1J <t 8 --

-211 -111 211 311 411 511 611 78 811 ell

Figure 3.7: Absolute profit frequencies of the first moving weak player in percents of the surplus. (Symmetric games are excluded)

3.4.2 COMPARISON OF DIFFERENT POINT-SOLUTION CONCEPTS

In this section different point solution concepts will be compared with the help of the

mean absolute deviation score measure (2.18) defined in section 2.1. An infinite division of the

payoffs was not allowed to the subjects, therefore if a theory predicts an impossible distribu

tion the theoretical solution is rounded in favor of the strong player. There are many inge

nious theories applicable to the type of two-person games underlying this paper, so only some

representatives can be quoted.

EQUAL SURPLUS: Many theorists propose this solution, which is nothing else than

SCHELLING's [1960] "split the difference". ZEUTHEN [1930], NASH [1950, 1953], KALAl


and SMORODINSKY [1975], RAIFFA [1953], HARSANYI [1956], and SHAPLEY [1953]

agree with this solution point for the type of games discussed within the framework of this

chapter.

ES () v(12)-v(I)-v(2) Xl = vI + 2

ES () v(12)-v(l}=v(2) X2 = v 2 + 2 (3.8)

EQUALITY NORM: Subjects will always divide the coalition value into equal parts

(3.9)

OUTSIDE OPTION PRINCIPLE: This non-cooperative approach to the analysis of

sequential bargaining is proposed by SUTTON and SHAKED [1984]. Their model proves that

the presence of an "outside option" has no effect, if it lies below the payoff the player would

receive in bargaining without the outside option, but if the outside option exceeds this payoff,

then the payoff is the value of the outside option.

SUTTON, SHAKED, and BINMORE [1985] have tested this theory in an experimental

setup. The results obtained by their study are in agreement with the predictions of

non-cooperative game theory.

Though their bargaining rules are different from those used in the experiments evaluated

here, the OUTSIDE OPTION PRINCIPLE will be included in our comparisons. Even if

SUTTON and SHAKED derived the principle as the result of a non-cooperative bargaining

theory, it also has a cooperative interpretation.

In our context the OUTSIDE OPTION PRINCIPLE does not mean anything else than

the EQUALITY NORM with the exception that individual rationality is taken into account.


OOP () OOP X2 = v 12 -Xl (3.10)

It is a remarkable result that this theory is not invariant with respect to strategic

equivalence.

PARITY NORM: GAMSON [1961) assumes that there is a norm; the parity norm which

specifies that rewards be divided in direct proportion to the resources v(i), i=I,2:

PAl. () v(l} Xl = v 12 • v(I)+v(2) (3.11)

PAl. () v(2} X2 = v 12 . v(I)+v(2)

CHERTKOFF's EXPECTANCY THEORY: CHERTKOFF [1970) proposes a theory

that assumes that each person wants to maximize his share of the reward and expects his

share in a coalition to be halfway between parity and an equal division of the prize. Two

versions of this theory will be tested. The first version uses the real EQUALITY NORM

while the second uses the OUTSIDE OPTION PRINCIPLE.

Expl PAl. + EI =~. v(l~ +~ 1) Xl Xl

Xl = 2 v(I)+v 2)

xEXPI = xPu + EI

=~. v( 2~ +~ 2 X2 (3.12) 2 2 v(I)+v 2)

{>q21. V (1/ + vil2) Ifv(l»~ PAl. + xOOP 2) xEXP2 Xl !

v(I)+v 2) ,

I 2 vQ2L v(l~ +~ if (1)<~ v(I)+v 2) , v -

r~21. */ + v(12) 1fv(1)~ xEXP2 x~u + x~op v(I)+v 2) ,

(3.13) 2 2

vQ2} • V(2~ + v(12}-v(l} if (1)<~ v(I)+v 2) 2 ' v -


NAA EXPECTATION: In order to compare these theories with the NAA an expectancy

theory similar to CHERTKOFF's theory will be developed. Therefore the proposed bounds

will be used instead of EQUALITY and PARITY NORMS.

For the case of v(1»v(2) one has:

[max max ] lU!E_1 . A1 • A1

X - ~ v(12) max max + v(12) max aU 1 A1 + A2 A1 + A2

_v(l2) [vp2)-v~2) + 2fv~12)-V~2)~ ] and -~ 2v( 2)-v( )-v(2) 3v I )-v(1 -v 2)

[max att ] lUU 1 A2 A2

X = ~ v(12). max max + v(12). max atE 2 A1 + A2 A1 + A2

_vQ2) [V~12)-VP) + V(12~-v(1l+v(2~ ] - 2v( 2)-v( )-v(2) 3v( I )-v( ) -v( ) ,

while for v(1)=v(2)

max xlU!E = v(12). A 1 = vQ2) and

1 Amax Amax 1 + 2

max xlU!E = v(12). A 2 = VQ2)

2 A max Amax 1 + 2

(3.14)

An example for the predictions of the different solution concepts is shown in figure 3.8.

The results of the laboratory experiments (see table 3.5) show that the EQUAL SUR

PLUS theory is the best predictor, while the EQUALITY NORM is the worst prediction.

3.4 Evaluation of two Pilot Experiments

Player 1

v(l)

PARITY NORM

EQUAL SURPLUS CHERTKOFF', EXPECTANCY THEORY I

NEGOTIATION AGREEMENT AREA EXPECTATION CHERTKOFF', EXPECTANCY THEORY 2

OUTSIDE OPTION PRINCIPLE EQUALITY NORM

v(12) '--____________ ""--_ Player 2

v(2)

Figure 3.8: Graphical representation of the different theories.

Table 3.5: Comparison of the different point-solution concepts. (Mean absolute deviation scores)

Theory Overall session 1 session 2

EQUAL SURPLUS 0.1397 0.1570 0.1202 NAA EXPECTATION 0.1450 0.1535 0.1354 CHERTKOFF 2 0.1604 0,1592 0.1618 PARITY NORM 0.2742 0,3547 0.1833 OUTSIDE OPTION PRINC. 0,3645 0.3363 0.3942 CHERTKOFF 1 0.4581 0.4767 0.4371 EQUALITY NORM 1.0949 1.2275 0.9449

37


Whether these results are significant or not will have to be tested after more replications are

available.

While the solution concepts discussed above only depend on the structure of the game,

the next step is to look whether it is possible to improve the prediction if more information,

namely the revealed aspirations are taken into account. Therefore it will be proposed that the

subjects divide the coalition value v(12) in proportion of the first demands if they are

credible. A demand is credible if the offer to the opponent yields more then the BATNA.

Instead of an incredible demand the maximal aspiration level will be used. Hence,

. rev max x~ev = v(12) mtn [AI , A I ]

. [Arev Amax] . [Arev Ama, mtn I , I + mln 2, 2 J and

min [A r2ev , A m2 ax] xr2ev = v(12) ----::====~-~--L_ --='::.L.......L..,_ =-==-. rev max . rev max..

mtn[A I ,AI ] + mln[A2 ,A 2 J (3.15)

This theory is more successful than the EQUAL SURPLUS solution. In the average over the

mean absolute deviation scores over both studies a value of 0.1198 is reached

(session 1: 0.1441; session 2: 0.0923). One has to conclude that this is a strong hint, that nego

tiators agree on a distribution depending on revealed aspirations. Unfortunately these

revealed aspirations cannot be observed ex ante. Therefore it seems to be reasonable not to

use point solution theories but area theories depending on boundaries deduced from possible

aspiration levels.

3-4:.3 COMPARISON OF DIFFERENT AREA THEORIES

For the comparison of different area theories the success measure (2.17) introduced by

SELTEN and KRISCHKER [1983] will be used. Though no specific area theories are known

for two-person games until now, some can be constructed from point solution concepts in


order to provide a competition for the NEGOTIATION AGREEMENT AREA. All theories

specify preliminary lower bounds Xi for the players i = 1, 2. Final lower bounds are then

computed according to (3.7) (Ui = max [v(1)+1, A int (Xi/A)] ), which takes the prominence

level into account. The theories predict that no player will receive a payoff below his final

bound Ui. Bounds for different theories will be distinguished by different upper indices. The

lower bound for one player's payoff implies an upper bound for the other player's payoff.

JUSTICE NORMS: It will be assumed that no player agrees in a coalition if his reward

is lower than the worst equity norm:

xi = max [V(l),.]

x~ = v(2) + v(12)-V~1)-V(2) (3.16)

PARITY AND EQUALITY: The basic assumption of the "bargaining theory" proposed

by KOMORITA and CHERTKOFF [1973] is that those who are "strong" will expect and

demand a share of the reward based on parity norms, while those who are "weak" will

demand equality. Therefore the players should receive at least:

X~E = max [V(l),.] and

PE () v(2) X2 = v 12 • v(I)+v(2) . (3.17)

PARITY AND EQUAL SURPLUS: The "strong" player in this theory demands as much

as in the concept above, but the weak player demands an equal surplus solution. Therefore

the lower bounds are:

X~ES = v(l) + V(12)-;(1)-V(2) and

PES () v(2) X2 = v 12 . v(I)+v(2) (3.18)


EQUAL SURPLUS: The negotiators distribute the prize nearly as in the equal surplus

concept, but in contrast to the equal surplus concept used in the chapter above, here the

prominence level will be taken into account.

x~S = v(l) + V(12)-v~1)--V(2)

x~S = v(2) + V(12)-V~1)-V(2) (3.19)

NEGOTIATION AGREEMENT AREA: This theory is based on the idea that the

strong player can always choose his maximal aspiration level as his first demand. Player 2, if

he is weaker, may choose a first demand as low as his payoff obtained by an equal split of the

surplUS. The agreement is then determined by the principle of equal relative concessions. The

lower bound for player 1 is attained, if player 2 plays tough and demands his maximal aspira-

tion too,

NU. () vP2)-V~2) Xl = v 12 • 2v( 2)-v( )-v(2) (3.20)

Player 2, if he is weaker, may think that tough playing is very risky because a break~ff of

the negotiation must be taken into account, so he may start with a first demand lower than

his maximal aspiration level, but he will at least ask for an equal split of the surplus. This

yields the following lower bound for his agreement payoff:

XHA _ v(12). V(12~-v(1~+V(2~ 2 - 3v(1 )-v( )-v( r (3.21)

If both players are equally strong (v(1)=v(2», than the coalition value v(12) will be distri

buted in proportion to the maximal aspirations (x~u. = x~u.).

The prominence level calculated from the negotiation protocols is 5 in both data sets

(significance level 0.01). The results of the average success measure over the four independent

subject groups prove that the NAA concept is more successful than other theories (see

table 3.6).

3.5 Summary on Two-Person Games 41

Table 6: Comparison of different area theories with the success measure

Theory 1 Subject group

2 3 4 Average

NAA 0.3523 0.5120 0.3022 0.3791 0.3864 PARITY AND EQUAL 0.0764 0.2202 0.4523 0.3259 0.2687

SURPLUS PARITY AND EQUALITY 0.1928 0.2408 0.2240 0.4004 0.2645 EQUAL SURPLUS 0.1651 0.2423 0.0948 0.4540 0.2390 JUSTICE NORMS 0.2844 0.2629 0.1335 0.5285 0.2348

The main problem of two-person games in characteristic function games seems to be the

high dispersion of the agreed shares. The most hits (87.98%) are explained by the so called

PARITY and EQUALITY theory but the predicted area is 61.53% of the area of possible

outcomes, so that the success measure performs poorly. The smallest area is predicted by the

EQUAL SURPLUS theory but unfortunately only 35.11% of the outcomes are explained, so

the best compromise seems to be the NAA concept. Further tests not reported here have been

done with varying a permissible percentage of deviation in the EQUAL SURPLUS concept,

but a predictive success as high as obtained by the NAA concept could not be reached.

3.5 SUMMARY ON TWO-PERSON GAMES

Chapter 3 was concerned with the presentation of a new descriptive theory for experi

mental two-person characteristic function games. The NEGOTIATION AGREEMENT

AREA was found to be more successful in prediction than other theories. However, the data

set is very small and there have to be more replications of the experiment in order to confirm

our new descriptive theory. Especially more has to be found out about the first move

advantage.

Up to now our theory is only applicable to two-person characteristic function games

with non-negative payoffs to the one-person coalitions, but a version which takes negative


values to the one-person coalitions into account will be available soon. Moreover, there are

some suggestions for a generalization to other types of two-person bargaining games. How

ever, this study on two-person games was mainly concerned with lower bounds for the

outcome, derived from the described proportional division scheme. The data on three-person

characteristic :function games suggested the appropriateness of these bounds. The tWo-peIson

experiments had the purpose to examine their predictive power in the more basic two-person

context.

4. THREE-PERSON BARGAINING GAMES

Many theories for three-person bargaining games in characteristic function form have

been developed, but most of them do not seem to have much relevance for the explanation of

laboratory experiments, because most of the theories are normative rather than descriptive.

This chapter is concerned with the presentation of a new descriptive theory, called

PROPORTIONAL DIVISION PAYOFF BOUNDS, which takes findings of experimental

research on two-person characteristic function games with positive payoffs to the on~perBon

coalitions into account. This theory is a modification and generalization of SELTEN's [1987]

EQUAL DMSION PAYOFF BOUNDS, a theory which was found to be more successful in

the prediction of experimental results than different versions of the BARGAINING SET

[AUMANN and MASCHLER 1964] for a large data basis [SELTEN and UHLICH 1988].

For the purpose of developing the PROPORTIONAL DIVISION PAYOFF BOUNDS

additional experiments had to be conducted at the Bonn Laboratory of Experimental Econo

mics, therefore a computer program, which will be described in section 4.1, had to be written.

Altogether 34 data sets with 1099 plays of different games were obtained by experiments with

the help of the "NEGOTIATIONS 3" computer program. Now our data basis is enlarged to

49 data sets with 3088 plays of different games. 15 data sets were reported by other resear

chers and will be reevaluated in this chapter. The experimental design and the data sets of all

reported studies and of our series will be described in detail in section 4.2.

To get an impression of the predictive power of the PROPORTIONAL DMSION

PAYOFF BOUNDS we shall compare this theory with the CORE [GILLIES 1953], the

BARGAINING SET, the EQUAL DIVISION PAYOFF BOUNDS, and a modified version of

the EQUAL EXCESS THEORY, which was originall, devel~~d by KOMORITA [1979]. All

theories will be described in section 4.3.

44 4. Three-Person Bargaining Games

Section 4.4 is concerned with the evaluation and reevaluation of all 49 data sets. More

over, the ORDER OF STRENGTH HYPOTHESIS, introduced by SELTEN [1972] as part of

the EQUAL SHARE ANALYSIS, will be tested in our enlarged data basis. This hypothesis

was found to improve predictive success of all theories examined in an earlier paper [SELTEN

and UHLICH 1988]. Further, the effects of the subjects' experience on the results will be

discussed.

4.1 PROGRAMS FOR COMPUTER-CONTROLLED THREE-PERSON

BARGAINING GAMES

One of the reasons for the multiplicity of cooperative solution concepts in game theory

can be seen in the fact, that a game in characteristic function form is an insufficient descrip

tion of real game situations. Various methods in which proposals can be made and agreements

can be reached by different bargaining rules may result in different theories.

First we have to look at the communication conditions. If negotiations take place with

free communication, personal skills might improve a negotiator's performance. The possibility

to transmit reasons for proposed agreements, or the ability to persuade others may be of

crucial importance. Restricting free verbal communication to a more formalized interaction

with written verbal messages up to an interaction by a limited set of messages, like proposals

and decisions to accept or reject proposals of other negotiators will reduce the influence of

personal skills and tactical tricks.

Another important distinction is, whether the negotiations are face to face or not. People

probably find it easier to act tough if they are not looking at the other negotiator. Face to

face contact may facilitate the development of trust.


However, there is a wide field of intermediate conditions between free verbal face to face

communication and restricted communication via letters, computers or by telephone.

Moreover, the negotiations may be influenced by different bargaining rules. Most researchers

use a bargaining procedure with at least two stages. In the first stage players are free to make

proposals at any time in this stage to other players. If a proposed coalition and the

distribution of the payoff is accepted by all members of the coalition, then the second stage is

reached. The present agreement has to survive for a certain length of time, but all players are

free to negotiate for a better outcome within or outside of that coalition. The game ends if no

other proposal is accepted by all members of the other proposed coalition within that length

of time. If a new coalition is accepted again, the agreement again has to survive for the

certain length of time. A game theoretic analysis of the described procedure seems to be very

complicated. For the investigation of stationary equilibrium points in a noncooperative model

of characteristic function bargaining, SELTEN [1981] introduced a much simpler set of

bargaining rules. The rules describe a game with perfect information:

1. Initiator: At the beginning of the game, a randomly selected player becomes the ini

tiator. He must decide whether he wants to make a proposal. If he does not want to do this,

he must shift the initiative to another player; this other player then becomes the new

initiator.

I. Proposal: If the initiator does not shift the initiative, he must propose a genuine coali

tion, where he is a member, and a payoff division of the coalition value among the members

of the coalition. The other members in this coalition are called receivers of the proposal. The

initiator must select one of the receivers. This receiver becomes the responder.

S. Responder: The responder can either accept or reject the proposal. If he rejects, he

becomes the new initiator. The old proposal is erased, and he can make a new one or he can

shift the initiative. If all receivers have accepted the proposal the game ends. Otherwise the

responder must select a new responder among the receivers who have not yet accepted.


-I. End: If the proposal is accepted by all receivers the game ends and all members of the

proposed coalition receive the proposed payoffs. All the other players get the payoffs of their

one-person coalitions. A problem arises in the case of an infinite play, because there is no

natural end. In this case SELTEN's model assigns their one-person payoffs to all players. We

never observed this problem in an experimental game. In very long negotiations there was at

least one subject who gave up and accepted a low payoff.

NO

YES

Initiator proposes coalition C and division of u(C) and selects respondcr among receivc!$.

Proposal becomes final agrccmcnl.

Did all receivers accept?

Rcspondcr selects new respondcr among receivcrs who bavc nol yClacxeplcd.

Figure 4.1: Flow chart of the bargaining process [SELTEN 1981 p. 139]


All experiments conducted at the Bonn Laboratory of Experimental Economics use the

bargaining rules introduced by SELTEN. However, whereas SELTEN in his theoretical paper

permitted the formation of one-person coalitions, our laboratory procedure excluded this

possibility. To avoid influences by personal skills, friendship between negotiators etc., because

such things cannot be described by a formal model, the communication was restricted to

proposals only. All participants were seated in separate cubicles. Therefore, a set of computer

programs called "NEGOTIATIONS 3" was developed in order to establish anonymous com

munication among the subjects. The set includes four programs. The first one, called Ma

ster-Program, is the coordinator program that controls the whole experiment. The second

program called Terminal-Program is used by the subjects in their separate cubicles to com

municate with each other and sends data to the Master-Program for data recording. A spe

cial keyboard driver is necessary to get full control over the keyboard. The last program is

used for predefinition of game parameters.

4.1.1 SYSTEM REQUIREMENTS

To use the described software package in an experiment with three subjects, four IBM

PC, XT or AT (or closely compatibles) computers are required. The maximal configuration

to run the experiments with up to 15 subjects simultaneously requires 16 computers. The set

of programs support CGA-, Hercules-, EGA-, and VGA-Adapter. To run the Master-Pro

gram one computer must have 640 KB of memory, one disk drive, and a hard disk. The other

computers only need 256 KB of memory and one disk drive to run the Terminal-Program.

This software package was designed to run in an IBM local area network environment with

the IBM PC Network Program l.x, but it may run in different network environments as long

as it is possible to share a hard disk or a RAM disk with different computers. The operating

system has to be MS/PC-DOS 3.1 or a later version.


4.1.2 THE THREE-PERSON BARGAINING MASTER-PROGRAM

The Master-Program is used to coordinate and control the whole experiment. In the

current version it is designed to coordinate experiments with up to 15 subjects simultaneous

ly. This is done, because our laboratory consists of three rooms with 16 cubicles in them, so

there is a limited capacity for participants in one session of an experiment. The IBM PC

Network supports up to 72 terminals, but for speed considerations a high number of

controlled terminals is not advisable. Also the current version only supports three-person

characteristic function games.

After program initialization, all participants which are in a stand-by mode are grouped.

Then a characteristic function will be selected from the data file produced by the program for

predefinition of game parameters and will be attached to a group of three players. One player

called initiator, is selected randomly. At every time in the experiment, if three players are in

the stand-by mode they will only be grouped for a new game, if they have not played in this

grouping before. If one group ends a game, all players within this group have to wait until an

other group ends a game and a new grouping is possible. A problem arises, if one group

negotiates a long time and other subjects have to wait very long. In this case negotiations will

be interrupted after one hour. Now a new grouping with new games is possible. If there is no

further game in the data file, the interrupted games will be attached again. No game will be

terminated by the experimenter. A game ends only if an agreement is reached. The experi

ment ends if there is no further characteristic function in the data file, or after a specified

number of games. But all games have to end normally. The protocol of the negotiations is

written to a file on the fixed disk after every shift, reject, or accept command of the players,

so that in case of a technical problem not all data are lost. A further procedure for security is

implemented, for the case that a participant switches the system off, disconnects it from the

main power, or terminates the Terminal-Program in an unusual way. If this happens, all

terminals of players in the group with the problem have to be switched off before the experi-


menter can initialize them again with the Master-Program. After reinitialization all termi

nals have to be switched on again. Only the last proposal is lost, but the initiator, who is the

same as before the problem occurred, can make the same proposal again. At the end of the

experiment a list of all participants with their payoffs in cash is printed or shown on the

screen. The point to cash rate can be selected during the initialization of the program.

4.1.3 THE THREE-PERSON BARGAINING TERMINAL-PROGRAM

This is the computer program, running on all terminals, which the subjects use to

transmit proposals to other subjects. The bargaining rules, introduced by SELTEN [1981],

which are described in detail in section 4.1! are implemented in this program.

Before starting this programs, the Master-Program must have been started. Then the

keyboard driver has to be started. The keyboard driver is necessary to lock critical keys such

as "[ALT] [CTRL] [DEL]" and "[CTRLj [BREAKj". If these keys are not locked, subjects are

able to terminate the program. After the Terminal-Program is started, the participants see

five different windows on the screen (figure 4.2). At the beginning of each game, subjects see

this screen with a greeting in window 5. They are asked to press any key to enter the

stand-by mode, if they are ready to start the negotiations in the next game. If any three

participants, that not have played before in this grouping, are in stand-by mode, they receive

a game. The game will be displayed in window 1 (see figure 4.3). The whole characteristic

function is shown in a graphical representation. Subjects have to be introduced to the

meaning of the values in this representation. A translation of the instructions to the subjects

can be found in appendix B.

50

IIINDOIoI 1 IIINDOIoI 2

IIINDOIoI 4

IIINDOIoI 5

Figure 4.2: Screen of the Terminal-Program.

v(2) 2

1

vel)

v(23)

4. Three-Person Bargaining Games

IIINDOIoI 3

v(3)

3

Figure 4.3: Graphical representation of a characteristic function game. The single num hers are the player positions. The position assigned to a specific player will blink in the experiment. The v( • ) will be replaced by the payoffs of the game.


One player will be randomly selected as initiator. He has the option to make a proposal

or shift the initiati'l1e. IT the initiator chooses to shift, he has to select the next initiator.

Otherwise window 3 opens. In this "send-window' the initiator has to select potential part

ners for a desired coalition and to distribute the payoff among the members of the coalition.

The program now checks, that not more than the payoff of this coalition is distributed, and

that the proposal is rational for this coalition. Coalition rationality means, that a genuine

coalition should not receive less than the coalition value. The program does not check for

individual rationality, so it is possible, that a player receives less than his one-person value,

but not less than zero. IT the initiator has confirmed his proposal, he is asked to select a

responder among the receivers if a grand coalition is selected. IT a two-person coalition is

proposed, the only receiver automatically becomes responder. The responder receives this

proposal in window 2 (the "recei'l1e-window"). Now the responder has the option to accept or

to reject. IT the responder rejects the proposal, all players receive a message in their

"receive-window". This message contains the proposal, the number of the player who was the

initiator, and the number of the player who rejects the proposal. The responder who rejects

the proposal now becomes the new initiator. The old proposal is erased, and the initiator can

make a new one or shift the initiative. For the case that the responder accepts the proposal,

the program checks whether all members of the proposed coalition have accepted. IT that is

not true, the proposal will be send to the last receiver, who now becomes responder. The

game ends, if the proposal is accepted by all receivers. The members of the coalition receive

the proposed payoff, while the other players receive their one-person values. The result is

shown in window 3 for 30 seconds. Then in window 5 the players are asked not to enter the

stand-by mode if they are not ready for the next game. In the current version window 4 only

is needed for the message, that an interrupted game starts again. In a further version this

window will be used to transmit written verbal messages.


4.2 EXPERIMENTAL DESIGN

The experiments conducted at the Bonn Laboratory of Experimental Economics, which

will be reported below are all played with the help of our computer program "NEGOTIA

TIONS 3", therefore the bargaining rules are identical in all series. Some experiments were

designed to get impressions of subject's behavior if they play specific types of games or to

replicate findings in studies of other researchers, but most of the experiments were designed

such that there is a systematic variation over all types of games. The differences in our series

will be described in detail in section 4.2.2 .

4.2.1 EXPERIMENTAL PROCEDURE

In 26 experimental sessions 1099 plays of different games were played with our compu

ter-controlled procedure, which is described above. Subjects were 312 graduate and undergra

duate students mostly of economics and law. All subjects were recruited around the campus

to participate in a negotiation game. The games were played for points with a fixed point to

cash ratio. The earnings in the four to five hour sessions (including 30 minutes for the

introduction) differed between DM 10,- and DM 60,- . The complete instructions, given to

the subjects, are described in appendix B.

4.2.2 THE DATA BASE

This section is concerned with a description of all data sets in our data basis. First we

shall describe data sets of different researchers, which will be reevaluated within this study

and then the data obtained at the Bonn Laboratory of 'Experimental Economics will be

4.2 Experimental Design

described. A complete listing of the results of series 1 to 7 is given in appendix C.

Hens, Momper, OHmann and Schmauch [1985] I:

53

This set contains 30 plays of 15 different superadditive games and 30 plays of 15

non-superadditive games with nonzero values for the one-person coalition. All games

were played with free communication condition. The games were played for points

converted to cash at a fixed rate. The time limit for negotiations in one game was 105

minutes. Subjects were introduced to the rules and the experiment in a 2-hour session.

Hens, Momper, Ostmann and Schmauch [1985] IT:

The setup of this experiment is identical to the one above except, that there was restric

ted communication via computer terminals. Subjects were free to communicate with

written verbal messages. All participants played the games before with the free com

munication condition.

Hens, Momper, Ostmann and Schmauch [1985] m:

This data set of 128 plays of 32 different superadditive and non-superadditive games

with nonzero payoffs for the one-person coalitions was played face-to-face. Communi

cation was not restricted. The games were played for points converted to cash at a fixed

rate.

Kahan and Rapoport [1974]:

240 plays of five non-superadditive games with zero payoffs for the one-person coalition

are reported. Three different communication conditions were used with the Coalitions

program. Since the REMARK keyword was disabled we shall call these conditions as

restricted communication. Games were played for points converted to cash at a fixed

rate per point. 12 tetrads of players played four iterations of the set of five games. Play

ers within tetrads were rotated through positions and the "observer role", so that at

any time no one knew who was resting and who was in which game role. There was no

time restriction for the negotiations. The subjects were introduced to the experimental

apparatus and rules of the game in a 3-hour training session consisting of written in

structions, verbal elaborations, and three separate experimenter-guided practice games.



160 plays of five superadditive and 160 plays of five non-superadditive games with

nonzero values for the one-person coalitions. All games were played with a restricted

communication condition. Instead of playing in tetrads, as in Kahan and Rapoport

[1974], players were pa1red in 16 triads, whose members were shumed between sessions

of game playing. The other experimental setup is identical to Kahan and Rapoport

[1974].


The aim of this study was to obtain results of games with two weak and one strong

player. The study reports the outcomes of 90 plays of 5 superadditive games with zero

payoffs for the one-person coalitions. All games were played under restricted

communication conditions with the Coalitions-II program. For half of the plays the

REMARK keyword was enabled. This is a special case of a restricted communication

condition, because written verbal messages could be send to other negotiators. All

subjects were experienced in playing 4-person characteristic function games. The other

conditions were as in Kahan and Rapoport [1974].

Leopold-Wildburger [1985]:

This data set contains 54 plays (partially unpublished) of 9 superadditive games with

zero payoffs for the one-person coalition. All games were played face-to-face with free

communication. The negotiation time was not restricted. All subjects knew that no one

would play twice against the other and that each of them would be in the role of player

A, B an C respectively only once. Payoffs are in Austrian Schilling.

Maschler [1978]:

78 plays of different superadditive and non-superadditive games with zero payoffs for

one-person coalitions were reported. All games were played face-to-face with free

communication. There was no cash reward for the subjects, only three prizes were

offered to the first winners of the contest, in which also games not included here have

been played.


Medlin [1976]:

This study reports 160 plays of 8 superadditive games with zero payoffs for the

one-person coalitions. All games were played with restricted communication conditions

with the Coalitions program (see Kahan and Rapoport [1974]).

Mumighan and Roth [1977]:

Six different restricted communication communication conditions were tested with 432

plays of one superadditive game with zero payoffs for the one-person coalition. This

study contains 36 independent subject groups. The Subjects were introduced shortly

with written instructions. The participants did not receive any monetary payoffs. All of

them did receive credit toward a course requirement for participating.

Popp [1986]:

The reported data set contains 60 plays of three different games with zero payoffs to the

one-person coalitions. Subjects were 180 male students of economics and law, who were

payed a sum of DM 7.50, and additionally the payoff gained by the bargaining. All

subjects only played once and had no experience in coalition experiments. Half of the

triads were given a suitable text to establish a cooperative motivational orientation.

Moreover, two communication conditions were applied, therefore we shall split the data

into two data sets. In the first data set, called Popp I, subjects could communicate by

means of an intercommunication system. A player not included in communication could

neither hear anything nor perceive the offers of the communicators. The number of

bargaining rounds was not restricted, but each was limited to two minutes. In the

second data set, called Popp II verbal communication was not allowed.

Rapoport and Kahan [1976]:

This study is an extension of Kahan and Rapoport [1974], adding a nonzero valued

grand coalition to the characteristic function of the five games. 160 plays of these

superadditive games with 8 independent subject groups were reported.


Riker [1967]:

In this study one game is replicated 93 times. The non-iluperadditive game had a zero

valued grand coalition and zero payoffs for the one-person coalitions. The payoff was in

American cent. Communication was free only within dyads. However since the 3-person

coalition value was zero the study will be classified as free communication. At all there

were three independent subject groups.

Selten and Stoecker:

This unpublished data set contains 54 plays of 27 superadditive games with zero payoffs

for the one-person coalitions. The payoff was in German Pfennig. The bargaining rules

underlying this experiment were as described in section 4.1, with the exception that the

formation of one-person coalitions was permissible. The subjects were seated in

different rooms. Formalized messages were transmitted by intercom via a central

operator. No direct communication among the players was possible.

All following series were conducted at the Bonn Laboratory of Experimental Economics at

the University of Bonn with the set ofthe "NEGOTIATIONS 3" computer programs.

Series 1.1-1.2:

The aim of this series was a test of the new set of the "NEGOTIATIONS" computer

programs. The data set contains 56 different superadditive games with zero payoffs for

the one-person coalitions. Values of the two-person coalitions were varied systemati

cally. In the first session, due to a technical problem only 36 games could be played.

Altogether 92 results could be obtained in the two sessions. Subjects were undergra

duate students mostly of economics.

Series 2.1-2.2:

For this series a new data set was developed with games which were not zero-normali

zed. The normalized two-person coalition values v(ij)-v(i)-v(j) were varied systemati

cally. The payoffs for the one-person coalitions were varied randomly such that the sum


of the values for the on~person coalitions was a fixed number, the same one in every

game. The on~person coalition value then was added to every coalition, were that

player was a member. So the zero-normalized games are identical to those of series 1.

In two sessions of this series the participants were graduate students and professors of

economics. Due to the fact of time constraints not the whole set of games could be

played. Altogether 88 results could be obtained.

Series 3.1-3.2:

The data set used in this series was created similarly to the data set of series 2. In two

sessions 85 plays of different games could be observed. Subjects were undergraduate

students mostly of a course in game theory.

Series 4.1-4.6:

This series was conducted to replicate the results of Kahan and Rapoport [1980]. There

fore 10 superadditive games with zero payoffs for the on~person coalitions similar to

those played by Kahan and Rapoport were chosen. In three sessions with two indepen

dent subject groups in each, 120 results could be obtained. Subjects were undergraduate

students mostly of economics and law

Series 5.1-5.6:

The the same basic games as in series 4 with a higher valued grand coalition were used.

The procedure is identical to series 4.

Series 6.1-6.4:

The aim of this series was a test of different strength hypotheses. Therefore 10 very

problematic superadditive games with a zero payoff for only one on~person coalition

were created. In two sessions with two independent subject groups in each session 80

results could be observed. Subjects were students mostly of economics and law.

Series 7.1-7.12:

The aim of this series was to get data on the whole range of possible variations in


superadditive games with smaller steps as in the series above. Using a systematic

procedure of variation of the coalition values 125 different games with positive payoffs

for the one-person coalitions were constructed. In 12 sessions 514 results could be

obtained. Subjects were students of economics and law. In the first six sessions all

subjects were unexperienced, while in the following six sessions all subjects had

participated in 3-person characteristic functions games before.

4.3 THEORIES OF COALITION FORMATION

In this section a new descriptive theory for three-person characteristic function bargain-

ing, called PROPORTIONAL DIVISION PAYOFF BOUNDS will be presented. Further a

short description of the EQUAL DIVISION PAYOFF BOUNDS [SELTEN 1987] and a modi

fied version of the EQUAL EXCESS THEORY, which was originally introduced by KOMO

RITA [1979] will be given. Moreover, other theories such as the CORE [GILLIES 1953,

AUMANN and DREZE 1974] and the BARGAINING SET [AUMANN and MASCHLER

1964] which seem to have descriptive power will be explained.

4.3.1 THE CORE

One of the most popular solution concepts for characteristic function games is the CORE

introduced by GILLIES [1953] for superadditive games. This concept was generalized to non

superadditive games by AUMANN and DREZE [1974]

A payoff vector x that gives each player at least as much as he can guarantee himself

and gives all players together v(N) is called an imputation. The generalized CORE is easy to i

define as the set of all payoff configurations (Xh"" Xn; Ch .• ·., 'Cm) with

4.3 Theories of Coalition Formation

L Xi ~ v( C) for all C £ N ieC

59

(4.1)

In the following we shall restrict ourselves to zer<r-normalized three-person games.

Whenever talking about zer<r-normalized games, we shall use g instead of v(123) for the value

of the grand coalition, a for v(12), b for v(13), and c for v(23). The zer<r-normalization does

not matter to this concept, because the CORE is invariant with respect to strategic equiva

lence. However, for some games the core may be empty. Without loss of generality we shall

assume that a ~ b ~ c, which is always achievable by a suitable renumbering of players. For

the case that a > b + c the core is always non-empty, but if a 5 b + c holds the core is

empty unless g ~ ; (a + b + c). If the core is non-empty it is found in the coalition structure

{123} or {12,3} depending on whether or not g ~ a, respectively. If g = a the core is in both

coalition structures. However, if g < a the core is found in coalition {12} with Xl ~ b and

X2 ~ c and (Xl + X2 = a).

4.3.2 BARGAINING SET

The BARGAINING SET [AUMANN and MASCHLER, 1964] is one of the most impor-

tant theories for characteristic function games. Unlike some other normative solution con

cepts it seems to be at least in partial agreement with experimental results. In the literature

many versions of bargaining sets are presented, but in the special case of three-person games

they coincide.

The definition of the bargaining set is based on concepts named objections and counter

objections. A configuration is considered stable in the sense of the bargaining set, if for every

objection there is a counterobjection. A detailed definition can be found in the in the

literature [e.g. ROSENMULLER 1981, OWEN 1982, KAHAN and RAPOPORT 1984] and


will not be repeated here.

In the context of this study we shall restrict ourselves to the description of the bargain-

ing set for zero-normalized three-person games. For non-zero normalized games one has to

find the bargaining set of the zero-normalized game and to apply the inverse of the zero-nor

malization mapping.

In the following the notational conventions for zero-normalized three-person games will

be used (see 2.13). Without loss of generality it will be assumed that a ~ b ~ c, which can

always be achieved by a suitable renumbering of the players.

Three numbers qh q2, and q3 called quotas are important for the BARGAINING SET.

The quotas are characterized by the property

qi + Clj = v(ij)

for every permutation i,j,k of 1,2,3, so we have

q _ a+b-e l--r

q -a+b+c 3= 2

(4.2)

(4.3)

To calculate the BARGAINING SET some case distinctions have to be made, therefore

a complete calculation scheme is given in Table 4.1.

The BARGAINING SET does do not predict a coalition structure but only a region of

payoff vectors for every coalition structure. Not even the case that only one-person coalitions

form is excluded. We refer to this case as null-structure. It seems to be fair to add the predic

tion to the bargaining sets that at least one coalition with more then one member will be

formed. This definition of the BARGAINING SET without null-structure was used in earlier

studies by SELTEN [1987] and SELTEN and UHLICH [19~8J.

Tab

le 4

.1:

Bar

gain

ing

set

for

zero

-nor

mal

ized

thr

ee-p

erso

n ga

mes

a ~

b+

c

-(O

,O,O

j 1,

2,3)

12

(q1,

q2,O

j 12

,3)

13

(q1,

0,q3

j 13

,2)

23

(0,Q

2,Q

3j 1

,23)

(g,O

,Oj

123)

if g

<b

-e

(X1,X

2,Oj

123)

w

ith

Xi=Q

i+g-

-q~-

-q2

123

if 2

a-b-

e> g~

b-<:

,(Xt,X

2,X

3i 1

23)

wit

h X

i=Q

i-gt

+g2

+g3

-g

3 -

if ¥

a+b

+c)

> g

~ 2

a-b-

e

(X1,X

2,X3i

123

) w

ith

Xt+

X2

~ a,

Xt+

X3

~ b,

X

2+X

3 ~

c, X

t+X

2+X

3 =

g

if g

~ ¥

a+b

+c)

a>

b+

c

(O,O

,Oj

1,2,

3)

(X1,X

2,Oj

12,3

)

(b,O

,Oj

13,2

)

(O,C

,Oj

1,23

)

(g,O

,Oj

123)

if g

<b

-e

(X1,

X2,

0) if a>g~b-e

(X1,X

2,X3i

123

)

if g

~ a

wit

h Xt~b,

X2~C

, Xt+x2~a

wit

h Xi=Qi+g-q~--q2

wit

h X

t+X

2 ~

a, X

t+X

3 ~

b

X2+

X3

~ c,

Xt+

X2+

X3

= g

~

c,., ~

po" 2l ::l.

CD

rn S o o ~

.... g' i ~ g' C)

I-'


One cannot expect that experimental results will exactly hit the bargaining set in view of

the subjects' preference for round numbers, therefore on has to admit deviations up to the

prominence level d for each player's payoff separately. Unless specified otherwise, when

talking about the BARGAINING SET we mean BARGAINING SET without null-structure

and with deviations up to d.

In the discussion of his experiments, MASCHLER [1963, 1978] proposed to apply the

BARGAINING SET not to the characteristic function in its original form, but to two power

transformations of this function. Enlarging the original BARGAINING SET by all configura

tions of the BARGAINING SETS for the power transformations, which are also

configurations for the original characteristic function, we obtain the UNITED BARGAINING

SET [SELTEN 1987], but for the sake of shortness it will be not examined here.

4.3.3 EQUAL EXCESS THEORY

The EQUAL EXCESS THEORY [KOMORITA 1979] in its original form, has been

proposed to account for coalition formation and payoff division. The principal basis of this

theory is that an individual's bargaining strength in negotiations is based on the alternative

coalitions the person can form, and that the members of a potential coalition are most likely

to agree on a division based on sharing equally the excess of what can be gained by the coali

tion, relative to the total outcomes if each chooses his best alternative. The theory has the

form of an iterative process of expectation formation. At each stage of this process each

player has a payoff expectation for each coalition in which he is a member. The theory

assumes an unspecified finite number of iterations of the process. The main problem of the

theory is the lack of a prediction for this stage number. The modification of the EQUAL

EXCESS THEORY introduced here is an area theory, which avoids this problem.


Two assumptions specify the preferences of the individuals and their expectations during

the various stages of the process of expectation formation:

1) The initial expectation of player i in coalition C is given by the equal shares:

E~(C) = Tbf1 for all C, i e C (4.4)

2) In the first stage the players refer to the best alternative they expected to receive in

the initial stage 0, and use this as their starting point. An alternative coalition T to a

coalition C has to be a valid threat in the sense that the consensus of at least one

player of C is not needed in T. Therefore the best alternative expected payoff of

player i to a coalition C in stage r is defined as

A~(C) = max ( E~(T) I ieT, T#C, [(C\{i})nT = 0] V [TeC] ). (4.5)

The remainder of the coalition value over the sum of the best alternatives of the

members should be divided evenly among the members. This will be repeated for

every stage r, where the reference values for the best alternatives are derived from

stage r-I. The expectations change over successive stages according to

Ei(C) = Ai (T) + TCT v(C) - ~ Aj (T) r r-l 1 ( ~ r-l )

jeC

(4.6)

It could be argued, that the definition of the best alternative expected payoff of player i

should not include players in the alternative coalition T, who are also members of C, because

there consensus is needed to form T and therefore T cannot be a valid threat. Of course play

ers in C may not have valid alternative coalitions as individual players, but they may have

valid alternatives as subsets of players. Consider a three-person game where in negotiations

among all members of the grand coalition not only the solo coalitions are valid threats but

also each pair of players can threaten the third player to form a two-person coalition. Such

threats are highly credible, therefore subsets of players of C are ;Valid alternative coalitions.


In contrast to the original theory we shall determine bounds for the payoffs of the

players, which are given if we use the minimum of the maximal expectations of each stage

from 0 to the asymptote as lower bound and the maximum of the maximal expectations as

upper bounds including deviations up to the prominence level Ii. Hence,

r r inf Ei(C) -Ii S Xi(C) S sup Ei(C) + Ii reIN reIN

,ieC. (4.7)

Of course no specific coalition structure can be predicted, but if an agreement on coalition C

is reached, then our theory predicts that the payoffs of the members of C will be in the range

described by (4.7).

Unfortunately no general proof of convergence is known to the author, but for three-per

son quota games in the limit the quotas are reached in the two-person coalitions and in the

grand coalition each players' expectation converges to a value equidistant from his quota. For

all other games within this study the sequence converges to an asymptotic value. The same

was observed by KAHAN and RAPOPORT, Am. [1984, p.153].

4.3.4 EQUAL DMSION PAYOFF BOUNDS

The theory of EQUAL DIVISION PAYOFF BOUNDS [SELTEN 1983,1987] intends to

reflect the limited rationality of human decision making. Players in SELTEN's theory do not

behave as maximizers but they try to obtain at least as much as their lowest acceptable aspi

ration level. The theory of EQUAL DIVISION PAYOFF BOUNDS is an attempt to describe

commonsense reasons that influence the aspiration levels of the players, because experimental

subjects who do not perform complicated computations must base their strategic reasoning on

simple commonsense arguments.

Originally the theory was proposed to explain ~rf~ental results of superadditive

zero-normalized three-person games, but it may also be'applied to non-superadditive games


without any changes and to games with positive payoffs for one-person coalitions, simply by

the computation of EQUAL DIVISION PAYOFF BOUNDS for the strategically equivalent

zero-normalized game and a subsequent retransformation of the bounds [SELTEN and

UHLICH 1988].

The theory of EQUAL DIVISION PAYOFF BOUNDS specifies three numbers Uh U2,

and U3 to be interpreted as the lowest reasonable aspiration levels for players I, 2, and 3,

respectively. This does not mean that a player may not have higher aspiration level, but that

no player will have an aspiration level lower than his payoff bound Ui.

The bounds can be derived from the game through an analysis of the strategic situation

of the players in the order of strength beginning with the most powerful player using the idea

that players use various equal shares for the formation of aspiration levels.

Again using the notational convention for zero-normalized games it will be assumed

without loss of generality that a ~ b ~ c, which can always be achieved by a suitable renum

bering of the players. In an intuitively obvious sense for games with zero payoffs to the

one-person coalitions player 1 is stronger then player 2, if we have b > c and player 2 is

stronger than 3, if we have a > b. For b = c (a = b) players 1 and 2 (2 and 3) are equally

strong. To express the relationships stronger and equaUy strong the symbols > and - will be

used. In view of the convention on the renumbering of the players there are only four possible

orders of strength:

1>2>3 for a>b>c

IN2>3 for a>b=c

1>2N3 for a=b>c

IN2N3 for a=b=c (4.8)


Only a summary explanation of EQUAL DIVISION PAYOFF BOUNDS can be given

here, but the reasoning leading from tentative bounds to preliminary bounds and final bounds

is simple for every particular game. The tentative bounds depend on coalition shares, substi

tution shares, completion shares, and a competitive bound for player 3. The computation of

equal division payoff bounds is described in table 4.2.

Coalition shares are equal shares of the coalition values v(C). They are relevant for the

strongest players in every coalition. Since no other member of C is stronger, a strongest play

er should receive at least the equal share of v(C), if C is formed.

The substitution share is only relevant for player 2. Assume 1 >- 2 >- 3 and suppose that

player 1 and 3 form a coalition, then player 2 is able to throw 3 out, because player 1 can get

at most b in {13}, but in {12} both can get a - b more than that. Player 2 can claim at least

half of this additional amount.

Completion shares are defined by the assumption that two players j and k may form a

coalition, but if {123} is permissible then there may be an additional amount g-v(jk) if player

i completes the grand coalition. Player i can claim g-v~jk) , which becomes one of the tenta

tive bounds.

Player 3's competitive bound becomes relevant, if the highest tentative bounds of player 1

and 2, which are the maximum of all tentative bounds, sum up to less or equal a, and player

3 is weaker than player 2. In this very bad position he must fear that the most attractive

two-person coalition {12} forms, if no agreement on the grand coalition is reached. If player 3

wants to bargain about another two-person coalition, where he is a member, he must be

willing to offer to both other players what they maximally can get in {12}. The minimum of

both differences between the coalition values and his offers becomes the competitive bound w.


Table 4.2: Equal division payoff bounds

Tentative bounds for 1 and 2

ti = max [~, t] t2 = ti

[ c a-b ~] t2 = max 2"---r-' ~

Competitive bound w

hi = a - t2 h2 = a - tt

w = min [ b - hi , C - h2 ]

Player 3's tentative bound

ta = t2

Preliminary bounds

Pi = ti

Pi = t and P2 = Pa = ~ - i

Final bounds

[ Pi 1 Ui = max "1, ll. int A

for b=c

for b>c

fora=b

for a > b and ti + h ~ a

for a > b and ti + h > a

for i = 1,2,3

if g = a or ti + t2 + ta ~ g

for i = I, 2 and Pa = g - a

if tt + t2 + ta > g > a > b ~ c

fori = 1,2,3

if ti + h + ta > g > a = b = c

if ti + t2 + ta > g > a = b > c

for i = I, 2, 3

67


Aspirations based on tentative bounds become preliminary bounds in most of the cases,

but if the tentative bounds sum up to more than g in spite of the fact that {123) yields more

than {12} there is a strong tendency to form the grand coalition and aspirations must be

reduced generally by the weakest player. Only if player 2 and 3 are equally strong, player 1

has to reduce his aspirations.

The subjects tend to propose agreements involving round numbers. Therefore the final

bounds are determined as follows: The preliminary bounds are rounded to the next lower

number divisible by the prominence level. If this is a value lower or equal to zero the final

bound is one smallest money unit.

4.3.5 PROPORTIONAL DMSION PAYOFF BOUNDS

The theory of PROPORTIONAL DIVISION PAYOFF BOUNDS is closely connected to

SELTEN's [1983, 19871 theory of EQUAL DIVISION PAYOFF BOUNDS. In contrast to

SELTEN's theory, which was originally restricted to superadditive zero-normalized three

person games, the PROPORTIONAL DIVISION PAYOFF BOUNDS take non-negative

values to the one-person coalition explicitly into account. Though the generalization of the

EQUAL DIVISION PAYOFF BOUNDS to non-superadditive games does not meet serious

obstacles, the generalization to games with non-negative values to the one-person coalition is

very problematic, because subjects do not behave in the same way as in the strategically

equivalent zero-normalized game. Therefore two basic elements of the EQUAL DIVISION

PAYOFF BOUNDS have to be replaced. First, the crude power comparisons among players

have to be refined. Moreover, for some purposes the equal division concept has to be replaced

by a concept, which takes differences in the payoffs to one-person coalitions into account.


In the chapter above we already mentioned that a descriptive theory of bargaining

should ignore neither power nor justice norms. In the case of two-person games in characte

ristic function form it is easy to see which player is more powerful. One simply has to com

pare the one-person coalition values, because the relative power of both players depends

primarily upon how attractive to each is the option of not reaching an agreement. In the case

of three-person games one has to extend this argument. For the comparison of the power of

two players in a three-person game the players have the additional option to form a coalition

with the third player. Therefore the relative power of the two players now depends upon how

attractive to each are the options of not reaching an agreement with the player in compari

son. Such power comparisons will lead to an order of strength, which is a basic element of the

PROPORTIONAL DIVISION PAYOFF BOUNDS. Our new descriptive theory will analyze

the strategic situation of each player in the order of player power, because a weaker player

has to take aspirations of stronger players into account.

Another basic element of the theory depends on the fact that in asymmetric game situa

tions generally more than one justice norm has to be taken into account. Therefore for some

purposes we shall replace SELTEN's equal division concept from the EQUAL DMSION

PAYOFF BOUNDS by a proportional division concept, which depends on a proportional

division of the prize with respect to the aspiration levels. The proportional division scheme is

no justice norm, but the result of a negotiation process. As mentioned in the context of the

NEGOTIATION AGREEMENT AREA, we believe that the outcome of a negotiation pro

cess depends on the first demands of the players. These first demands are aspiration levels,

which depend on the strategic situation of the players. Strong players have very high de

mands, while weaker players may propose fair divisions of the prize. Almost always the first

demands are not compatible. This conflict can only be resolved by reduction of the players

demands. We suppose that the relative concession of a negotiator is nearly equal to the

relative concession of the opponent. There may be deviations from this rule due to the fact


that subjects prefer round numbers for proposals, or due to the fact that there is a smallest

money unit, which cannot be further subdivided. Moreover, there may exist soctological or

psychological differences between different subjects which lead to short-term deviations from

the process rule. Further, subjects may have strategic reasons to deviate. If the concessions

are very small, it may happen that one negotiator makes a larger concession, since otherwise

the process is very slow. Too small concessions of the opponent my be answered by a very

high demand in order to express the threat of the break-off of the negotiations with the

opponent. This may not occur in games with free verbal face to face communication con

ditions, since a threat can be expressed verbally.

With respect to the negotiation process there is still work in progress. Therefore we shall

not yet give a formal description of the observed processes. We assume that on average the

relative concessions of the negotiators are equal. This leads to a proportional division of the

prize according to the first demands. The first demands may be different for different subjects

and we do not know the demands before the negotiations start, but we can use reasonable

boundaries, which depend on power and justice considerations for the first demands in order

to compute lower bounds for the result of the negotiation process.

Moreover, some additional principles will be used in the theory of PROPORTIONAL

DIVISION PAYOFF BOUNDS, but we shall never use concepts such as "subjectively expec':

ted utility maximization". Therefore this theory can be seen as a theory, which takes boun

dedly rational behavior of human decision makers explicitly into account.

ORDER OF STRENGTH

Consider a zero-normalized three-person characteristic function game r(N,Q,v). With

out loss of generality it will be assumed that we have v(12) ~ v(13) ~ v(23), which always can

be achieved by a suitable numbering. As stated by iISHER and URY [1981], the


negotiating power of a player depends on his BATNA (Best Alternative to a Negotiated

Agreement), therefore in an intuitively obvious sense player 1 is stronger than player 2, if we

have v(13) > v(23), because player l's alternative to form a coalition with player 3 is more

profitable than player 2's alternative to form a coalition with player 3. Obviously within the

coalition of player 1 and player 2, the stronger one should receive at least not less than the

weaker player. Such power comparisons among all players lead to a unique ordering of all

players, which is SELTEN's idea of the order of strength in the EQUAL DMSION PAY

OFF BOUNDS. Now consider the case that one-person coalitions are profitable and we have

v(13»v(23), but v(1)<v(2) then it is not obvious that player 1 is stronger than 2 because the

best alternative to a coalition of player 1 and 2 may be the coalition with player 3 or no coali

tion. In an earlier paper [SELTEN and UHLICH 1988] we have assumed that the order of

strength is based on the comparison of the zero normalized two-person coalitions, such that

player 1 is stronger than player 2, if v(13)-v(1)-v(3) > v(23)-v(2)-v(3), though the theory of

EQUAL DIVISION PAYOFF BOUNDS using this order of strength was more successful in

the prediction of the results in 1711 plays of different games than different versions of the

BARGAINING SET, we got some hints, that this order of strength may be not the right one

to apply.

However, for the PROPORTIONAL DIVISION PAYOFF BOUNDS we shall use an

order of strength which depends on the comparison of the best alternative coalitions. For the

comparison of player 1 and 2 we have to find the best alternatives for both. Player 1 has two

alternative coalitions: the coalition with player 3 with the value v(13) and the one-person

coalition with the value v(l). The best alternative is defined as the maximum over the alter

native maximal aspirations, hence player I's best alternative is max[v(13)-v(3), v(l)].

Analogously player 2's best alternative is max[v(23)-v(3), v(2)]. The idea behind the order of

strength proposed here depends on pairwise comparisons of the players best alternatives. If

the best alternatives of two players are equal, then the second best alternatives will be


compared. We say that a three-person game is partially s'Uperadditive in the two-person coali

tions if we have v(ij) ~ v{i) + v(j) for every two-person coalition. It may happen that the

maximal aspirations from a specific coalition is negative in games without partial superaddi

tivity in the two-person coalitions. Up to now it is not clear, whether negative maximal aspi

rations can serve as alternatives. However, in the present study all games are partially super

additive in the two-person coalitions. Obviously, the best alternative is always found in the

alternative two-person coalition, and the second best alternative is the one-person coalition.

In our context the order of strength can be simply characterized by the following three pro

perties:

Property 1: If

Property 2: If

Property 3: If

v(ik) > v(jk) then i is stronger than j (i >- j).

v(ik) = v(jk) and v(i) > v(j)

then i is stronger than j (i >- j).

v(ik) = v(jk) and v(i) = v(j)

then i and j are equally strong (i '" j).

(4.9)

(4.10)

(4.11)

Lemma: Properties 1 to 3 uniquely determine a complete transitive order of strength for three

person games.

Proof: We have to make four case distinctions:

i) v(12) > v(13) > v(23)

With property 1 we have 1 >- 2 >- 3.

ii) v(12) = v(13) > v(23)

With property 1 we have 1 >- 2 and 1 >- 3. Intransitivity is excluded.

iii) v(12) > v(13) = v(23)

With property 1 we have 1 >- 3 and 2 >- 3. Intransitivity is excluded.

iv) v(12) = v(13) =v(23)

The order of strength is determined by the v(i). Intransitivity is excluded.


Without loss of generality it will be assumed in the following that the order of strength

obeys 1 t 2 t 3, which is always achievable be a suitable renumbering.

TENTATIVE BOUNDS

According to SELTEN's EQUAL DIVISION PAYOFF BOUNDS three types of bounds

for the payoffs of the players have to be defined. This section will discuss the tentative

bounds, which are derived from the order of strength and a proportional division scheme.

In chapter 2 we have introduced an area theory for two-person games, where the lower

bound for the strong player was determined by a proportional division of the prize according

to the maximal aspirations of both players. Since it can be observed in experimental data of

three-person games, that there are negotiation sequences of only two persons, it seems to be

reasonable to use coalition shares as tentative bounds. Coalition shares are defined for all

coalitions, which do not have genuine subcoalitions. Hence, the coalition shares are found in

all two- and one-person coalitions. The coalition shares are proportional shares of the strong

est member according to the order of strength of each two-person coalition and of course the

value of the one-person coalition. Consider a profitable coalition {ij}, where j is not stronger

than i, then v{ij) - v(j) is player i's maximal aspiration and v{ij) - v{i) is player j's maximal

aspiration. A division of v{ij) proportional to these maximal aspirations yields the following

coalition share of player i:

(4.12)

This coalition share is a tentative bound of player i.

If we have v(12) > v(13) > v(23) and v{ij) ~ v{i)+v(j) for i,j E N, then the order of

strength is 1 >- 2 >- 3, and we have three tentative bounds for player 1, because player 1 is a

strongest member in coalition {12} and {13} and of course i~ lUs solo coalition. Player two is

a strongest member only in coalition {23}, therefore he has only one coalition share in addi-


tion of his solo coalition. Since player 3 is never strongest member of a tw<>-person coalition

only his one-person value can serve as coalition share.

The next shares to be defined are the completion shares, which can serve as tentative

bounds for all three players. Consider two players j and k agree to form the permissible ge

nuine coalition {jk}, then player i is in a bad position. The only way to improve his outcome

is to propose a grand coalition, if it is attractive, but he is faced with the subcoalition {jk}

and therefore in a weak position. From ARISTOTLE we know that the weaker party seeks

for equality and justice, hence player i demands at least an equal split of the surplus,

v(i) + V(123)-v(i)-m3[v(jk),v(j)+v(k)], (4.13)

while the subcoalition demands the maximal aspiration

v(123) - v(i) (4.14)

If the prize will be distributed proportional to the demands, then the completion share

Ci = v(i) + v(123)-v(i )-m3[v(jk),v(j)+v(k)] ----------"---------v(123)

v(123) -v(i )+v(i)+ v( 123)-v(i )-max[ v (jk),v(j)+v(k)] 3

=

(4.15)

is a tentative bound for player i. Since all players are confronted with the situation that a

block forms, such completion shares are tentative bounds for all players.

Moreover, each player has eventually the possibility to be a substitute for another play

er. Suppose coalitions {ij} and {ik} are profitable. Without the help of player j player i can

not do anything better than to form a coalition with player k, where he can claim at most

v(ik)-v(k) if this amount is not negative. If v(ij)-(v(ik)-v(k» is greater than zero, then

player j may demand at least an equal share of this increment' and the substitution share


v{ ij)-{ v~ ik )-v{k)) (4.16)

becomes a tentative bound of player j. Analogous, if coalitions {jk} and {ik} are profitable,

then player k cannot do anything better without the help of player j than to form a coalition

with i, hence a further substitution share

V{jk)-{V~ik)-V{i)) (4.17)

becomes a tentative bound for player j. Such tentative bounds are available for all players, if

there is an increment from substitution arguments. This completes the discussion of tentative

bounds with exception of the weakest player.

According to our convention of numbering, the weakest player is player 3, if 2 )- 3, but

before player 3's further bound can be discussed, the highest tentative bounds tl and t2 of

player I and 2 have to be defined. The highest tentative bound ti of player i (i=I,2) is the

maximum of player i's tentative bounds. For the case that player I and player 2 are equally

strong, we have t2=h

Assume that tl+t2 ~ v(12) then player 3 must fear that coalition {12} forms, even if

v(123) > v(12), therefore he may be willing to make an attractive offer to each of the other

players. If player 2 does not reduce his aspiration level below his highest tentative bound t2

then player I cannot claim more than hi = v(12)-h, respectively player 2 cannot claim more

than h2 = v(12)-tl' hence player 3 may offer hi to player I, which leaves him v(13)-hl or h2

to player 2, which leaves him v(23)-h2. Of course player 3 has to reduce his aspiration level

to the minimum of both amounts, which defines player 3's competitive bound

(4.18)

and becomes one of his tentative bounds. As the highest tentative bounds of player I and 2,

the highest tentative bound of player 3 is the maximum over all of his tentative bounds.


PRELIMINARY BOUNDS

In general the highest tentative bounds ti become preliminary bounds Pi, but sometimes

there are reasons that aspiration levels according to the highest tentative bounds have to be

reduced. Assume that the grand coalition is attractive (see 2.8) but tl+t2+t3 > v(123), then

it seems to be plausible that one or more players have to reduce their aspiration levels in

order to make coalition {123} possible. The determination of players who have to reduce their

aspiration levels to a certain level is not easy even to the subjects, therefore a reasonable way

to allocate the prize is a proportional distribution according to the highest tentative bounds.

Moreover, it can happen that two players i and j are equally strong (i N j) and their highest

tentative bounds sum up to more than v(ij). In this situation both have to lower their

aspiration levels down to an equal share of v(ij), since they get into a competition in order to

form a coalition with player k, or they try to establish a grand coalition, where they can

claim the completion share. Therefore, both players reduce their aspiration levels below their

highest tentative bounds to the maximum of the alternatives, but not below their one-person

coalition value.

{c.

Let mi = 1

v(i)

if {123} at tractive

othewise

where Ci is the completion share defined by (4.15).

The preliminary bounds are defined as:

Pi = max [ mi'~' v(i) ]

ti otherwise

for i = 1,2,3

if {123} attractive and t 1+t2+t3> v(123)

if ti+tj > v(ij) and H and not iNk

tl+h+t3 ~ v(123)

(4.19)


FINAL BOUNDS

The final bounds are concerned with the phenomenon that subjects prefer "round

numbers". This will be taken into account through a reduction of the preliminary bounds to a

sufficiently round number. Therefore all Pi will be divided by the prominence level A within

the data set (see section 2.2) and the integer part of the result will be multiplied with A. This

is the final bound unless individual rationality is violated. Moreover it seems to be plausible

that no one joins a coalition if his outcome is at least not more or equal to his one-person

coalition value plus one smallest money unit r. Hence the final bounds are defined as

Ui = max [veil + r, A int ~] for i=l, 2, 3 (4.20)

The theory of PROPORTIONAL DIVISION PAYOFF BOUNDS predicts, that a

coalition C with v(C) ~ 1', Ui is formed if a coalition with this property is permissible;

iEC moreover the final payoffs Xi of the members of a genuine resulting coalition C will not be

below their final bounds:

Xi ~ Ui for every i E C (4.21)

CALCULATION SCHEME

1. Determination of the order of strength according to 4.9 - 4.11.

Renumber the players such that 1 ~ 2 ~ 3.

2. Calculation of player l's highest tentative bound

Tentative bounds are:

i) v~12l-v(2l 2v(I2 v I)-v 2) • v(12) if v(12)-v(1 )-v(2) > 0 (4.22)

ii) v~13l-v(3~ 2v(I3 -v I)-v 3) • v(13) if v(1:})-v(1)-v(3) > 0 (4.23)

iii) vel) (4.24)


iv)

v)

vi)

IV(123l+2v~ll-max[v~23l,v~2l+vf3l) ·v(123) if {123} attractive mt= 4v(12 )-v 1 -max v 23 ,v 2 +v 3

v(l) otherwise

V(12)-v~23)+V(3)

V(13)-v~23)+V(2)

if v(23)-v(3) > 0

if v(23)-v(2) > 0

The highest tentative bound tt of player 1 is the maximum over i) to vi).

For 1 N 2 player 2 has the same highest tentative bound as player 1.

3. Calculation of player 2's highest tentative bound for 1 >- 2.


i) 2V(23~~!~2)f!~3)' v(23) if v(23)-v(2)-v(3) > 0

ii) v(2)

iii ) IV( 1231+2v(2)-max[v(13),V(1)+V(3)). v(123) m2= 4v(12S )-v(2J-max v(13J,v(lJ+v(3)

v(2) otherwise

if {123} attractive

iv) V(12)-V~ 13)+v(3) if v(13)-v(3) > 0

v) V(23)-V~ 13)+v(1) if v(13)-v(1) > 0

The highest tentative bound t2 of player 2 is the maximum over i) to v).

For 2 N 3 player 3 has the same highest tentative bound as player 2.

4. Calculation of player 3's highest tentative bound for 2 >- 3.


i) v(3)

ii) min [v(13)-v(12)+t2' v(23)-v(12)+ttl

(4.25)

(4.26)

(4.27)

(4.28)

( 4.29)

(4.30)

(4.31)

(4.32)

(4.33)

(4.34)


iii) IV( 123~+2vf3l-max[vf12l,vf1l+vf2l]· v(123) m3= 4v(12 ) -v 3 -max v 12 ,v 1 +v 2

v(3)

if {123} attractive

otherwise

iv) V(23)-V~12)+V(1) ifv(12)-v(1) > 0

v) V(13)-V~ 12)+v(2) if v(12)-v(2) > 0

The highest tentative bound t3 of player 3 is the maximum over i) to v).

5. Determination of preliminary bounds

i) Coalition {123} is attractive and tl+h+t3 > v(123)

for i=l, 2, 3

ii) Condition i) is not fulfilled but tl+t2 > v(12) and 1 '" 2 >- 3

Pl = max [ mtz vQ2), v(l) ]

P2 = max [ m2, vQ2), v(2) ]

P3 = t3

iii) Condition i) is not fulfilled, but t2 + t3 > v(23) and 1 >- 2", 3

P2 = max [ m2, V(~3), v(2) ]

P3 = max [ m3, vQ3), v(3) ]

iv) Conditions i), ii) and iii) are not fulfilled

for i = 1, 2, 3

79

(4.35)

(4.36)

(4.37)

(4.38)

(4.39)

(4.40)

( 4.41)

(4.42)

(4.43)

(4.44)

( 4.45)


6. The final bounds

Ui = max [v(i) + 1,6. int E!] for i = 1, 2, 3 (4.46)

All calculations are in terms of permissible coalitions, that means tentative bounds referring

to values of non-permissible coalitions must be left out in the determination of highest

tentative bounds.

Lemma on preliminary bounds

For superadditive games we have

P! + P2 + P3 ~ v(123)

Proof:

( 4.(7)

Consider the case 5.i). Inequality (4.47) holds in view of (4.38). Now consider the case

5.ii). In viw of t! + t2 + t3 ~ v(123) and P3 = t3 it is sufficient to show P! ~ t! and P2 ~ t2·

Since 1 and 2 are equally strong we have v(13) = v(23) and v(l) = v(2). Therefore we have

m! = m2 in view of (4.15). Moreover t! = t2 holds. In view of t! + t2 ~ v{l2) and t! + t2 + t3

~ v(123) inequality (4.(7) holds if v(12)/2 is the maximum of mt, v(12)/2, and v{l). If the,

maximum is mt, then m! ~ t! holds since m! is the maximum of two tentative bounds c! and

v{l). Therefore in this case (4.(7) holds. If v{l) is is the maximum of mt, v(12)/2, and v{l)

then v{l) ~ tt, since v{l) is a tentative bound. We can conclude that (4.(7) holds in case 5.ii).

The case 5.iii) is analogous to 5.ii). Inequality (4.(7) holds here, too. In the case 5.iv)

inequality (4.(7) holds in view oft! + t2 +t3 ~ v(123).


AN EXAMPLE

The computation of the final bounds may look very complicated, but for a specific game

the calculations are very simple. Consider the following example:

40

o 100 100

The triangle describes a three-person characteristic function game, where the values at the

corners are the one-person coalition values, the values at the sides are the two-person

coalition values, and the payoff in the middle is available for the grand coalition.

The computation of the bounds follows the order of strength, therefore the coalition

values have to be ordered such that 1 t 2 t 3.

v(123) = 240

v(12) = 140

v(13) = 100

v(23) = 60

v(l) = 100

v(2) = 40

v(3) = 0

The tentative bounds of player 1 will be calculated by (4.22) to (4.27)

(4.22) not fulfilled


(4.24) 100

(4.25) 240+2.100-max(60z40) 4·240 IOO-max 60,40 . 240 = 114

82

(4.26)

(4.27)

14~0+0 = 40

1O~0+40 = 40

4. Three-Person Bargaining Games

Player l's highest tentative bound is the maximum of (4.22) to (4.27), hence tl = 144.

Player 2's tentative bound will be calculated by (4.28) to (4.33)

60-0 (4.28) 2.60=4O::U· 60 = 45

(4.29) 40

( ) 240+2·40-max[100,100] 4.30 4· 24 O=4O=max[lOO,lOOj . 240 = 64.39

(4.31) 140-~00+0 = 20


The maximum of (4.28) to (4.32) is player 2's highest tentative bound t2 = 64.39.

Player 3's highest tentative bound is the maximum of (4.33) to (4.37):

(4.33) 0


(4.35) 240+0-m ax [14061406-4·2404=max[14 ,14 J . 240 = 29.27


(4.37) 100-140+40 =0 2

hence t3 = 29.27.

In this example the highest tentative bounds of all players are found in the grand

coalition. Further corrections are not necessary and the highest tentative bounds become

preliminary bounds:

PI = 114

P2 = 64.39

P3 = 29.27


Assume that the prominence level within the data set was found to be 10, and the

smallest money unit '1 is equal to 1, then the final bounds are :

UI = max [101, 10 • intlM] = 110,

U2 = max [ 41, 10 • int6fo39] = 60,

U3 = max [ 1, 10 . int2~o 2~ = 20.

The theory of PROPORTIONAL DIVISION PAYOFF BOUNDS predicts that no mem-

ber of a genuine resulting coalition C will receive a payoff Xi which is lower than Ui. Since we

have

UI+U2 > v(12),

UI+U3 > v(13),

U2+U3 > v(23),

only the grand-coalition is predicted in this example with Xl ~ 110, X2 ~ 60, and X3 ~ 20.

This game was played within series 2.2 where the subjects agreed to form the grand

coalition with the final payoff allocation Xl = 125, X2 = 65, and X3 = 50. Our prediction turn

ed out to be correct.

COMPARISON OF PROPORTIONAL DMSION PAYOFF BOUNDS AND EQUAL

DMSION PAYOFF BOUNDS.

The PROPORTIONAL DIVISION PAYOFF BOUNDS are designed for the explanation

of experimental results and can be seen as an extension of SELTEN's theory of the EQUAL

DIVISION PAYOFF BOUNDS which are originally developed for the explanation of zero

normalized superadditive three-person games. Though the PROPORTIONAL DIVISION

PAYOFF BOUNDS are restricted to three-person games too, now there are no restrictions to

superadditivity and to zero-normalization. The EQUAL DIV1SION PAYOFF BOUNDS may


be applied to non-superadditive games and non~ro payoffs to the one-person coalition with

only slight modifications.We have shown that this slightly modified version is more successful

than other theories [SELTEN and UHLICH 1988], but we shall see in the next sections, that

the PROPORTIONAL DIVISION PAYOFF BOUNDS will improve the success of prediction

in a wide range of data. Especially the prediction of payoffs in games with non~ro payoffs to

the one-person coalition is more successful, because SELTEN's EQUAL DIVISION PAYOFF

BOUNDS coincide with PROPORTIONAL DIVISION PAYOFF BOUNDS for most of the

zero-normalized superadditive games. Even if the bounds predicted by both theories are often

identical for zero-normalized superadditive games, they are caused by different arguments.

The lower bounds in the theory of EQUAL DIVISION PAYOFF BOUNDS can be seen as

aspiration levels. In the tradition of limited rationality theory going back to SIMON [1957]

and SAUERMANN and SELTEN [1962] aspiration levels are lower bounds on goal variables.

The theory of EQUAL DIVISION PAYOFF BOUNDS is an attempt to describe common

sense reasons that influence the aspiration levels of the players. The players are portrayed as

satisficing rather than maximizing. A satisficer tries to obtain at least as much as his aspira

tion levels. We agree with most of this, but especially if the games are not zero-normalized,

then the structure of the game may be too complex for the subjects in order to use common

sense arguments for the determination of such minimal aspiration levels. In contrast to SEL

TEN's theory, our lower bounds can be seen as a result of a negotiation process. Even if our

theory looks more complicated than SELTEN's, all calculations, which may be done by the

subjects, are very simple.


4.4 EXPERIMENTAL RESULTS

This chapter is concerned with the evaluation and reevaluation of 49 different experi

ments on three-person characteristic function games. There are 3088 plays of different games

in our data base. 15 data sets with 1989 plays of different games are reported by different

researchers and 34 data sets with 1099 plays of different games were obtained at the Bonn

Laboratory of Experimental Economics. A detailed description of all data sets is given in

section 4.2.2. The results of all experiments conducted at Bonn are listed in appendix C.

All comparisons of predictive power of the theories under consideration will be made

with the help of the success measure M (see section 2.1). Our comparisons apply the two

tailed Wilcoxon matched pairs signed rank test to the success measures. Whenever we say a

result is not significant, we mean that it is not significant on the 10% level.

4.4.1 OVERALL COMPARISONS

In most of the 49 data sets examined here, experimental subjects participated in several

plays of characteristic function games. Therefore it is not justified to look at each of the 3088

plays as an independent observation, even if the same pair of subjects never meets twice.

Though it is possible to locate independent subject groups, in the sense that there was no

interaction between members of different groups, we shall use each data set as only one obser

vation in order to avoid unbalanced weights. For example the data set of MURNIGHAN and

ROTH [1977] contains 36 independent subject groups with 432 plays of only one game, ther~

fore it seems to be reasonable to use this data set as only one observation.

Table 4.3 shows areas, hit rates, and success measures of the BARGAINING SET (BS),

the modified EQUAL EXCESS THEORY (EE), the EQUAL DIVISION PAYOFF BOUNDS

(EDPB), and the PROPORTIONAL DIVISION PAYOFF BOUNDS (PDPB) for every data

set.

Tab

le 4

.3:

Res

ults

of

all d

ata

sets

Bar

gain

ing

Set

E

qual

Exc

ess

The

ory

Dat

a se

ts

hit

rat

e ar

ea

mea

sure

h

it r

ate

area

m

easu

re

Hen

s et

. al

. I

85

0.90

00

0.34

71

0.55

29

0.91

67

0.41

05

0.50

62

Hen

s et

. al

. II

85

0.70

00

0.22

21

0.47

79

0.85

00

0.31

58

0.53

42

Hen

s et

. al

. II

I 85

0.

9062

0.

5126

0.

3936

0.

9921

0.

5298

0.

4623

K

ahan

k la

popo

rt 7

4 0.

5875

0.

2779

0.

3096

0.

8083

0.

3662

0.

4421

K

ahan

k R

apop

ort

77

0.57

82

0.26

78

0.31

04

0.82

82

0.32

46

0.50

36

Kah

an k

lap

opor

t 80

0.

0556

0.

1337

-0.

0781

0.

9556

0.

3293

0.

6263

L

eopo

ld-V

ildbu

rger

85

0.79

63

0.14

41

0.65

22

0.85

18

0.27

33

0.57

85

lasc

hler

78

0.48

71

0.33

31

0.15

40

0.94

87

0.47

38

0.47

49

led

lin

76

0.66

87

0.08

76

0.58

11

0.87

50

0.19

20

0.68

30

lurn

igha

n k

loth

77

0.03

70

0.03

07

0.00

63

0.96

53

0.60

39

0.36

14

Popp

I

0.03

33

0.01

33

0.02

00

0.96

67

0.13

18

0.83

49

Popp

II

0.00

00

0.00

64 -

0.00

64

0.76

67

0.12

42

0.64

25

Rap

opor

t k

Kah

an 7

6 0.

5062

0.

0798

0.

4264

0.

8187

0.

1771

0.

6416

li

ker

67

0.40

86

0.26

16

0.14

70

0.66

66

0.34

30

0.32

36

Sel

ten

k St

oeck

er

0.70

37

0.18

53

0.51

84

0.94

44

0.27

55

0.66

89

Ser

ies

1.1

0.63

88

0.20

25

0.43

63

0.97

22

0.42

04

0.55

18

Ser

ies

1.2

0.67

86

0.20

21

0.47

65

0.98

21

0.41

83

0.56

38

Ser

ies

2.1

0.76

74

0.38

00

0.38

74

0.86

05

0.41

04

0.45

01

Ser

ies

2.2

0.86

66

0.40

23

0.46

43

0.88

89

0.39

70

0.49

19

Ser

ies

3.1

0.64

29

0.28

76

0.35

53

0.85

71

0.25

32

0.60

39

Ser

ies

3.2

0.95

35

0.57

17

0.38

18

0.97

67

0.50

27

0.47

40

Ser

ies

4.1

0.10

00

0.16

18 -

0.06

18

0.95

00

0.37

69

0.57

31

Ser

ies

4.2

0.10

00

0.09

01

0.00

99

0.95

00

0.30

57

0.64

43

Ser

ies

4.3

0.35

00

0.14

27

0.20

73

0.95

00

0.36

66

0.58

34

Ser

ies

4.4

0.00

00

0.04

22 -

0.04

22

0.95

00

0.25

76

0.69

24

Ser

ies

4.5

0.25

00

0.09

01

0.15

99

1.00

00

0.30

57

0.69

43

Ser

ies

4.6

0.30

00

0.16

18

0.13

82

1.00

00

0.37

69

0.62

31

Eq.

Div

. Pa

yoff

Bou

nds

hit

rat

e ar

ea

mea

sure

0.90

00

0.14

95

0.75

05

0.81

66

0.13

36

0.68

30

0.89

85

0.24

66

0.65

19

0.93

33

0.16

77

0.76

56

0.69

38

0.23

72

0.45

66

0.72

22

0.34

33

0.37

89

0.94

45

0.20

47

0.73

98

0.91

03

0.15

28

0.75

75

0.93

12

0.19

77

0.73

35

0.91

44

0.30

70

0.60

74

0.76

67

0.18

62

0.58

05

0.30

00

0.17

83

0.12

17

0.90

62

0.18

06

0.72

56

0.68

82

0.12

43

0.56

39

0.90

74

0.13

05

0.77

69

0.86

11

0.09

42

0.76

69

0.92

85

0.08

98

0.83

87

0.72

09

0.09

39

0.62

70

0.66

67

0.08

10

0.58

57

0.83

33

0.14

68

0.68

65

0.93

02

0.36

35

0.56

67

0.90

00

0.34

89

0.55

11

0.85

00

0.30

90

0.54

10

0.95

00

0.34

45

0.60

55

0.90

00

0.27

95

0.62

05

0.95

00

0.30

90

0.64

10

1.00

00

0.34

89

0.65

11

Prop

. D

iv.

Payo

ff B

ound

s

hit

rat

e ar

ea

mea

sure

0.93

34

0.18

71

0.74

63

0.88

33

0.15

47

0.72

86

0.92

97

0.28

85

0.64

12

0.93

33

0.16

77

0.76

56

0.79

06

0.26

02

0.53

04

0.72

22

0.34

33

0.37

89

0.94

45

0.21

43

0.73

02

0.91

02

0.15

69

0.75

33

0.91

88

0.19

76

0.72

12

0.91

44

0.30

70

0.60

74

0.76

67

0.18

62

0.58

05

0.30

00

0.17

93

0.12

07

0.90

62

0.18

06

0.72

56

0.68

82

0.12

43

0.56

39

0.94

44

0.16

06

0.78

38

0.86

11

0.10

29

0.75

82

0.92

86

0.09

80

0.83

06

0.90

70

0.22

55

0.68

15

0.82

22

0.20

64

0.61

58

0.88

10

0.21

28

0.66

82

0.88

38

0.35

59

0.52

79

0.90

00

0.34

89

0.55

11

0.85

00

0.30

90

0.54

10

0.95

00

0.31

16

0.63

84

0.90

00

0.27

95

0.62

05

0.95

00

0.30

90

0.64

10

1.00

00

0.34

89

0.65

11

00

0

)

~

J-3

I:l" ... 1 ~ '" o =:l t:d e: 0

tI

~.

S.

~

Q ~ '"

(Tab

le 4

.3 c

ontin

ued)

----------

Bar

gain

ing

Set

E

qual

Exc

ess

The

ory

Dat

a se

ts

hit

rat

e ar

ea

mea

sure

h

it r

ate

area

m

easu

re

Ser

ies

5.1

0.35

00

0.17

95

0.17

05

0.95

00

0.36

58

0.58

42

Ser

ies

5.2

0.20

00

0.17

95

0.02

05

0.95

00

0.36

58

0.58

42

Ser

ies

5.3

0.35

00

0.17

95

0.17

05

1.00

00

0.36

58

0.63

42

Ser

ies

5.4

0.15

00

0.17

95 -

0.02

95

1.00

00

0.36

58

0.63

42

Ser

ies

5.5

0.45

00

0.16

03

0.28

97

0.95

00

0.36

58

0.58

42

Ser

ies

5.6

0.25

00

0.17

95

0.07

05

0.90

00

0.36

58

0.53

42

Ser

ies

6.1

0.60

00

0.07

41

0.52

59

1.00

00

0.12

78

0.87

22

Ser

ies

6.2

0.35

00

0.07

41

0.27

59

1.00

00

0.12

78

0.87

22

Ser

ies

6.3

0.15

00

0.03

77

0.11

23

0.95

00

0.09

01

0.85

99

Ser

ies

6.4

0.45

00

0.07

41

0.37

59

1.00

00

0.12

78

0.87

22

Ser

ies

7.1

0.64

28

0.26

19

0.38

09

0.57

15

0.29

23

0.27

92

Ser

ies

7.2

0.66

67

0.26

45

0.40

22

0.61

91

0.24

69

0.37

22

Ser

ies

7.3

0.80

49

0.36

62

0.43

87

0.82

92

0.37

32

0.45

60

Ser

ies

7.4

0.72

09

0.26

46

0.45

63

0.67

44

0.30

42

0.37

02

Ser

ies

7.5

0.70

46

0.26

12

0.44

34

0.54

55

0.24

42

0.30

13

Ser

ies

7.6

0.77

50

0.25

95

0.51

55

0.75

00

0.29

06

0.45

94

Ser

ies

7.7

0.88

64

0.38

49

0.50

15

0.75

00

0.38

47

0.36

53

Ser

ies

7.8

0.65

11

0.26

11

0.39

00

0.55

82

0.24

22

0.31

60

Ser

ies

7.9

0.75

00

0.37

00

0.38

00

0.79

55

0.36

47

0.43

08

Ser

ies

7.10

0.

6818

0.

2682

0.

4136

0.

5000

0.

3018

0.

1982

S

erie

s 7.

11

0.75

00

0.26

12

0.48

88

0.54

55

0.24

42

0.30

13

Ser

ies

7.12

0.

8140

0.

2640

0.

5500

0.

5581

0.

2894

0.

2687

Ave

rage

0.

5136

0.

2131

0.

3004

0.

8549

0.

3165

0.

5384

Eq.

Div

. Pa

yoff

Bou

nds

hit

rat

e ar

ea

mea

sure

0.95

00

0.27

92

0.67

08

0.95

00

0.27

92

0.67

08

1.00

00

0.27

92

0.72

08

1.00

00

0.27

92

0.72

08

0.85

00

0.27

92

0.57

08

0.85

00

0.27

92

0.57

08

1.00

00

0.22

84

0.77

16

1.00

00

0.22

84

0.77

16

0.75

00

0.20

76

0.54

24

1.00

00

0.22

84

0.77

16

0.52

38

0.12

47

0.39

91

0.71

43

0.14

41

0.57

02

0.70

73

0.18

33

0.52

40

0.53

49

0.12

76

0.40

73

0.54

55

0.14

40

0.40

15

0.67

50

0.14

09

0.53

41

0.79

55

0.17

00

0.62

55

0.67

44

0.14

53

0.52

91

0.72

73

0.18

20

0.54

53

0.59

10

0.12

53

0.46

57

0.79

55

0.14

40

0.65

15

0.81

39

0.13

83

0.67

56

0.81

98

0.20

58

0.61

40

Prop

. D

iv.

Payo

ff B

ound

s

hit

rat

e ar

ea

mea

sure

0.95

00

0.29

30

0.65

70

0.95

00

0.29

30

0.65

70

1.00

00

0.29

30

0.70

70

1.00

00

0.29

30

0.70

70

0.85

00

0.26

03

0.58

97

0.85

00

0.29

30

0.55

70

1.00

00

0.18

06

0.81

94

1.00

00

0.18

06

0.81

94

0.75

00

0.16

52

0.58

48

1.00

00

0.18

06

0.81

94

0.78

57

0.17

09

0.61

48

0.92

86

0.16

49

0.76

37

0.80

49

0.22

78

0.57

71

0.69

77

0.17

58

0.52

19

0.81

82

0.16

30

0.65

52

0.60

00

0.16

78

0.43

22

0.95

45

0.22

81

0.72

64

0.86

04

0.16

10

0.69

94

0.90

91

0.22

43

0.68

48

0.84

09

0.17

34

0.66

75

0.90

91

0.16

30

0.74

61

0.88

38

0.16

64

0.71

74

0.87

07

0.22

11

0.64

95

..... ~ g CD

::!

. I e. w

~ "" 00

-'

l


A graphical representation of the average result over the 49 data sets is given in figure

4.4. It can be seen that the PROPORTIONAL DIVISION PAYOFF BOUNDS are more

successful in prediction than other theories. As table 4.4 shows, all comparisons are highly

significant with respect to the Wilcoxon test applied to the success measures of the 49 data

sets.

Q.II

0.11

0.7

... 0.11 ~ • '" Q!I ~

% o.. IU

Q2

0.1

0.0

115 II:

THEORIES

Figure 4.4: Overall comparison

Table 4.4: Significance levels of the overall comparison

Comparison

PDPB-EDPB PDPB-EE PDPB-BS EDPB-EE EDPB-BS EE -BS

Significance

0.0169 0.0002

< 0.0001 0.0017

< 0.0001 < 0.0001


From an earlier study [SELTEN and UHLICH 1988], we already knew that the EQUAL

DIVISION PAYOFF BOUNDS are more successful than the different versions of the

BARGAINING SET. Moreover, it was proved that the UNITED BARGAINING SET, which

will not be examined here, was more successful than the BARGAINING SET, but the

comparison with the EQUAL DIVISION PAYOFF BOUNDS yields a highly significant

difference with respect to the Wilcoxon test in favor of the EQUAL DIVISION PAYOFF

BOUNDS, so there are no reasons to suppose, that this result could change in a wider data

base.

Even if we have theories, which are powerful in the prediction of the results on average,

we have to examine whether this depends on specific types of games. Therefore a classifica

tion seems to be necessary, even if we restrict ourselves to three-person games. From the

literature several terms which serve to differentiate types of games as superadditive and non

superadditive games, constant-sum and non-constant-sum games, quota games, market

games, and veto games are known, but for experimental purposes a sharper differentiation is

necessary. An interesting index, which expresses the power structure in a three-person game,

is the ratio of the sum of the quotas of the two weak players to the quota of the powerful one

[KAHAN and RAPOPORT 1980]. However, this index is restricted to quota games and may

be applicable for games with zero-payoffs to the one-person coalitions. It could be argued

that games with positive payoffs to the one-person coalitions have to be transformed to their

strategically equivalent zero-normalized form in order to apply the index for a classification

of quota games, but from several experimental studies there is strong evidence that the beha

vior of players is not invariant with respect to strategic equivalence. Therefore it is not clear

whether we have to use the quotas from the original game or from the normalized game.

In order to take all games of our data base into account, we shall propose a classification

which is not restricted to quota games or superadditivity and which does not need the hypo

thesis of strategic equivalence. Our classification contains 13 types of games which depend on

the order of the normalized quotas of the game when the power structure of the game is


1 t 2 t 3 accordi~ to the order of strength used for the PROPORTIONAL DMSION PAY

OFF BOUNDS (see section 4.3.5), which is always achievable by a suitable renumbering of

the player numbers. A list of all types is given in table 4.5

Table 4.5: Classification of three-person characteristic function games

Type Order of quotas*

1 ql> q2> q3 2 ql = q2> q3 3 ql> q2 = q3 4 ql = q2 = q3 5 ql> q3 > q2 6 ql = q3 > q2 7 q2> ql > q3 8 q2 > ql = q3 9 q2 > q3 > ql

10 q2 = q3 > ql 11 q3> ql > q2 12 q3> ql = q2 13 q3> q2> ql

* Quotas for the normalized game. A quota may be negative.

Every game with 1 t 2 t 3 belongs to exactly one of these classes. This can be seen as

follows. Ambiguities could arise only in cases i N j for at least one pair of players. For i N j we

have v(ik) = v(jk) and v(i) = v(j); this has the consequence that the normalized quotas qi

and Qj are equal; which means that the exchange of the player numbers i and j does not.

change the type of the game.

Table 4.6 shows hit rates, areas, success measures and the number of observations in

each class for the BS, EE, EDPB, and PDPB. A graphical representation of the average

results from table 4.6 is given in figure 4.5. It can be seen, that according to the classification,

the predictive power of the PROPORTIONAL DMSION PAYOFF BOUNDS increases

while the predictive power of EQUAL DMSION PAYOFF BOUNDS decreases in compari

son to the average success measures over the independent data sets. The significance levels of

all comparisons with respect to the Wilcoxon test are listed an table 4.7, but we have to use

these values carefully, because the results for the different, types of games are not independent

Tab

le 4

.6:

Res

ults

of a

ll ty

pes

of g

ames

Bar

gain

ing

Set

E

qual

Exc

ess

The

ory

Type

O

bsv.

h

it r

ate

area

m

easu

re

hit

rat

e ar

ea

mea

sure

1

1700

0.

5147

0.

2025

0.

3122

0.

8806

0.

3136

0.

5670

2

52

0.92

31

0.29

17

0.63

14

0.92

31

0.43

89

0.48

42

3 58

6 0.

1416

0.

1000

0.

0417

0.

9625

0.

4707

0.

4917

4

20

0.95

00

0.34

43

0.60

57

0.95

00

0.35

53

0.59

47

5 14

6 0.

7192

0.

3204

0.

3988

0.

8630

0.

3714

0.

4916

6

2 1.

0000

0.

2039

0.

7961

0.

5000

0.

2507

0.

2493

7

119

0.72

27

0.29

76

0.42

51

0.71

43

0.40

98

0.30

45

8 18

0.

8333

0.

3808

0.

4525

0.

8889

0.

3870

0.

5019

9

145

0.76

55

0.34

56

0.41

99

0.82

76

0.31

71

0.51

05

10

8 1.

0000

0.

4049

0.

5951

1.

0000

0.

1613

0.

8387

11

74

0.

8649

0.

3306

0.

5343

0.

5946

0.

2795

0.

3151

12

14

0.

6429

0.

2556

0.

3872

0.

5000

0.

1888

0.

3112

13

20

4 0.

7255

0.

2772

0.

4483

0.

4902

0.

1939

0.

2963

Ave

rage

0.

7541

0.

2889

0.

4653

0.

7765

0.

3183

0.

4582

Eq.

D

iv.

Pay

off

Bou

nds

hit

rat

e ar

ea

mea

sure

0.

8624

0.

2112

0.

6512

0.

7500

0.

0152

0.

7348

0.

9027

0.

2747

0.

6280

0.

7000

0.

0279

0.

6721

0.

7192

0.

1940

0.

5252

0.

5000

0.

0020

0.

4980

0.

7143

0.

1984

0.

5159

0.

8889

0.

2252

0.

6637

0.

6414

0.

1452

0.

4962

0.

6250

0.

0732

0.

5518

0.

6622

0.

1231

0.

5391

0.

8571

0.

1991

0.

6580

0.

6618

0.

1678

0.

4940

0.72

96

0.14

28

0.58

68

Pro

p.

Div

. P

ayof

f B

ound

s

hit

rat

e ar

ea

mea

sure

0.

8735

0.

2192

0.

6543

0.

8846

0.

1036

0.

7810

0.

9181

0.

2841

0.

6340

0.

9500

0.

0900

0.

8600

0.

9178

0.

2556

0.

6622

0.

5000

0.

1102

0.

3898

0.

8067

0.

2388

0.

5679

0.

8333

0.

2485

0.

5849

0.

7448

0.

1677

0.

5771

0.

8750

0.

0441

0.

8309

0.

7973

0.

1580

0.

6393

0.

9286

0.

2742

0.

6544

0.

7745

0.

1752

0.

5993

0.83

11

0.18

22

0.64

89

,j:>.

~ g (1

) ::1. S a e. ~

<Il

~

<Il

co ,....


from each other in the sense that some subjects may have played games of different types.

If we look at the game types 6,8, and 12 in table 4.6, it can be seen that the PROPOR

TIONAL DIVISION PAYOFF BOUNDS are not the best predictor for all types of games in

comparison to the EQUAL DIVISION PAYOFF BOUNDS, but we have to be aware, that

this depends only on a small number of observations .

• 0

•• oa

0>

0 .... .. ..... G'

Q.)

C12

G'

... '" II: .". .....

THrOllllCS

Figure 4.5: Overall comparison of classified games

Table 4.7: Significance levels of the overall comparisons according to the different types of games

Comparison

PDPB-EDPB PDPB-BS PDPB-EE EDPB-BS EDPB-EE BS-EE

Significance

0.0592 0.0211 0.0021 0.0253 0.0253

not significant


A surprising result is, that the success measures of the BARGAINING SET and the

modified version of the EQUAL EXCESS THEORY are nearly equal, while the success mea

sures differed highly in the comparison over all independent data sets in favor of the EQUAL

EXCESS THEORY. This result seems to depend on a special type of games where

ql> q2 = q3 (class 3). Excluding this type in the comparison of EQUAL EXCESS THEORY

and the BARGAINING SET yields an average success measure of 0.5006 for BS and 0.4554

for EE, therefore it can be said that the BARGAINING SET is not an adequate theory for all

types of games. However, this may have caused MASCHLER [1963, 1978] to propose that the

BARGAINING SET should not be applied to the characteristic function in its original form,

but to a power transformation of this function. 432 plays of the 586 plays of games in class 3

were obtained by a study of MURNIGHAN AND ROTH [1977] who played the game v(123)

= v(12) = v(13) = 100 and v(23) = v(1) = v(2) = v(3) = o. In this case the BARGAINING

SET coincides with the CORE and predicts a payoff of 100 for player 1 if a genuine coalition

forms. Even if deviations up to a prominence level /). are taken into account, this seems not

to be a reasonable prediction of the outcome. SELTEN [1987] already mentioned, that in view

of the extreme character of this game, one should expect power transformations to be rele

vant. Though the UNITED BARGAINING SET achieves a success measure of 0.2909, the

BARGAINING SET theory for this class of games is not very impressive. Such great differen

ces in the prediction for different types of games cannot be observed for other theories. A

comparison of the standard deviations of the success measures for the different types of games

shows the smallest value for the EQUAL DIVISION PAYOFF BOUNDS (EDPB: 0.0799,

PDPB: 0.1180, EE: 0.1565, BS: 0.1745).

We have proved that the predictive power of the PROPORTIONAL DIVISION PAY

OFF BOUNDS is greater than that of other theories in a comparison over all independent

data sets as well as in a comparison according to our classification of the power structure

within the games. In the next section we shall compare the theories for games with zer<rpay

offs and positive payoffs payoffs to the one-person coalitions separately. Experimental results

of games with negative payoffs to the one-person coalitions are not available to us.


4.4.2 GAMES WITH ZERO AND POSITIVE PAYOFFS TO THE ONE-PERSON

COALITIONS

With the exception of the EQUAL EXCESS THEORY and the PROPORTIONAL

DIVISION PAYOFF BOUNDS, all other solution concepts being examined here are

invariant with respect to strategic equivalence, which means that there is no need to

distinguish between games with zero and non-zero payoffs to the on~person coalitions,

because one ought to be able to transform these games to their zero-normalized strategic

equivalents with a subsequent retransformation after the application of the solution concepts.

Though it was not SELTEN's [1987] intention to propose the EQUAL DIVISION PAYOFF

BOUNDS for games with positive payoffs to the on~person coalition, we found in an earlier

study [SELTEN and UHLICH 1988] that the EQUAL DIVISION PAYOFF BOUNDS, even

if they are applied to the zero-normalized strategic equivalent transformation of games with

positive payoffs to the on~person coalitions, are more successful than different versions of the

BARGAINING SET.

However, this transformation has implications for non-invariant properties such as

equality and the order of player power expressed by the quotas. For example a basic idea of

the EQUAL DMSION PAYOFF BOUNDS are coalition shares, which are equal splits of th,e

coalition values, but this concept, which enhances the intuitive value of this theory, changes

if the theory is applied to a zero-normalized transformation of a game with positive payoffs

to the on~person coalitions to a "split the difference" concept after retransformation. More

over, the computation of the bounds depends on the order of player power, which depends on

comparisons of the coalition values of a game with zero-payoffs to the on~person coalitions.

This gives an order in an intuitively obvious sense and equals the order expressed by the quo

tas, but for games with positive payoffs to the on~person coalitions, it does not seem to be

clear whether the normalized game or the original game should be used to determine the

order of strength. If the game theoretic assumption of stra.t~gic equivalence is accepted, then


the zero-normalized game is the right one to use, but as mentioned above, there is strong

empirical and theoretical evidence against this hypothesis. Because of these potential

problems we shall now analyze games with zero payoffs and positive payoffs to the

one-person coalitions separately.

Figure 4.6 shows average hit rates, areas and success measures for the BS, EE, EDPB,

and PDPB over 25 independent data sets with different plays of games with zero payoffs to

the one-person coalitions.

,D

I~I ... ""

QS04

'" % 0.'

cu Q.2

0.1II1II 0.'

II'l a: ~ I'tJ'U

T"[O~'[S

Figure 4.6: Comparison of games with zero payoffs to the one-person coalitions

It can be seen that the BARGAINING SET performs poorly while the EQUAL DIVI

SION PAYOFF BOUNDS and the PROPORTIONAL DIVISION PAYOFF BOUNDS are

nearly equal in the prediction, which depends on the fact that both theories coincide for for

most of the games with zero payoffs to the one-person coalitions. In 114 plays of different

zero-normalized games both theories predict different low:er bounds. In these cases I


the PROPORTIONAL DIVISION PAYOFF BOUNDS predict on average a greater area

than the EQUAL DIVISION PAYOFF BOUNDS. While the hit rate is the same in both

theories, some hits are in different plays. Among the 25 data sets containing zero-normalized

games, there are only 14 which contain at least one of the 114 plays. We can compare both

theories with the help of the success measures for these data sets, where only the 114 plays

are taken into account. The differences are significant (0.0057) with respect to the Wilcoxon

test in favor of the EQUAL DIVISION PAYOFF BOUNDS. However, the number of plays in

the independent observations is small. The result may change if more plays of this type are

available.

A comparison over all data sets with plays of games with zero payoffs to the one-person

coalitions yields no significant difference between both theories. Of course, this is partially

due to the 11 data sets were both theories agree for all games played.

The EQUAL EXCESS THEORY seems to be a successful theory for this type of games

too, but EQUAL DIVISION PAYOFF BOUNDS and PROPORTIONAL DIVISION PA Y-

OFF BOUNDS are significantly more successful than other theories. The significance levels

with respect to the Wilcoxon test are given in table 4.8.

Table 4.8: Significance levels of the comparison over games with zero payoffs to the one-person coalitions

Comparison

EDPB-PDPB EDPB-EE EDPB-BS PDPB-EE PDPB-BS EE-BS

Significance

not significant 0.0851

< 0.0001 0.0758

< 0.0001 < 0.0001

However, the poor performance of the BARGAINING SET may be caused by the fact,

that a lot of games of this type have an empty core, or the core is a singleton, and for theories


which in such cases predict only small areas as the BARGAINING SET one cannot expect

many hits.

Figure 4.7 shows average hit rates, areas, and success measures for the BS, EE, EDPB,

and PDPB over 24 independent data sets with different plays of games with positive payoffs

to the one-person coalitions. The significance levels with respect to the Wilcoxon test are

given in table 4.9.

It can be seen that the theory of PROPORTIONAL DIVISION PAYOFF BOUNDS has

very high predictive power for games with positive payoffs to the one-person coalitions in

comparison to the other theories examined here. This indicates that proportional divisions of

demands, based on player power and justice norms, rather than equality considerations are

adequate for games with positive payoffs to one-person coalitions. As we have already seen

for two-person games, theories based on "split the difference" concepts are less successful in

prediction than concepts which take the status quo into account. Of course, if the status quo

expressed by the payoffs to the one-person coalitions is equal for all players, especially if they

.0

Og

< .. O!I

i: a4 0.":5S

D.3

112

0..

os a: ~ ~

TH[ORI[ 5

FIGURE 4.7: Comparison over games with positive payoffs to the one-person coalitions


are zero, then concepts based on equality considerations as the EQUAL DIVISION PAYOFF

BOUNDS are very impressive predictors, therefore the PROPORTIONAL DIVISION P AY

OFF BOUNDS coincide with the EQUAL DIVISION PAYOFF BOUNDS for most of such

cases.

Table 4.9: Significance levels of the comparisons over games with positive payoffs to the one-person coalitions

Comparison

PDPB-EDPB PDPB-EE PDPB-BS EDPB-EE EDPB-BS EE-BS

Significance

0.0005 0.0006

< 0.0001 0.0064

< 0.0001 not significant

However, some theories are more successful in prediction of the results of games with

positive payoffs and others are more successful for games with zero payoffs to the one-person

coalitions. In comparison over data sets (each data set taken as one observation), the PRO

PORTIONAL DIVISION PAYOFF BOUNDS are never significantly less successful than any

other theory in both types of games. The same cannot be said about any other theory examin

ed here.

4.4.3 ADDITIONAL HYPOTHESES

From an earlier study [SELTEN and UHLICH 1988] we know that the predictive success

of various theories can be improved by some simple additional hypotheses as the ORDER OF

STRENGTH and EXHAUSTIVITY. These hypotheses were introduced by SELTEN [1972]

as parts of the EQUAL SHARE ANALYSIS.


The study mentioned above contained 11 data sets with 1711 plays of different three

person characteristic function games which all are included within this study . We found a

very strong regularity in the results of the experiments on zero-normalized characteristic

function games, that within a coalition, which is actually formed, a stronger player does not

receive less than a weaker player. Adding this ORDER OF STRENGTH hypothesis in a

version which permits deviations up to the prominence level ll. to the EQUAL DMSION

PAYOFF BOUNDS and to different versions of the BARGAINING SET yield a significant

improvement of the predictive power of all theories under consideration. However, for games

with positive payoffs to the one-person coalitions no improvement was found. This was due

to the fact that a version of the ORDER OF STRENGTH hypothesis was applied which was

invariant with respect to strategic equivalence, since it was based on the zero-normalized

game.

The aim of this section is a replication of the above mentioned study with an enlarged

data base, since more plays of games with positive payoffs to one-person coalitions are avail

able now. Moreover, two additional theories, the EQUAL EXCESS THEORY and the PRO

PORTIONAL DIVISION PAYOFF BOUNDS will be taken into account. One obtains the

combined theories BSO, EEO, EDPBO, and PDPBO if the ORDER OF STRENGTH hypo

thesis is added to the original theories. Whenever a theory has to be applied to the zero nor

malization of a game, the ORDER OF STRENGTH hypothesis will be also applied to the

zero-normalization. Further the ORDER OF STRENGTH hypothesis alone will be tested.

This theory, abbreviated by 0, predicts that within a coalition actually formed a stronger

player receives at most ll. less than a weaker player. For example if coalition C of player i

and j is formed and i t j holds for i,j e C than Xi ~ Xj - ll. holds for the payoffs Xi and Xj of i

and j respectively. 0 will be always applied to the original game.

Table 4.10 gives an overview over the results of each data set. The first part shows all

data sets with plays of games with zero payoffs to the one-person coalitions and the second

part is concentrated to plays of games with positive payoffs to the one-person coalitions.

Tab

le 4

.10:

Com

pari

son

of th

eori

es c

ombi

ned

with

the

OR

DE

R O

F S

TR

EN

GT

H h

ypot

hesi

s

Fir

st p

art:

Gam

es w

ith

v(i)

=

0

Ord

er o

f st

reng

th

Bar

gain

ing

set

Equ

al E

xces

s T

heor

y Eq

. D

iv.

Payo

ff B

ound

s O

rder

of

stre

ngth

O

rder

of

stre

ngth

O

rder

of

stre

ngth

Dat

a se

ts

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

Kah

an k

Rap

opor

t 74

0.

9875

0.

3191

0.

6684

0.

5875

0.

0779

0.

5096

0.

8041

0.

1449

0.

6592

0.

9333

0.

1660

0.

7673

K

ahan

k R

apop

ort

80

0.96

67

0.37

80

0.58

87

0.05

56

0.12

56 -

0.07

00

0.92

22

0.28

56

0.63

66

0.68

89

0.32

16

0.36

73

Leo

pold

-Wild

burg

er 8

5 1.

0000

0.

3687

0.

6313

0.

7963

0.

1348

0.

6615

0.

8518

0.

2384

0.

6134

0.

9445

0.

1999

0.

7446

la

schl

er 7

8 0.

9872

0.

3637

0.

6235

0.

4871

0.

1530

0.

3341

0.

9487

0.

2716

0.

6771

0.

9103

0.

1519

0.

7584

le

dli

n 7

6 0.

9937

0.

3603

0.

6334

0.

6687

0.

0865

0.

5822

0.

8688

0.

1673

0.

7015

0.

9313

0.

1928

0.

7385

lu

rnig

han

k R

oth

77

0.97

46

0.34

99

0.62

47

0.03

70

0.03

02

0.00

68

0.96

07

0.33

35

0.62

72

0.91

44

0.30

70

0.60

74

Popp

I

1.00

00

0.33

62

0.66

38

0.03

33

0.01

24

0.02

09

0.96

67

0.12

78

0.83

89

0.76

67

0.18

20

0.58

47

Popp

II

0.93

33

0.33

20

0.60

13

0.00

00

0.00

55 -

0.00

55

0.73

33

0.12

03

0.61

30

0.30

00

0.17

43

0.12

57

Rap

opor

t k

Kah

an 7

6 0.

9625

0.

3594

0.

6031

0.

5062

0.

0798

0.

4264

0.

8000

0.

1536

0.

6464

0.

8875

0.

1780

0.

7095

R

iker

67

0.98

93

0.31

42

0.67

51

0.40

86

0.06

08

0.34

78

0.66

66

0.12

70

0.53

96

0.68

82

0.12

31

0.56

51

Sel

ten

k St

oeck

er

0.98

15

0.38

50

0.59

65

0.70

37

0.14

69

0.55

68

0.92

60

0.24

14

0.68

46

0.90

74

0.12

26

0.78

48

Ser

ies

1.1

1.00

00

0.45

28

0.54

72

0.63

89

0.16

54

0.47

35

0.97

22

0.32

43

0.64

79

0.86

11

0.09

18

0.76

93

Ser

ies

1.2

0.98

21

0.43

80

0.54

41

0.66

07

0.15

54

0.50

53

0.98

21

0.32

17

0.66

04

0.91

07

0.08

69

0.82

38

Ser

ies

4.1

0.95

00

0.40

36

0.54

64

0.10

00

0.15

60 -

O. 0

560

0.95

00

0.33

00

0.62

00

0.90

00

0.33

37

0.56

63

Ser

ies

4,2

0.95

00

0.37

49

0.57

51

0.10

00

0.08

83

0.01

17

0.95

00

0.27

19

0.67

81

0.85

00

0.29

62

0.55

38

Ser

ies 4.~

0.95

00

0.36

44

0.58

56

0.35

00

0.13

96·

0.21

04

0.95

00

0.32

19

0.62

81

0.95

00

0.33

37

0.61

63

Ser

ies

4.4

0.95

00

0.36

09

0.58

91

0.00

00

0.04

22 -

0.04

22

0.95

00

0.23

75

0.71

25

0.90

00

0.26

94

0.63

06

Ser

ies

4.5

0.95

00

0.37

49

0.57

51

0.25

00

0.08

83

0.16

17

0.95

00

0.27

19

0.67

81

0.95

00

0.29

62

0.65

38

Ser

ies

4.6

1.00

00

0.40

36

0.59

64

0.30

00

0.15

60

0.14

40

1.00

00

0.33

00

0.67

00

1.00

00

0.33

37

0.66

63

Ser

ies

5.1

0.95

00

0.40

22

0.54

78

0.35

00

0.16

88

0.18

12

0.95

00

0.32

27

0.62

73

0.95

00

0.26

01

0.68

99

Ser

ies

5.2

0.95

00

0.40

22

0.54

78

0.20

00

0.16

88

0.03

12

0.95

00

0.32

27

0.62

73

0.95

00

0.26

01

0.68

99

Ser

ies

5.3

1.00

00

0.40

22

0.59

78

0.35

00

0.16

88

0.18

12

1.00

00

0.32

27

0.67

73

1.00

00

0.26

01

0.73

99

Ser

ies

5.4

1.00

00

0.40

22

0.59

78

0.15

00

0.16

88 -

0.01

88

1.00

00

0.32

27

0.67

73

1.00

00

0.26

01

0.73

99

Ser

ies

5.5

0.90

00

0.35

37

0.54

63

0.40

00

0.14

96

0.25

04

0.90

00

0.32

27

0.57

73

0.85

00

0.26

01

0.58

99

Ser

ies

5.6

0.95

00

0.40

22

0.54

78

0.25

00

0.16

88

0.08

12

0.90

00

0.32

27

0.57

73

0.85

00

0.26

01

0.58

99

Ave

rage

0.

9703

0.

3762

0.

5952

0.

3353

0.

1159

0.

2194

0.

9141

0.

2623

0.

6519

0.

8718

0.

2289

0.

6429

Prop

. D

iv.

Payo

ff B

ound

s O

rder

of

stre

ngth

hit

rat

e ar

ea

mea

sure

0.93

33

0.16

60

0.76

73

0.68

89

0.32

16

0.36

73

0.94

44

0.20

92

0.73

52

0.91

02

0.15

53

0.75

49

0.91

88

0.19

14

0.72

74

0.91

44

0.30

70

0.60

74

0.76

67

0.18

20

0.58

47

0.30

00

0.17

52

0.12

48

0.88

75

0.17

80

0.70

95

0.68

82

0.12

31

0.56

51

0.94

44

0.14

62

0.79

82

0.86

11

0.09

85

0.76

26

0.91

07

0.09

27

0.81

80

0.90

00

0.33

37

0.56

63

0.85

00

0.29

62

0.55

38

0.95

00

0.29

80

0.65

20

0.90

00

0.26

94

0.63

06

0.95

00

0.29

62

0.65

38

1.00

00

0.33

37

0.66

63

0.95

00

0.27

37

0.67

63

0.95

00

0.27

37

0.67

63

1.00

00

0.27

37

0.72

63

1.00

00

0.27

37

0.72

63

0.85

00

0.24

15

0.60

85

0.85

00

0.27

37

0.57

63

0.87

27

0.23

13

0.64

14

.... Cl

Cl

~

1-3 ! ~ g to JJ ~. s. ~

Q S fJl

Tab

le 4

.10:

Com

pari

son

of t

heor

ies

com

bine

d w

ith t

he O

RD

ER

OF

ST

RE

NG

TH

hyp

othe

sis

Seco

nd p

art:

Gam

es w

ith

v{i)

> 0

--------

--

----

---

Ord

er o

f st

reng

th

Bar

gain

ing

set

Equ

al E

xces

s Th

eory

Eq

. D

iv.

Payo

ff B

ound

s O

rder

of

stre

ngth

O

rder

of

stre

ngth

O

rder

of

stre

ngth

Dat

a se

ts

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

Hen

s et

. al

. I

85

0.96

66

0.47

20

0.49

46

0.90

00

0.29

75

0.60

25

0.91

67

0.35

11

0.56

56

0.90

00

0.14

38

0.75

62

Hen

s et

. al

. II

85

0.93

33

0.42

24

0.51

09

0.63

33

0.17

86

0.45

47

0.85

00

0.27

20

0.57

80

0.81

66

0.12

92

0.68

74

Hen

s et

. al

. II

I 85

0.

9688

0.

5351

0.

4337

0.

8672

0.

4454

0.

4218

0.

9922

0.

4328

0.

5594

0.

8593

0.

2280

0.

6313

K

ahan

k R

apop

ort

77

0.90

00

0.47

64

0.42

36

0.56

87

0.16

67

0.40

20

0.80

00

0.29

47

0.50

53

0.69

06

0.23

29

0.45

77

Ser

ies

2.1

0.95

35

0.59

03

0.36

32

0.76

74

0.28

47

0.48

27

0.86

04

0.37

45

0.48

59

0.69

77

0.09

23

0.60

54

Ser

ies

2.2

0.93

34

0.58

69

0.34

65

0.86

67

0.31

03

0.55

64

0.88

89

0.36

19

0.52

70

0.60

00

0.07

94

0.52

06

Ser

ies

3.1

0.83

33

0.42

82

0.40

51

0.64

29

0.19

67

0.44

62

0.76

19

0.21

37

0.54

82

0.76

19

0.14

21

0.61

98

Ser

ies

3.2

1.00

00

0.55

53

0.44

47

0.95

35

0.45

37

0.49

98

0.97

67

0.39

31

0.58

36

0.93

02

0.35

37

0.57

65

Ser

ies

6.1

1.00

00

0.25

09

0.74

91

0.60

00

0.07

40

0.52

60

1.00

00

0.10

71

0.89

29

1.00

00

0.22

53

0.77

47

Ser

ies

6.2

1.00

00

0.25

09

0.74

91

0.35

00

0.07

40

0.27

60

1.00

00

0.10

71

0.89

29

1.00

00

0.22

53

0.77

47

Ser

ies

6.3

0.90

00

0.23

99

0.66

01

0.15

00

0.03

77

0.11

23

0.90

00

0.07

89

0.82

11

0.75

00

0.20

62

0.54

38

Ser

ies

6.4

1.00

00

0.25

09

0.74

91

0.45

00

0.07

40

0.37

60

1.00

00

0.10

71

0.89

29

1.00

00

0.22

53

0.77

47

Ser

ies

7.1

0.92

85

0.60

09

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oj:>

.

~ g CD

~. ~ £ ~

"" e. <+ "" ~

o ~


Within this section we are mainly interested whether improvements in prediction can be

obtained by the addition of the ORDER OF STRENGTH hypothesis to the original theories.

Therefore figure 4.8 shows average hit rates, areas, and success measures over all data sets of

the combined theories in comparison to the results already described in section 4.4.1. The

average values are given in table 4.11.

It can be seen, that all theories with exception of the EQUAL EXCESS THEORY have

a smaller success measure if the ORDER OF STRENGTH hypothesis is added to them.

ON

~ Q.II

~ ~ ~ D.II -- ;§ ~ ~ ~ ~ Q7

~ ~ ~ ~ ~ WI

~ f-- , f-eu

~ ~ Q4 ~

~ ~ f-

CI.l t- t-Q2 ~ .. ~

Q' ~ I-

u.u

THEORIES

Figure 4.8: Comparison of theories with and without ORDER OF STRENGTH

Tab

le 4

.11:

Com

pari

son

of th

eori

es w

ith

and

wit

hout

the

OR

DE

R O

F

ST

RE

NG

TH

HY

PO

TH

ES

IS

--------------

---------

Com

paris

on

0 BS

BS

o EE

EE

o

hit

rat

e 0.

9576

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4679

0.

8549

0.

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O

vera

ll

area

0.

4489

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2131

0.

1697

0.

3165

0.

2638

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easu

re

0.50

86

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04

0.29

82

0.53

84

0.57

74

hit

rat

e 0.

9703

0.

3381

0.

3353

0.

9229

0.

9141

G

ames

w

ith v

(i)

= 0

ar

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82

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59

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ames

with

v(i

) >

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54

mea

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4997

---

.. --~

---_

.. _

---

------------

EDPB

ED

PBo

0.81

98

0.79

99

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58

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72

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40

0.60

27

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46

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18

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89

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57

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29

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28

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49

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14

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42

0.59

14

0.56

08

-----

PDPB

0.87

07

0.22

11

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95

0.87

55

0.24

20

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35

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56

0.19

94

0.66

62

PDPB

o

0.85

30

0.20

65

0.64

65

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27

0.23

13

0.64

14

0.83

24

0.18

07

0.65

17

.... ~ g ctl ::l. I eo if a. '" I-'

o ~


Figures 4.9 and 4.10 show the results of the two subsamples for games with and without

zero payoffs to the one-person coalitions. These results completely confirm our earlier find

ings. For games with zero payoffs to the one-person coalitions there is a significant (a = 0.01) improvement of all theories if the ORDER OF STRENGTH hypothesis is added to

them. If the combined theories are applied to games with positive payoffs to the one-person

coalitions (see figure 4.10) we have lower success measures in comparison to the original

theories for BSO, EDPBO, and PDPBO, but the difference is not significant (a = 0.01).

Moreover, we have confirmed the earlier finding that the ORDER OF STRENGTH

hypothesis alone is significantly (a = 0.0001) better than the sophisticated game theoretic

solution concept BS, if applied to games with zero payoffs to the one.person coalitions.

ID~==-----------------------'

OJI

aeH~----t~··~~~~~~

a7~~----;~~~~~

ael-l~------fo.'lool'lH

a. DO~~~~~~~~~~U-~~~

o IISOSOa: aDlIJ'8!IF1lOPIJ'III'CPIlO

THEOR IES

Figure 4.9: Comparison of theories with and without the ORDER OF STRENGTH hypothesis for games with zero payoffs to the one-person coalitions

4.4 Experimental Results

'.0

011

0&

07

06

0.'

().J

02

0.'

00

I--

I--

f--

I-

~

~ ~ -~ ~~- ~ ~

~ ~ ~ ~ ~ P"'" ~

~ ~ ~ f-:;:s

~ ~ ~ f-

f-

f-

l-

f-

OBSlI!DlI:lIDlD'BlD'IDFtFUf'tPOO

THeOR,eS

Figure 4.10: Comparison of theories with and without the ORDER OF STRENGTH hypothesis for games with positive payoffs to the one-person coalitions

105

A remarkable result is that the EQUAL EXCESS THEORY combined with the ORDER

OF STRENGTH hypothesis is more successful than the PROPORTIONAL DIVISION PAY

OFF BOUNDS combined with the ORDER OF STRENGTH hypothesis, if applied to games

with zero payoffs to the one-person coalitions. However, the differences between the most

successful theories EEO, EDPBO,and PDPBO are not significant, but the surprising result of

our version of the EQUAL EXCESS THEORY for this type of games suggests that more

work should be done on this theory. The predicted upper and lower boundaries for every

player in each coalition seem to be appropriate, but the process of expectation formation

underlying this theory does not seem to be plausible.

Up to now we have discussed solution concepts based on the characteristic function it

self, but in fact a game in characteristic function form is an insufficient description of real

game situations. One may believe that various methods in which proposals can be made and

agreements can be reached by different bargaining rules should significantly influence the


outcome of a game. In order to confirm this conjecture it would be desirable that some speci

fic types of games are played under different bargaining rules, but a study like this is not

known to the author. However, our impression is that at least the EQUAL DIVISION PAY

OFF BOUNDS and the PROPORTIONAL DIVISION PAYOFF BOUNDS are not very

sensitive with respect to bargaining rules.

A significant influence on the outcome of characteristic function games was found [SEL

TEN and UHLICH 1988] in a comparison of formalized anonymous communication and free

verbal face to face interaction. The difference concerns the hypothesis of EXHAUSTIVITY

[SELTEN 1972], which requires that the union of several coalitions, which are formed, does

not yield a payoff which is greater than the sum of the payoffs for the coalitions in the union.

Of course the union has to be a permissible coalition. It was found that for games with free

verbal face to face communication adding this hypothesis to the EQUAL DIVISION PAY

OFF BOUNDS and different versions of the BARGAINING SET improves the predictive

success of all theories. The application of these combined theories to games with restricted

formal communication worsens the predictive success significantly. The same results were

found for the PROPORTIONAL DIVISION PAYOFF BOUNDS but will not be reported

here. Some reasons for these results are easy to imagine. It seems to be more difficult to form

larger coalitions under restricted formal communication conditions than under free face to ,

face communication, which offers the possibility to transmit reasons for proposed agreements,

and the opportunity to persuade others. Moreover, negotiators probably find it easier to act

tough if they are not looking at the other persons. On the other hand face to face contact may

facilitate the development of trust.

However, there is a wide field of intermediate conditions between free verbal face to face

communication and restricted communication via computers or by telephone. Because the

reasons for the salient effects of different communication conditions are not completely clear,

more experimentation on this question needs to be done.


4AA THE RELEVANCE OF THE CORE IN GAMES WITH A TIDCK CORE

This section is concerned with the evaluation of games which have a thick CORE in

order to compare the predictive power of the CORE-concept with other solution theories. It

seems not to be a fair comparison if games with an empty CORE will be taken into account.

Moreover, we shall restrict ourselves to games which have a thick CORE, which means that

games have to be superadditive and the following condition holds:

v(123) > fr (v(12) + v(13) + v(23».

Hence, the CORE will always be found in the grand coalition and the CORE is not a

singleton. Under these conditions, ·one should expect that predictions of the CORE are highly

successful. But figure 4.11, which is a graphical representation of the average results from

table 4.12, shows that the predictions of the CORE are only more successful than those of the

BARGAINING SET.

Q11----------t\~

.,.1---~1l'(II

Q.

c -'MCO'"es

Figure 4.11: Comparison of theories for games with a thick CORE

Tab

le 4

.12:

Com

pari

son

of t

heor

ies

for

gam

es w

ith

a th

ick

core

Cor

e B

arga

inin

g S

et

Equa

l E

xces

s T

heor

y

Dat

a se

ts

hit

rat

e ar

ea

mea

sure

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e ar

ea

mea

sure

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sure

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s et

. al

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s et

. al

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hit

rat

e ar

ea

mea

sure

0.96

66

0.17

48

0.79

18

0.80

00

0.13

89

0.66

11

0.98

44

0.30

17

0.68

27

0.97

22

0.20

52

0.76

70

0.93

33

0.12

92

0.80

41

1.00

00

0.20

15

0.79

85

0.93

18

0.14

88

0.78

30

0.79

17

0.08

31

0.70

86

0.92

50

0.08

64

0.83

86

0.90

32

0.18

67

0.71

65

0.78

79

0.17

03

0.61

76

0.90

00

0.19

63

0.70

37

0.87

10

0.34

11

0.52

99

1.00

00

0.43

17

0.56

83

0.50

00

0.33

67

0.16

33

1.00

00

0.43

17

0.56

83

1.00

00

0.28

79

0.71

21

0.75

00

0.33

67

0.41

33

1.00

00

0.43

17

0.56

83

0.95

00

0.29

30

0.65

70

0.95

00

0.29

30

0.65

70

1.00

00

0.29

30

0.70

70

1.00

00

0.29

30

0.70

70

0.85

00

0.26

03

0.58

97

0.85

00

0.29

30

0.55

70

1.00

00

0.19

13

0.80

87

1.00

00

0.19

13

0.80

87

1.00

00

0.17

59

0.82

41

1.00

00

0.19

13

0.80

87

.....

o 00

~

t-3 [ t CD

... § C:j ~ e. s. ~

Q i

(Tab

le 4

.12

cont

inue

d)

Cor

e B

arga

inin

g S

et

Dat

a se

ts

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

Ser

ies

7.1

0.32

36

0.08

06

0.24

30

0.73

53

0.30

19

0.43

34

Ser

ies

7.2

0.44

12

0.07

81

0.36

31

0.70

59

0.30

31

0.40

28

Ser

ies

7.3

0.32

43

0.08

58

0.23

85

0.81

09

0.38

53

0.42

56

Ser

ies

7.4

0.48

57

0.08

02

0.40

55

0.77

14

0.30

40

0.46

74

Ser

ies

7.5

0.41

67

0.07

57

0.34

10

0.72

22

0.29

69

0.42

53

Ser

ies

7.6

0.44

44

0.07

05

0.37

39

0.83

33

0.27

76

0.55

57

Ser

ies

7.7

0.55

55

0.09

70

0.45

85

0.91

67

0.43

00

0.48

67

Ser

ies

7.8

0.62

86

0.07

48

0.55

38

0.74

28

0.29

78

0.44

50

Ser

ies

7.9

0.42

50

0.08

55

0.33

95

0.75

00

0.38

81

0.36

19

Ser

ies

7.10

0.

6111

0.

0819

0.

5292

0.

8055

0.

3073

0.

4982

S

erie

s 7.

11

0.58

34

0.07

57

0.50

77

0.77

78

0.29

69

0.48

09

Ser

ies

7.12

0.

5897

0.

0714

0.

5183

0.

8205

0.

2812

0.

5393

K

ahan

k l

apop

ort

74

Kah

an k

Rap

opor

t 77

K

ahan

'k R

apop

ort

80

lurn

igha

B !

Rot

h 77

N

o

p I

a y

s Po

pp I

Po

pp I

I la

popo

rt k

Kah

an 7

6 R

iker

67

Ave

rage

0.

4074

0.

0561

0.

3513

0.

5917

0.

2604

0.

3313

_._

-

Equ

al E

xces

s T

heor

y Eq

. D

iv.

Payo

ff B

ound

s Pr

op.

Div

. Pa

yoff

Bou

nds

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

0.52

94

0.29

86

0.23

08

0.52

94

0.10

90

0.42

04

0.76

47

0.15

81

0.60

66

0.58

82

0.24

21

0.34

61

0.70

59

0.12

41

0.58

18

0.94

12

0.15

60

0.78

52

0.83

78

0.37

75

0.46

03

0.67

57

0.18

02

0.49

55

0.78

38

0.22

70

0.55

68

0.62

86

0.31

31

0.31

55

0.48

57

0.11

30

0.37

27

0.71

43

0.16

45

0.54

98

0.50

00

0.23

91

0.26

09

0.50

00

0.12

51

0.37

49

0.83

33

0.15

41

0.67

92

0.77

78

0.29

50

0.48

28

0.66

67

0.13

85

0.52

82

0.63

88

0.16

92

0.46

96

0.69

44

0.39

10

0.30

34

0.75

00

0.15

15

0.59

85

0.97

22

0.21

58

0.75

64

0.54

28

0.23

64

0.30

64

0.65

71

0.12

62

0.53

09

0.85

71

0.15

15

0.70

56

0.80

00

0.36

77

0.43

23

0.72

50

0.17

89

0.54

61

0.95

00

0.22

32

0.72

68

0.50

00

0.30

99

0.19

01

0.66

67

0.11

06

0.55

61

0.86

11

0.16

19

0.69

92

0.47

23

0.23

91

0.23

32

0.80

56

0.12

51

0.68

05

0.94

44

0.15

41

0.79

03

0.53

84

0.29

33

0.24

51

0.79

48

0.13

58

0.65

90

0.87

18

0.16

75

0.70

43

of

the

req

uir

ed

t

y p

e

0.85

16

0.33

70

0.51

45

0.82

57

0.20

00

0.62

56

0.89

63

0.22

44

0.67

20

"" ;.,. g (1) ::I. ~ e. ~

rn e. c

+

rn

~

o to


This result is not reliable, because each of the series 4.1 to 6.4 for example contain only a

small number of observations of the required type. Therefore the average over the indepen

dent data sets should not be used for a comparison. Looking at figure 4.12, which is a graphi

cal representation of averages obtained from table 4.13 shows a different result if we classify

the games in 13 classes as described in section 4.4.1 (table 4.5).

'0

09

08

07

.... as

'" .. 05

: o. 03

02

O.

c as IE

TH(O~'(S

Figure 4.12: Comparison of theories for games with a thick CORE according to our

classification

Now the predictions of the CORE are significantly more successful than the predictions

of the BARGAINING SET and the EQUAL EXCESS THEORY. However, even if we con

struct ideal conditions for the CORE, it performs poorly in comparison to the PROPORTIO

NAL DIVISION PAYOFF BOUNDS. All significance levels are listed in table 4.14.

Tal

e 4.

13:

Com

pari

son

of th

eori

es f

or c

lass

ifie

d ga

mes

Cor

e B

arga

inin

g S

et

Type

O

bsv.

h

it r

ate

area

m

easu

re h

it r

ate

area

m

easu

re

1 46

3 0.

5227

0.

0600

0.

4627

0.

6674

0.

2708

0.

3966

2

39

0.71

79

0.10

98

0.60

82

0.89

74

0.33

49

0.56

25

3 62

0.

5323

0.

0853

0.

4470

0.

5968

0.

3021

0.

2947

4

13

0.92

31

0.19

68

0.72

62

1.00

00

0.47

23

0.52

77

5 99

0.

5859

0.

0942

0.

4917

0.

7677

0.

3884

0.

3793

6

2 1.

0000

0.

0154

0.

9846

1.

0000

0.

2039

0.

7961

7

73

0.43

84

0.06

01

0.37

82

0.80

82

0.30

73

0.50

09

8 13

0.

6154

0.

0923

0.

5231

0.

7692

0.

4063

0.

3629

9

73

0.61

64

0.10

46

0.51

18

0.82

19

0.42

51

0.39

68

10

8 0.

8750

0.

0806

0.

7944

1.

0000

0.

4049

0.

5951

11

,

74

0.55

41

0.08

15

0.47

26

0.86

49

0.33

06

0.53

43

12

~

10

0.40

00

0.06

71

0.33

29

0.60

00

0.30

45

0.29

55

13

113

0.51

33

0.07

95

0.43

37

0.80

53

0.30

82

0.49

71

Ave

rage

0.

6380

0.

0867

0.

5513

0.

8153

0.

3430

0.

4723

Equ

al E

xces

s T

heor

y Eq

. D

iv.

Pay

off

Bou

nds

hit

rat

e ar

ea

mea

sure

hit

rat

e ar

ea

mea

sure

0.92

66

0.37

72

0.54

93

0.86

83

0.19

54

0.67

29

0.89

74

0.46

34

0.43

40

0.74

36

0.01

79

0.72

57

0.93

55

0.40

46

0.53

09

0.79

03

0.17

78

0.61

26

1.00

00

0.45

54

0.54

46

0.69

23

0.00

47

0.68

76

0.83

84

0.38

80

0.45

03

0.65

66

0.16

57

0.49

08

0.50

00

0.25

07

0.24

93

0.50

00

0.00

20

0.49

80

0.63

01

0.37

37

0.25

65

0.75

34

0.15

95

0.59

39

0.84

62

0.39

10

0.45

52

0.92

31

0.21

87

0.70

44

0.80

82

0.31

10

0.49

73

0.78

08

0.12

28

0.65

81

1.00

00

0.16

13

0.83

87

0.62

50

0.07

32

0.55

18

0.59

46

0.27

95

0.31

51

0.66

22

0.12

31

0.53

91

0.40

00

0.19

08

0.20

92

0.80

00

0.22

37

0.57

63

0.34

51

0.14

83

0.19

68

0.71

68

0.13

57

0.58

11

0.74

79

0.32

27

0.42

52

0.73

17

0.12

46

0.60

71 P

rop.

D

iv.

Pay

off

Bou

nds

hit

rat

e ar

ea

mea

sure

0.92

87

0.23

11

0.69

76

0.87

18

0.09

86

0.77

32

0.93

55

0.21

77

0.71

78

1.00

00

0.09

13

0.90

87

0.91

92

0.25

81

0.66

11

0.50

00

0.11

02

0.38

98

0.78

08

0.19

16

0.58

92

0.84

62

0.24

78

0.59

83

0.83

56

0.14

57

0.68

99

0.87

50

0.04

41

0.83

09

0.79

73

0.15

80

0.63

93

0.90

00

0.27

63

0.62

37

0.84

07

0.13

20

0.70

87

0.84

85

0.16

94

0.67

91

~

..... ~ CD

::1

. S a e:..

ffii g. '" .... .... ....


Table 4.14: Significance levels of the comparisons over classified. games with a thick CORE

Comparison

PDPB-EDPB PDPB-C PDPB-BS PDPB-EE EDPB-C EDPB-BS EDPB-EE C-BS C-EE BS-EE

Significance

0.0592 0.0253 0.0210 0.0021

not significant 0.0253 0.0119 0.0360 0.0303

not significant

In the section above we discussed. significant effects on the outcome of plays, if they are

played under free verbal face to face communication conditions. Our interpretation was, that

it seems to be more difficult to form larger coalitions under restricted formal communication

conditions than under free face to face communication. Comparing the success measures of

the CORE for the first five data sets from table 4.12, which were obtained. from experimental

procedures using unrestricted. communication conditions, with the remaining data sets sug

gests that the predictions of the CORE are highly successful if communications are unrestric

ted and it performs poorly under restricted communication conditions. It should be noted.

that the data set "Hens et. al. II" was obtained by a computerized procedure but communica

tion with free verbal written messages was possible. However, up to now it is not completely

clear whether the high success measures of the first five data sets depend on communication

conditions or on the size of the Core. More experimentation with various communication

conditions and variation of the size of the CORE is necessary to answer this question.


4.4.5 EXPERIENCE OF NEGOTIATORS IN BARGAINING GAMES

The aim of this section is to examine whether the subjects' experience effects the results

of experimental three-person games. All subjects, which have participated in series 7.7 to

7.12, where experienced subjects in the sense that they had participated in one of the other

experimental sessions 1.1 to 7.6 before. We shall compare the results of series 7.7 to 7.12 with

those of series 7.1 to 7.6. The games played in both sub samples are identical, but one set is

played by experienced subjects and the other is played by unexperienced subjects. Each sub

sample consists of 6 independent data sets .

Figure 4.13 shows the average success measures for all theories in comparison. If a

column refers to the abbreviation" IU", we mean that the success measures for the specified

theory was obtained by the group of unexperienced players, and "IE" stands for experienced

players.

0

7

011

011

G

G6

Q5

I-P::;;; I-

I-

II-

~

~ CI!iD'II

b.m ~ ~

THeORI[S

Figure 4.13: Experienced and unexperienced negotiators

1170911

------


It can be seen that with exception of the EQUAL EXCESS THEORY all theories are

more successful in prediction if games were played by experienced negotiators. However, the

differences for the BARGAINING SET and the EQUAL EXCESS THEORY are not signifi

cant, while the success measures for the EQUAL DIVISION PAYOFF BOUNDS and the

PROPORTIONAL DIVISION PAYOFF BOUNDS significantly increase with respect to the

Wilcoxon test (Significance levels: EDPB!E-EDPB!U 0.0935; PDPB!E-PDPB!U 0.05917).

Table 4.15 shows, that experienced players have an increased tendency to form the grand

coalition instead of a two-person coalition. Therefore, one should expect that the average

gain for each player position should increase, due to the fact that within the grand coalition

more money is to be distributed.

Table 4.15: Relative coalition frequencies

Coalition Unexperienced Experienced

123 34.40% 49.01% 12 26.99% 18.47% 13 18.90% 16.83% 23 19.71% 15.69%

The grade of efficiency will be defined as a fraction the three players on average gain~d

of the most valuable grand coalition. From table 4.16 it can be seen, that the grade of effi

ciency increases if experienced subjects negotiate, but the additional reward is mainly gained

by the strongest player.

Table 4.16: Grade of efficiency

Player 1 Player 2 Player 3 Total

Unexperienced

45.32 28.95 19.56 93.83

Experienced

47.83 28.94 18.85 95.62

4.5 Summary on Three-Person Games 115

However, a theory like our version of the EQUAL EXCESS THEORY, which has a

worsened predictive success if experienced players are acting in contrast to games played by

unexperienced subjects, seems to be very doubtful for descriptive purposes. The great area,

predicted by that theory may have the advantage that mistakes and clumsiness of unex

perienced subjects is taken into'account. Nevertheless other theories are more successful.

4.5 SUMMARY ON THREE-PERSON GAMES

Chapter 4 was mainly concerned with the presentation of the new descriptive theory of

PROPORTIONAL DIVISION PAYOFF BOUNDS. In comparison with other descriptive

and normative theories the PROPORTIONAL DMSION PAYOFF BOUNDS were found to

be significantly more successful in prediction if our whole data basis is taken into account.

Even if the construction of our new theory is partially based on an ex post analysis of 3088

plays of different three-person characteristic function games, there are no reasons to believe

that the prediction will be worse for further experiments. In contrast to normative theories,

deSCriptive theories can only be constructed in the light of data.

The comparison of the predictive success of different theories has to be done carefully,

because the results may depend on the types of games under consideration. First of all an

appropriate measure as the success measure M [SELTEN and KRISCHKER 1983, SELTEN

1989], which takes the size of the predicted area into account, has to be used. Moreover, if

subjects have participated in several plays of characteristic function games it is not justified

to look at each play as an independent observation, even if the same pair of subjects never

meets twice. Though it may be possible to locate independent subject groups, in the sense

that there is no interaction between members of different groups, within one data set, we may

get unreliable weights if a certain type of games is played often. This obviously occurs in the

study of MURNIGHAN and ROTH [1977]. Their data set contains 36 independent subject


groups in 432 plays of only one game. Therefore one could consider to use only each data set

as an independent observation, but even this will not protect us from getting wrong results.

Many researchers conduct experiments to answer specific questions and therefore only certain

types of games are constructed for the experiment, but our issue is a comparison of different

theories and we need equal weights on all possible parameter constellations of three-person

characteristic function games. However, to achieve this is a very tedious task, and it may be

sufficient to have an appropriate classification. Our proposed classification contains 13 classes

of games which depend on the order of strength and the normalized quotas. Using the aver

ages over all plays of a specific class as only one observation shows that the PROPORTION

AL DIVISION PAYOFF BOUNDS are significantly more successful than other theories. This

was also found if each data set was used as one observation. The results for the BARGAI

NING SET and our version of the EQUAL EXCESS THEORY are different. While in com

parison over all data sets the EQUAL EXCESS THEORY was found to be Significantly more

successful than the BARGAINING SET, there was no significant difference if the classifica

tion was used.

An exceptional superiority of the PROPORTIONAL DIVISION PAYOFF BOUNDS

can be seen in the predictive power for games with positive payoffs to one-person coalitions.

The differences in the success measure in comparison to other theories are highly significant.

This may depend on the fact that the other theories, with exception of the EQUAL EXCESS

THEORY, have to be applied to the so called "strategic equivalent transformation" of the

games, but the behavior of players is not invariant with respect to strategic equivalence.

Therefore theories designed for zero-normalized games should only be applied to such games.

For games with zero-payoffs to the one-person coalitions the theory of EQUAL DIVISION

PAYOFF BOUNDS was found to be more successful than other theories, but the differences

between EQUAL DIVISION PAYOFF BOUNDS and PROPORTIONAL DIVISION PAY

OFF BOUNDS are very small. While both theories coincide for most zero-normalized games,

in 114 plays they predict different lower bounds. Among t1te 25 data sets containing zero-nor-

4.5 Summary on Three-Person Games 117

malized games, there are only 14 which contain at least one of the 114 plays. A comparison of

both theories over these data sets, where only the 114 plays are taken into account, yields a

significant difference in favor of the EQUAL DMSION PAYOFF BOUNDS. While the hit

rate is the same for both theories (some hits are in different plays), the EQUAL DMSION

PAYOFF BOUNDS predict a smaller area on average over the considered plays. This result

may be different if more plays of this type are available. However, for zero-normalized games

both theories are significantly more successful than the BARGAINING SET and our version

of the EQUAL EXCESS THEORY. It would be desirable to replace the theory of EQUAL

DIVISION PAYOFF BOUNDS by the more general PROPORTIONAL DIVISION PAY

OFF BOUNDS in order to have only one theory for different types of games.

The predictive power of all theories under consideration can be improved, if they are

combined with the order of strength hypothesis and applied to games with zero-payoffs to the

one-person coalitions. For games with positive payoffs to one-person coalitions the predictive

power of all theories, with exception of our version of the EQUAL EXCESS THEORY, wors

ens. This may depend on the fact, that such games are too complex for the subjects. It seems

to be difficult to recognize the individual power. However, for games with zero-payoffs to the

one-person coalitions a remarkable result was found for the EQUAL EXCESS THEORY

combined with the order of strength hypothesis. This theory was more successful than

EQUAL DIVISION PAYOFF BOUNDS and PROPORTIONAL DMSION PAYOFF

BOUNDS. Though the differences are not significant, the result suggests that more work

should be done on this theory. Another remarkable result, which already was found earlier,

confirms that the very simple theory of the ORDER OF STRENGTH alone is more success

ful than the sophisticated BARGAINING SET.

The evaluation of a subsample of the complete data basis, which contained only plays of

three-person characteristic function games with a thick core, shows that even if normative

theories as the CORE and the BARGAINING SET have thej.rtbest chances, the predictive

power of both is significantly lower than the predictive power of the descriptive PROPOR-


TIONALDIVISION PAYOFF BOUNDS.

We already mentioned that the structure of games with positive payoffs to one-person

coalitions may be too complex for an analysis of the subjects, therefore one should expect that

the predictive power of theories improves, if subjects have experience in characteristic func

tion bargaining. Such an improvement was found to be significant for the EQUAL DIVISION

PAYOFF BOUNDS and the PROPORTIONAL DMSION PAYOFF BOUNDS. Even if the

differences for the BARGAINING SET and our version of the EQUAL EXCESS THEORY

are not significant, it is interesting that only the predictive power of the EQUAL EXCESS

THEORY decreases for experienced subjects. This may be interpreted as due to the fact that

the big predicted area for the two-person coalitions takes mistakes and clumsiness of unexper

ienced players into account, but for experienced players, who have an increased tendency to

form the more efficient grand coalition, the boundaries do not seem to be appropriate.

5. SUMMARY AND CONCLUSION

The aim of this book was an extensive examination of experimental three-person charac

teristic function bargaining. We can summarize our study as follows:

1) We had to develop two sets of computer programs for the experimentation of two

and three-person characteristic function bargaining games. Moreover, a program for the

evaluation of the data was developed. All programs are written in Turbo-Pascal and consist

together of nearly 17000 lines of source code.

2) In 26 experimental sessions 1099 plays of different three-person characteristic function

games and in two sessions 59 plays of different two-person characteristic function games were

obtained. Moreover, we collected data obtained by different researchers. Our data base on

three-person characteristic function games now consists of 3088 plays of different games.

3) A new descriptive theory for experimental two-person characteristic function games

could be presented. The NEGOTIATION AGREEMENT AREA derives lower bounds for the

outcome from a proportional division scheme. Comparisons with other solution concepts show

the appropriateness of these bounds.

4) We presented the new descriptive theory of the PROPORTIONAL DMSION PAY

OFF BOUNDS for experimental three-person characteristic function games. As in our theory

for two-person games, bounded1y rational behavior in human decision making is explicitly

taken into account. Important to both theories are justice norms and power considerations.

"Subjectively expected utility maximization" does not enter as an explanatory principle.

5) For the comparison of our theory with other descriptive and normative theories, a

short description ofthe CORE, the BARGAINING SET, and the EQUAL DMSION PAY

OFF BOUNDS has been given. Moreover, we presented a modified version of the EQUAL

EXCESS THEORY.

120 5. Summary and Conclusion

6) In order to avoid unbalanced weights of too many observations of specific types of

games, we proposed a classification of three-person characteristic function games. Theories

should be able to explain the results of games in all classes.

7) The theory of PROPORTIONAL DIVISION PAYOFF BOUNDS is significantly

more successful in the prediction of the outcome than other theories if all games are grouped

according to our proposed classification. The same is true over all 49 data sets if each data set

is used as only one observation. While the PROPORTIONAL DIVISION PAYOFF

BOUNDS are significantly more successful than other theories if only games with positive

payoffs to one-person coalitions are taken into account, there is no significant difference

between EQUAL DIVISION PAYOFF BOUNDS and PROPORTIONAL DIVISION PAY

OFF BOUNDS for games with zer<rpayoffs to one-person coalitions. The last result depends

on the fact that both theories coincide for most of the zer<rnormalized games. This result

changes if only those plays of games are taken into account, where both theories predict dif

ferent lower bounds. In this case a Significant advantage of the EQUAL DIVISION PAYOFF

BOUNDS over the PROPORTIONAL DIVISION PAYOFF BOUNDS was found. However,

the number of plays in the independent observations is small. The result may change if more

plays of this type are available. Over all zer<rnormalized games the PROPORTIONAL DI

VISION PAYOFF BOUNDS and the EQUAL DIVISION PAYOFF BOUNDS are signifi

cantly more successful than the BARGAINING SET and our version of the EQUAL EXCESS

THEORY. The predictive power of all theories can be improved if they are combined with

the order of strength hypothesis and applied to zer<rnormalized games. For this type of

games our version of the EQUAL EXCESS THEORY combined with the order of strength

hypothesis is the best predictor, but the differences in the success measures between all de

scriptive theories are not significant. The normative theories (CORE, BARGAINING SET)

pedorm poorly, even if only games with a thick core are taken into account. Moreover, the

simple theory of the ORDER OF STRENGTH HYPOTHESIS alone is more successful than

the sophisticated BARGAINING SET. Experienced players have greater tendencies to form

121

the more valuable grand coalition instead of two-person coalitions. The predictive power of

theories, with the exception of the EQUAL EXCESS THEORY, increases if experienced

players are acting.

Our new descriptive theories for two- and three-person characteristic function games are

compared with some other theories. Of course, there are more th~ries in the literature, but

the computation of the success measures is very tedious even with a personal computer. How

ever, we think the most interesting theories are taken into account.

The purpose of this work was primarily concerned with the formulation of a descriptive

theory for three-person characteristic function games. Normally this should include a theory

for two-person games, but we decided to analyze these types of games separately to get clear

results. Therefore the players, according to our bargaining rules, were not allowed to form a

one-person coalition, such that the three-person game reduces to a two-person game. When

there will be more data available for two-person games, and if these data confirm the pre

dictive power of the NEGOTIATION AGREEMENT AREA, a next step will be the com

plete implementation into the PROPORTIONAL DIVISION PAYOFF BOUNDS and an

experimental testing of bargaining rules which allow the formation of one-person coalitions.

Between two- and three-person games, there is a basic qualitative difference in the nature of

the con:flict situation. This difference concerns the possibility of coalition formation, by which

a group may achieve control over outcomes. When moving to games with n>3 the differences

may be smaller. However, it is a very difficult task to construct theories for n-person games

about how individuals do behave in contrast to theories about how individuals ought to be

have.

A main weakness of nearly all existing theories can be seen in the fact that not enough

attention is paid to the bargaining process leading to a given outcome. A detailed consider

ation of such processes may lead to an understanding of the variability of the outcomes of

characteristic function games.

APPENDIX

A. INTRODUCTION TO THE RULES AND THE EXPERIMENTAL APPARATUS OF

A TWO-PERSON BARGAINING EXPERIMENT

Immediately after these instructions we will enter the laboratory, there you will find

numbered cubicles with a personal computer in each of them. Everybody has a sheet in front

of you for your own purposes. On the top of the sheet you will find a number. In the labora

tory you have to enter the cubicles with your number.

Each of you will participate in xx different situations. From all participants we shall

randomly select groups with two subjects in each. After the end of negotiations you will be

grouped in a new dyad. Therefore every new situation can be seen as independent from the

others. All negotiations will be completely anonymous. You will not be told, who the other

negotiators are. Sometimes experiments are conducted where the subjects play against com

puters. You are not playing against a computer. The personal computers will only be used to

communicate with other negotiators.

The description of our bargaining situation is very easy. Two players have to negotiate

about the allocation of a certain money payoff, which we will call coalition value. The negO

tiations end if the proposed allocation of the value made by one player is accepted by the

other player, or if one player decides to break off the negotiations, then each player receives a

guaranteed payoff.

To make these completely clear we shall now go through an example. At the beginning of

the negotiations the screen of your computer display will look like figure A.1 (The payoffs

will be different.)

A. Introduction to the Rules of Two-Person Experiments 123

TWO PERSON BARGAINING GAIlE - TERMINAL PROGRM :

- aJRRENT PROPOSAl - -PAYOffS- I D::: I Proponl My Her/Hla MY HER/HIS No:11o'ho1 lIIere lIIere

73 115 277 PLAYER : ,

HISTORY COALITION GAIlE Ho. : 1

I ACCEPT II SHIft II PROPOSE I I BREAK Offl

PgUp/PgDn : PIIiI8 Up/Dn In} HanelEnd : Beglmlng/End of hlatory tIl : Scroll Up/Dn In "1+ : Select action <-oJ : Enter Esc: : Esc:ape Del : Delete last digit Space Bar: Delete whole II1lUt

FigureA.l

In the window entitled lip A YOFFS", the coalition value and your guaranteed payoffs are

displayed. The top left window shows the outstanding proposal and the window below lists

the complete history of the negotiations. The bottom right window describes all available

keys and their actions. A menu bar can be found in the middle of the right half of the screen.

You have four options if you are in the "decide state". The current state is displayed in the

top right window. The four options are "ACCEPT, SmFT, PROPOSE, and BREAK OFF".

If you want to select a specific option, use the II-I, 1-11 keys to highlight the desired option

and press the E:J key. Of course the "ACCEPT" option is only available if there is an out

standing proposal by the other player.

Suppose you are randomly selected to make a proposal. A pop up window appears and

you may enter your demand. You have to enter only your demand, the remainder of the

coalition value will automatically be offered to the other player (figure A.2)

124 Appendix

--! TWO PERSOII BARGAINING GAME • TERMINAL PROGRAM :

r-- CURRENT PROPOSAL - ,-PAYOFFS- [O:;:J prcposal My Her/His MY HER/HIS NO.] 111107 share share

73 115 277 PLAYER : 1

HISTORY COALITION GAME No. : 1

I ACCEPT II

SHIFT II PROPOSE II BREAK OFF I

My Her/Hfs pIOn fn} share share ina/End of history

UplOn in 250 27 action

Esc : Escape Del : Delete last digit Space Bar : Delete whole Input

Figure A.2

We suppose a demand of 250 points. The remaining 21 points will be offered to the other

player. Now the [2J key has to be pressed and a new window, the "confirmation-window",

appears (figure A.3). You have to confirm every decision I

TIIO PERSOII BARGAINING GAME • TERMINAL PROGRAM :

.-- QJRRENT PROPOSAL - ,-PAYOFFS-

I D:;: I prrsal My Her/His MY HER/HIS No. 111107 share share

73 115 277 PLAYER : 1


I ACCEPT II

SHIFT I

PROPOSE IBREAK OFFI

Are you My Her/His pi sure 7

share share in history U <V,n>

250 27 a

Esc : Escape Del : Delete last digit Space Bar : Delete whole input

Figure A.3

A. Introduction to the Rules of Two-Person Experiments 125

Now your proposal will be send to the other player and you get into the "wait-iitate".

T\IO PERSON BARCAINIIIG GAME • TERMINAL PROGRM

;-- aJRRENT PltOPOSAL - r-PAYOFFS-

[S:::J Prcpoul ICy Her/His MY HER/HIS No~l Who? share share

73 115 t I 250 27 277

~ HISTORY COALITION GAME No.: t

t I 250 27 II PROPOSE II BREAK OFF I I ACCEPT II SHIFT

PsUp/PgDn : Page Up/On In} Home/End : &eglmlng/End of history , / 1 : Scroll Up/On In .. / ... : Select action <-l : Enter Esc : Escape Del : Delete list dlgl t Space aIr : Delete whole l."ut

Figure A.4

The screen of the other player will look like figure A.5. He is the new initiator and is free

to select any of his options, but it may not be reasonable to accept the proposal, because he

T\IO PERSON BARGAINING GAME • TERMINAL PROGRM :

r--- CURRENT PROPOSAL - r-PAYOFFS- I D::: I ProposIl My Her/His MY HER/HIS No.1 Who? shIre shIre

115 73 1 She/He 27 250 277 PUYER : 2


1 She/He 27 250 I ACCEPT II SHIFT II PltOPOSE II BREAK OFF I PgUp/PgDn PIge Up/On In} Home/End aeglmlnglEnd of history , / 1 Scroll Upton In ./ ... Select action <-l Enter Esc Escape Del Delete lIst digit Space aIr Delete whole lI"flUt

Figure A.S

126 Appendix

will receive 115 points, if the negotiations break down. We suppose he does not want to make

an own proposal, but he shifts the initiative to the first player.

Now the first player is initiator again and he makes a concession of 10 points.

TWO PERSON BARGAINING GAME • TERMiNAl PROGRNI

r-- aJRRENT PROPOSAL - ~ PAY 0 F F S --- I 0:;: I Proposal My I Her/Hia MY HER/HIS NO:/""'07 share share

73 115 2 she/he has shifted 217 PLAYER : 1

HISTORY COM.ITIOII GAME No. : 1

1 I 250.! 27 2 shethe has shifted I ACCEPT II SHIFT II PRWOSE I I BREAK OFFI

P~p/PfI)n : Page UpIDn in} Home/End : Begimil'llllEnd of hiatory t / , : Scroll IIpIDn in .. /~ : Select action c-J : Enter Esc : Escape Del : Delete laat dilit space Bar : Delete whole flllUt

Figure A.6

However, it may take some time to achieve an agreement. We suppose that the other

player broke off the negotiations and both players receive only their guaranteed payoffs.

A. Introduction to the Rules of Two-Person Experiments

T\IO PERSON BARGAINING GAME - TERMINAL PROGRAM •

roo-- CURRENT PROPOSAL - r-- RESULT -[S:::J Proposal My Her/Hfs MY HER/HIS

NO:/IIIIO? share share 73 115

3 I 240 37 PLAYER : 1 He hn aborted

HISTORY GAME 110. : 1

, I 250 J 27 I ACCEPT II II PROPOSE II BREAK OFF I 2 Ihelhe ha. shifted SHIFT 3 I 240 37

PgUptpgOn : Page Up/On In} Home/End : Begimlng/End of t / 1 : Scroll ,Upton In .. / ~ : Select 8Ctlon <-oJ : Enter Esc : Esc:.pe Del : Delete lnt digit Space Bar : Delete whole 1rf)Ut

Figure A.7

Are there any questions?

There are no time constraints.

history

Try to maximize your own profit. Each point is worth x.xx DM.

Good luck.

127

128 Appendix

B. INTRODUCTION TO THE RULES AND THE EXPERIMENTAL APPARATUS OF

A THREE-PERSON BARGAINING EXPERIMENT

Immediately after this instructions we will enter the laboratory, there you will find num

bered cubicles with a personal computer in each of them. Everybody has a sheet in front of

you for your private bookkeeping. On the first page you will find a number. In the laboratory

you have to enter the cubicle with your number.

Each of you will participate in xx different situations. From all participants we will

randomly select groups with three subjects in each, who will negotiate with each other. After

the end of negotiations you will be grouped in a new triad and there will be at least one

subject in it, you never met before. No triad will meet again. Therefore every new situation

can be seen as independent from the others. All negotiations will be completely anonymous.

You will not be told, who the other negotiators are. Sometimes experiments are conducted

where the subjects play against computers. You are not playing against a computer. The

personal computers will only be used to communicate with other negotiators. The bargaining

situation will be described by a triangle (figure B.I). The numbers in brackets are the names

of the other negotiators. This will be always the numbers I, 2, and 3.

FigureB.I

30 (2)

(1)

20

110 50 (3)

B. Introduction to the Rules of Three-Person Experiments 129

The other numbers are payoffs to certain coalitions. A value in the middle of the triangle is a

payoff which can be gained by a coalition of negotiators 1,2 and 3. Values of the sides are

payoffs for coalitions of two negotiators. For example 90 points can be gained by a coalition

of 1 and 3, 110 points for 2 and 3, and 150 for 1 and 2. If at the end of the negotiation a

two-person coalition forms, then the third negotiator who is not member of the coalition

receives the value at his comer. The aim of the negotiations is to form a coalition, in which

you can ma.ximize your profit. The negotiations end, if all members of a proposed coalition

accept a proposed distribution of the coalition value. Each point of the agreed share will be

worth x.xx DM for you.

Now let us have a look at an example. At the beginning of the experiment the screen of

your personal computer will look like figure B.2.

Dear participant I You are negotiator in a three-person bargaining situation. As soon as the next turn begins, all coalition values will be displayed In the top left window. Try to maximize your profitl lie wish good luck.

Press any key to start negotiations I

Figure B.2

If you press any key then one of the three players will have a screen looking like figure

B.3. Assume player 2 is randomly selected to be the initiator. The other players have to wait

(figure B.4)

130 Appendix

(1 ) 20 I \

I \ I \

150 I 220 \ 90 I \

I \ 30 110 50

(2) (3)

You are the initiatorl Do you want to I118ke a proposal <yIn> ?

Your decesion :

Figure B.3

(1) 20 I \

I \ I \

150 I 220 \90 I \

I \ 30 110 50

(2) (3)

Please be patient

One of the other two players was decided to be the initiator.

Figure B.4


The top left window shows the actual bargaining situation. Your player number will

blink on the screen. Player 2 now has two options. If he decides "n", which means that he

does not want to make a proposal, then he is asked to select another player to be the initia

tor. We will assume player 2 wants to make a proposal. As figure B.5 shows, player 2 now

can choose a desirable coalition. In the top right window a list of the other players is display

ed. Player 2 has to decide for each player, whether he wants him to be a member or not. In

our example player 2 wants to form a grand coalition.

(1 ) 20 Potent i al menbers : I \

I \ Player 1 : I \

150 I 220 \90 Player 3 : I \

I \ 30 110 50

(2) (3)

You have decided to propose a coal it ion. There are two other players. You may propose two di fferent two-person coalitions or a grand coalition. For each player you have to decide whether he should be a menber of the coalition or not.

Enter "y" for des i red menbers and "n" for no menber!

Figure B.5

Immediately after the decision for the last player the content of the top right window changes

(figure B.6).

132 Appendix

(1) Your decision _ to fol'1l 20 coali tion : I \

I \ < 1 23> I \

150 I 220 \90 There are 220 points to I \ allocatel

I \ 30 110 50 Is that what you Intended

(2) (3) to do <yIn> 7

Figure B.6

Player 2 gets an overview of his decisions. If he wants to change his mind or if there are any

mistakes "n" should be answered to get back to the last step, but if he answers "y", then he

can distribute the coalition value to all members of the coalition (figure B.7).

(1) 20 Your proposal : I \

I \ 1 receives : 60 I \ 2 receives : 120

150 I 220 \90 3 receives : 40 I \

I \ Responder : 1 30 110 50

(2) (3)

NOlI you have to propose an allocation of the coal ition value.

If you are proposing a grand coalition, then you have additionally to select one of the other members to be the first rece! ver (responder) of your. proposal.

Figure B.7


Player 2 is free to choose any values unless they are non-negative. Due to the smallest money

unit of one point a further subdivision is not possible. Moreover, all values have to add up to

the coalition value. If there are any errors, you will be prompted. Assume for a moment that

player 2 has decided to propose a two-person coalition, than immediately after the last value

is entered, the proposal will be sent to the other member of the coalition. In our case player 2

proposes a gran~ coalition and there are two other members, therefore he has to select one of

the other players to be the first receiver of your proposal. We call him responder. Figure B.7

shows player 2's decisions.

Now let us have a look on player l's screen. In the top middle window the proposal of

player 2 has arrived (Figure B.8). In our example player 1 agrees.

(1) Proposal of player 2 20 I \

I \ 1 receives: 60.00 I \ 2 receives : 120.00

150 I 220 \90 3 receives: 40.00 I \

I \ 30 110 50 Do you agree ? <yIn>

(2) (3)

Figure B.8

Automatically player 3 becomes the responder, and as we see on his screen, the proposal

of player 2 arrived and in addition he is informed that player 1 agrees with the proposal (fi

gure B.9). If player 3 agrees too, the negotiations end, and each player receives the allocated

payoffs.

134 Appendix

(1) P .. oposal of playe .. 2 20 Playe .. 1 ag .. ees I \

I \ 1 .. ecelves : 60.00 I \ 2 .. ecelves : 120.00

150 I 220 \90 3 .. ecelves : 40.00 I \

I \ 30 110 50 00 you a" .. ee ? cy/rP

(2) (3)

I FigureB.9

However, player 3 thinks that there is a good chance to get more than 40 units, therefore

he enters "n" and rejects the proposal. Now players 1 and 2 receives the message that player

3 rejects the proposal (figure B.lO).

(1) 20 MESSAGE I \

I \ Player 1 60.00 I \ Player 2 : 120.00

150 I 220 \90 Player 3 : 40.00 I \

I \ P .. oposal of playe .. 2 30 110 50 Rejected by playe .. 3

(2) (3)

Figure B.lO


According to our rules a rejector becomes the new initiator, therefore the screen of player

3 looks like figure B.11. We already mentioned the options of an initiator.

(1) 20 I \

I \ I \

150 I 220 \90 I \

I \ 30 110 50

(2) (3)

You are the initiatorl Do you want to make a proposal <yIn> ?

Your decesion :

Figure B.11

Suppose player 3 has decided to make an proposal, but he wants to form a two-person

coalition with player 1 (figure B.12).

(1) 20 Potential menbers : I \

I \ Player 1 : y I \

150 I 220 \90 Player 2 : n I \

I \ 30 110 50

(2) (3)

You have decided to propose a coalition. There are two other players. You may propose two different two-person coalitions or a grand coalition. For each player you have to decide whether he should be a merrber of the coalition or not.

Enter My" for desired "menbers and Un" for no menberl

Figure B.12

136 Appendix

Player 3 checks his decisions (figure B.13).

(1) Your decision was to forti 20 coalition : I \

I \ < 1 3 > I \

150 I 220 \90 There are 90 points to I \ allocatel

I \ 30 110 50 I s that what you Intended

(2) (3) to do <yIn> ?

Figure B.I3

and proposes a distribution of the coalition value. Since there is only one possible responder,

player 3 will not be asked to select one (figure B.14)

(1) 20 Your proposa l : I \

I \ 1 receives : 40 I \ 3 receives : 50

150 I 220 \ 90 I \

/ \ 30 110 50

(2) (3)

Now you have to propose an allocation of the coalition value.

If you are proposing a grand coalition, then you have edditionally to select one of the other menbers to be the first receiver (responder) of your proposal.

Figure B.I4


It will not be necessary to look at the screen of player 1 again. We will assume that

player 1 received player 3's proposal and agrees. This ends the negotiations, because all mem

bers of the coalition accepted the allocation of the coalition value. With other words, only

members of the proposed coalition have to accept, therefore player 2 will not be asked. Player

2 receives his one-person coalition value. Figure B.IS shows the result of this negotiations.

(1) 20 RESULT I \

I \ Player 1 : 40.00 I \ Player 2: 30.00

150 I 220 \90 Playerr 3 : 50.00 I \

I \ 30 110 50

(2) (3)

Now you 1liiY have a short break, If necessary I In 60 seconds you will be ~ted to press tInY key if you are ready to get Into new negotiations. There are no time constraints, but we hope you will not walt too long.

Figure B.IS

If there is any need for a short break, then the best time is now. Figure B.IS will be

displayed 60 seconds, the figure B.2 will be displayed again. If you are ready for new negotia

tions press any key.

In our example we did not mention the case, that a two-person coalition is proposed, but

the responder does not agree. Even if one player is not involved in the negotiations of two

players, he will receive all messages of a rejector, and of course he will be informed about the

results.

138 Appendix

Are there any questions?

Now we have to discuss some technical details. Whenever you press a key, the corres

ponding character will blink on the screen, until you have pressed the EJ key. Now your

decision is accepted. As long as your decision blinks, it can be erased with the I +-1 key. If

there is more than one decision within a window, you have to accept each decision. However,

the complete window can be erased with the I ESC I key.

There are no time constraints.

Try to maximize your own profit. Each point is worth x.xx DM.

Good luck.

C. Listing of all Results 139

C. LISTING OF ALL RESULTS

In this appendix all results obtained by experiments at the Bonn Laboratory of Experimental

Economics are listed. The first column shows which coalition formed:

0: No agreement is reached

1 : Player one is excluded; coalition C12 has formed

2 : Player two is excluded; coalition C13 has formed

3 : Player three is excluded; coalition C23 has formed

4: No player is excluded; coalition C123 has formed.

Columns two to eight show the characteristic function:

v(123), v(12), v(13), v(23), v(l), v(2), v(3).

Columns nine to eleven show the final payoffs :

xl. X2, X3·

Series 1.1

3 100 80 60 20 0 0 0 40.00 40.00 0.00 3 100 80 80 80 0 0 0 40.00 40.00 0.00 3 100 100 60 20 0 0 0 50.00 50.00 0.00 4 100 80 0 0 0 0 0 47.00 41.00 12.00 4 100 80 40 20 0 0 0 45.00 45.00 10.00 4 100 0 0 0 0 0 0 34.00 33.00 33.00 2 100 100 20 20 0 0 0 19.00 0.00 1.00 3 100 60 60 40 0 0 0 30.00 30.00 0.00 2 100 100 80 60 0 0 0 60.00 0.00 20.00 3 100 80 40 0 0 0 0 38.00 42.00 0.00 4 100 40 40 0 0 0 0 34.00 33.00 33.00 4 100 40 40 20 0 0 0 34.00 33.00 33.00 3 100 100 20 0 0 0 0 59.00 41.00 0.00 2 100 80 80 20 0 0 0 45.00 0.00 35.00 4 100 20 20 20 0 0 0 34.00 33.00 33.00 1 100 100 80 80 0 0 0 0.00 50.00 30.00 3 100 100 60 60 0 0 0 50.00 50.00 0.00 2 100 100 100 20 0 0 0 66.00 0.00 34.00 4 100 40 20 0 0 0 0 35.00 30.00 35.00 2 100 80 60 40 0 0 0 30.00 0.00 30.00 3 100 100 40 0 0 0 0 50.00 50.0p 0.00 2 100 100 100 0 0 0 0 60.00 0;.00 40.00 2 100 100 60 0 0 0 0 59.00 0.00 1.00 4 100 60 60 60 0 0 0 33.00 34.00 33.00

140 Appendix

4 100 20 0 0 0 0 0 35.00 35.00 30.00 3 100 60 20 0 0 0 0 30.00 30.00 0.00 2 100 80 80 40 0 0 0 46.00 0.00 34.00 3 100 100 80 40 0 0 0 50.00 50.00 0.00 2 100 100 100 40 0 0 0 60.00 0.00 40.00 3 100 100 0 0 0 0 0 51.00 49.00 0.00 3 100 100 100 60 0 0 0 69.00 31.00 0.00 2 100 100 80 20 0 0 0 65.00 0.00 15.00 3 100 80 60 0 0 0 0 43.00 37.00 0.00 3 100 100 100 100 0 0 0 50.00 50.00 0.00 4 100 40 40 40 0 0 0 34.00 33.00 33.00 4 100 80 20 0 0 0 0 80.00 10.00 10.00

Series 1.2

3 100 80 0 0 0 0 0 40.00 40.00 0.00 3 100 80 80 80 0 0 0 40.00 40.00 0.00 3 100 100 60 20 0 0 0 50.00 50.00 0.00 4 100 80 40 20 0 0 0 45.00 40.00 15.00 4 100 0 0 0 0 0 0 33.00 33.00 34.00 3 100 80 60 20 0 0 0 47.00 33.00 0.00 3 100 100 20 20 0 0 0 50.00 50.00 0.00 4 100 60 60 40 0 0 0 33.00 34.00 33.00 3 100 100 20 0 0 0 0 50.00 50.00 0.00 3 100 80 40 40 0 0 0 40.00 40.00 0.00 4 100 40 20 0 0 0 0 38.00 37.00 25.00 3 100 100 80 60 0 0 0 52.00 48.00 0.00 3 100 100 40 20 0 0 0 50.00 50.00 0.00 4 100 80 40 0 0 0 0 42.00 42.00 16.00 4 100 40 40 20 0 0 0 34.00 33.00 33.00 4 100 20 20 20 0 0 0 34.00 34.00 32.00 3 100 80 80 20 0 0 0 50.00 30.00 0.00 4 100 40 40 0 0 0 0 54.00 20.00 26.00 4 100 80 20 0 0 0 0 54.00 40.00 6.00 1 100 100 80 80 0 0 0 0.00 50.00 30.00 4 100 40 0 0 0 0 0 34.00 34.00 32.00 4 100 40 20 20 0 0 0 37.00 37.00 26.00 3 100 100 60 60 0 0 0 50.00 50.00 0.00 3 100 80 60 40 0 0 0 45.00 35.00 0.00 2 100 100 100 20 0 0 0 65.00 0.00 35.00 4 100 60 20 0 0 0 0 50.00 40.00 10.00 2 100 100 100 0 0 0 0 60.00 0.00 40.00 3 100 100 40 0 0 0 0 35.00 65.00 0.00 3 100 100 60 0 0 0 0 50.00 50.00 0.00 4 100 60 60 60 0 0 0 33.00 33.00 34.00 4 100 20 0 0 0 0 0 34.00 34.00 32.00 4 100 80 20 20 0 0 0 50.00 40.00 10.00 4 100 60 40 40 0 0 0 36.00 34.00 30.00 4 100 60 40 0 0 0 0 50.00 30.00 20.00 2 100 100 80 40 0 0 0 60.00 0.00 20.00 3 100 80 80 40 0 0 0 43.00 37.00 0.00 3 100 100 0 0 0 0 0 50.00 ,50.00 0.00 3 100 100 100 40 0 0 0 73.0(); 27.00 0.00 3 100 80 60 0 0 0 0 56.00 24.00 0.00 4 100 60 0 0 0 0 0 42 .. 00 42.00 16.00


3 100 100 100 60 0 0 0 60.00 40.00 0.00 2 100 100 SO 20 0 0 0 70.00 0.00 10.00 4 100 40 40 40 0 0 0 33.00 33.00 34.00 2 100 100 100 100 0 0 0 50.00 0.00 50.00 2 100 60 20 20 0 0 0 10.00 0.00 10.00 3 100 100 SO 0 0 0 0 69.00 31.00 0.00 3 100 100 60 40 0 0 0 50.00 50.00 0.00 3 100 100 100 SO 0 0 0 5S.00 42.00 0.00 1 100 100 40 40 0 0 0 0.00 39.00 1.00 2 100 SO SO 0 0 0 0 60.00 0.00 20.00 3 100 SO SO 60 0 0 0 4S.00 32.00 0.00 4 100 60 60 0 0 0 0 3S.00 31.00 31.00 3 100 SO 60 60 0 0 0 42.00 3S.00 0.00 4 100 60 40 20 0 0 0 41.00 30.00 29.00 4 100 20 20 0 0 0 0 3S.00 31.00 31.00 4 100 60 60 20 0 0 0 50.00 25.00 25.00

Series 2.1

1 240 lS0 100 200 20 60 60 20.00 120.00 SO.OO 2 240 240 220 20 120 20 0 200.00 20.00 20.00 4 240 60 lS0 160 20 o 120 70.00 40.00 130.00 3 240 200 60 200 20 80 40 75.00 125.00 40.00 2 240 140 100 140 40 60 40 50.00 60.00 50.00 4 240 60 140 180 20 20 100 60.00 60.00 120.00 4 240 160 120 180 40 80 20 60.00 120.00 60.00 4 240 40 180 200 0 20 120 20.00 50.00 170.00 2 240 100 200 140 20 40 80 80.00 40.00 120.00 4 240 80 100 100 40 40 60 80.00 80.00 80.00 3 240 220 200 140 80 40 20 140.00 80.00 20.00 3 240 160 180 120 60 0 80 110.00 50.00 80.00 4 240 100 140 60 100 0 40 130.00 80.00 30.00 4 240 80 100 180 0 60 80 30.00 92.00 118.00 2 240 220 120 100 40 100 0 90.00 100.00 30.00 4 240 160 200 60 120 0 20 165.00 25.00 50.00 4 240 80 180 80 60 0 80 105.00 25.00 110.00 4 240 100 100 160 0 40 100 30.00 95.00 115.00 1 240 220 80 180 20 120 0 20.00 162.00 18.00 1 240 180 100 180 20 100 20 20.00 50.00 130.00 4 240 140 100 160 o 120 20 45.00 130.00 65.00 3 240 180 120 120 40 40 60 90.00 90.00 60.00 3 240 240 60 180 20 120 0 70.00 170.00 0.00 3 240 200 100 100 60 40 40 110.00 90.00 40.00 3 240 140 140 120 60 0 80 90.00 50.00 80.00 4 240 180 40 120 20 100 20 50.00 160.00 30.00 3 240 200 120 100 SO 20 40 120.00 80.00 40.00 2 240 160 220 80 SO 20 40 140.00 20.00 80.00 4 240 100 200 100 100 0 40 135.00 10.00 95.00 4 240 200 220 60 120 0 20 180.00 20.00 40.00 1 240 220 160 140 80 60 0 SO.OO 120.00 20.00 4 240 100 80 180 0 60 80 15.00 95.00 130.00 3 240 200 80 160 40 80 20 80.00 120·90 20.00 1 240 40 lS0 220 20 20 100 20.00 40.00 180.00 2 240 220 240 40 120 0 20 180.00 0.00 60.00 3 240 140 220 140 40 20 SO 95.00 45.00 SO.OO

142 Appendix

1 240 140 140 40 120 20 0 120.00 30.00 10.00 4 240 100 140 120 20 80 40 77.00 80.00 83.00 4 240 100 180 160 20 20 100 60.00 50.00 130.00 1 240 140 120 240 0 80 60 0.00 130.00 110.00 2 240 160 80 180 40 100 0 60.00 100.00 20.00 2 240 140 120 200 20 100 20 42.00 100.00 78.00 2 240 220 180 180 60 60 20 120.00 60.00 60.00 2 240 160 180 160 40 40 60 87.00 40.00 93.00 4 240 180 20 120 20 120 0 55.00 160.00 25.00

Series 2.2

3 240 180 100 200 20 60 60 60.00 120.00 60.00 1 240 160 180 160 40 40 60 40.00 80.00 80.00 1 240 180 100 180 20 100 20 20.00 60.00 120.00 1 240 40 180 220 20 20 100 20.00 60.00 160.00 2 240 160 220 160 60 20 60 110.00 20.00 110.00 4 240 100 80 180 0 60 80 20.00 100.00 120.00 1 240 160 80 180 40 100 0 40.00 145.00 35.00 3 240 200 120 100 80 20 40 110.00 90.00 40.00 1 240 80 220 220 20 o 120 20.00 50.00 170.00 2 240 220 120 100 40 100 0 100.00 100.00 20.00 2 240 100 200 140 20 40 80 85.00 40.00 115.00 4 240 100 200 100 100 0 40 120.00 40.00 80.00 2 240 140 100 160 o 120 20 45.00 120.00 55.00 2 240 160 200 60 120 0 20 150.00 0.00 50.00 4 240 160 160 20 140 0 0 170.00 35.00 35.00 4 240 100 140 120 20 80 40 65.00 100.00 75.00 1 240 140 120 120 120 20 0 120.00 80.00 40.00 4 240 200 80 100 80 60 0 100.00 120.00 20.00 4 240 80 100 100 40 40 60 80.00 80.00 80.00 2 240 60 180 160 20 o 120 40.00 0.00 140.00 1 240 140 120 240 0 80 60 0.00 130.00 110.00 2 240 200 120 80 100 40 0 110.00 40.00 10.00 3 240 160 240 120 60 0 80 132.00 28.00 80.00 4 240 160 180 120 60 0 80 115.00 45.00 80.00 1 240 160 120 180 40 80 20 40.00 120.00 60.00 2 240 40 180 200 0 20 120 20.00 20.00 160.00 4 240 100 140 60 100 0 40 125.00 50.00 65.00 4 240 220 80 180 20 120 0 65.00 165.00 10.00 3 240 200 220 60 120 0 20 165.00 35.00 20.00 3 240 200 60 200 20 80 40 75.00 125.00 40.00 2 240 220 180 180 60 60 20 110.00 60.00 70.00 4 240 180 40 120 20 100 20 70.00 140.00 30.00 3 240 200 100 240 o 100 40 60.00 140.00 40.00 3 240 180 120 120 40 40 60 90.00 90.00 60.00 2 240 140 140 120 60 0 80 60.00 0.00 80.00 4 240 80 220 120 20 o 120 60.00 20.00 160.00 3 240 220 240 40 120 0 20 200.00 20.00 20.00 2 240 140 220 140 40 20 80 100.00 20.00 120.00 4 240 140 140 40 120 20 0 170.00 40.00 30.00 1 240 140 120 200 20 100 20 20.00 1115. 00 85.00 4 240 80 180 80 60 0 80 110.00 20.00 110.00 4 240 140 100 140 40 60 40 70.00 100.00 70.00 1 240 80 100 180 0 60 80 0.00 80.00 100.00


3 240 200 100 100 60 40 40 100.00 100.00 40.00 2 240 160 220 80 80 20 40 140.00 20.00 80.00 4 240 180 20 120 20 120 0 60.00 160.00 20.00

Series 3.1

1 80 61 46 57 23 6 19 23.00 29.00 28.00 3 80 65 57 50 14 19 15 40.00 25.00 15.00 3 80 56 41 71 9 15 24 19.00 37.00 24.00 1 80 37 61 50 22 15 11 22.00 20.00 30.00 1 80 66 61 33 39 3 6 39.00 18.00 15.00 2 80 33 57 38 22 7 19 28.00 7.00 29.00 1 80 53 67 60 20 13 15 20.00 25.00 35.00 1 80 52 47 65 11 17 20 11.00 35.00 30.00 2 80 47 51 46 10 13 25 24.00 13.00 27.00 4 80 36 57 51 21 15 12 25.00 20.00 35.00 2 80 40 54 50 14 26 8 25.00 26.00 29.00 1 80 49 69 58 22 11 15 22.00 29.00 29.00 4 80 48 43 45 19 13 16 30.00 25.00 25.00 3 80 59 48 37 11 20 17 33.00 26.00 17.00 2 80 56 57 43 9 19 20 28.00 19.00 29.00 4 80 36 63 41 19 13 16 32.00 16.00 32.00 2 80 58 41 65 7 39 2 22.00 39.00 19.00 4 80 49 52 47 13 16 19 27.00 25.00 28.00 1 80 42 37 57 23 11 14 23.00 27.00 30.00 4 80 50 44 34 14 20 14 31.00 25.00 24.00 1 80 73 50 45 19 22 7 19.00 35.00 10.00 3 80 35 59 54 2 5 41 18.00 17.00 41.00 3 80 29 63 36 24 5 19 24.00 5.00 19.00 3 80 59 47 50 30 5 13 35.00 24.00 13.00 4 80 41 49 42 6 11 31 22.00 22.00 36.00 1 80 50 41 57 19 11 18 19.00 28.00 29.00 4 80 51 46 27 41 2 5 45.00 18.00 17.00 4 80 61 54 29 39 2 7 44.00 19.00 17.00 3 80 61 42 57 23 6 19 29.00 32.00 19.00 4 80 60 55 37 35 5 8 30.00 25.00 25.00 3 80 52 50 38 10 18 20 32.00 20.00 20.00 4 80 50 45 29 31 7 10 38.00 21.00 21.00 2 80 39 61 48 4 11 33 21.00 11.00 40.00 3 80 70 65 41 39 3 6 48.00 22.00 6.00 4 80 49 48 43 13 16 19 28.00 26.00 26.00 2 80 53 46 49 23 10 15 26.00 10.00 20.00 4 80 47 42 23 33 6 9 48.00 18.00 14.00 2 80 49 45 38 14 19 15 22.00 19.00 23.00 3 80 55 50 43 9 14 25 26.00 29.00 25.00 4 80 57 37 30 18 23 7 31.00 33.00 16.00 1 80 62 57 65 15 15 18 15.00 33.00 32.00 1 80 52 45 59 13 27 8 13.00 37.00 22.00 4 80 49 42 53 23 10 15 30.00 20.00 30.00

144 Appendix

Series 3.2

1 80 62 57 65 15 15 18 15.00 31.00 34.00 3 80 58 58 48 12 18 18 29.00 29.00 18.00 1 80 53 42 65 11 34 3 11.00 40.00 25.00 3 80 60 51 49 31 5 12 33.00 27.00 12.00 1 80 52 45 59 13 27 8 13.00 33.00 26.00 3 80 54 47 39 33 5 10 35.00 19.00 10.00 4 80 43 49 44 8 11 29 17.00 27.00 36.00 4 80 49 42 53 23 10 15 27.00 26.00 27.00 2 80 59 50 43 13 18 17 22.00 18.00 28.00 4 80 50 43 59 13 29 6 21.00 38.00 21.00 4 80 58 51 63 13 25 10 17.00 31.00 32.00 2 80 44 51 61 15 29 4 24.00 29.00 27.00 4 80 37 30 69 7 18 23 10.00 35.00 35.00 1 80 39 68 53 23 12 13 23.00 18.00 35.00 4 80 55 41 60 8 35 5 21.00 38.00 21.00 2 80 57 59 52 12 17 19 30.00 17.00 29.00 4 80 43 42 35 13 18 17 29.00 27.00 24.00 4 80 49 42 45 23 10 15 30.00 25.00 25.00 1 80 41 69 54 22 11 15 22.00 15.00 39.00 3 80 71 38 31 17 22 9 35.00 36.00 9.00 3 80 41 42 53 15 26 7 15.00 26.00 7.00 2 80 45 63 52 20 13 15 27.00 13.00 36.00 4 80 53 40 35 13 16 19 27.00 27.00 26.00 1 80 49 35 68 12 13 23 12.00 34.00 34.00 2 80 38 63 43 17 17 14 32.00 17.00 31.00 4 80 25 48 43 5 8 35 16.00 16.00 48.00 3 80 59 39 34 14 17 17 29.00 30.00 17.00 4 80 42 33 61 11 15 22 16.00 32.00 32.00 4 80 31 68 41 23 8 17 33.00 10.00 37.00 4 80 35 45 40 16 19 13 26.00 26.00 28.00 1 80 59 52 41 11 20 17 11.00 21.00 20.00 3 80 56 55 45 11 17 20 28.00 28.00 20.00 2 80 64 48 36 12 20 16 30.00 20.00 18.00 1 80 45 71 56 24 9 15 24.00 17.00 39.00 2 80 55 54 47 9 14 25 28.00 14.00 26.00 2 80 46 63 51 17 17 14 30.00 17.00 33.00 2 80 56 57 39 9 19 20 32.00 19.00 25.00 4 80 43 49 40 8 15 25 26.00 24.00 30.00 1 80 55 49 64 12 31 5 12.00 38.00 26.00 4 80 53 41 58 6 39 3 20.00 40.00 20.00 2 80 42 51 51 17 21 10 25.00 21.00 26.00 4 80 41 54 45 7 14 27 26.00 26.00 28.00 3 80 66 48 38 14 20 14 32.00 34.00 14.00

Series 4.1

3 180 166 49 163 0 0 0 83.00 83.00 0.00 1 180 49 163 166 0 0 0 0.00 76.00 90.00 2 180 165 179 24 0 0 0 90.00 0.00 89.00 1 180 24 165 179 0 0 0 0.00 pO.OO 69.00 3 180 180 63 163 0 0 0 90.00. 90.00 0.00 2 180 63 163 180 0 0 0 81.qO 0.00 82.00 2 180 36 162 162 0 0 0 37.00 0.00 125.00


3 180 162 162 36 0 0 0 112.00 50.00 0.00 2 180 157 171 8 0 0 0 81.00 0.00 90.00 1 180 8 157 171 0 0 0 0.00 76.00 95.00 3 180 160 176 16 0 0 0 120.00 40.00 0.00 1 180 176 16 160 0 0 0 0.00 90.00 70.00 3 180 158 166 54 0 0 0 97.00 61.00 0.00 2 180 158 166 54 0 0 0 130.00 0.00 36.00 1 180 64 147 173 0 0 0 0.00 70.00 103.00 3 180 173 64 147 0 0 0 83.00 90.00 0.00 1 180 42 158 164 0 0 0 0.00 64.00 100.00 2 180 42 158 164 0 0 0 45.00 0.00 113.00 3 180 164 166 30 0 0 0 130.00 34.00 0.00 1 180 30 164 166 0 0 0 0.00 26.00 140.00

Series 4.2

3 180 166 49 163 0 0 0 30.00 136.00 0.00 2 180 49 163 166 0 0 0 100.00 0.00 63.00 2 180 165 179 24 0 0 0 90.00 0.00 89.00 1 180 24 165 179 0 0 0 0.00 89.00 90.00 3 180 180 63 163 0 0 0 90.00 90.00 0.00 1 180 63 163 180 0 0 0 0.00 50.00 130.00 1 180 36 162 162 0 0 0 0.00 62.00 100.00 3 180 162 162 36 0 0 0 83.00 79.00 0.00 1 180 8 157 171 0 0 0 0.00 86.00 85.00 1 180 157 171 8 0 0 0 0.00 4.00 4.00 1 180 176 16 160 0 0 0 0.00 80.00 80.00 3 180 160 176 16 0 0 0 120.00 40.00 0.00 2 180 158 166 54 0 0 0 100.00 0.00 66.00 2 180 158 166 54 0 0 0 100.00 0.00 66.00 1 180 64 147 173 0 0 0 0.00 63.00 110.00 1 180 173, 64 147 0 0 0 0.00 99.00 48.00 2 180 42 158 164 0 0 0 47.00 0.00 111.00 2 180 42 158 164 0 0 0 48.00 0.00 110.00 3 180 164 166 30 0 0 0 100.00 64.00 0.00 1 180 30 164 166 0 0 0 0.00 50.00 116.00

Series 4.3

2 180 30 164 166 0 0 0 72.00 0.00 92.00 2 180 164 166 30 0 0 0 83.00 0.00 83.00 2 180 8 157 171 0 0 0 42.00 0.00 115.00 3 180 171 8 157 0 0 0 65.00 106.00 0.00 2 180 49 163 166 0 0 0 78.00 0.00 85.00 2 180 49 163 166 0 0 0 63.00 0.00 100.00 1 180 176 16 160 0 0 0 0.00 112.00 48.00 1 180 176 16 160 0 0 0 0.00 95.00 65.00 1 180 63 163 180 0 0 0 0.00 75.00 105.00 3 180 163 180 63 0 0 0 130.00 33.00 0.00 1 180 164 42 158 0 0 0 0.00 110.00 48.00 1 180 42 158 164 0 0 0 0.00 34.0q 130.00 2 180 147 173 64 0 0 0 123.00 0.00 50.00 2 180 64 147 173 0 0 0 20.00 0.00 127.00 2 180 165 179 24 0 0 0 138.00 0.00 41.00

146 Appendix

2 180 179 24 165 0 0 0 12.00 0.00 12.00 3 180 162 36 162 0 0 0 30.00 132.00 0.00 2 180 162 36 162 0 0 0 18.00 0.00 18.00 4 180 158 166 54 0 0 0 0.00 90.00 90.00 3 180 166 54 158 0 0 0 40.00 126.00 0.00

Series 4.4

2 180 30 164 166 0 0 0 80.00 0.00 84.00 3 180 164 166 30 0 0 0 84.00 80.00 0.00 1 180 171 8 157 0 0 0 0.00 87.00 70.00 1 180 8 157 171 0 0 0 0.00 85.00 86.00 1 180 49 163 166 0 0 0 0.00 70.00 96.00 1 180 49 163 166 0 0 0 0.00 65.00 101.00 1 180 176 16 160 0 0 0 0.00 90.00 70.00 3 180 176 16 160 0 0 0 69.00 107.00 0.00 1 180 63 163 180 0 0 0 0.00 105.00 75.00 3 180 163 180 63 0 0 0 123.00 40.00 0.00 3 180 164 42 158 0 0 0 59.00 105.00 0.00 1 180 42 158 164 0 0 0 0.00 42.00 122.00 2 180 147 173 64 0 0 0 120.00 0.00 53.00 2 180 64 147 173 0 0 0 30.00 0.00 117.00 1 180 179 24 165 0 0 0 0.00 105.00 60.00 2 180 165 179 24 0 0 0 154.00 0.00 25.00 1 180 162 36 162 0 0 0 0.00 115.00 47.00 1 180 162 36 162 0 0 0 0.00 122.00 40.00 1 180 166 54 158 0 0 0 0.00 80.00 78.00 2 180 158 166 54 0 0 0 130.00 0.00 36.00

Series 4.5

1 180 49 163 166 0 0 0 0.00 84.00 82.00 3 180 166 49 163 0 0 0 68.00 98.00 0.00 3 180 162 162 36 0 0 0 101.00 61.00 0.00 1 180 36 162 162 0 0 0 0.00 68.00 94.00 1 180 176 16 160 0 0 0 0.00 130.00 30.00 1 180 176 16 160 0 0 0 0.00 75.00 85.00 3 180 147 173 64 0 0 0 120.00 27.00 0.00 2 180 173 64 147 0 0 0 32.00 0.00 32.00 3 180 165 179 24 0 0 0 150.00 15.00 0.00 1 180 24 165 179 0 0 0 0.00 34.00 145.00 2 180 163 180 63 0 0 0 90.00 0.00 90.00 3 180 180 63 163 0 0 0 45.00 135.00 0.00 2 180 54 158 166 0 0 0 28.00 0.00 130.00 1 180 166 54 158 0 0 0 0.00 118.00 40.00 3 180 42 158 164 0 0 0 21.00 21.00 0.00 2 180 158 164 42 0 0 0 139.00 0.00 25.00 2 180 30 164 166 0 0 0 20.00 0.00 144.00 3 180 164 166 30 0 0 0 85.00 79.00 0.00 2 180 157 171 8 0 0 0 151.00 0.00 20.00 3 180 8 157 171 0 0 0 3.00 5.00 0.00


Series 4.6

1 180 49 163 166 0 0 0 0.00 70.00 96.00 3 180 166 49 163 0 0 0 80.00 86.00 0.00 2 180 162 162 36 0 0 0 110.00 0.00 52.00 2 180 36 162 162 0 0 0 37.00 0.00 125.00 1 180 176 16 160 0 0 0 0.00 120.00 40.00 1 180 176 16 160 0 0 0 0.00 130.00 30.00 2 180 147 173 64 0 0 0 90.00 0.00 83.00 1 180 173 64 147 0 0 0 0.00 102.00 45.00 1 180 24 165 179 0 0 0 0.00 59.00 120.00 3 180 165 179 24 0 0 0 155.00 10.00 0.00 2 180 163 180 63 0 0 0 115.00 0.00 65.00 2 180 180 63 163 0 0 0 32.00 0.00 31.00 2 180 54 158 166 0 0 0 38.00 0.00 120.00 3 180 166 54 158 0 0 0 38.00 128.00 0.00 3 180 158 164 42 0 0 0 136.00 22.00 0.00 2 180 42 158 164 0 0 0 30.00 0.00 128.00 2 180 164 166 30 0 0 0 130.00 0.00 36.00 1 180 30 164 166 0 0 0 0.00 21.00 145.00 2 180 157 171 8 0 0 0 131.00 0.00 40.00 2 180 8 157 171 0 0 0 7.00 0.00 150.00

Series 5.1

1 210 166 30 164 0 0 0 0.00 104.00 60.00 1 210 164 166 30 0 0 0 0.00 15.00 15.00 1 210 8 157 171 0 0 0 0.00 83.00 88.00 4 210 8 157 171 0 0 0 29.00 29.00 152.00 3 210 162 36 162 0 0 0 62.00 100.00 0.00 3 210 162 36 162 0 0 0 36.00 126.00 0.00 2 210 16 160 176 0 0 0 11.00 0.00 149.00 4 210 176 16 160 0 0 0 40.00 150.00 20.00 3 210 173 64 147 0 0 0 60.00 113.00 0.00 2 210 147 173 64 0 0 0 108.00 0.00 65.00 1 210 49 163 166 0 0 0 0.00 46.00 120.00 4 210 163 166 49 0 0 0 120.00 45.00 45.00 2 210 163 180 63 0 0 0 60.00 0.00 120.00 3 210 180 63 163 0 0 0 37.00 143.00 0.00 2 210 158 164 42 0 0 0 120.00 0.00 44.00 3 210 165 179 24 0 0 0 110.00 55.00 0.00 2 210 158 164 42 0 0 0 121.00 0.00 43.00 2 210 165 179 24 0 0 0 130.00 0.00 49.00 4 210 54 158 166 0 0 0 37.00 43.00 130.00 4 210 166 54 158 0 0 0 47.00 120.00 43.00

Series 5.2

4 210 164 166 30 0 0 0 45.00 82.00 83.00 3 210 166 30 164 0 0 0 76.00 90.00 0.00 2 210 8 157 171 0 0 0 71.00 O,'()O 86.00 2 210 8 157 171 0 0 0 45.00 :0.00 112.00 1 210 162 36 162 0 0 0 0.00 90.00 72.00 1 210 162 36 162 0 0 0 0.00 115.00 47.00

148 Appendix

1 210 16 160 176 0 0 0 0.00 41.00 135.00 1 210 176 16 160 0 0 0 0.00 113.00 47.00 3 210 173 64 147 0 0 0 73.00 100.00 0.00 4 210 147 173 64 0 0 0 110.00 35.00 65.00 2 210 163 166 49 0 0 0 120.00 0.00 46.00 4 210 49 163 166 0 0 0 50.00 40.00 120.00 2 210 163 180 63 0 0 0 100.00 0.00 80.00 2 210 158 164 42 0 0 0 94.00 0.00 70.00 3 210 180 63 163 0 0 0 38.00 142.00 0.00 3 210 158 164 42 0 0 0 118.00 40.00 0.00 4 210 165 179 24 0 0 0 130.00 35.00 45.00 2 210 165 179 24 0 0 0 136.00 0.00 43.00 2 210 54 158 166 0 0 0 60.00 0.00 98.00 1 210 166 54 158 0 0 0 0.00 123.00 35.00

Series 5.3

1 210 16 160 176 0 0 0 0.00 76.00 100.00 2 210 16 160 176 0 0 0 50.00 0.00 110.00 2 210 164 166 30 0 0 0 110.00 0.00 56.00 2 210 30 164 166 0 0 0 54.00 0.00 110.00 3 210 166 54 158 0 0 0 66.00 100.00 0.00 1 210 166 54 158 0 0 0 0.00 95.00 63.00 2 210 42 158 164 0 0 0 43.00 0.00 115.00 3 210 158 164 42 0 0 0 100.00 58.00 0.00 2 210 157 171 8 0 0 0 120.00 0.00 51.00 4 210 8 157 171 0 0 0 27.00 42.00 141.00 1 210 64 147 173 0 0 0 0.00 53.00 120.00 4 210 147 173 64 0 0 0 120.00 25.00 65.00 1 210 166 49 163 0 0 0 0.00 109.00 54.00 1 210 163 166 49 0 0 0 0.00 25.00 24.00 2 210 165 179 24 0 0 0 135.00 0.00 44.00 3 210 179 24 165 0 0 0 49.00 130.00 0.00 1 210 63 163 180 0 0 0 0.00 45.00 135.00 3 210 180 63 163 0 0 0 55.00 125.00 0.00 4 210 162 36 162 0 0 0 40.00 130.00 40.00 4 210 36 162 162 0 0 0 50.00 50.00 110.00

Series 5.4

2 210 16 160 176 0 0 0 71.00 0.00 89.00 2 210 16 160 176 0 0 0 70.00 0.00 90.00 2 210 164 166 30 0 0 0 83.00 0.00 83.00 1 210 30 164 166 0 0 0 0.00 60.00 106.00 3 210 166 54 158 0 0 0 66.00 100.00 0.00 3 210 166 54 158 0 0 0 66.00 100.00 0.00 1 210 42 158 164 0 0 0 0.00 64.00 100.00 4 210 158 164 42 0 0 0 114.00 46.00 50.00 2 210 157 171 8 0 0 0 81.00 0.00 90.00 4 210 8 157 171 0 0 0 30.00 40.00 140.00 1 210 64 147 173 0 0 0 0.00 ?3.00 100.00 3 210 147 173 64 0 0 0 100.00 47.00 0.00 3 210 163 166 49 0 0 0 113.00 50.00 0.00 1 210 166 49 163 0 0 0 0.00 110.00 53.00


2 210 165 179 24 0 0 0 127.00 0.00 52.00 3 210 179 24 165 0 0 0 71.00 108.00 0.00 3 210 180 63 163 0 0 0 70.00 110.00 0.00 1 210 63 163 180 0 0 0 0.00 75.00 105.00 4 210 36 162 162 0 0 0 45.00 45.00 120.00 1 210 162 36 162 0 0 0 0.00 108.00 54.00

Series 5.5

3 210 162 162 36 0 0 0 82.00 80.00 0.00 3 210 162 36 162 0 0 0 91.00 71.00 0.00 3 210 171 8 157 0 0 0 71.00 100.00 0.00 1 210 157 171 8 0 0 0 0.00 4.00 4.00 3 210 158 164 42 0 0 0 95.00 63.00 0.00 4 210 164 42 158 0 0 0 40.00 130.00 40.00 3 210 166 54 158 0 0 0 83.00 83.00 0.00 2 210 54 158 166 0 0 0 58.00 0.00 100.00 1 210 30 164 166 0 0 0 0.00 40.00 126.00 2 210 164 166 30 0 0 0 120.00 0.00 46.00 1 210 63 163 180 0 0 0 0.00 50.00 130.00 3 210 63 163 180 0 0 0 31.00 32.00 0.00 3 210 160 176 16 0 0 0 130.00 30.00 0.00 2 210 16 160 176 0 0 0 30.00 0.00 130.00 2 210 165 179 24 0 0 0 130.00 0.00 49.00 3 210 24 165 179 0 0 0 0.00 24.00 0.00 3 210 173 64 147 0 0 0 43.00 130.00 0.00 2 210 147 173 64 0 0 0 120.00 0.00 53.00 3 210 163 166 49 0 0 0 135.00 28.00 0.00 4 210 163 166 49 0 0 0 130.00 40.00 40.00

Series 5.6

4 210 162 36 162 0 0 0 70.00 70.00 70.00 3 210 162 162 36 0 0 0 134.00 28.00 0.00 3 210 157 171 8 0 0 0 137.00 20.00 0.00 1 210 164 42 158 0 0 0 0.00 28.00 130.00 2 210 158 164 42 0 0 0 110.00 0.00 54.00 3 210 166 54 158 0 0 0 60.00 106.00 0.00 2 210 164 166 30 0 0 0 100.00 0.00 66.00 2 210 54 158 166 0 0 0 35.00 0.00 123.00 3 210 171 8 157 0 0 0 40.00 131.00 0.00 4 210 30 164 166 0 0 0 36.00 36.00 138.00 1 210 63 163 180 0 0 0 0.00 60.00 120.00 2 210 63 163 180 0 0 0 43.00 0.00 120.00 1 210 16 160 176 0 0 0 0.00 56.00 120.00 2 210 160 176 16 0 0 0 110.00 0.00 66.00 4 210 165 179 24 0 0 0 120.00 40.00 50.00 2 210 24 165 179 0 0 0 25.00 0.00 140.00 4 210 147 173 64 0 0 0 110.00 40.00 60.00 3 210 173 64 147 0 0 0 20.00 153.00 0.00 1 210 163 166 49 0 0 0 0.00 24.0p 25.00 4 210 163 166 49 0 0 0 120.00 45.00 45.00

150 Appendix

Series 6.1

1 320 200 230 240 62 0 0 62.00 120.00 120.00 1 320 240 200 230 0 0 62 0.00 121.00 109.00 2 320 180 210 220 70 0 0 90.00 0.00 120.00 1 320 180 210 220 70 0 0 70.00 109.00 111.00 1 320 185 215 225 68 0 0 ' 68.00 110.00 115.00 1 320 185 215 225 68 0 0 68.00 110.00 115.00 3 320 250 200 240 0 0 58 125.00 125.00 58.00 1 320 250 200 240 0 0 58 0.00 135.00 105.00 2 320 255 265 215 0 52 0 133.00 52.00 132.00 2 320 215 255 265 52 0 0 105.00 0.00 150.00 1 320 255 205 245 0 0 56 0.00 140.00 105.00 3 320 255 205 245 0 0 56 125.00 130.00 56.00 1 320 260 210 250 0 0 54 0.00 140.00 110.00 3 320 260 210 250 0 0 54 120.00 140.00 54.00 3 320 230 190 220 0 0 66 105.00 125.00 66.00 2 320 190 220 230 66 0 0 95.00 0.00 125.00 1 320 205 235 245 60 0 0 60.00 115.00 130.00 1 320 245 205 235 0 0 60 0.00 140.00 95.00 1 320 195 225 235 64 0 0 64.00 105.00 130.00 2 320 225 235 195 0 64 0 125.00 64.00 110.00

Series 6.2

3 320 240 200 230 0 0 62 120.00 120.00 62.00 1 320 200 230 240 62 0 0 62.00 115.00 125.00 2 320 180 210 220 70 0 0 85.00 0.00 125.00 1 320 180 210 220 70 0 0 70.00 110.00 110.00 1 320 185 215 225 68 0 0 68.00 110.00 115.00 2 320 185 215 225 68 0 0 100.00 0.00 115.00 3 320 250 200 240 0 0 58 120.00 130.00 58.00 3 320 250 200 240 0 0 58 125.00 125.00 58.00 1 320 215 255 265 52 0 0 52.00 130.00 135.00 3 320 255 265 215 0 52 0 140.00 115.00 0.00 3 320 255 205 245 0 0 56 125.00 130.00 56.00 3 320 255 205 245 0 0 56 127.00 128.00 56.00 3 320 260 210 250 0 0 54 120.00 140.00 54.00 1 320 260 210 250 0 0 54 0.00 140.00 110.00 3 320 230 190 220 0 0 66 114.00 116.00 66.00 2 320 190 220 230 66 0 0 105.00 0.00 115.00 3 320 245 205 235 0 0 60 120.00 125.00 60.00 2 320 205 235 245 60 0 0 105.00 0.00 130.00 3 320 225 235 195 0 64 0 125.00 100.00 0.00 2 320 195 225 235 64 0 0 100.00 0.00 125.00

Series 6.3

3 320 230 190 220 0 0 66 115.00 115.00 66.00 3 320 230 190 220 0 0 66 115.00 115.00 66.00 1 320 235 195 225 0 0 64 0.00 110.00 115.00 2 320 195 225 235 64 0 0 107.00 , 10•00 118.00 1 320 260 210 250 0 0 54 0.00 125.00 125.00 1 320 260 210 250 0 0 54 0.00 131.00 119.00


3 320 230 240 200 0 62 0 115.00 115.00 0.00 1 320 200 230 240 62 0 0 62.00 125.00 115.00 3 320 225 185 215 0 0 68 115.00 110.00 68.00 3 320 225 185 215 0 0 68 100.00 125.00 68.00 2 320 200 240 250 58 0 0 105.00 0.00 135.00 2 320 200 240 250 58 0 0 110.00 0.00 130.00 3 320 220 180 210 0 0 70 105.00 115.00 70.00 1 320 220 180 210 0 0 70 0.00 112.00 98.00 1 320 245 205 235 0 0 60 0.00 110.00 125.00 1 320 205 235 245 60 0 0 60.00 120.00 125.00 3 320 255 205 245 0 0 56 120.00 135.00 56.00 1 320 205 245 255 56 0 0 56.00 110.00 145.00 3 320 255 265 215 0 52 0 140.00 115.00 0.00 3 320 255 265 215 0 52 0 150.00 105.00 0.00

Series 6.4

3 320 230 190 220 0 0 66 115.00 115.00 66.00 3 320 230 190 220 0 0 66 115.00 115.00 66.00 1 320 195 225 235 64 0 0 64.00 118.00 117.00 1 320 235 195 225 0 0 64 0.00 118.00 107.00 1 320 260 210 250 0 0 54 0.00 135.00 115.00 3 320 260 210 250 0 0 54 130.00 130.00 54.00 1 320 200 230 240 62 0 0 62.00 120.00 120.00 2 320 230 240 200 0 62 0 132.00 62.00 108.00 1 320 225 185 215 0 0 68 0.00 115.00 100.00 1 320 225 185 215 0 0 68 0.00 125.00 90.00 1 320 200 240 250 58 0 0 58.00 120.00 130.00 1 320 200 240 250 58 0 0 58.00 105.00 145.00 1 320 220 180 210 0 0 70 0.00 111.00 99.00 4 320 220 180 210 0 0 70 100.00 120.00 100.00 3 320 245 205 235 0 0 60 105.00 140.00 60.00 2 320 205 235 245 60 0 0 95.00 0.00 140.00 3 320 255 205 245 0 0 56 105.00 150.00 56.00 2 320 205 245 255 56 0 0 100.00 0.00 145.00 2 320 255 265 215 0 52 0 132.00 52.00 133.00 3 320 255 265 215 0 52 0 155.00 100.00 0.00

Series 7.1

1 240 130 110 110 70 56 14 70.00 60.00 50.00 3 240 230 110 170 56 84 0 95.00 135.00 0.00 1 240 90 130 170 42 14 84 42.00 45.00 125.00 4 240 150 90 150 70 56 14 105.00 90.00 45.00 1 240 30 170 210 14 o 126 14.00 25.00 185.00 3 240 190 190 150 84 14 42 123.00 67.00 42.00 3 240 190 190 50 126 0 14 145.00 45.00 14.00 1 240 130 150 110 112 0 28 112.00 25.00 85.00 1 240 150 150 130 98 0 42 98.00 50.00 80.00 3 240 130 130 150 14 42 84 55.00 75.00 84.00 1 240 110 150 170 42 28 70 42.00 70.00 100.00 3 240 110 130 130 14 56 70 50.00 60100 70.00 2 240 130 130 130 42 84 14 85.00 84.00 45.00 1 240 110 150 150 0 70 70 0.00 ·75.00 75.00

152 Appendix

3 240 150 150 150 0 56 84 56.00 94.00 84.00 4 240 110 110 170 14 42 84 40.00 80.00 120.00 3 240 190 190 90 56 42 42 95.00 95.00 42.00 2 240 110 190 110 56 42 42 120.00 42.00 70.00 1 240 90 130 150 14 28 98 14.00 50.00 100.00 1 240 190 110 190 14 84 42 14.00 120.00 70.00 4 240 90 150 70 84 0 56 40.00 70.00 130.00 4 240 70 130 170 14 28 98 50.00 70.00 120.00 3 240 190 110 170 0 98 42 55.00 135.00 42.00 4 240 50 150 170 28 14 98 50.00 50.00 140.00 2 240 150 130 150 56 84 0 71.00 84.00 59.00 4 240 130 130 210 28 56 56 30.00 105.00 105.00 3 240 170 170 150 70 28 42 110.00 60.00 42.00 3 240 150 150 110 56 14 70 100.00 50.00 70.00 3 240 230 110 170 42 98 0 96.00 134.00 0.00 1 240 190 70 150 o 112 28 0.00 120.00 30.00 1 240 190 150 150 84 56 0 84.00 95.00 55.00 3 240 170 110 150 56 42 42 90.00 80.00 42.00 4 240 70 150 210 14 42 84 28.00 86.00 126.00 4 240 150 150 50 126 14 0 137.00 64.00 39.00 4 240 170 50 170 28 112 0 55.00 145.00 40.00 1 240 210 10 190 o 140 0 0.00 165.00 25.00 4 240 130 130 130 14 56 70 60.00 85.00 95.00 3 240 150 150 130 28 42 70 66.00 84.00 70.00 3 240 170 130 150 0 70 70 55.00 115.00 70.00 1 240 170 170 190 42 56 42 42.00 95.00 95.00 2 240 170 190 150 56 28 56 95.00 28.00 95.00 4 240 170 130 150 56 70 14 83.00 105.00 52.00 4 240 50 110 170 0 42 98 43.00 69.00 128.00

Series 7.2

2 240 170 190 150 56 28 56 81.00 28.00 109.00 2 240 50 210 210 0 14 126 42.00 14.00 168.00 3 240 170 130 150 56 70 14 80.00 90.00 14.00 4 240 130 170 110 84 0 56 97.00 56.00 87.00 2 240 110 150 170 0 42 98 35.00 42.00 115.00 4 240 90 150 150 28 o 112 65.00 45.00 130.00 3 240 50 110 170 0 42 98 5.00 45.00 98.00 1 240 30 210 210 28 o 112 28.00 55.00 155.00 1 240 150 130 150 70 70 0 70.00 110.00 40.00 1 240 210 150 170 56 70 14 56.00 115.00 55.00 4 240 130 130 70 70 56 14 95.00 85.00 60.00 1 240 90 170 190 28 o 112 28.00 60.00 130.00 4 240 130 130 130 84 42 14 95.00 85.00 60.00 1 240 150 150 150 56 56 28 56.00 85.00 65.00 3 240 150 170 190 42 28 70 80.00 70.00 70.00 4 240 130 110 130 42 56 42 78.00 84.00 78.00 2 240 210 130 130 56 84 0 95.00 84.00 35.00 2 240 190 110 190 o 126 14 55.00 126.00 55.00 4 240 90 130 150 0 14 126 45.00 50.00 145.00 1 240 190 110 170 42 84 14 42.00 128.00 42.00 2 240 90 150 150 0 70 70 50.00. '70.00 100.00 3 240 130 190 210 14 28 98 50.00 80.00 98.00 4 240 70 130 150 14 56 70 60.00 80.00 100.00

c. Listing of all Results 153

4 240 130 190 130 84 0 56 105.00 40.00 95.00 4 240 130 150 110 42 0 98 84.00 53.00 103.00 1 240 210 110 170 70 70 0 70.00 130.00 40.00 3 240 150 150 90 70 14 56 95.00 55.00 56.00 4 240 170 170 90 126 14 0 140.00 62.00 38.00 1 240 130 190 110 112 0 28 112.00 40.00 70.00 1 240 130 150 110 84 14 42 84.00 50.00 60.00 2 240 150 150 150 28 70 42 65.00 70.00 85.00 1 240 70 210 210 28 14 98 28.00 50.00 160.00 3 240 150 170 170 0 56 84 60.00 90.00 84.00 4 240 130 130 70 98 14 28 134.00 53.00 53.00 4 240 30 150 150 0 28 112 40.00 60.00 140.00 3 240 110 150 150 0 28 112 45.00 65.00 112.00 2 240 190 130 130 42 98 0 85.00 98.00 45.00 2 240 150 150 150 84 14 42 95.00 14.00 55.00 1 240 170 150 150 84 42 14 84.00 80.00 70.00 4 240 110 150 150 14 56 70 60.00 85.00 95.00 4 240 90 150 150 0 28 112 42.00 73.00 125.00 4 240 90 150 170 0 o 140 46.00 47.00 147.00

Series 7.3

2 240 130 190 70 70 0 70 100.00 0.00 90.00 3 240 150 170 110 70 0 70 100.00 50.00 70.00 1 240 90 150 170 0 o 140 0.00 60.00 110.00 2 240 170 130 170 0 98 42 60.00 98.00 70.00 3 240 170 90 170 28 98 14 60.00 110.00 14.00 3 240 190 110 150 o 126 14 64.00 126.00 14.00 2 240 150 170 150 14 70 56 75.00 70.00 95.00 1 240 150 150 110 84 0 56 84.00 55.00 55.00 3 240 210 90 90 84 56 0 130.00 80.00 0.00 4 240 170 10 170 o 140 0 50.00 140.00 50.00 1 240 110 150 230 0 28 112 0.00 75.00 155.00 4 240 130 130 110 84 14 42 105.00 60.00 75.00 1 240 70 150 150 56 0 84 56.00 50.00 100.00 2 240 150 170 150 84 0 56 95.00 0.00 75.00 3 240 110 170 190 0 28 112 50.00 60.00 112.00 4 240 170 90 150 14 98 28 55.00 120.00 65.00 4 240 150 170 110 112 0 28 115.00 50.00 75.00 4 240 130 90 130 14 98 28 55.00 125.00 60.00 4 240 150 130 150 70 14 56 100.00 60.00 80.00 3 240 130 170 210 0 42 98 40.00 90.00 98.00 4 240 150 150 130 70 14 56 100.00 90.00 50.00 1 240 150 150 150 70 70 0 70.00 80.00 70.00 1 240 90 130 190 0 56 84 0.00 85.00 105.00 4 240 170 170 50 98 14 28 128.00 56.00 56.00 1 240 110 150 170 42 0 98 42.00 40.00 130.00 3 240 210 30 170 o 140 0 43.00 167.00 0.00 2 240 110 190 110 98 0 42 120.00 0.00 70.00 1 240 150 90 150 28 70 42 28.00 95.00 55.00 3 240 150 110 130 28 42 70 67.00 83.00 70.00 1 240 170 150 150 84 56 0 84.00 90.(10 60.00 3 240 170 150 170 42 56 42 80.00 90100 42.00 4 240 150 150 150 56 42 42 86.00 77.00 77.00 2 240 130 150 170 56 28 56 70.00 ' 28.00 80.00

154 Appendix

1 240 150 150 150 84 14 42 84.00 75.00 75.00 4 240 150 90 110 84 56 0 110.00 80.00 50.00 1 240 130 230 110 112 0 28 112.00 25.00 85.00 3 240 170 190 170 70 0 70 100.00 70.00 70.00 3 240 190 110 130 56 56 28 96.00 94.00 28.00 1 240 210 70 150 28 98 14 28.00 125.00 25.00 4 240 90 130 130 14 70 56 55.00 95.00 90.00 4 240 190 110 150 56 84 0 80.00 120.00 40.00 1 240 70 130 150 0 42 98 0.00 45.00 105.00 3 240 170 130 170 14 70 56 65.00 105.00 56.00 2 240 110 230 90 98 0 42 130.00 0.00 100.00

Series 7.4

3 240 130 110 110 70 56 14 70.00 60.00 14.00 2 240 110 130 130 14 56 70 65.00 56.00 65.00 1 240 90 130 170 42 14 84 42.00 80.00 90.00 1 240 230 110 170 56 84 0 56.00 135.00 35.00 4 240 190 190 50 126 0 14 160.00 34.00 46.00 4 240 150 90 150 70 56 14 90.00 80.00 70.00 3 240 190 190 150 84 14 42 100.00 90.00 42.00 3 240 130 130 150 14 42 84 50.00 80.00 84.00 1 240 30 170 210 14 o 126 14.00 50.00 160.00 2 240 150 150 130 98 0 42 98.00 0.00 52.00 4 240 150 150 150 0 56 84 75.00 80.00 85.00 4 240 130 150 110 112 0 28 124.00 40.00 76.00 1 240 110 150 170 42 28 70 42.00 50.00 120.00 4 240 130 130 130 42 84 14 80.00 90.00 70.00 4 240 90 130 150 14 28 98 50.00 70.00 120.00 2 240 190 190 90 56 42 42 95.00 42.00 95.00 3 240 190 110 190 14 84 42 60.00 130.00 42.00 4 240 110 110 170 14 42 84 65.00 85.00 90.00 4 240 90 150 70 84 0 56 90.00 50.00 100.00 1 240 150 130 150 56 84 0 56.00 110.00 40.00 2 240 110 190 110 56 42 42 105.00 42.00 85.00 3 240 190 110 170 0 98 42 50.00 140.00 42.00 4 240 110 150 150 0 70 70 39.00 88.00 113.00 4 240 70 130 170 14 28 98 45.00 80.00 115.00 4 240 130 130 210 28 56 56 30.00 105.00 105.00 4 240 50 150 170 28 14 98 60.00 60.00 120.00 3 240 190 70 150 o 112 28 48.00 142.00 28.00 3 240 150 150 110 56 14 70 90.00 60.00 70.00 3 240 230 110 170 42 98 0 100.00 130.00 0.00 3 240 190 150 150 84 56 0 94.00 96.00 0.00 1 240 170 170 150 70 28 42 70.00 73.00 77.00 3 240 170 110 150 56 42 42 80.00 90.00 42.00 4 240 150 150 50 126 14 0 135.00 57.00 48.00 3 240 170 50 170 28 112 0 50.00 120.00 0.00 1 240 70 150 210 14 42 84 14.00 75.00 135.00 4 240 150 150 130 28 42 70 78.00 79.00 83.00 1 240 210 10 190 o 140 0 0.00 170.00 20.00 1 240 170 170 190 42 56 42 42.00 105.00 85.00 4 ·240 130 130 130 14 56 70 48.00 96.00 96.00 3 240 170 130 150 0 70 70 66.00 104.00 70.00 3 240 170 190 150 56 28 56 105.00 65.00 56.00


4 240 170 130 150 56 70 14 76.00 108.00 56.00 1 240 50 110 170 0 42 98 0.00 57.00 113.00 2 240 50 210 210 0 14 126 50.00 14.00 160.00

Series 7.5

3 240 170 190 150 56 28 56 100.00 70.00 56.00 2 240 170 130 150 56 70 14 83.00 70.00 47.00 1 240 50 110 170 0 42 98 0.00 65.00 105.00 1 240 50 210 210 0 14 126 0.00 50.00 160.00 3 240 110 150 170 0 42 98 26.00 84.00 98.00 1 240 130 170 110 84 0 56 84.00 37.00 73.00 2 240 130 130 70 70 56 14 105.00 56.00 25.00 1 240 150 130 150 70 70 0 70.00 100.00 50.00 1 240 210 150 170 56 70 14 56.00 110.00 60.00 2 240 30 210 210 28 o 112 55.00 0.00 155.00 4 240 90 150 150 28 o 112 55.00 48.00 137.00 1 240 130 130 130 84 42 14 84.00 85.00 45.00 1 240 190 110 170 42 84 14 42.00 125.00 45.00 4 240 150 150 150 56 56 28 90.00 90.00 60.00 1 240 90 170 190 28 o 112 28.00 40.00 150.00 2 240 150 170 190 42 28 70 62.00 28.00 108.00 2 240 210 130 130 56 84 0 100.00 84.00 30.00 2 240 190 110 190 o 126 14 50.00 126.00 60.00 3 240 130 190 210 14 28 98 30.00 100.00 98.00 4 240 130 110 130 42 56 42 62.00 116.00 62.00 2 240 90 150 150 0 70 70 32.00 70.00 118.00 4 240 90 130 150 0 14 126 50.00 60.00 130.00 1 240 130 190 130 84 0 56 84.00 44.00 86.00 4 240 130 130 70 98 14 28 110.00 60.00 70.00 4 240 70 130 150 14 56 70 54.00 93.00 93.00 3 240 210 110 170 70 70 0 75.00 135.00 0.00 3 240 130 150 110 42 0 98 65.00 65.00 98.00 4 240 150 150 150 28 70 42 65.00 95.00 80.00 4 240 170 170 90 126 14 0 138.00 55.00 47.00 4 240 130 190 110 112 0 28 125.00 45.00 70.00 4 240 130 150 110 84 14 42 108.00 60.00 72.00 4 240 150 150 90 70 14 56 114.00 38.00 88.00 1 240 70 210 210 28 14 98 28.00 70.00 140.00 3 240 150 170 170 0 56 84 50.00 100.00 84.00 4 240 30 150 150 0 28 112 40.00 60.00 140.00 2 240 150 150 150 84 14 42 84.00 14.00 66.00 2 240 190 130 130 42 98 0 80.00 98.00 50.00 4 240 110 150 150 14 56 70 59.00 80.00 101.00 4 240 90 150 150 0 28 112 50.00 60.00 130.00 4 240 110 150 150 0 28 112 50.00 75.00 115.00 3 240 90 150 170 0 o 140 45.00 45.00 140.00 1 240 170 150 150 84 42 14 84.00 85.00 65.00 3 240 150 170 110 70 0 70 90.00 60.00 70.00 2 240 190 110 150 o 126 14 50.00 126.00 60.00

156 Appendix

Series 7.6

4 240 90 150 170 0 o 140 60.00 60.00 120.00 4 240 150 170 110 70 0 70 90.00 60.00 90.00 3 240 130 190 70 70 0 70 65.00 65.00 70.00 2 240 150 170 150 14 70 56 85.00 70.00 85.00 4 240 170 130 170 0 98 42 80.00 80.00 80.00 3 240 190 110 150 o 126 14 60.00 130.00 14.00 1 240 150 150 110 84 0 56 84.00 49.00 61.00 1 240 170 90 170 28 98 14 28.00 123.00 47.00 3 240 170 10 170 o 140 0 20.00 150.00 0.00 4 240 170 90 150 14 98 28 60.00 60.00 120.00 4 240 150 170 150 84 0 56 98.00 52.00 90.00 1 240 110 150 230 0 28 112 0.00 88.00 142.00 3 240 210 90 90 84 56 0 103.00 107.00 0.00 4 240 130 130 110 84 14 42 92.00 68.00 80.00 1 240 70 150 150 56 0 84 56.00 50.00 100.00 1 240 150 170 110 112 0 28 112.00 53.00 57.00 3 240 110 170 190 0 28 112 45.00 65.00 112.00 3 240 130 170 210 0 42 98 30.00 100.00 98.00 4 240 150 130 150 70 14 56 80.00 80.00 80.00 4 240 170 170 50 98 14 28 140.00 49.00 51.00 4 240 130 90 130 14 98 28 43.00 119.00 78.00 4 240 150 150 130 70 14 56 80.00 80.00 80.00 1 240 90 130 190 0 56 84 0.00 85.00 105.00 4 240 210 30 170 o 140 0 30.00 180.00 30.00 4 240 110 150 170 42 0 98 62.00 53.00 125.00 4 240 150 150 150 70 70 0 94.00 94.00 52.00 2 240 110 190 110 98 0 42 110.00 0.00 80.00 2 240 130 150 170 56 28 56 61.00 28.00 89.00 4 240 150 90 150 28 70 42 52.00 114.00 74.00 3 240 170 150 170 42 56 42 74.00 96.00 42.00 1 240 170 150 150 84 56 0 84.00 90.00 60.00 3 240 150 110 130 28 42 70 64.00 86.00 70.00 4 240 150 150 150 56 42 42 80.00 80.00 80.00 4 240 130 230 110 112 0 28 125.00 6.00 109.00 1 240 150 150 150 84 14 42 84.00 70.00 80.00 3 240 170 190 170 70 0 70 105.00 65.00 70.00 4 240 190 110 130 56 56 28 97.00 97.00 46.00 3 240 210 70 150 28 98 14 75.00 135.00 14.00 1 240 150 90 110 84 56 0 84.00 80.00 30.00 4 240 70 130 150 0 42 98 45.00 75.00 120.00 4 240 90 130 130 14 70 56 48.00 99.00 93.00 4 240 190 110 150 56 84 0 80.00 122.00 38.00 4 240 170 130 170 14 70 56 65.00 75.00 100.00 3 240 110 230 90 98 0 42 105.00 5.00 42.00

Series 7.7

3 240 230 110 170 56 84 0 105.00 125.00 0.00 4 240 130 110 110 70 56 14 90.00 80.00 70.00 2 240 190 190 50 126 0 14 155.00 0.00 35.00 1 240 150 90 150 70 56 14 70.00 95.00 55.00 3 240 190 190 150 84 14 42 115.00 75.00 42.00 1 240 30 170 210 14 o 126 14.00 50.00 160.00


4 240 90 130 170 42 14 84 70.00 40.00 130.00 4 240 130 150 110 112 0 28 125.00 55.00 60.00 4 240 150 150 130 98 0 42 120.00 40.00 80.00 3 240 150 150 150 0 56 84 50.00 100.00 84.00 4 240 110 130 130 14 56 70 65.00 75.00 100.00 1 240 110 150 170 42 28 70 42.00 60.00 110.00 4 240 130 130 150 14 42 84 63.00 77.00 100.00 4 240 90 130 150 14 28 98 55.00 65.00 120.00 4 240 130 130 130 42 84 14 80.00 100.00 60.00 4 240 110 150 150 0 70 70 50.00 90.00 100.00 3 240 190 190 90 56 42 42 110.00 80.00 42.00 3 240 190 110 190 14 84 42 55.00 135.00 42.00 4 240 150 130 150 56 84 0 85.00 120.00 35.00 2 240 110 190 110 56 42 42 110.00 42.00 80.00 4 240 110 110 170 14 42 84 49.00 77.00 114.00 3 240 150 150 110 56 14 70 80.00 70.00 70.00 2 240 90 150 70 84 0 56 90.00 0.00 60.00 3 240 190 110 170 0 98 42 65.00 125.00 42.00 4 240 70 130 170 14 28 98 30.00 70.00 140.00 1 240 130 130 210 28 56 56 28.00 105.00 105.00 4 240 50 150 170 28 14 98 60.00 60.00 120.00 3 240 230 110 170 42 98 0 85.00 145.00 0.00 1 240 170 170 150 70 28 42 70.00 75.00 75.00 4 240 190 70 150 o 112 28 50.00 140.00 50.00 3 240 190 150 150 84 56 0 110.00 80.00 0.00 4 240 170 110 150 56 42 42 90.00 90.00 60.00 1 240 70 150 210 14 42 84 14.00 70.00 140.00 4 240 150 150 50 126 14 0 152.00 47.00 41.00 3 240 210 10 190 o 140 0 25.00 185.00 0.00 4 240 170 50 170 28 112 0 55.00 160.00 25.00 3 240 150 150 130 28 42 70 75.00 75.00 70.00 1 240 170 170 190 42 56 42 42.00 100.00 90.00 4 240 130 130 130 14 56 70 75.00 75.00 90.00 2 240 170 130 150 0 70 70 50.00 70.00 80.00 4 240 170 130 150 56 70 14 90.00 110.00 40.00 3 240 170 190 150 56 28 56 100.00 70.00 56.00 2 240 50 210 210 0 14 126 50.00 14.00 160.00 4 240 50 110 170 0 42 98 28.00 63.00 149.00

Series 7.8

4 240 170 130 150 56 70 14 90.00 115.00 35.00 2 240 50 210 210 0 14 126 42.00 14.00 168.00 2 240 170 190 150 56 28 56 99.00 28.00 91.00 4 240 130 170 110 84 0 56 120.00 95.00 25.00 4 240 110 150 170 0 42 98 36.00 78.00 126.00 4 240 50 110 170 0 42 98 25.00 75.00 140.00 1 240 150 130 150 70 70 0 70.00 125.00 25.00 1 240 30 210 210 28 o 112 28.00 40.00 170.00 4 240 130 130 70 70 56 14 90.00 75.00 75.00 1 240 210 150 170 56 70 14 56.00 115.00 55.00 4 240 190 110 170 42 84 14 70.00 125,.00 45.00 1 240 90 150 150 28 o 112 28.00 .30.00 120.00 4 240 150 150 150 56 56 28 80.00 80.00 80.00 4 240 90 170 190 28 o 112 47.00. 48.00 145.00

158 Appendix

4 240 130 110 130 42 56 42 78.00 84.00 78.00 3 240 150 170 190 42 28 70 90.00 60.00 70.00 4 240 130 130 130 84 42 14 97.00 85.00 58.00 3 240 210 130 130 56 84 0 100.00 110.00 0.00 3 240 190 110 190 o 126 14 40.00 150.00 14.00 3 240 130 190 210 14 28 98 55.00 75.00 98.00 4 240 90 130 150 0 14 126 40.00 57.00 143.00 4 240 70 130 150 14 56 70 50.00 80.00 110.00 4 240 130 150 110 42 0 98 90.00 45.00 105.00 4 240 130 190 130 84 0 56 107.00 38.00 95.00 1 240 210 110 170 70 70 0 70.00 120.00 50.00 4 240 170 170 90 126 14 0 148.00 57.00 35.00 4 240 130 130 70 98 14 28 144.00 35.00 61.00 2 240 90 150 150 0 70 70 51.00 70.00 99.00 2 240 150 150 150 28 70 42 60.00 70.00 90.00 4 240 150 150 90 70 14 56 110.00 55.00 75.00 4 240 130 190 110 112 0 28 125.00 45.00 70.00 1 240 150 150 150 84 14 42 84.00 60.00 90.00 4 240 130 150 110 84 14 42 104.00 58.00 78.00 3 240 150 170 170 0 56 84 60.00 90.00 84.00 2 240 70 210 210 28 14 98 50.00 14.00 160.00 2 240 190 130 130 42 98 0 90.00 98.00 40.00 4 240 30 150 150 0 28 112 50.00 60.00 130.00 4 240 110 150 150 0 28 112 42.00 78.00 120.00 4 240 110 150 150 14 56 70 50.00 73.00 117.00 1 240 170 150 150 84 42 14 84.00 93.00 57.00 3 240 90 150 170 0 o 140 45.00 45.00 140.00 2 240 90 150 150 0 28 112 20.00 28.00 130.00 4 240 150 170 110 70 0 70 100.00 55.00 85.00 2 240 190 110 150 o 126 14 40.00 126.00 70.00

Series 7.9

3 240 90 150 170 0 o 140 45.00 45.00 140.00 2 240 190 110 150 o 126 14 50.00 126.00 60.00 2 240 130 190 70 70 0 70 95.00 0.00 95.00 3 240 150 170 110 70 0 70 95.00 55.00 70.00 2 240 150 170 150 14 70 56 75.00 70.00 95.00 3 240 170 130 170 0 98 42 50.00 120.00 42.00 4 240 150 150 110 84 0 56 120.00 60.00 60.00 1 240 170 90 170 28 98 14 28.00 110.00 60.00 4 240 170 90 150 14 98 28 55.00 130.00 55.00 1 240 150 170 150 84 0 56 84.00 50.00 100.00 4 240 130 130 110 84 14 42 110.00 40.00 90.00 4 240 210 90 90 84 56 0 125.00 105.00 10.00 4 240 170 10 170 o 140 0 35.00 170.00 35.00 3 240 110 170 190 0 28 112 40.00 70.00 112.00 1 240 110 150 230 0 28 112 0.00 95.00 135.00 4 240 70 150 150 56 0 84 80.00 40.00 120.00 3 240 170 170 50 98 14 28 130.00 40.00 28.00 1 240 150 170 110 112 0 28 112.00 45.00 65.00 1 240 150 130 150 70 14 56 70.00 7p.00 80.00 4 240 130 90 130 14 98 28 51.00 125.00 64.00 4 240 130 170 210 0 42 98 60.00 50.00 130.00 3 240 150 150 130 70 14 56 85.00 65.00 56.00


4 240 90 130 190 0 56 84 32.00 90.00 118.00 3 240 210 30 170 o 140 0 40.00 170.00 0.00 4 240 150 150 150 70 70 0 85.00 85.00 70.00 4 240 110 150 170 42 0 98 50.00 70.00 120.00 4 240 150 110 130 28 42 70 40.00 90.00 110.00 3 240 150 90 150 28 70 42 46.00 104.00 42.00 1 240 130 150 170 56 28 56 56.00 61.00 109.00 1 240 170 150 150 84 56 0 84.00 90.00 60.00 4 240 150 150 150 56 42 42 80.00 80.00 80.00 4 240 110 190 110 98 0 42 125.00 35.00 80.00 3 240 150 150 150 84 14 42 100.00 50.00 42.00 3 240 170 150 170 42 56 42 76.00 94.00 42.00 2 240 130 230 110 112 0 28 145.00 0.00 85.00 3 240 170 190 170 70 0 70 100.00 70.00 70.00 4 240 190 110 130 56 56 28 100.00 100.00 40.00 3 240 210 70 150 28 98 14 65.00 145.00 14.00 4 240 90 130 130 14 70 56 60.00 90.00 90.00 4 240 70 130 150 0 42 98 60.00 70.00 110.00 1 240 150 90 110 84 56 0 84.00 90.00 20.00 4 240 190 110 150 56 84 0 90.00 115.00 35.00 2 240 170 130 170 14 70 56 60.00 70.00 70.00 2 240 110 230 90 98 0 42 150.00 0.00 80.00

Series 7.10

3 240 230 110 170 56 84 0 105.00 125.00 0.00 4 240 110 130 130 14 56 70 45.00 65.00 130.00 4 240 190 190 50 126 0 14 145.00 45.00 50.00 1 240 90 130 170 42 14 84 42.00 57.00 113.00 3 240 190 190 150 84 14 42 125.00 65.00 42.00 4 240 130 110 110 70 56 14 105.00 80.00 55.00 1 240 150 90 150 70 56 14 70.00 100.00 50.00 3 240 130 130 150 14 42 84 40.00 90.00 84.00 4 240 30 170 210 14 o 126 25.00 45.00 170.00 4 240 90 130 150 14 28 98 60.00 50.00 130.00 1 240 150 150 130 98 0 42 98.00 50.00 80.00 3 240 150 150 150 0 56 84 65.00 85.00 84.00 4 240 110 150 170 42 28 70 60.00 75.00 105.00 4 240 130 150 110 112 0 28 115.00 45.00 80.00 3 240 190 110 190 14 84 42 55.00 135.00 42.00 2 240 190 190 90 56 42 42 130.00 42.00 60.00 4 240 130 130 130 42 84 14 87.00 105.00 48.00 4 240 110 150 150 0 70 70 60.00 80.00 100.00 4 240 110 110 170 14 42 84 35.00 81.00 124.00 2 240 110 190 110 56 42 42 100.00 42.00 90.00 4 240 150 130 150 56 84 0 83.00 100.00 57.00 2 240 70 130 170 14 28 98 30.00 28.00 100.00 4 240 150 150 110 56 14 70 105.00 58.00 77.00 4 240 90 150 70 84 0 56 105.00 42.00 93.00 1 240 130 130 210 28 56 56 28.00 105.00 105.00 2 240 190 110 170 0 98 42 45.00 98.00 65.00 4 240 50 150 170 28 14 98 50.00 70;00 120.00 4 240 170 170 150 70 28 42 110.00 80.00 50.00 3 240 190 70 150 o 112 28 55.00 ,135.00 28.00 2 240 190 150 150 84 56 0 110.00 56.00 40.00

160 Appendix

3 240 230 110 170 42 98 0 95.00 135.00 0.00 3 240 170 110 150 56 42 42 85.00 85.00 42.00 4 240 150 150 50 126 14 0 160.00 40.00 40.00 4 240 70 150 210 14 42 84 20.00 70.00 150.00 4 240 210 10 190 o 140 0 15.00 210.00 15.00 3 240 150 150 130 28 42 70 65.00 85.00 70.00 4 240 170 50 170 28 112 0 50.00 140.00 50.00 2 240 170 170 190 42 56 42 85.00 56.00 85.00 3 240 170 130 150 0 70 70 55.00 115.00 70.00 4 240 170 130 150 56 70 14 75.00 100.00 65.00 4 240 130 130 130 14 56 70 50.00 89.00 101.00 3 240 170 190 150 56 28 56 120.00 50.00 56.00 4 240 50 110 170 0 42 98 40.00 71.00 129.00 4 240 50 210 210 0 14 126 57.00 57.00 126.00

Series 7.11

3 240 170 190 150 56 28 56 100.00 70.00 56.00 2 240 50 210 210 0 14 126 35.00 14.00 175.00 3 240 170 130 150 56 70 14 80.00 90.00 14.00 3 240 50 110 170 0 42 98 5.00 45.00 98.00 4 240 110 150 170 0 42 98 40.00 75.00 125.00 4 240 130 170 110 84 0 56 110.00 45.00 85.00 4 240 90 150 150 28 o 112 65.00 45.00 130.00 4 240 30 210 210 28 o 112 40.00 20.00 180.00 1 240 210 150 170 56 70 14 56.00 120.00 50.00 4 240 150 130 150 70 70 0 90.00 110.00 40.00 2 240 190 110 170 42 84 14 70.00 84.00 40.00 1 240 90 170 190 28 o 112 28.00 45.00 145.00 4 240 130 130 70 70 56 14 100.00 90.00 50.00 2 240 150 150 150 56 56 28 100.00 56.00 50.00 4 240 130 110 130 42 56 42 75.00 90.00 75.00 1 240 150 170 190 42 28 70 42.00 60.00 130.00 4 240 130 130 130 84 42 14 105.00 75.00 60.00 1 240 130 190 210 14 28 98 14.00 70.00 140.00 2 240 190 110 190 o 126 14 50.00 126.00 60.00 2 240 210 130 130 56 84 0 100.00 84.00 30.00 3 240 90 130 150 0 14 i26 30.00 60.00 126.00 4 240 70 130 150 14 56 70 40.00 85.00 115.00 4 240 130 150 110 42 0 98 85.00 48.00 107.00 4 240 130 190 130 84 0 56 110.00 40.00 90.00 3 240 210 110 170 70 70 0 85.00 125.00 0.00 4 240 170 170 90 126 14 0 145.00 65.00 30.00 4 240 90 150 150 0 70 70 38.00 85.00 117.00 4 240 130 130 70 98 14 28 130.00 50.00 60.00 3 240 150 150 90 70 14 56 103.00 47.00 56.00 2 240 150 150 150 28 70 42 65.00 70.00 85.00 1 240 150 150 150 84 14 42 84.00 60.00 90.00 4 240 130 190 110 112 0 28 130.00 30.00 80.00 1 240 70 210 210 28 14 98 28.00 60.00 150.00 4 240 130 150 110 84 14 42 110.00 65.00 65.00 2 240 150 170 170 0 56 84 70.00 513. 00 100.00 2 240 190 130 130 42 98 0 90.00 ; 98.00 40.00 4 240 30 150 150 0 28 112 30.00 60.00 150.00 4 240 110 150 150 14 56 70 45.00 80.00 115.00

c. Listing of all Results 161

4 240 110 150 150 0 28 112 45.00 70.00 125.00 2 240 170 150 150 84 42 14 95.00 42.00 55.00 4 240 90 150 150 0 28 112 40.00 70.00 130.00 4 240 90 150 170 0 o 140 45.00 45.00 150.00 2 240 190 110 150 o 126 14 50.00 126.00 60.00 4 240 150 170 110 70 0 70 110.00 40.00 90.00

Series 7.12

2 240 190 110 150 o 126 14 25.00 126.00 85.00 4 240 130 190 70 70 0 70 105.00 35.00 100.00 4 240 90 150 170 0 o 140 48.00 48.00 144.00 3 240 170 130 170 0 98 42 85.00 85.00 42.00 3 240 150 170 110 70 0 70 100.00 50.00 70.00 2 240 150 170 150 14 70 56 75.00 70.00 95.00 4 240 150 150 110 84 0 56 110.00 45.00 85.00 4 240 170 10 170 o 140 0 40.00 160.00 40.00 4 240 170 90 150 14 98 28 50.00 120.00 70.00 3 240 210 90 90 84 56 0 110.00 100.00 0.00 4 240 130 130 110 84 14 42 110.00 45.00 85.00 4 240 170 90 170 28 98 14 53.00 140.00 47.00 1 240 110 150 230 0 28 112 0.00 80.00 150.00 1 240 110 170 190 0 28 112 0.00 60.00 130.00 1 240 150 170 150 84 0 56 84.00 65.00 85.00 4 240 70 150 150 56 0 84 78.00 50.00 112.00 1 240 150 170 110 112 0 28 112.00 40.00 70.00 3 240 130 170 210 0 42 98 40.00 90.00 98.00 4 240 170 170 50 98 14 28 140.00 50.00 50.00 4 240 150 150 150 70 70 0 100.00 100.00 40.00 1 240 150 130 150 70 14 56 70.00 55.00 95.00 4 240 130 90 130 14 98 28 50.00 125.00 65.00 1 240 110 150 170 42 0 98 42.00 40.00 130.00 4 240 90 130 190 0 56 84 50.00 80.00 110.00 4 240 210 30 170 o 140 0 40.00 180.00 20.00 3 240 150 110 130 28 42 70 70.00 80.00 70.00 4 240 150 150 130 70 14 56 107.00 51.00 82.00 1 240 110 190 110 98 0 42 98.00 30.00 80.00 1 240 130 150 170 56 28 56 56.00 70.00 100.00 4 240 150 90 150 28 70 42 56.00 112.00 72.00 4 240 150 150 150 56 42 42 84.00 78.00 78.00 3 240 170 150 170 42 56 42 80.00 90.00 42.00 1 240 170 150 150 84 56 0 84.00 105.00 45.00 1 240 150 150 150 84 14 42 84.00 65.00 85.00 1 240 170 190 170 70 0 70 70.00 70.00 100.00 4 240 150 90 110 84 56 0 120.00 80.00 40.00 2 240 130 230 110 112 0 28 130.00 0.00 100.00 4 240 190 110 130 56 56 28 100.00 100.00 40.00 4 240 90 130 130 14 70 56 65.00 95.00 80.00 4 240 210 70 150 28 98 14 76.00 140.00 24.00 4 240 70 130 150 0 42 98 26.00 76.00 138.00 4 240 190 110 150 56 84 0 83.00 117.00 40.00 4 240 170 130 170 14 70 56 40.00 120.1'0 80.00 4 240 110 230 90 98 0 42 141.00 14>.00 89.00

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Vol. 285: Toward Interactive and Intelligent Decision Support Systems. Volume 1. Proceedings, 1986. Edited by Y. Sawaragi, K. Inoue and H. Nakayama. XII, 445 pages. 1987.

Vol. 286: Toward Interactive and Intelligent Decision Support Systems. Volume 2. Proceedings, 1986. Ed"rted by Y. Sawaragi, K. Inoue and H. Nakayama. XII, 450 pages. 1987.

Vol. 287: Dynamical Systems. Proceedings, 1985. Edited by A. B. Kurzhanski and K. Sigmund. VI, 215 pages. 1987.

Vol. 286: G. D. Rudebusch, The Estimstion of Macroeconomic DiSequilibrium Models with Regime Classification Infomiaticn. VII, 128 pages. 1987.

Vol. 289: B. R. Meijboom, Planning in Decentralized Firms. X, 168 pages. 1987.

Vol. 290: D.A. Carison, A. Haurie, Infinite Horizon Optimal Control. XI, 254 pages. 1987./

Vol. 291: N. TakahUhi, Design of Adaptive Organizations. VI, 140 pages. 1987.

Vol. 292: I. Tchijov, L Tomaszewicz (Eds.), Input·Output Modeling. Proceedings, 1985. VI, 195 pages. 1987.

Vol. 293: D. Batten, J. Casti, B. Johansson (Eds.), Economic Evolu· tion and Structural Adjustment. Proceedings, 1985. VI, 382 pages. 1987.

Vol. 294: J. Jahn, W. Krabs (Eds.), Recent Advances and Historical Development of Vector Optimization. VII, 405 pages. 1987.

Vol. 295: H. Meister, The Purification Problem for Constrained Games with Incomplete Information. X, 127 pages. 1987.

Vol. 296: A. BOrsch-Supan, Econometric Analysis of Discrete Choice. VIII, 211 pages. 1987.

Vol. 297: V. Fedorov, H. Liiuter (Eds.), Model-Oriented Oats Analysis. Procsedings, 1987. VI, 239_ pages. 1988.

Vol. 298: S. H. Chew, Q. Zheng, Integral Global Optimization. VII, 179 pages. 1 988.

Vol. 299: K. Marti, Descent Directions and Efficient Solutions in Discretely Distributed Stochastic Programs. XIV, 178 pages. 1988.

Vol. 300: U. Derigs, Programming in Networks and Graphs. XI, 315 pages. 1988.

Vol. 301: J. Kacprzyk, M. Roubens (Eds.), Non-Conventional Preference Relations in Decision Making. VII, 155 pages. 1988.

Vol. 302: H.A. Eisel!, G. Pederzoli (Eds.), Advances in Optimization and Control. Proceedings, 1985. VIII, 372 pages. 1988.

Vol. 303: F.X. Diebold, Empirical Modeling of Exchange Rate Dynamics. VII, 143 pages. 1988.

Vol. 304: A. Kurzhanski, K. Neumann, D. Pallaschke (Eds.), Optimization, Parallel Processing and Applications. Proceedings, 1987. VI, 292 pages. 1988.

Vol. 305: G.-J.C.Th. van Schijndel, Dynamic Firm and Investor Behaviour under Progressive Perscnal Taxation. X, 215 pages. 1988.

Vol. 306: Ch. Klein, A Static Microeconomic Model of Pure Competition. VIII, 139 pages. 1988.

Vol. 307: T. K. Dijkstra (Ed.), On Model Uncertainty and its Ststistical Implications. VII, 138 pages. 1988.

Vol. 308: J. R. Daduna, A. Wren (Eds.), Computer-Aided Transit Scheduling. VIII, 339 pages. 1988.

Vol. 309: G. Ricci, K. Velupillai (Eds.), Growth Cycles and Munisectoral Economics: the Goodwin Tradition. III, 126 pages. 1988.

Vol. 310: J. Kacprzyk, M. Fedrizzi (Eds.), Combining Fuzzy Imprecision with Probabilistic Uncertainty in Decision Making. IX, 399 pages. 1988.

Vol. 311: R. Fare, Fundamentals of Production Theory. IX, 163 pages. 1988.

Vol. 312: J. Krishnakumar, Estimation of Simunaneous Equation Models wijh Error Components Structure. X, 357 pages. 1988.

Vol. 313: W. Jammemegg, Sequential Binary Investment Decisions. VI, 156 pages. 1988.

Vol. 314: R. Tietz, W. Albers, R. Selten (Eds.), Bounded Rational Behavior in Experimental Garnes and Markets. VI, 368 pages. 1988.

Vol. 315: I. Orishimo, G.J.D. Hewings, P. Nijksmp (Eds.), Information Technology: Social and Spatial Perspectives. Proceedings, 1986. VI, 268 pages. 1988.

Vol. 316: R. L. Basmann, D.J. Slottje, K. Hayes, J. D. Johnson, D.J. Molina, The Generalized Fechner-Thurstone Direct Utilijy Function and Some of ijs Uses. VIII, 159 pages. 1988.

Vol. 317: L. Bianco, A. La Bella (Eds.), Freight Transport Planning and Logistics. Proceedings, 1987. X, 568 pages. 1988.

Vol. 318: T. Doup, Simplicial Algomhms on the Simplotope. VIII, 282 pages. 1988.

Vol. 319: D.T. Luc, Theory of Vector Optimization. VIII, 173 pages. 1989.

Vol. 320: D. van der Wjst, Financial Structure in Small Business. VII, 181 pages. 1989.

Vol. 321: M. Di Matteo, R.M. Goodwin, A. Vereelli (Eds.), Technplogical and Social Factors in Long Term Fluctuations. Proceedings. IX, 442 pages. 1989.

Vol. 322: T. Kollintzas (Ed.), The Rational Expectations Equilibrium Inventory Model. XI, 269 pages. 1989.

Vol. 323: M.B.M. de Koster, Capacijy Oriented Analysis and Design of Production Systems. XII, 245 pages. 1989.

Vol. 324: I. M. Bomze, B. M. Piitscher, Game Theoretical Foundations of Evolutionary Stsbilijy. VI, 145 pages. 1989.

Vol. 325: P. Ferri, E. Greenberg, The Labor Market and Business Cycle Theories. X, 183 pages. 1989.

Vol. 326: Ch. Sauer, Anemative Theories of Output, Unemployment, and Inflation in Germany: 1980-1985. XIII, 206 pages. 1989.

Vol. 327: M. Tawada, Production Structure and Intemational Trade. V, 132 pages. 1989.

Vol. 328: W. Gilih, B. Kalkofen, Unique Solutions for Strategic Garnes. VII, 200 pages. 1989.

Vol. 329: G. Tillmann, Equijy, Incentives, and Taxation. VI, 132 pages. 1989.

Vol. 330: P. M. Kort, Optimal Dynamic Investment Policies of a Value Maximizing Firm. VII, 185 pages. 1989.

Vol. 331: A. Lewandowski, A.P. Wierzbicki (Eds.), Aspiration Based Decision Support Systems. X, 400 pages. 1989.

Vol. 332: T.R. Gulledge, Jr., L.A. Litteral (Eds.), Cost Analysis Applications of Economics and Operations Research. Proceedings. VII, 422 pages. 1989.

Vol. 333: N. Dellaert, Production to Order. VII, 158 pages. 1989.

Vol. 334: H.-W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion. XI, 248 pages. 1989.

Vol. 335: A. G. Lockett, G.lslei (Eds.), Improving Decision Making in Organisations. Proceedings. IX, 606 pages. 1989.

Vol. 336: T. Puu, Nonlinear Economic Dynamics. VII, 119 pages. 1989.

Vol. 337: A. Lewandowski, I. Sta.nchev (Eds.), Methodology and Software for Interactive Decision Support. VIII, 309 pages. 1989.

Vol. 338: J. K. Ho, R. P. Sundarraj, DECOMP: an Implementation of Dantzig-Wolfe Decomposition for Linear Programming. VI, 206 pages. 1989.

Vol. 339: J. Terceiro Lomba, Estimation of Dynamic Econometric Models with Errors in Variables. VIII, 116 pages. 1990.

Vol. 340: T. Vasko, R. Ayres, L. Fontvieille (Eds.), Life Cycles and Long Waves. XIV, 293 pages. 1990.

Vol. 341: G. R. Uhlich, Descriptive Theories of Bargaining. IX, 165 pages. 1990.

J.C. Willems (Ed.)

From Data to Model 1989. VII, 246 pp. 35 figs. 10 tabs. Hardcover DM 98,- ISBN 3-540-51571-2

This book consists of 5 chapters. The general theme is to develop a mathematical framework and a language for modelling dynamical systems from observed data. Two chapters study the statistical aspects of approximate linear time-series analysis. One chapter develops worst case aspects of system identification. Finally, there are two chapters on system approximation. The first one is a tutorial on the Hankel-norm approximation as an approach to model simplification in linear systems. The second one gives a philosophy for setting up numerical algorithms from which a model optimally fits an observed time series.

P.Hackl (Ed.)

Statistical Analysis and Forecasting of Economic Structural Change 1989. XIX, 488 pp. 98 figs. 60 tabs. Hardcover DM 178,- ISBN 3-540-51454-6

This book treats methods and problems of the statistical analysis of economic data in the context of structural change. It documents the state of the art, gives insights into existing methods, and describes new developments and trends. An introductory chapter gives a survey of the book and puts the following chapters into a broader context. The rest of the volume is organized in three parts: a) Identification of Structural Change; b) Model Building in the Presence of Structural Change; c) Data Analysis and Modeling.

Springer-Vedag

C. D. Aliprantis, D. J. Brown, O. Burkinshaw

EXm~nceandOpHmaH~ of Competitive Equilibria 1989. XII, 284 pp. 38 figs. Hardcover DM 110,- ISBN 3-540-50811-2

Contents: The Arrow-Debreu Model. - Riesz Spaces of Commodities and Prices. - Markets with Infinitely Many Commodities. - Production with Infinitely Many Commodities. -The Overlapping Generations Model. -References. - Index.

B.L.Golden, E.A. Wasil, P. T.Harker (Eds.)

The Analytic Hierarchy Process Applications and Studies

With contributions by numerous experts 1989. VI, 265 pp. 60 figs. 74 tabs. Hardcover DM 110,- ISBN 3-540-51440-6

The book is divided into three sections. In the first section, a detailed tutorial and an extensive annotated bibliography serve to introduce the methodology. The second section includes two papers which present new methodological advances in the theory of the AHP. The third section, by far the largest, is dedicated to applications and case studies; it contains twelve chapters. Papers dealing with project selection, electric utility planning, govemmental decision making, medical decision making, conflict analysis, strategic planning, and others are used to illustrate how to successfully apply the AHP. Thus, this book should serve as a useful text in courses dealing with decision making as well as a valuable reference for those involved in the application of decision analysis techniques.

Berlin Heidelberg New York London Paris Tokyo Hong Kong

Descriptive Theories of Bargaining: An Experimental Analysis of Two- and Three-Person Characteristic Function Bargaining

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