A THEORY OF FAIRNESS, COMPETITION, AND ...web.stanford.edu/~niederle/Fehr.Schmidt.1999.QJE.pdfA THEORY OF FAIRNESS, COMPETITION, AND COOPERATION

A THEORY OF FAIRNESS COMPETITIONAND COOPERATION

ERNST FEHR AND KLAUS M SCHMIDT

There is strong evidence that people exploit their bargaining power incompetitive markets but not in bilateral bargaining situations There is also strongevidence that people exploit free-riding opportunities in voluntary cooperationgames Yet when they are given the opportunity to punish free riders stablecooperation is maintained although punishment is costly for those who punishThis paper asks whether there is a simple common principle that can explain thispuzzling evidence We show that if some people care about equity the puzzles canbe resolved It turns out that the economic environment determines whether thefair types or the selsh types dominate equilibrium behavior

I INTRODUCTION

Almost all economic models assume that all people areexclusively pursuing their material self-interest and do not careabout lsquolsquosocialrsquorsquo goals per se This may be true for some (maybemany) people but it is certainly not true for everybody By now wehave substantial evidence suggesting that fairness motives affectthe behavior of many people The empirical results of KahnemanKnetsch and Thaler [1986] for example indicate that customershave strong feelings about the fairness of rmsrsquo short-run pricingdecisions which may explain why some rms do not fully exploittheir monopoly power There is also a lot of evidence suggestingthat rmsrsquo wage setting is constrained by workersrsquo views aboutwhat constitutes a fair wage [Blinder and Choi 1990 Agell andLundborg 1995 Bewley 1995 Campbell and Kamlani 1997]According to these studies a major reason for rmsrsquo refusal to cutwages in a recession is the fear that workers will perceive pay cutsas unfair which in turn is expected to affect work morale ad-versely There are also many well-controlled bilateral bargainingexperiments which indicate that a nonnegligible fraction of the

We would like to thank seminar participants at the Universities of Bonnand Berlin Harvard Princeton and Oxford Universities the European SummerSymposium on Economic Theory 1997 at Gerzensee (Switzerland) and the ESAconference in Mannheim for helpful comments and suggestions We are particu-larly grateful to three excellent referees and to Drew Fudenberg and John Kagelfor their insightful comments The rst author also gratefully acknowledgessupport from the Swiss National Science Foundation (project number 1214-0510097) and the Network on the Evolution of Preferences and Social Norms ofthe MacArthur Foundation The second author acknowledges nancial support bythe German Science Foundation through grant SCHM 119614-1

r 1999 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics August 1999

817

subjects do not care solely about material payoffs [Guth and Tietz1990 Roth 1995 Camerer and Thaler 1995] However there isalso evidence that seems to suggest that fairness considerationsare rather unimportant For example in competitive experimen-tal markets with complete contracts in which a well-denedhomogeneous good is traded almost all subjects behave as if theyare only interested in their material payoff Even if the competi-tive equilibrium implies an extremely uneven distribution of thegains from trade equilibrium is reached within a few periods[Smith and Williams 1990 Roth Prasnikar Okuno-Fujiwara andZamir 1991 Kachelmeier and Shehata 1992 Guth Marchandand Rulliere 1997]

There is similarly conicting evidence with regard to coopera-tion Reality provides many examples indicating that people aremore cooperative than is assumed in the standard self-interestmodel Well-known examples are that many people vote pay theirtaxes honestly participate in unions and protest movements orwork hard in teams even when the pecuniary incentives go in theopposite direction1 This is also shown in laboratory experiments[Dawes and Thaler 1988 Ledyard 1995] Under some conditions ithas even been shown that subjects achieve nearly full cooperationalthough the self-interest model predicts complete defection [Isaacand Walker 1988 1991 Ostrom and Walker 1991 Fehr andGachter 1996]2 However as we will see in more detail in SectionIV there are also those conditions under which a vast majority ofsubjects completely defect as predicted by the self-interest model

There is thus a bewildering variety of evidence Some piecesof evidence suggest that many people are driven by fairnessconsiderations other pieces indicate that virtually all peoplebehave as if completely selsh and still other types of evidencesuggest that cooperation motives are crucial In this paper we askwhether this conicting evidence can be explained by a singlesimple model Our answer to this question is affirmative if one iswilling to assume that in addition to purely self-interestedpeople there are a fraction of people who are also motivated byfairness considerations No other deviations from the standard

1 On voting see Mueller [1989] Skinner and Slemroad [1985] argue that thestandard self-interest model substantially underpredicts the number of honesttaxpayers Successful team production in eg Japanese-managed auto factoriesin North America is described in Rehder [1990] Whyte [1955] discusses howworkers establish lsquolsquoproduction normsrsquorsquo under piece-rate systems

2 Isaac and Walker and Ostrom and Walker allow for cheap talk while inFehr and Gachter subjects could punish each other at some cost

QUARTERLY JOURNAL OF ECONOMICS818

economic approach are necessary to account for the evidence Inparticular we do not relax the rationality assumption3

We model fairness as self-centered inequity aversion Ineq-uity aversion means that people resist inequitable outcomes iethey are willing to give up some material payoff to move in thedirection of more equitable outcomes Inequity aversion is self-centered if people do not care per se about inequity that existsamong other people but are only interested in the fairness of theirown material payoff relative to the payoff of others We show thatin the presence of some inequity-averse people lsquolsquofairrsquorsquo and lsquolsquocoopera-tiversquorsquo as well as lsquolsquocompetitiversquorsquo and lsquolsquononcooperativersquorsquo behavioralpatterns can be explained in a coherent framework A maininsight of our examination is that the heterogeneity of preferencesinteracts in important ways with the economic environment Weshow in particular that the economic environment determinesthe preference type that is decisive for the prevailing behavior inequilibrium This means for example that under certain competi-tive conditions a single purely selsh player can induce a largenumber of extremely inequity-averse players to behave in acompletely selsh manner too Likewise under certain conditionsfor the provision of a public good a single selsh player is capableof inducing all other players to contribute nothing to the publicgood although the others may care a lot about equity We alsoshow however that there are circumstances in which the exis-tence of a few inequity-averse players creates incentives for amajority of purely selsh types to contribute to the public goodMoreover the existence of inequity-averse types may also induceselsh types to pay wages above the competitive level Thisreveals that in the presence of heterogeneous preferences theeconomic environment has a whole new dimension of effects4

There are a few other papers that formalize the notion offairness5 In particular Rabin [1993] argues that people want tobe nice to those who treat them fairly and want to punish thosewho hurt themAccording to Rabin an action is perceived as fair if

3 This differentiates our model from learning models (eg Roth and Erev[1995]) that relax the rationality assumption but maintain the assumption that allplayers are only interested in their own material payoff The issue of learning isfurther discussed in Section VII below

4 Our paper is therefore motivated by a concern similar to the papers byHaltiwanger and Waldman [1985] and Russell and Thaler [1985] While theseauthors examine the conditions under which nonrational or quasi-rational typesaffect equilibrium outcomes we analyze the conditions under which fair typesaffect the equilibrium

5 Section VIII deals with them in more detail

A THEORY OF FAIRNESS COMPETITION AND COOPERATION 819

the intention that is behind the action is kind and as unfair if theintention is hostile The kindness or the hostility of the intentionin turn depends on the equitability of the payoff distributioninduced by the action Thus Rabinrsquos model as our model is basedon the notion of an equitable outcome In contrast to our modelhowever Rabin models the role of intentions explicitly Weacknowledge that intentions do play an important role and that itis desirable to model them explicitly However the explicit model-ing of intentions comes at a cost because it requires the adoption ofpsychological game theory that is much more difficult to applythan standard game theory In fact Rabinrsquos model is restricted totwo-person normal form games which means that very importantclasses of games like eg market games and n-person publicgood games cannot be analyzed Since a major focus of this paperis the role of fairness in competitive environments and theanalysis of n-person cooperation games we chose not to modelintentions explicitly This has the advantage of keeping the modelsimple and tractable We would like to stress however thatmdashalthough we do not model intentions explicitlymdashit is possible tocapture intentions implicitly by our formulation of fairness prefer-ences We deal with this issue in Section VIII

The rest of the paper is organized as followed In Section II wepresent our model of inequity aversion Section III applies thismodel to bilateral bargaining and market games In Section IVcooperation games with and without punishments are consideredIn Section V we show that on the basis of plausible assumptionsabout preference parameters the majority of individual choices inultimatum and market and cooperation games considered in theprevious sections are consistent with the predictions of our modelSection VI deals with the dictator game and with gift exchangegames In Section VII we discuss potential extensions and objec-tions to our model Section VIII compares our model with alterna-tive approaches in the literature Section IX concludes

II A SIMPLE MODEL OF INEQUITY AVERSION

An individual is inequity averse if he dislikes outcomes thatare perceived as inequitable This denition raises of course thedifficult question of how individuals measure or perceive thefairness of outcomes Fairness judgments are inevitably based ona kind of neutral reference outcome The reference outcome that isused to evaluate a given situation is itself the product of compli-


cated social comparison processes In social psychology [Festinger1954 Stouffer 1949 Homans 1961 Adams 1963] and sociology[Davis 1959 Pollis 1968 Runciman 1966] the relevance of socialcomparison processes has been emphasized for a long time Onekey insight of this literature is that relative material payoffs affectpeoplersquos well-being and behavior As we will see below without theassumption that at least for some people relative payoffs matter itis difficult if not impossible to make sense of the empiricalregularities observed in many experiments There is moreoverdirect empirical evidence for the importance of relative payoffsAgell and Lundborg [1995] and Bewley [1998] for example showthat relative payoff considerations constitute an important con-straint for the internal wage structure of rms In addition Clarkand Oswald [1996] show that comparison incomes have a signi-cant impact on overall job satisfaction They construct a compari-son income level for a random sample of roughly 10000 Britishindividuals by computing a standard earnings equation Thisearnings equation determines the predicted or expected wage ofan individual with given socioeconomic characteristics Then theyexamine the impact of this comparison wage on overall jobsatisfaction Their main result is thatmdashholding other thingsconstantmdashthe comparison income has a large and signicantlynegative impact on overall job satisfaction

Strong evidence for the importance of relative payoffs is alsoprovided by Loewenstein Thompson and Bazerman [1989] Theseauthors asked subjects to ordinally rank outcomes that differ inthe distribution of payoffs between the subject and a comparisonperson On the basis of these ordinal rankings the authorsestimate how relative material payoffs enter the personrsquos utilityfunction The results show that subjects exhibit a strong androbust aversion against disadvantageous inequality for a givenown income xi subjects rank outcomes in which a comparisonperson earns more than xi substantially lower than an outcomewith equal material payoffs Many subjects also exhibit anaversion to advantageous inequality although this effect seems tobe signicantly weaker than the aversion to disadvantageousinequality

The determination of the relevant reference group and therelevant reference outcome for a given class of individuals isultimately an empirical question The social context the saliencyof particular agents and the social proximity among individualsare all likely to inuence reference groups and outcomes Because


in the following we restrict attention to individual behavior ineconomic experiments we have to make assumptions aboutreference groups and outcomes that are likely to prevail in thiscontext In the laboratory it is usually much simpler to denewhat is perceived as an equitable allocation by the subjects Thesubjects enter the laboratory as equals they do not know any-thing about each other and they are allocated to different roles inthe experiment at random Thus it is natural to assume that thereference group is simply the set of subjects playing against eachother and that the reference point ie the equitable outcome isgiven by the egalitarian outcome

More precisely we assume the following First in addition topurely selsh subjects there are subjects who dislike inequitableoutcomes They experience inequity if they are worse off inmaterial terms than the other players in the experiment and theyalso feel inequity if they are better off Second however weassume that in general subjects suffer more from inequity that isto their material disadvantage than from inequity that is to theirmaterial advantage Formally consider a set of n players indexedby i [ 1 n and let x 5 x1 xn denote the vector of mone-tary payoffs The utility function of player i [ 1 n is given by

(1) Ui(x) 5 xi 2 a i

1

n 2 1 ojTHORN i

max xj 2 xi0

2 b i

1

n 2 1 ojTHORN i

max xi 2 xj0

where we assume that b i a i and 0 b i 1 In the two-playercase (1) simplies to

(2) Ui(x) 5 xi 2 a i max xj 2 xi0 2 b i max xi 2 xj0 i THORN j

The second term in (1) or (2) measures the utility loss fromdisadvantageous inequality while the third term measures theloss from advantageous inequality Figure I illustrates the utilityof player i as a function of xj for a given income xi Given his ownmonetary payoff xi player irsquos utility function obtains a maximumat xj 5 xi The utility loss from disadvantageous inequality (xj xi)is larger than the utility loss if player i is better off than playerj(xj x i)6

6 In all experiments considered in this paper the monetary payoff functionsof all subjects were common knowledge Note that for inequity aversion to be


To evaluate the implications of this utility function let usstart with the two-player case For simplicity we assume that theutility function is linear in inequality aversion as well as in xiThis implies that the marginal rate of substitution betweenmonetary income and inequality is constant This may not be fullyrealistic but we will show that surprisingly many experimentalobservations that seem to contradict each other can be explainedon the basis of this very simple utility function already Howeverwe will also see that some observations in dictator experimentssuggest that there are a nonnegligible fraction of people whoexhibit nonlinear inequality aversion in the domain of advanta-geous inequality (see Section VI below)

Furthermore the assumption a i $ b i captures the idea that aplayer suffers more from inequality that is to his disadvantageThe above-mentioned paper by Loewenstein Thompson and

behaviorally important it is not necessary for subjects to be informed about thenal monetary payoffs of the other subjects As long as subjectsrsquo material payofffunctions are common knowledge they can compute the distributional implica-tions of any (expected) strategy prole ie inequity aversion can affect theirdecisions

FIGURE IPreferences with Inequity Aversion


Bazerman [1989] provides strong evidence that this assumptionis in general valid Note that a i $ b i essentially means that asubject is loss averse in social comparisons negative deviationsfrom the reference outcome count more than positive deviationsThere is a large literature indicating the relevance of loss aversionin other domains (eg Tversky and Kahneman [1991]) Hence itseems natural that loss aversion also affects social comparisons

We also assume that 0 b i 1 b i $ 0 means that we rule outthe existence of subjects who like to be better off than others Weimpose this assumption here although we believe that there aresubjects with b i 07 The reason is that in the context of theexperiments we consider individuals with b i 0 have virtually noimpact on equilibrium behavior This is in itself an interestinginsight that will be discussed extensively in Section VII Tointerpret the restriction b i 1 suppose that player i has a highermonetary payoff than player j In this case b i 5 05 implies thatplayer i is just indifferent between keeping one dollar to himselfand giving this dollar to player j If b i 5 1 then player i isprepared to throw away one dollar in order to reduce his advan-tage relative to player j which seems very implausible This is whywe do not consider the case b i $ 1 On the other hand there is nojustication to put an upper bound on a i To see this suppose thatplayer i has a lower monetary payoff than player j In this caseplayer i is prepared to give up one dollar of his own monetarypayoff if this reduces the payoff of his opponent by (1 1 a i) a i

dollars For example if a i 5 4 then player i is willing to give upone dollar if this reduces the payoff of his opponent by 125 dollarsWe will see that observable behavior in bargaining and publicgood games suggests that there are at least some individuals withsuch high a rsquos

If there are n 2 players player i compares his income withall other n 2 1 players In this case the disutility from inequalityhas been normalized by dividing the second and third term by n 21 This normalization is necessary to make sure that the relativeimpact of inequality aversion on player irsquos total payoff is indepen-dent of the number of players Furthermore we assume forsimplicity that the disutility from inequality is self-centered in thesense that player i compares himself with each of the other

7 For the role of status seeking and envy see Frank [1985] and Banerjee[1990]


players but he does not care per se about inequalities within thegroup of his opponents

III FAIRNESS RETALIATION AND COMPETITION ULTIMATUM

AND MARKET GAMES

In this section we apply our model to a well-known simplebargaining gamemdashthe ultimatum gamemdashand to simple marketgames in which one side of the market competes for an indivisiblegood As we will see below a considerable body of experimentalevidence indicates that in the ultimatum game the gains fromtrade are shared relatively equally while in market games veryunequal distributions are frequently observed Hence any alterna-tive to the standard self-interest model faces the challenge toexplain both lsquolsquofairrsquorsquo outcomes in the ultimatum game and lsquolsquocompeti-tiversquorsquo and rather lsquolsquounfairrsquorsquo outcomes in market games

A The Ultimatum Game

In an ultimatum game a proposer and a responder bargainabout the distribution of a surplus of xed size Without loss ofgenerality we normalize the bargaining surplus to one Theresponderrsquos share is denoted by s and the proposerrsquos share by 1 2s The bargaining rules stipulate that the proposer offers a share s[ [01] to the responder The responder can accept or reject s Incase of acceptance the proposer receives a (normalized) monetarypayoff x1 5 1 2 s while the responder receives x2 5 s In case of arejection both players receive a monetary return of zero Theself-interest model predicts that the responder accepts any s [(01] and is indifferent between accepting and rejecting s 5 0Therefore there is a unique subgame perfect equilibrium in whichthe proposer offers s 5 0 which is accepted by the responder8

By now there are numerous experimental studies from differ-ent countries with different stake sizes and different experimen-tal procedures that clearly refute this prediction (for overviews

8 Given that the proposer can choose s continuously any offer s 0 cannot bean equilibrium offer since there always exists an s8 with 0 s8 s which is alsoaccepted by the responder and yields a strictly higher payoff to the proposerFurthermore it cannot be an equilibrium that the proposer offers s 5 0 which isrejected by the responder with positive probability In this case the proposer woulddo better by slightly raising his pricemdashin which case the responder would acceptwith probability 1 Hence the only subgame perfect equilibrium is that theproposer offers s 5 0 which is accepted by the responder If there is a smallestmoney unit e then there exists a second subgame perfect equilibrium in which theresponder accepts any s [ [ e 1] and rejects s 5 0 while the proposer offers e


see Thaler [1988] Guth and Tietz [1990] Camerer and Thaler[1995] and Roth [1995]) The following regularities can be consid-ered as robust facts (see Table I) (i) There are virtually no offersabove 05 (ii) The vast majority of offers in almost any study is inthe interval [04 05] (iii) There are almost no offers below 02 (iv)Low offers are frequently rejected and the probability of rejectiontends to decrease with s Regularities (i) to (iv) continue to hold forrather high stake sizes as indicated by the results of Cameron[1995] Hoffman McCabe and Smith [1996] and Slonim and Roth[1997] The 200000 rupiahs in the second experiment of Cameron(see Table I) are eg equivalent to three monthsrsquo income for theIndonesian subjects Overall roughly 60ndash80 percent of the offersin Table I fall in the interval [04 05] while only 3 percent arebelow a share of 02

To what extent is our model capable of accounting for thestylized facts of the ultimatum game To answer this questionsuppose that the proposerrsquos preferences are represented by (a 1b 1)while the responderrsquos preferences are characterized by ( a 2 b 2)The following proposition characterizes the equilibrium outcomeas a function of these parameters

PROPOSITION 1 It is a dominant strategy for the responder toaccept any offer s $ 05 to reject s if

s s8( a 2) a 2(1 1 2 a 2) 05

and to accept s s8( a 2) If the proposer knows the preferencesof the responder he will offer

(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05

in equilibrium If the proposer does not know the preferencesof the responder but believes that a 2 is distributed accordingto the cumulative distribution function F( a 2) where F( a 2)has support [ a a ] with 0 a a ` then the probability(from the perspective of the proposer) that an offer s 05 isgoing to be accepted is given by

(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )


Hence the optimal offer of the proposer is given by

(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05

TABLE IPERCENTAGE OF OFFERS BELOW 02 AND BETWEEN 04 AND 05

IN THE ULTIMATUM GAME

Study(Payment method)

Number ofobservations

Stake size(country)

Percentage ofoffers with

s 02


04 s 05

Cameron [1995](All Ss Paid)

35 Rp 40000(Indonesia)

0 66

Cameron [1995](all Ss paid)


5 57

FHSS [1994](all Ss paid)

67 $5 and $10(USA)

0 82

Guth et al [1982](all Ss paid)

79 DM 4ndash10(Germany)

8 61

Hoffman McCabeand Smith [1996](All Ss paid)

24 $10(USA)

0 83

Hoffman McCabeand Smith [1996](all Ss paid)

27 $100(USA)

4 74

KahnemanKnetsch andThaler [1986](20 of Ss paid)

115 $10(USA)

75a

Roth et al [1991](random pay-ment method)

116b approx $10(USA SloveniaIsrael Japan)

3 70

Slonim and Roth[1997](random pay-ment method)

240c SK 60(Slovakia)

04d 75



8d 69

Aggregate result ofall studiese

875 38 71

a percentage of equal splits b only observations of the nal period c observations of all ten periodsd percentage of offers below 025 e without Kahneman Knetsch and Thaler [1986]


Proof If s $ 05 the utility of a responder from accepting s isU2(s) 5 s 2 b 2(2s 2 1) which is always positive for b 2 1 and thusbetter than a rejection that yields a payoff of 0 The point is thatthe responder can achieve equality only by destroying the entiresurplus which is very costly to him if s $ 05 ie if the inequalityis to his advantage For s 05 a responder accepts the offer onlyif the utility from acceptance U2(s) 5 s 2 a 2(1 2 2s) is nonnega-tive which is the case only if s exceeds the acceptance threshold

s8( a 2) a 2(1 1 2 a 2) 05

At stage 1 a proposer never offers s 05 This would reduce hismonetary payoff as compared with an offer of s 5 05 which wouldalso be accepted with certainty and which would yield perfectequality If b 1 05 his utility is strictly increasing in s for all s 05 This is the case where the proposer prefers to share hisresources rather than to maximize his own monetary payoff so hewill offer s 5 05 If b 1 5 05 he is just indifferent between givingone dollar to the responder and keeping it to himself ie he isindifferent between all offers s [ [srsquo( a 2) 05] If b 1 05 theproposer would like to increase his monetary payoff at the expenseof the responder However he is constrained by the responderrsquosacceptance threshold If the proposer is perfectly informed aboutthe responderrsquos preferences he will simply offer s8( a 2) If theproposer is imperfectly informed about the responderrsquos type thenthe probability of acceptance is F(s(1 2 2s)) which is equal to oneif s $ a (1 1 2 a ) and equal to zero if s a (1 1 a ) Hence in thiscase there exists an optimal offer s [ (s8(a ) s8( a )]

QEDProposition 1 accounts for many of the above-mentioned facts

It shows that there are no offers above 05 that offers of 05 arealways accepted and that very low offers are very likely to berejected Furthermore the probability of acceptance F(s(1 2 2s))is increasing in s for s s8( a ) 05 Note also that the acceptancethreshold s8( a 2) 5 a 2(1 1 2 a 2) is nonlinear and has some intui-tively appealing properties It is increasing and strictly concave ina 2 and it converges to 05 if a 2 ` Furthermore relatively smallvalues of a 2 already yield relatively large thresholds For examplea 2 5 13 implies that s8( a 2) 5 02 and a 2 5 075 implies that s8( a 2) 503

In Section V we go beyond the predictions implied by Proposi-tion 1 There we ask whether there is a distribution of preferences


that can explain not just the major facts of the ultimatum gamebut also the facts in market and cooperation games that will bediscussed in the next sections

B Market Game with Proposer Competition

It is a well-established experimental fact that in a broad classof market games prices converge to the competitive equilibrium[Smith 1982 Davis and Holt 1993] For our purposes the interest-ing fact is that convergence to the competitive equilibrium can beobserved even if that equilibrium is very lsquolsquounfairrsquorsquo by virtually anyconceivable denition of fairness ie if all of the gains from tradeare reaped by one side of the market This empirical feature ofcompetition can be demonstrated in a simple market game inwhich many price-setting sellers (proposers) want to sell one unitof a good to a single buyer (responder) who demands only one unitof the good9

Such a game has been implemented in four different coun-tries by Roth Prasnikar Okuno-Fujiwara and Zamir [1991]suppose that there are n 2 1 proposers who simultaneouslypropose a share si [ [01] i [ 1 n 2 1 to the responder Theresponder has the opportunity to accept or reject the highest offers 5 maxi si If there are several proposers who offered s one ofthem is randomly selected with equal probability If the responderrejects s no trade takes place and all players receive a monetarypayoff of zero If the responder accepts s her monetary payoff is sand the successful proposer earns 1 2 s while unsuccessfulproposers earn zero If players are only concerned about theirmonetary payoffs this market game has a straightforward solu-tion the responder accepts any s 0 Hence for any si s 1there exists an e 0 such that proposer i can strictly increase thismonetary payoff by offering s 1 e 1 Therefore any equilibriumcandidate must have s 5 1 Furthermore in equilibrium aproposer i who offered si 5 1 must not have an incentive to lowerhis offer Thus there must be at least one other player j whoproposed sj 5 1 too Hence there is a unique subgame perfect

9 We deliberately restrict our attention to simple market games for tworeasons (i) the potential impact of inequity aversion can be seen most clearly insuch simple games (ii) they allow for an explicit game-theoretic analysis Inparticular it is easy to establish the identity between the competitive equilibriumand the subgame perfect equilibrium outcome in these games Notice that someexperimental market games like eg the continuous double auction as developedby Smith [1962] have such complicated strategy spaces that no completegame-theoretic analysis is yet available For attempts in this direction seeFriedman and Rust [1993] and Sadrieh [1998]


equilibrium outcome in which at least two proposers make an offerof one and the responder reaps all gains from trade10

Roth et al [1991] have implemented a market game in whichnine players simultaneously proposed si while one player acceptedor rejected s Experimental sessions in four different countrieshave been conducted The empirical results provide ample evi-dence in favor of the above prediction After approximately ve tosix periods the subgame perfect equilibrium outcome was reachedin each experiment in each of the four countries To what extentcan our model explain this observation

PROPOSITION 2 Suppose that the utility functions of the playersare given by (1) For any parameters (a i b i) i [ 1 n there is a unique subgame perfect equilibrium outcome inwhich at least two proposers offer s 5 1 which is accepted bythe responder

The formal proof of the proposition is relegated to theAppendix but the intuition is quite straightforward Note rstthat for similar reasons as in the ultimatum game the respondermust accept any s $ 05 Suppose that he rejects a lsquolsquolowrsquorsquo offer s 05 This cannot happen on the equilibrium path either since inthis case proposer i can improve his payoff by offering si 5 05which is accepted with probability 1 and gives him a strictlyhigher payoff Hence on the equilibrium path s must be acceptedConsider now any equilibrium candidate with s 1 If there is oneplayer i offering si s then this player should have offeredslightly more than s There will be inequality anyway but bywinning the competition player i can increase his own monetarypayoff and he can turn the inequality to his advantage A similarargument applies if all players offer si 5 s 1 By slightlyincreasing his offer player i can increase the probability ofwinning the competition from 1(n 2 1) to 1 Again this increaseshis expected monetary payoff and it turns the inequality towardthe other proposers to his advantage Therefore s 1 cannot bepart of a subgame perfect equilibrium Hence the only equilib-rium candidate is that at least two sellers offer s 5 1 This is asubgame perfect equilibrium since all sellers receive a payoff of 0and no player can change this outcome by changing his actionThe formal proof in the Appendix extends this argument to the

10 Note that there are many subgame perfect equilibria in this gameAs longas two sellers propose s 5 1 any offer distribution of the remaining sellers iscompatible with equilibrium


possibility of mixed strategies This extension also shows that thecompetitive outcome must be the unique equilibrium outcome inthe game with incomplete information where proposers do notknow each othersrsquoutility functions

Proposition 2 provides an explanation for why markets in allfour countries in which Roth et al [1991] conducted this experi-ment quickly converged to the competitive outcome even thoughthe results of the ultimatum game that have also been done inthese countries are consistent with the view that the distributionof preferences differs across countries11

C Market Game with Responder Competition

In this section we apply our model of inequity aversion to amarket game for which it is probably too early to speak ofwell-established stylized facts since only one study with a rela-tively small number of independent observations [Guth March-and and Rulliere 1997] has been conducted so far The gameconcerns a situation in which there is one proposer but manyresponders competing against each other The rules of the gameare as follows The proposer who is denoted as player 1 proposesa share s [ [01] to the responders There are 2 n responderswho observe s and decide simultaneously whether to accept orreject s Then a random draw selects with equal probability one ofthe accepting responders In case all responders reject s allplayers receive a monetary payoff of zero In case of acceptance ofat least one responder the proposer receives 1 2 s and therandomly selected responder gets paid s All other respondersreceive zero Note that in this game there is competition in thesecond stage of the game whereas in subsection IIIB we havecompeting players in the rst stage

The prediction of the standard model with purely selshpreferences for this game is again straightforward Respondersaccept any positive s and are indifferent between accepting andrejecting s 5 0 Therefore there is a unique subgame perfectequilibrium outcome in which the proposer offers s 5 0 which isaccepted by at least one responder12 The results of Guth March-and and Rulliere [1997] show that the standard model captures

11 Rejection rates in Slovenia and the United States were signicantlyhigher than rejection rates in Japan and Israel

12 In the presence of a smallest money unit e there exists an additionalslightly different equilibrium outcome the proposer offers s 5 e which is acceptedby all the responders To support this equilibrium all responders have to rejects 5 0 We assume however that there is no smallest money unit


the regularities of this game rather well The acceptance thresh-olds of responders quickly converged to very low levels13 Althoughthe game was repeated only ve times in the nal period theaverage acceptance threshold is well below 5 percent of theavailable surplus with 71 percent of the responders stipulating athreshold of exactly zero and 9 percent a threshold of s8 5 002Likewise in period 5 the average offer declined to 15 percent ofthe available gains from trade In view of the fact that proposershad not been informed about respondersrsquo previous acceptancethresholds such low offers are remarkable In the nal period alloffers were below 25 percent while in the ultimatum game suchlow offers are very rare14 To what extent is this apparentwillingness to make and to accept extremely low offers compatiblewith the existence of inequity-averse subjects As the followingproposition shows our model can account for the above regularities

PROPOSITION 3 Suppose that b 1 (n 2 1)n Then there exists asubgame perfect equilibrium in which all responders acceptany s $ 0 and the proposer offers s 5 0 The highest offer sthat can be sustained in a subgame perfect equilibrium isgiven by

(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix

The rst part of Proposition 3 shows that responder competi-tion always ensures the existence of an equilibrium in which allthe gains from trade are reaped by the proposer irrespective of theprevailing amount of inequity aversion among the respondersThis result is not affected if there is incomplete information aboutthe types of players and is based on the following intuition Giventhat there is at least one other responder j who is going to acceptan offer of 0 there is no way for responder i to affect the outcomeand he may just as well accept this offer too However note thatthe proposer will offer s 5 0 only if b 1 (n 2 1)n If there are n

13 The gains from trade were 50 French francs Before observing the offer seach responder stated an acceptance threshold If s was above the threshold theresponder accepted the offer if it was below she rejected s

14 Due to the gap between acceptance thresholds and offers we conjecturethat the game had not yet reached a stable outcome after ve periods The strongand steady downward trend in all previous periods also indicates that a steadystate had not yet been reached Recall that the market game of Roth et al [1991]was played for ten periods


players altogether than giving away one dollar to one of theresponders reduces inequality by 1 1 [1(n 2 1)] 5 n(n 2 1)dollars Thus if the nonpecuniary gain from this reduction ininequality b 1[n(n 2 1)] exceeds the cost of 1 player 1 prefers togive money away to one of the responders Recall that in thebilateral ultimatum game the proposer offered an equal split ifb 1 05 An interesting aspect of our model is that an increase inthe number of responders renders s 5 05 less likely because itincreases the threshold b 1 has to pass

The second part of Proposition 3 however shows that theremay also be other equilibria Clearly a positive share s can besustained in a subgame perfect equilibrium only if all responderscan credibly threaten to reject any s8 s When is it optimal tocarry out this threat Suppose that s 05 has been offered andthat this offer is being rejected by all other responders j THORN i In thiscase responder i can enforce an egalitarian outcome by rejectingthe offer as well Rejecting reduces not only the inequality towardthe other responders but also the disadvantageous inequalitytoward the proposer Therefore responder i is willing to reject thisoffer if nobody else accepts it and if the offer is sufficiently smallie if the disadvantageous inequality toward the proposer issufficiently large More formally given that all other respondersreject responder i prefers to reject as well if and only if the utilityof acceptance obeys

(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0

This is equivalent to

(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

Thus an offer s 0 can be sustained if and only if (10) holdsfor all responders It is interesting to note that the highestsustainable offer does not depend on all the parameters a i and b i

but only on the inequity aversion of the responder with the lowestacceptance threshold s8i In particular if there is only one re-sponder with a i 5 0 Proposition 3 implies that there is a uniqueequilibrium outcome with s 5 0 Furthermore the acceptancethreshold is decreasing with n Thus the model makes theintuitively appealing prediction that for n ` the highest


sustainable equilibrium offer converges to zero whatever theprevailing amount of inequity aversion15

D Competition and Fairness

Propositions 2 and 3 suggest that there is a more generalprinciple at work that is responsible for the very limited role offairness considerations in the competitive environments consid-ered above Both propositions show that the introduction ofinequity aversion hardly affects the subgame perfect equilibriumoutcome in market games with proposer and responder competi-tion relative to the prediction of the standard self-interest modelIn particular Proposition 2 shows that competition betweenproposers renders the distribution of preferences completelyirrelevant It does not matter for the outcome whether there aremany or only a few subjects who exhibit strong inequity aversionBy the same token it also does not matter whether the playersknow or do not know the preference parameters of the otherplayers The crucial observation in this game is that no singleplayer can enforce an equitable outcome Given that there will beinequality anyway each proposer has a strong incentive to outbidhis competitors in order to turn part of the inequality to hisadvantage and to increase his own monetary payoff A similarforce is at work in the market game with responder competitionAs long as there is at least one responder who accepts everythingno other responder can prevent an inequitable outcome There-fore even very inequity-averse responders try to turn part of theunavoidable inequality into inequality to their advantage byaccepting low offers It is thus the impossibility of preventinginequitable outcomes by individual players that renders inequityaversion unimportant in equilibrium

The role of this factor can be further highlighted by thefollowing slight modication of the market game with proposercompetition suppose that at stage 2 the responder may accept anyof the offers made by the proposers he is not forced to take thehighest offer Furthermore there is an additional stage 3 at whichthe proposer who has been chosen by the responder at stage 2 candecide whether he wants to stick to his offer or whether he wantsto withdrawmdashin which case all the gains from trade are lost for all

15 Note that the acceptance threshold is affected by the reference group Forexample if each responder compares his payoff only with that of the proposer butnot with those of the other responders then the acceptance threshold increases foreach responder and a higher offer may be sustained in equilibrium


parties This game would be an interesting test for our theory ofinequity aversion Clearly in the standard model with selshpreferences these modications do not make any difference forthe subgame perfect equilibrium outcome Also if some playershave altruistic preferences in the sense that they appreciate anyincrease in the monetary payoff of other players the resultremains unchanged because altruistic players do not withdrawthe offer at stage 3 With inequity aversion the outcome will beradically different however A proposer who is inequity aversemay want to destroy the entire surplus at stage 3 in order toenforce an egalitarian outcome in particular if he has a high a i

and if the split between himself and the responder is uneven Onthe other hand an even split will be withdrawn by proposer i atstage 3 only if b i (n 2 1)(n 2 2) Thus the responder may preferto accept an offer si 5 05 rather than an offer sj 05 because thelsquolsquobetterrsquorsquo offer has a higher chance of being withdrawn This in turnreduces competition between proposers at stage 1 Thus whilecompetition nullies the impact of inequity aversion in theordinary proposer competition game inequity aversion greatlydiminishes the role of competition in the modied proposercompetition game This change in the role of competition is causedby the fact that in the modied game a single proposer can enforcean equitable outcome

We conclude that competition renders fairness considerationsirrelevant if and only if none of the competing players can punishthe monopolist by destroying some of the surplus and enforcing amore equitable outcome This suggests that fairness plays asmaller role in most markets for goods16 than in labor marketsThis follows from the fact that in addition to the rejection of lowwage offers workers have some discretion over their work effortBy varying their effort they can exert a direct impact on therelative material payoff of the employer Consumers in contrasthave no similar option available Therefore a rm may bereluctant to offer a low wage to workers who are competing for ajob if the employed worker has the opportunity to respond to alow wage with low effort As a consequence fairness consider-

16 There are some markets for goods where fairness concerns play a role Forexample World Series or NBA playoff tickets are often sold far below themarket-clearing price even though there is a great deal of competition amongbuyers This may be explained by long-term prot-maximizing considerations ofthe monopolistwho interacts repeatedly with groups of customers who care for fairticket prices On this see also Kahnemann Knetsch and Thaler [1986]


ations may well give rise to wage rigidity and involuntaryunemployment17

IV COOPERATION AND RETALIATION COOPERATION GAMES

In the previous section we have shown that our model canaccount for the relatively lsquolsquofairrsquorsquo outcomes in the bilateral ultima-tum game as well as for the rather lsquolsquounfairrsquorsquo or lsquolsquocompetitiversquorsquooutcomes in games with proposer or responder competition Inthis section we investigate the conditions under which coopera-tion can ourish in the presence of inequity aversion We showthat inequity aversion improves the prospects for voluntarycooperation relative to the predictions of the standard model Inparticular we show that there is an interesting class of conditionsunder which the selsh model predicts complete defection whilein our model there exist equilibria in which everybody cooperatesfully But there are also other cases where the predictions of ourmodel coincide with the predictions of the standard model

We start with the following public good game There are n $ 2players who decide simultaneously on their contribution levelsgi [ [0 y] i [ 1 n to the public good Each player has anendowment of y The monetary payoff of player i is given by

(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1

where a denotes the constant marginal return to the public goodG S j5 1

n gj Since a 1 a marginal investment into G causes amonetary loss of (1 2 a) ie the dominant strategy of a com-pletely selsh player is to choose gi 5 0 Thus the standard modelpredicts gi 5 0 for all i [ 1 n However since a 1n theaggregate monetary payoff is maximized if each player choosesgi 5 y

Consider now a slightly different public good game thatconsists of two stages At stage 1 the game is identical to theprevious game At stage 2 each player i is informed about thecontribution vector ( g1 gn) and can simultaneously impose apunishment on the other players ie player i chooses a punish-ment vector pi 5 ( pi1 pin) where pij $ 0 denotes thepunishment player i imposes on player j The cost of this

17 Experimental evidence for this is provided by Fehr Kirchsteiger andRiedl [1993] and Fehr and Falk [forthcoming] We deal with these games in moredetail in Section VI


punishment to player i is given by c S j 5 1n pij 0 c 1 Player i

however may also be punished by the other players whichgenerates an income loss to i of S j 5 1

n pji Thus the monetary payoffof player i is given by

(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij

What does the standard model predict for the two-stagegame Since punishments are costly playersrsquo dominant strategyat stage 2 is to not punish Therefore if selshness and rationalityare common knowledge each player knows that the second stageis completely irrelevant As a consequence players have exactlythe same incentives at stage 1 as they have in the one-stage gamewithout punishments ie each playerrsquos optimal strategy is stillgiven by gi 5 0 To what extent are these predictions of thestandard model consistent with the data from public good experi-ments For the one-stage game there are fortunately a largenumber of experimental studies (see Table II) They investigatethe contribution behavior of subjects under a wide variety ofconditions In Table II we concentrate on the behavior of subjectsin the nal period only since we want to exclude the possibility ofrepeated games effects Furthermore in the nal period we havemore condence that the players fully understand the game thatis being played18

The striking fact revealed by Table II is that in the nalperiod of n-person cooperation games (n 3) without punishmentthe vast majority of subjects play the equilibrium strategy ofcomplete free riding If we average over all studies 73 percent ofall subjects choose gi 5 0 in the nal period It is also worthmentioning that in addition to those subjects who play exactly theequilibrium strategy there are very often a nonnegligible fractionof subjects who play lsquolsquoclosersquorsquo to the equilibrium In view of the factspresented in Table II it seems fair to say that the standard modellsquolsquoapproximatesrsquorsquo the choices of a big majority of subjects ratherwell However if we turn to the public good game with punish-ment there emerges a radically different picture although thestandard model predicts the same outcome as in the one-stage

18 This point is discussed in more detail in Section V Note that in some of thestudies summarized in Table II the group composition was the same for all Tperiods (partner condition) In others the group composition randomly changedfrom period to period (stranger condition) However in the last period subjects inthe partner condition also play a true one-shot public goods game Therefore TableII presents the behavior from stranger as well as from partner experiments


game Figure II shows the distribution of contributions in the nalperiod of the two-stage game conducted by Fehr and Gachter[1996] Note that the same subjects generated the distribution inthe game without and in the game with punishment Whereas inthe game without punishment most subjects play close to com-plete defection a strikingly large fraction of roughly 80 percentcooperates fully in the game with punishment19 Fehr and Gachter

19 Subjects in the Fehr and Gachter study participated in both conditionsie in the game with punishment and in the game without punishment Theparameter values for a and n in this experiment are a 5 04 and n 5 4 It isinteresting to note that contributions are signicantly higher in the two-stagegame already in period 1 Moreover in the one-stage game cooperation stronglydecreases over time whereas in the two-stage game cooperation quickly convergesto the high levels observed in period 10

TABLE IIPERCENTAGE OF SUBJECTS WHO FREE RIDE COMPLETELY IN THE FINAL PERIOD OF A

REPEATED PUBLIC GOOD GAME

Study CountryGroupsize (n)

Marginalpecuniaryreturn (a)

Totalnumber

of subjects

Percentageof freeriders(gi 5 0)

Isaac and Walker [1988] USA 4and10 03 42 83Isaac and Walker [1988] USA 4and10 075 42 57Andreoni [1988] USA 5 05 70 54Andreoni [1995a] USA 5 05 80 55Andreoni [1995b] USA 5 05 80 66Croson [1995] USA 4 05 48 71Croson [1996] USA 4 05 96 65Keser and van Winden

[1996] Holland 4 05 160 84Ockenfels and

Weimann [1996] Germany 5 033 200 89Burlando and Hey

[1997] UKItaly 6 033 120 66Falkinger Fehr

Gachter andWinter-Ebmer[forthcoming] Switzerland 8 02 72 75

Falkinger FehrGachter andWinter-Ebmer[forthcoming] Switzerland 16 01 32 84

Total number of subjects in all experiments andpercentage of complete free riding 1042 73


report that the vast majority of punishments are imposed bycooperators on the defectors and that lower contribution levels areassociated with higher received punishments Thus defectors donot gain from free riding because they are being punished

The behavior in the game with punishment represents anunambiguous rejection of the standard model This raises thequestion whether our model is capable of explaining both theevidence of the one-stage public good game and of the public goodgame with punishment Consider the one-stage public good gamerst The prediction of our model is summarized in the followingproposition

PROPOSITION 4(a) If a 1 b i 1 for player i then it is a dominant strategy for

that player to choose gi 5 0(b) Let k denote the number of players with a 1 b i 1 0

k n If k(n 2 1) a2 then there is a unique equilib-rium with gi 5 0 for all i [ 1 n

(c) If k(n 2 1) (a 1 b j 2 1)( a j 1 b j) for all players j [1 n with a 1 b j 1 then other equilibria withpositive contribution levels do exist In these equilibria allk players with a 1 b i 1 must choose gi 5 0 while allother players contribute gj 5 g [ [0y] Note further that(a 1 b j 2 1)( a j 1 b j) a2

FIGURE IIDistribution of Contributions in the Final Period of the Public Good Game with

Punishment (Source Fehr and Gachter [1996])


The formal proof of Proposition 4 is relegated to the AppendixTo see the basic intuition for the above results consider a playerwith a 1 b i 1 By spending one dollar on the public good heearns a dollars in monetary terms In addition he may get anonpecuinary benet of at most b i dollars from reducing inequal-ity Therefore since a 1 b i 1 for this player it is a dominantstrategy for him to contribute nothing Part (b) of the propositionsays that if the fraction of subjects for whom gi 5 0 is a dominantstrategy is sufficiently high there is a unique equilibrium inwhich nobody contributes The reason is that if there are only afew players with a 1 b i 1 they would suffer too much from thedisadvantageous inequality caused by the free riders The proof ofthe proposition shows that if a potential contributor knows thatthe number of free riders k is larger than a(n 2 1)2 then he willnot contribute either The last part of the proposition shows that ifthere are sufficiently many players with a 1 b i 1 they cansustain cooperation among themselves even if the other players donot contribute However this requires that the contributors arenot too upset about the disadvantageous inequality toward thefree riders Note that the condition k(n 2 1) (a 1 b j 2 1)( a j 1 b j) is less likely to be met as a j goes up To put it differentlythe greater the aversion against being the sucker the moredifficult it is to sustain cooperation in the one-stage game We willsee below that the opposite holds true in the two-stage game

Note that in almost all experiments considered in Table IIa 12 Thus if the fraction of players with a 1 b i 1 is largerthan 14 then there is no equilibrium with positive contributionlevels This is consistent with the very low contribution levels thathave been observed in these experiments Finally it is worthwhilementioning that the prospects for cooperation are weakly increas-ing with the marginal return a

Consider now the public good game with punishment Towhat extent is our model capable of accounting for the very highcooperation in the public good game with punishment In thecontext of our model the crucial point is that free riding generatesa material payoff advantage relative to those who cooperate Sincec 1 cooperators can reduce this payoff disadvantage by punish-ing the free riders Therefore if those who cooperate are suffi-ciently upset by the inequality to their disadvantage ie if theyhave sufficiently high a rsquos then they are willing to punish thedefectors even though this is costly to themselves Thus thethreat to punish free riders may be credible which may induce


potential defectors to contribute at the rst stage of the gameThis is made precise in the following proposition

PROPOSITION 5 Suppose that there is a group of nrsquo lsquolsquoconditionallycooperative enforcersrsquorsquo 1 nrsquo n with preferences that obeya 1 b i $ 1 and

(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8

whereas all other players do not care about inequality iea i 5 b i 5 0 for i [ nrsquo 1 1 n Then the followingstrategies which describe the playersrsquo behavior on and off theequilibrium path form a subgame perfect equilibriumc In the rst stage each player contributes gi 5 g [ [0 y]c If each player does so there are no punishments in the

second stage If one of the players i [ nrsquo 1 1 ndeviates and chooses gi g then each enforcer j [1 nrsquo chooses pji 5 ( g 2 gi)(nrsquo 2 c) while all otherplayers do not punish If one of the lsquolsquoconditionally coopera-tive enforcersrsquorsquo chooses gi g or if any player chooses gi g or if more than one player deviated from g then oneNash-equilibrium of the punishment game is being played

Proof See Appendix

Proposition 5 shows that full cooperation as observed in theexperiments by Fehr and Gachter [1996] can be sustained as anequilibrium outcome if there is a group of nrsquo lsquolsquoconditionallycooperative enforcersrsquorsquo In fact one such enforcer may be enough(nrsquo 5 1) if his preferences satisfy c a i(n 2 1)(1 1 a i) and a 1b i $ 1 ie if there is one person who is sufficiently concernedabout inequality To see how the equilibrium works consider sucha lsquolsquoconditionally cooperative enforcerrsquorsquo For him a 1 b i $ 1 so he ishappy to cooperate if all others cooperate as well (this is why he iscalled lsquolsquoconditionally cooperativersquorsquo) In addition condition (13)makes sure that he cares sufficiently about inequality to hisdisadvantage Thus he can credibly threaten to punish a defector(this is why he is called lsquolsquoenforcerrsquorsquo) Note that condition (13) is lessdemanding if nrsquo or a i increases The punishment is constructedsuch that the defector gets the same monetary payoff as theenforcers Since this is less than what a defector would havereceived if he had chosen gi 5 g a deviation is not protable


If the conditions of Proposition 5 are met then there exists acontinuum of equilibrium outcomes This continuum includes thelsquolsquogood equilibriumrsquorsquo with maximum contributions but also the lsquolsquobadequilibriumrsquorsquo where nobody contributes to the public good In ourview however there is a reasonable renement argument thatrules out lsquolsquobadrsquorsquo equilibria with low contributions To see this notethat the equilibrium with the highest possible contribution levelgi 5 g 5 y for all i [ 1 n is the unique symmetric andefficient outcome Since it is symmetric it yields the same payofffor all players Hence this equilibrium is a natural focal point thatserves as a coordination device even if the subjects choose theirstrategies independently

Comparing Propositions 4 and 5 it is easy to see that theprospects for cooperation are greatly improved if there is anopportunity to punish defectors Without punishments all playerswith a 1 b i 1 will never contribute Players with a 1 b i 1 maycontribute only if they care enough about inequality to theiradvantage but not too much about disadvantageous inequality Onthe other hand with punishment all players will contribute ifthere is a (small) group of lsquolsquoconditionally cooperative enforcersrsquorsquoThe more these enforcers care about disadvantageous inequalitythe more they are prepared to punish defectors which makes iteasier to sustain cooperation In fact one person with a suffi-ciently high a i is already enough to enforce efficient contributionsby all other players

Before we turn to the next section we would like to point outan implication of our model for the Prisonerrsquos Dilemma (PD) Notethat the simultaneous PD is just a special case of the public goodgame without punishment for n 5 2 and gi [ 0 y i 5 12Therefore Proposition 4 applies ie cooperation is an equilib-rium if both players meet the condition a 1 b i 1 Yet if only oneplayer meets this condition defection of both players is the uniqueequilibrium In contrast in a sequentially played PD a purelyselsh rst mover has an incentive to contribute if he faces asecond mover who meets a 1 b i 1 This is so because the secondmover will respond cooperatively to a cooperative rst move whilehe defects if the rst mover defects Thus due to the reciprocalbehavior of inequity-averse second movers cooperation ratesamong rst movers in sequentially played PDs are predicted to behigher than cooperation rates in simultaneous PDs There is fairlystrong evidence in favor of this prediction Watabe Terai Haya-shi and Yamagishi [1996] and Hayashi Ostrom Walker and


Yamagishi [1998] show that cooperation rates among rst moversin sequential PDs are indeed much higher and that reciprocalcooperation of second movers is very frequent

V PREDICTIONS ACROSS GAMES

In this section we examine whether the distribution ofparameters that is consistent with experimental observations inthe ultimatum game is consistent with the experimental evidencefrom the other games It is not our aim here to show that ourtheory is consistent with 100 percent of the individual choicesThe objective is rather to offer a rst test for whether there is achance that our theory is consistent with the quantitative evi-dence from different games Admittedly this test is rather crudeHowever at the end of this section we make a number ofpredictions that are implied by our model and we suggest howthese predictions can be tested rigorously with some newexperiments

In many of the experiments referred to in this section thesubjects had to play the same game several times either with thesame or with varying opponents Whenever available we take thedata of the nal period as the facts to be explained There are tworeasons for this choice First it is well-known in experimentaleconomics that in interactive situations one cannot expect thesubjects to play an equilibrium in the rst period already Yet ifsubjects have the opportunity to repeat their choices and to betterunderstand the strategic interaction then very often rather stablebehavioral patterns that may differ substantially from rst-period-play emerge Second if there is repeated interaction between thesame opponents then there may be repeated games effects thatcome into play These effects can be excluded if we look at the lastperiod only

Table III suggests a simple discrete distribution of a i and b iWe have chosen this distribution because it is consistent with thelarge experimental evidence we have on the ultimatum game (seeTable I above and Roth [1995]) Recall from Proposition 1 that forany given a i there exists an acceptance threshold s8( a i) 5a i(1 1 2 a i) such that player i accepts s if and only if s $ s8( a i) Inall experiments there is a fraction of subjects that rejects offerseven if they are very close to an equal split Thus we (conserva-tively) assume that 10 percent of the subjects have a 5 4 whichimplies an acceptance threshold of s8 5 49 5 0444 Another


typically much larger fraction of the population insists on gettingat least one-third of the surplus which implies a value of a whichis equal to one These are at least 30 percent of the populationNote that they are prepared to give up one dollar if this reducesthe payoff of their opponent by two dollars Another say 30percent of the subjects insist on getting at least one-quarterwhich implies that a 5 05 Finally the remaining 30 percent ofthe subjects do not care very much about inequality and are happyto accept any positive offer ( a 5 0)

If a proposer does not know the parameter a of his opponentbut believes that the probability distribution over a is given byTable III then it is straightforward to compute his optimal offer asa function of his inequality parameter b The optimal offer is givenby

(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235

Note that it is never optimal to offer less than one-third of thesurplus even if the proposer is completely selsh If we look at theactual offers made in the ultimatum game there are roughly 40percent of the subjects who suggest an equal split Another 30percent offer s [ [04 05) while 30 percent offer less than 04There are hardly any offers below 025 This gives us the distribu-tion of b in the population described in Table III

Let us now see whether this distribution of preferences isconsistent with the observed behavior in other games Clearly wehave no problem in explaining the evidence on market games withproposer competition Any distribution of a and b yields thecompetitive outcome that is observed by Roth et al [1991] in all

TABLE IIIASSUMPTIONS ABOUT THE DISTRIBUTION OF PREFERENCES

DISTRIBUTION OF a rsquos ANDASSOCIATED ACCEPTANCETHRESHOLDS OF BUYERS

DISTRIBUTION OF b rsquos ANDASSOCIATED OPTIMAL OFFERS

OF SELLERS

a 5 0 30 percent s8 5 0 b 5 0 30 percent s 5 13a 5 05 30 percent s8(05) 5 14 b 5 025 30 percent s 5 49a 5 1 30 percent s8 (1) 5 13 b 5 06 40 percent s 5 12a 5 4 10 percent s8 (4) 5 49


their experiments Similarly in the market game with respondercompetition we know from Proposition 3 that if there is at leastone responder who does not care about disadvantageous inequal-ity (ie a i 5 0) then there is a unique equilibrium outcome withs 5 0 With ve responders in the experiments by Guth March-and and Rulliere [1997] and with the distribution of types fromTable III the probability that there is at least one such player ineach group is given by 1ndash075 5 83 percent This is roughlyconsistent with the fact that 71 percent of the players accepted anoffer of zero and 9 percent had an acceptance threshold of s8 5002 in the nal period

Consider now the public good game We know by Proposition4 that cooperation can be sustained as an equilibrium outcomeonly if the number k of players with a 1 b i 1 obeys k(n 2 1) a2 Thus our theory predicts that there is less cooperation thesmaller a which is consistent with the empirical evidence of Isaacand Walker [1988] presented in Table II20 In a typical treatmenta 5 05 and n 5 4 Therefore if all players believe that there is atleast one player with a 1 b i 1 then there is a uniqueequilibrium with gi 5 0 for all players Given the distribution ofpreferences of Table III the probability that there are four playerswith b 05 is equal to 044 5 256 percent Hence we shouldobserve that on average almost all individuals fully defect Asimilar result holds for most other experiments in Table II Exceptfor the Isaac and Walker experiments with n 5 10 a single playerwith a 1 b i 1 is sufficient for the violation of the necessarycondition for cooperation k(n 2 1) a2 Thus in all theseexperiments our theory predicts that randomly chosen groups arealmost never capable of sustaining cooperation Table II indicatesthat this is not quite the case although 73 percent of individualsindeed choose gi 5 0 Thus it seems fair to say that our model isconsistent with the bulk of individual choices in this game21

Finally the most interesting experiment from the perspectiveof our theory is the public good game with punishment While in

20 For a 5 03 the rate of defection is substantially larger than for a 5 075The Isaac and Walker experiments were explicitly designed to test for the effects ofvariations in a

21 When judging the accuracy of the model one should also take into accountthat there is in general a signicant fraction of the subjects that play close tocomplete free riding in the nal round A combination of our model with the viewthat human choice is characterized by a fundamental randomness [McKelvey andPalfrey 1995 Anderson Goeree and Holt 1997] may explain much of theremaining 25 percent of individual choices This task however is left for futureresearch


the game without punishment most subjects play close to com-plete defection a strikingly large fraction of roughly 80 percentcooperate fully in the game with punishment To what extent canour model explain this phenomenon We know from Proposition 5that cooperation can be sustained if there is a group of nrsquolsquolsquoconditionally cooperative enforcersrsquorsquo with preferences that satisfy(13) and a 1 b i $ 1 For example if all four players believe thatthere is at least one player with a i $ 15 and b i $ 06 there is anequilibrium in which all four players contribute the maximumamount As discussed in Section V this equilibrium is a naturalfocal point Since the computation of the probability that theconditions of Proposition 5 are met is a bit more cumbersome wehave put them in the Appendix It turns out that for the preferencedistribution given in Table III the probability that a randomlydrawn group of four players meets the conditions is 611 percentThus our model is roughly consistent with the experimentalevidence of Fehr and Gachter [1996]22

Clearly the above computations provide only rough evidencein favor of our model To rigorously test the model additionalexperiments have to be run We would like to suggest a fewvariants of the experiments discussed so far that would beparticularly interesting23

c Our model predicts that under proposer competition twoproposers are sufficient for s 5 1 to be the unique equilib-rium outcome irrespective of the playersrsquo preferences Thusone could conduct the proposer competition game with twoproposers that have proved to be very inequity averse inother games This would constitute a particularly toughtest of our model

c Most public good games that have been conducted hadsymmetric payoffs Our theory suggests that it will be moredifficult to sustain cooperation if the game is asymmetricFor example if the public good is more valuable to some ofthe players there will in general be a conict betweenefficiency and equality Our prediction is that if the game issufficiently asymmetric it is impossible to sustain coopera-tion even if a is very large or if players can use punishments

22 In this context one has to take into account that the total number ofavailable individual observations in the game with punishment is much smallerthan for the game without punishment or for the ultimatum game Futureexperiments will have to show whether the Fehr-Gachter results are the rule inthe punishment game or whether they exhibit unusually high cooperation rates

23 We are grateful to a referee who suggested some of these tests


c It would be interesting to repeat the public good experi-ment with punishments for different values of a c and nProposition 5 suggests that we should observe less coopera-tion if a goes down and if c goes up The effect of an increasein the group size n is ambiguous however For any givenplayer it becomes more difficult to satisfy condition (13) asn goes up On the other hand the larger the group thehigher is the probability that there is at least one personwith a very high a Our conjecture is that a moderatechange in the size of the group does not affect the amount ofcooperation

c One of the most interesting tests of our theory would be todo several different experiments with the same group ofsubjects Our model predicts a cross-situation correlationin behavior For example the observations from one experi-ment could be used to estimate the parameters of theutility function of each individual It would then be possibleto test whether this individualrsquos behavior in other games isconsistent with his estimated utility function

c In a similar fashion one could screen subjects according totheir behavior in one experiment before doing a public goodexperiment with punishments If we group the subjects inthis second experiment according to their observed inequal-ity aversion the prediction is that those groups with highinequality aversion will contribute while those with lowinequality aversion will not

VI DICTATOR AND GIFT EXCHANGE GAMES

The preceding sections have shown that our very simplemodel of linear inequality aversion is consistent with the mostimportant facts in ultimatum market and cooperation gamesOne problem with our approach however is that it yields tooextreme predictions in some other games such as the lsquolsquodictatorgamersquorsquo The dictator game is a two-person game in which onlyplayer 1 the lsquolsquodictatorrsquorsquo has to make a decision Player 1 has todecide what share s [ [01] of a given amount of money to pass onto player 2 For a given share s monetary payoffs are given by x1 51 2 s and x2 5 s respectively Obviously the standard modelpredicts s 5 0 In contrast in the experimental study of ForsytheHorowitz Savin and Sefton [1994] only about 20 percent ofsubjects chose s 5 0 60 percent chose 0 s 05 and again


roughly 20 percent chose s 5 05 In the study by Andreoni andMiller [1995] the distribution of shares is again bimodal but putsmore weight on the lsquolsquoextremesrsquorsquo approximately 40 percent of thesubjects gave s 5 0 20 percent gave 0 s 05 and roughly 40percent gave s 5 05 Shares above s 5 05 were practically neverobserved

Our model predicts that player 1 offers s 5 05 if b 1 05 ands 5 0 if b 1 05 Thus we should observe only very lsquolsquofairrsquorsquo or verylsquolsquounfairrsquorsquo outcomes a prediction that is clearly refuted by the dataHowever there is a straightforward solution to this problem Weassumed that the inequity aversion is piecewise linear Thelinearity assumption was imposed in order to keep our model assimple as possible If we allow for a utility function that is concavein the amount of advantageous inequality there is no problem ingenerating optimal offers that are in the interior of [005]

It is important to note that nonlinear inequity aversion doesnot affect the qualitative results in the other games we consid-ered This is straightforward in market games with proposer orresponder competition Recall that in the context of proposercompetition there exists a unique equilibrium outcome in whichthe responder receives the whole gains from trade irrespective ofthe prevailing amount of inequity aversion Thus it also does notmatter whether linear or nonlinear inequity aversion prevailsLikewise under responder competition there is a unique equilib-rium outcome in which the proposer receives the whole surplus ifthere is at least one responder who does not care about disadvan-tageous inequality Obviously this proposition holds irrespectiveof whether the inequity aversion of the other responders is linearor not Similar arguments hold for public good games with andwithout punishment Concerning the public good game withpunishment for example the existence of nonlinear inequityaversion obviously does not invalidate the existence of an equilib-rium with full cooperation It only renders the condition for theexistence of such an equilibrium ie condition (13) slightly morecomplicated

Another interesting game is the so-called trust- or giftexchange game [Fehr Kirchsteiger and Riedl 1993 Berg Dick-haut and McCabe 1995 Fehr Gachter and Kirchsteiger 1997]The common feature of trust- or gift exchange games is that theyresemble a sequentially played PD with more than two actions foreach player In some experiments the gift exchange game has beenembedded in a competitive experimental market For example a


slightly simplied version of the experiment conducted by FehrGachter and Kirchsteiger [1997] has the following structureThere is one experimental rm which we denote as player 1 andwhich can make a wage offer w to the experimental workersThere are 2 n workers who can simultaneously accept orreject w Then a random draw selects with equal probability one ofthe accepting workers Thereafter the selected worker has tochoose effort e from the interval [ee] 0 e e In case that allworkers reject w all players receive nothing In case of acceptancethe rm receives xf 5 ve 2 w where v denotes the marginalproduct of effort The worker receives xw 5 w 2 c(e) where c(e)denotes the effort costs and obeys c(e) 5 c8(e) 5 0 and c8 0 c9 0for e e Moreover v c8(e) so that e 5 e is the efficient effort levelThis game is essentially a market game with responder competi-tion in which an accepting responder has to make an effort choiceafter he is selected

If all players are pure money maximizers the prediction forthis game is straightforward Since the selected worker alwayschooses the minimum effort e the game collapses into a respondercompetition game with gains from trade equal to ve In equilib-rium the rm earns ve and w 5 0 Yet since v c8(e) there existmany (we)-combinations that would make both the rm and theselected worker better off In sharp contrast to this prediction andalso in sharp contrast to what is observed under respondercompetition without effort choices rms offer substantial wages tothe workers and wages do not decrease over time Moreoverworkers provide effort above e and there is a strong positivecorrelation between w and e

To what extent can our model explain this outcome Putdifferently why is it the case that under responder competitionwithout effort choice the respondersrsquo income converges toward theselsh solution whereas under responder competition with effortchoice wages substantially above the selsh solution can bemaintained From the viewpoint of our model the key fact isthatmdashby varying the effort choicemdashthe randomly selected workerhas the opportunity to affect the difference xf 2 xw If the rmoffers lsquolsquolowrsquorsquo wages such that xf xw holds at any feasible effortlevel the selected worker will always choose the minimum effortHowever if the rm offers a lsquolsquohighrsquorsquo wage such that at e theinequality xw xf holds inequity-averse workers with a suffi-ciently high b i are willing to raise e above e Moreover in thepresence of nonlinear inequity aversion higher wages will be


associated with higher effort levels The reason is that by raisingthe effort workers can move in the direction of more equitableoutcomes Thus our model is capable of explaining the apparentwage rigidity observed in gift exchange games Since the presenceof inequity-averse workers generates a positive correlation be-tween wages and effort the rm does not gain by exploiting thecompetition among the workers Instead it has an incentive topay efficiency wages above the competitive level

VII EXTENSIONS AND POSSIBLE OBJECTIONS

So far we ruled out the existence of subjects who like to bebetter off than others This is unsatisfactory because subjects withb i 0 clearly exist Fortunately however such subjects havevirtually no impact on equilibrium behavior in the games consid-ered in this paper To see this suppose that a fraction of subjectswith b i 5 0 exhibits b i 0 instead This obviously does not changerespondersrsquo behavior in the ultimatum game because for themonly a i matters It also does not change the proposer behavior inthe complete information case because both proposers with b i 5 0and those with b i 0 will make an offer that exactly matches theresponderrsquos acceptance threshold24 In the market game withproposer competition proposers with b i 0 are even more willingto overbid a going share below s 5 1 compared with subjects withb i 5 0 because by overbidding they gain a payoff advantagerelative to the other proposers Thus Proposition 2 remainsunchanged Similar arguments apply to the case of respondercompetition (without effort choices) because a responder with b i 0 is even more willing to underbid a positive share compared witha responder with b i 5 0 In the public good game withoutpunishment all players with a 1 b i 1 have a dominant strategyto contribute nothing It does not matter whether these playersexhibit a positive or a negative b i Finally the existence of typeswith b i 0 also leaves Proposition 5 unchanged25 If there aresufficiently many conditionally cooperative enforcers it does not

24 It may affect proposer behavior in the incomplete information casealthough the effect of a change in b i is ambiguous This ambiguity stems from thefact that the proposerrsquos marginal expected utility of s may rise or fall if b i falls

25 This holds true if for those with a negative b i the absolute value of b i isnot too large Otherwise defectors would have an incentive to punish thecooperators A defector who imposes a punishment of one on a cooperator gains[ 2 b i(n 2 1)](1 2 c) 0 in nonpecuniary terms and has material costs of c Thushe is willing to punish if b i $ [c(1 2 c)](n 2 1) holds This means that onlydefectors with implausibly high absolute values of b i are willing to punish For


matter whether the remaining players have b i 0 or not Recallthatmdashaccording to Proposition 5mdashstrategies that discipline poten-tial defectors make the enforcers and the defectors equally well offin material terms Hence a defector cannot gain a payoff advan-tage but is even worse off relative to a cooperating nonenforcerThese punishment strategies therefore are sufficient to disci-pline potential defectors irrespective of their b i-values

Another set of questions concerns the choice of the referencegroup As argued in Section II for many laboratory experimentsour assumption that subjects compare themselves with all othersubjects in the (usually relatively small) group is a naturalstarting point However we are aware of the possibility that thismay not always be an appropriate assumption26 There may wellbe interactive structures in which some agents have a salientposition that makes them natural reference agents Moreover thesocial context and the institutional environment in which interac-tions take place is likely to be important27 Bewley [1998] forexample reports that in nonunionized rms workers comparethemselves exclusively with their rm and with other workers intheir rm This suggests that only within-rm social comparisonsbut not across-rm comparisons affect the wage-setting processThis is likely to be different in unionized sectors because unionsmake across-rm and even across-sector comparisons BabcockWang and Loewenstein [1996] for example provide evidencethat wage bargaining between teachersrsquo unions and school boardsis strongly affected by reference wages in other school districts

An obvious limitation of our model is that it cannot explainthe evolution of play over time in the experiments discussedInstead our examination aims at the explanation of the stablebehavioral patterns that emerge in these experiments afterseveral periods It is clear that a model that solely focuses onequilibrium behavior cannot explain the time path of play Thislimitation of our model also precludes a rigorous analysis of the

example for c 5 05 and n 5 4 b i $ 3 is required For c 5 02 and n 5 4 b i stillhas to exceed 075

26 Bolton and Ockenfels [1997] develop a model similar to ours that differs inthe choice of the reference payoff In their model subjects compare themselves onlywith the average payoff of the group

27 A related issue is the impact of social context on a personrsquos degree ofinequity aversion It seems likely that a person has a different degree of inequityaversion when interacting with a friend in personal matters than in a businesstransaction with a stranger In fact evidence for this is provided by LoewensteinThompson and Bazerman [1989] However note that in all experiments consid-ered above interaction took place among anonymous strangers in a neutrallyframed context


short-run impact of equity considerations28 The empirical evi-dence suggests that equity considerations also have importantshort-run effects This is obvious in ultimatum games public goodgames with punishment and gift exchange games where equityconsiderations lead to substantial deviations from the selshsolution in the short and in the long run However they also seemto play a short-run role in market games with proposer orresponder competition or public good games without punishmentthat is in games in which the selsh solution prevails in the longrun In these games the short-run deviation from equilibrium istypically in the direction of more equitable outcomes29

VIII RELATED APPROACHES IN THE LITERATURE

There are several alternative approaches that try to accountfor persistent deviations from the predictions of the self-interestmodel by assuming a different motivational structure The ap-proach pioneered by Rabin [1993] emphasizes the role of inten-tions as a source of reciprocal behavior Rabinrsquos approach hasrecently been extended in interesting ways by Falk and Fisch-bacher [1998] and Dufwenberg and Kirchsteiger [1998] Andreoniand Miller [1995] is based on the assumption of altruistic motivesAnother interesting approach is Levine [1997] who assumes thatpeople are either spiteful or altruistic to various degrees Finallythere is the approach by Bolton and Ockenfels [1997] that is likeour model based on a kind of inequity aversion

The theory of reciprocity as developed by Rabin [1993] restson the idea that people are willing to reward fair intentions and topunish unfair intentions Like our approach Rabinrsquos model is alsobased on the notion of equity player j perceives player irsquosintention as unfair if player i chooses an action that gives j less

28 In the short-run minor changes in the (experimental) context can affectbehavior For example there is evidence that subjects contribute more in aone-shot PD if it is called lsquolsquocommunity gamersquorsquo than if it is called lsquolsquoWall Street GamersquorsquoUnder the plausible assumption that the community frame triggers more optimis-tic beliefs about other subjects inequity aversion our model is consistent with thisobservation

29 Such short-run effects also are suggested by the results of KahnemanKnetsch and Thaler [1986] and Franciosi et al [1995] Franciosi et al showthatmdashin a competitive experimental market (without effort choices)mdashequityconsiderations signicantly retard the adjustment to the (selsh) equilibriumUltimately however they do not prevent full adjustment to the equilibrium Notethat the retardation effect suggests that temporary demand shocks (eg after anatural disaster) may have no impact on prices at all because the shock vanishesbefore competitive forces can overcome the fairness-induced resistance to pricechanges


than the equitable material payoff The advantage of his model isthat the disutility of an unfair offer can be explicitly interpreted asarising from jrsquos judgment about irsquos unfair intention As a conse-quence player jrsquos response to irsquos action can be explicitly inter-preted as arising from jrsquos desire to punish an unfair intentionwhile our model does not explicitly suggest this interpretation ofjrsquos response On the other hand disadvantages of Rabinrsquos modelare that it is restricted to two-person normal form games and thatit gives predictions if it is applied to the normal form of importantsequential move games30

The lack of explicit modeling of intentions in our model doeshowever not imply that the model is incompatible with anintentions-based interpretation of reciprocal behavior In ourmodel reciprocal behavior is driven by the preference parametersa i and b i The model is silent as to why a i and b i are positiveWhether these parameters are positive because individuals caredirectly for inequality or whether they infer intentions fromactions that cause unequal outcomes is not modeled Yet thismeans that positive a irsquos and b irsquos can be interpreted as a directconcern for equality as well as a reduced-form concern forintentions An intentions-based interpretation of our preferenceparameters is possible because bad or good intentions behind anaction are in general inferred from the equity implications of theaction Therefore people who have a desire to punish a badintention behave as if they dislike being worse off relative to anequitable reference point and people who reward good intentionsbehave as if they dislike being better off relative to an equitablereference point As a consequence our preference parameters arecompatible with the interpretation of intentions-driven reciprocity

To illustrate this point further consider eg an ultimatumgame that is played under two different conditions [Blount 1995]

c In the lsquolsquorandomrsquorsquo condition the rst moverrsquos offer is deter-mined by a random device The responder knows how the

30 In the sequentially played Prisonerrsquos Dilemma Rabinrsquos model predictsthat unconditional cooperation by the second mover is part of an equilibrium iethe second mover cooperates even if the rst mover defects Moreover conditionalcooperation by the second mover is not part of an equilibrium The data in Watabeet al [1996] and Hayashi et al [1998] however show that unconditionalcooperation is virtually nonexistent while conditional cooperation is the ruleLikewise in the gift exchange game workers behave conditionally cooperativewhile unconditional cooperation is nonexistent The reciprocity approaches of Falkand Fischbacher [1998] and of Dufwenberg and Kirchsteiger [1998] do not sharethis disadvantage of Rabinrsquos model


offer is generated and that the proposer cannot be heldresponsible for it

c In the lsquolsquointentionrsquorsquo condition the proposer makes the offerhimself and the responder knows that this is the proposerrsquosdeliberate choice

In the intention condition the responder may not only bedirectly concerned about inequity He may also react to thefairness of the perceived intentions of the proposer In contrast inthe random condition it is only the concern for pure equity thatmay affect the responderrsquos behavior In fact Blount [1995] reportsthat there are responders who reject positive but unequal offers inboth conditions However the acceptance threshold is signi-cantly higher in the intention condition31 Recall from Proposition1 that there is a monotonic relationship between the acceptancethreshold and the parameter a i Thus this result suggests thatthe preference parameters do not remain constant across randomand intention condition Yet for all games played in the intentioncondition and hence for all games considered in the previoussections the preference parameters should be constant acrossgames

Altruism is consistent with voluntary giving in dictator andpublic good games It is however inconsistent with the rejectionof offers in the ultimatum game and it cannot explain the hugebehavioral differences between public good games with andwithout punishment It also seems difficult to reconcile theextreme outcomes in market games with altruism Levinersquosapproach can explain extreme outcomes in market games as wellas the evidence in the centipede game but it cannot explainpositive giving in the dictator game It also seems that Levinersquosapproach has difficulties in explaining that the same subjectsbehave very noncooperatively in the public good game withoutpunishment while they behave very cooperatively in the gamewith punishment

The approach by Bolton and Ockenfels [1997] is similar to ourmodel although there are some differences in the details Forexample in their model people compare their material payoff withthe material average payoff of the group In our view the appropri-ate choice of the reference payoff is ultimately an empirical

31 Similar evidence is given by Charness [forthcoming] for a gift exchangegame For further evidence in favor of intentions-driven reciprocity see Bolle andKritikos [1998] Surprisingly and in contrast to these studies Bolton Brandtsand Katok [1997] and Bolton Brandts and Ockenfels [1997] nd no evidence forintentions-driven reciprocity


question that cannot be solved on the basis of the presentlyavailable evidence There may well be situations in which theaverage payoff is the appropriate choice However in the contextof the public good game with punishment it seems to be inappro-priate because it cannot explain why cooperators want to punish adefector If there are say n 2 1 fully cooperating subjects and onefully defecting subject the payoff of each cooperator is below thegrouprsquos average payoff Cooperators can reduce this differencebetween own payoff and the grouprsquos average payoff by punishingone of the other players ie they are indifferent between punish-ing other cooperators and the defector

Bolton and Ockenfels [1997] assume that the marginaldisutility of small deviations from equality is zero Therefore ifsubjects are nonsatiated in their own material payoff they willnever propose an equal split in the dictator game Likewise theywillmdashin case of nonsatiation in material payoffsmdashnever proposean equal split in the ultimatum game unless a 2 5 ` for sufficientlymany responders Typically the modal offer in most ultimatumgame experiments is however the equal split In addition theassumption implies that complete free riding is the uniqueequilibrium in the public good game without punishment for alla 1 and all n $ 2 Their approach thus rules out equilibriawhere only a fraction of all subjects cooperate32

IX SUMMARY

There are situations in which the standard self-interestmodel is unambiguously refuted However in other situations thepredictions of this model seem to be very accurate For example insimple experiments like the ultimatum game the public goodgame with punishments or the gift exchange game the vastmajority of the subjects behave in a lsquolsquofairrsquorsquo and lsquolsquocooperativersquorsquomanner although the self-interest model predicts very lsquolsquounfairrsquorsquoand lsquolsquononcooperativersquorsquo behavior Yet there are also experimentslike eg market games or public good games without punish-ment in which the vast majority of the subjects behaves in arather lsquolsquounfairrsquorsquo and lsquolsquononcooperativersquorsquo waymdashas predicted by theself-interest model We show that this puzzling evidence can beexplained in a coherent framework ifmdashin addition to purelyselsh peoplemdashthere is a fraction of the population that cares for

32 Persistent asymmetric contributions are observed in Isaac Walker andWilliams [1994]


equitable outcomes Our theory is motivated by the psychologicalevidence on social comparison and loss aversion It is very simpleand can be applied to any game The predictions of our model areconsistent with the empirical evidence on all of the above-mentioned games Our theory also has strong empirical implica-tions for many other games Therefore it is an important task forfuture research to test the theory more rigorously against compet-ing hypotheses In addition we believe that future researchshould aim at formalizing the role of intentions explicitly for then-person case

A main insight of our analysis is that there is an importantinteraction between the distribution of preferences in a givenpopulation and the strategic environment We have shown thatthere are environments in which the behavior of a minority ofpurely selsh people forces the majority of fair-minded people tobehave in a completely selsh manner too For example in amarket game with proposer or responder competition it is verydifficult if not impossible for fair players to achieve a lsquolsquofairrsquorsquooutcome Likewise in a simultaneous public good game withpunishment even a small minority of selsh players can triggerthe unraveling of cooperation Yet we have also shown that aminority of fair-minded players can force a big majority of selshplayers to cooperate fully in the public good game with punish-ment Similarly our examination of the gift exchange gameindicates that fairness considerations may give rise to stable wagerigidity despite the presence of strong competition among theworkers Thus competition may or may not nullify the impact ofequity considerations If despite the presence of competitionsingle individuals have opportunities to affect the relative mate-rial payoffs equity considerations will affect market outcomeseven in very competitive environments In our view these resultssuggest that the interaction between the distribution of prefer-ences and the economic environment deserves more attention infuture research

APPENDIX

Proof of Proposition 2

We rst show that it is indeed a subgame perfect equilibriumif at least two proposers offer s 5 1 which is accepted by theresponder Note rst that the responder will accept any offer s $


05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0

To see this note that (A1) is equivalent to

(A2) (n 2 1) s $ b i(ns 2 1)

Since b i 1 this inequality clearly holds if

(A3) (n 2 1)s $ ns 2 1

which must be the case since s 1 Hence the buyer will accepts 5 1 Given that there is at least one other proposer who offers s 51 and given that this offer will be accepted each proposer gets amonetary payoff of 0 anyway and no proposer can affect thisoutcome Hence it is indeed optimal for at least one otherproposer to offer s 5 1 too

Next we show that this is the unique equilibrium outcomeSuppose that there is another equilibrium in which s 1 withpositive probability This is only possible if each proposer offerss 1 with positive probability Let si be the lowest offer of proposeri that has positive probability It cannot be the case that player iputs strictly positive probability on offers si [ [si sj) because theprobability that he wins with such an offer is zero To see this notethat in this case player i would get

(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1

On the other hand if proposer i chooses si [ (maxj THORN i sj05 1) thenthere is a positive probability that he will winmdashin which case hegets

(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1

Of course there may also be a positive probability that proposer idoes not win but in this case he again gets 2 a i(n 2 1) Thusproposer i would deviate It follows that it must be the case thatsi 5 s for all i

Suppose that proposer i changes his strategy and offers s 1e 1 in all states when his strategy would have required him to


choose s The cost of this change is that whenever proposer i wouldhave won with the offer s he now receives only 1 2 s 2 e Howeverby making e arbitrarily small this cost becomes arbitrarily smallThe benet is that there are now some states of the world whichhave strictly positive probability in which proposer i does win withthe offer s 1 e but in which he would not have won with the offer sThis benet is strictly positive and does not go to zero as e becomessmall Hence s 1 cannot be part of an equilibrium outcome

QED


We rst show that s 5 1 which is accepted by all respondersis indeed a subgame perfect equilibrium Note that any offer s $05 will be accepted by all responders The argument is exactly thesame as the one in the beginning of the proof of Proposition 1 Thefollowing Lemma will be useful

LEMMA 1 For any s 05 there exists a continuation equilibriumin which everybody accepts s

Given that all other players accept s player i prefers to accept aswell if and only if

(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is

which is equivalent to

(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0

Since we assume that b i 1 this inequality must hold h

Consider now the proposer Clearly it is never optimal to offers 05 Such an offer is always dominated by s 5 05 which yieldsa higher monetary payoff and less inequality On the other handwe know by Lemma 1 that for any s 05 there exists acontinuation equilibrium in which this offer is accepted by every-body Thus we only have to look for the optimal s from the point ofview of the proposer given that s will be accepted His payofffunction is

(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)


Differentiating with respect to s yields

(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1

which is independent of s and is smaller than 0 if and only if

(A10) b 1 (n 2 1)n

Hence if this condition holds it is an equilibrium that theproposer offers s 5 0 which is accepted by all responders We nowshow that the highest offer that can be sustained in a subgameperfect equilibrium is given by (8)

LEMMA 2 Suppose that s 05 has been offered There exists acontinuation equilibrium in which this offer is rejected by allresponders if and only if

(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n

Given that all other responders reject s responder i will reject s aswell if and only if

(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is

which is equivalent to (A11) Thus (A11) is a sufficient conditionfor a continuation equilibrium in which s is rejected by everybody

Suppose now that (A11) is violated for at least one i [2 n We want to show that in this case there is no continua-tion equilibrium in which s is rejected by everybody Note rst thatin this case responder i prefers to accept s if all other respondersreject it Suppose now that at least one other responder accepts sIn this case responder i prefers to accept s as well if and only if

(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s

The right-hand side of this inequality is smaller than 0 We knowalready that the left-hand side is greater than 0 since (A11) isviolated Therefore responder i prefers to accept s as well Weconclude that if (A11) does not hold for at least one i then at leastone responder will accept s Hence (A11) is also necessary h

If b 1 (n 2 1)n an equilibrium offer must be sustained by


the threat that any smaller offer s will be rejected by everybodyBut we know from Lemma 2 that an offer s may be rejected only if(A11) holds for all i Thus the highest offer s that can be sustainedin equilibrium is given by (8)

QED


(a) Suppose that 1 2 a b i for player i Consider an arbitrarycontribution vector ( g1 gi 2 1 gi 1 1 gn) of the other play-ers Without loss of generality we relabel the players such that i 51 and 0 g2 g3 gn If player 1 chooses g1 5 0 his payoffis given by

(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj

Note rst that if all other players choose gj 5 0 too then g1 5 0 isclearly optimal Furthermore player 1 will never choose g1 max gj Suppose that there is at least one player who chooses gj 0 If player 1 chooses g1 0 g1 [ [ gk gk1 1] k [ 2 n then hispayoff is given by

U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)

Hence gi 5 0 is indeed a dominant strategy for player i


(b) It is clearly an equilibrium if all players contributenothing because to unilaterally contribute more than zero reducesthe monetary payoff and causes disadvantageous inequalitySuppose that there exists another equilibrium with positivecontribution levels Relabel players such that 0 g1 g2 gn By part (a) we know that all k players with 1 2 a b i mustchoose gi 5 0 Therefore 0 5 g1 5 gk Consider player l k whohas the smallest positive contribution level ie 0 5 gl 2 1 gl gl 1 1 gn Player 1rsquos utility is given by

(A15) Ul( gl) 5 y 2 gl 1 agl 1 a oj5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

where Ul(0) is the utility player 1 gets if he deviates and choosesgl 5 0 Since a l $ b l l $ k 1 1 and b l 1 we have

(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0

player l prefers to deviate from the equilibrium candidate and to


choose gl 5 0 But this inequality is equivalent to

(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2

which is the condition given in the proposition(c) Suppose that the conditions of the proposition are satis-

ed We want to construct an equilibrium in which all k playerswith 1 2 a b i contribute nothing while all other n 2 k playerscontribute g [ [0 y] We only have to check that contributing g isindeed optimal for the contributing players Consider some playerj with 1 2 a b j If he contributes g his payoff is given by

(A19) Uj( g) 5 y 2 g 1 (n 2 k)ag 2 [ a j(n 2 1)] kg

It clearly does not pay to contribute more than g So suppose thatplayer j reduces his contribution level by D 0 Then his payoff is

Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

Thus a deviation does not pay if and only if

1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)


Thus if this condition holds for all (n 2 k) players j with 1 2 a b j then this is indeed an equilibrium It remains to be shown that(a 1 b j 2 1)( a j 1 b j) a2 Note that a j $ b j implies that(a 1 b j 2 1)( a j 1 b j) (a 1 b j 2 1)(2 b j) Furthermore

a 1 b j 2 1

2 b j

a

2Ucirc a 1 b j 2 1 b ja Ucirc a(1 2 b j) 1 2 b j Ucirc a 1

which proves our claimQED


Suppose that one of the players i [ n8 1 1 n choosesgi g If all players stick to the punishment strategies in stage 2then deviator i gets the same monetary payoff as each enforcer j [

1 n8 In this case monetary payoffs of i and j are given by

(A21) xi 5 y 2 gi 1 a[(n 2 1)g 1 gi] 2 n8g 2 gi

n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi

Thus given the punishment strategy of the enforcers devia-tors cannot get a payoff higher than what the enforcers getHowever they get a strictly lower payoff than the nonenforcerswho did not deviate We now have to check that the punishmentstrategies are credible ie that an enforcer cannot gain fromreducing his pij If an enforcer reduces pij by e he saves c e andexperiences less disadvantageous inequality relative to those(n 2 n8 2 1) players who chose g but do not punish This creates anonpecuniary utility gain of [ a i(n 2 n8 2 1) ce ](n 2 1) On theother hand the enforcer also has nonpecuniary costs because heexperiences now disadvantageous inequality relative to the defec-tor and a distributional advantage relative to the other (n8 2 1)enforcers who punish fully The latter generates a utility loss ofb i(n8 2 1) c e (n 2 1) whereas the former reduces utility bya i(1 2 c) e (n 2 1) Thus the loss from a reduction in pij is greater


than the gain if

(A23)

1

n 2 1[ a i(1 2 c) e 1 b i(n8 2 1)c e ] ce 1 a i(n 2 n8 2 1)

c e

n 2 1

holds Some simple algebraic manipulations show that condition(A23) is equivalent to condition (13) Hence the punishment iscredible

Consider now the incentives of one of the enforcers to deviatein the rst stage Suppose that he reduces his contribution by e 0 Ignoring possible punishments in the second stage for amoment player i gains (1 2 a) e in monetary terms but incurs anonpecuniary loss of b 1e by creating inequality to all otherplayers Since 1 2 a b i by assumption this deviation does notpay If his defection triggers punishments in the second stagethen this reduces his monetary payoff which cannot make himbetter off than he would have been if he had chosen gi 5 g Hencethe enforcers are not going to deviate at stage 1 either It is easy tosee that choosing gi g cannot be protable for any player eithersince it reduces the monetary payoff and increases inequality

QED

Computation of the Probability That There Are ConditionallyCooperative Enforcers

To compute the probability that in a randomly drawn groupof four there are subjects who obey condition (13) and a 1 b i $ 1we have to make an assumption about the correlation between a i

and b i We mentioned already that the empirical evidence sug-gests that these parameters are positively correlated For concrete-ness we assume that the correlation is perfect Thus in terms ofTable III all players with a 5 1 or a 5 4 are assumed to have b 506 This is clearly not fully realistic but it simplies the analysisdramatically

In the Fehr-Gachter [1996] experiment the relevant pa-rameters are a 5 04 n 5 4 and (roughly33)) c 5 02 The followingsummary states the conditions on a i and b i implied by Proposition5 for a group of n8 [ 1 4 conditionally cooperative enforcers

33 The cost function in Fehr and Gachter is actually convex so that we haveto slightly simplify their model Yet the vast majority of actual punishmentsoccurred where c 5 02


If one of these conditions holds cooperation can be sustained inequilibrium(i) n8 5 1 a i $ 15 and b i $ 06(ii) n8 5 2 a i $ 1 2 03 b i and b i $ 06(iii) n8 5 3 a i $ 075 2 05 b i and b i $ 06(iv) n8 5 4 a i $ 06 2 06 b i and b i $ 06Note that for each group n8 of conditionally cooperative enforcersthe conditions on a i and b i have to hold simultaneously Given thediscrete distribution of a and b of Table III this can only be thecase if

c there is at least one player with a i 5 4 and b i 5 06 orc there are at least two players with a i 5 1 and b i 5 06 orc both

Given the numbers of Table III it is not difficult to show that theprobability that one of these cases applies is equal to 6112percent

UNIVERSITY OF ZURICH

UNIVERSITY OF MUNICH

REFERENCES

Adams J Stacy lsquolsquoToward an Understanding of Inequityrsquorsquo Journal of Abnormal andSocial Psychology LXVII (1963) 422ndash436

Agell Jonas and Per Lundberg lsquolsquoTheories of Pay and Unemployment SurveyEvidence from Swedish Manufacturing Firmsrsquorsquo Scandinavian Journal ofEconomics XCVII (1995) 295ndash308

Anderson Simon P Jacob K Goeree and Charles H Holt lsquolsquoStochastic GameTheorymdashAdjustment to Equilibrium under Bounded Rationalityrsquorsquo Universityof Virginia Working Paper No 304 1997

Andreoni James lsquolsquoWhy Free RidemdashStrategies and Learning in Public GoodsExperimentsrsquorsquo Journal of Public Economics XXXXVII (1988) 291ndash304 lsquolsquoCooperation in Public-Goods Experiments Kindness or Confusionrsquorsquo Ameri-can Economic Review LXXXV (1995a) 891ndash904 lsquolsquoWarm Glow versus Cold Prickle The Effects of Positive and NegativeFraming on Cooperation in Experimentsrsquorsquo Quarterly Journal of EconomicsCX (1995b) 1ndash21

Andreoni James and John H Miller lsquolsquoGiving according to GARP An Experimen-tal Study of Rationality and Altruismrsquorsquo SSRI Working Paper University ofWisconsin Madison 1996

Babcock Linda Xianghong Wang and George Loewenstein lsquolsquoChoosing the WrongPond Social Comparisons in Negotiations That Reect a Self-Serving BiasrsquorsquoQuarterly Journal of Economics CXI (1996) 1ndash21

Banerjee Abhihit V lsquolsquoEnvyrsquorsquo in Dutta Bhaskar et al eds Economic Theory andPolicy Essays in Honour of Dipak Banerjee (Oxford Oxford University Press1990)

Berg Joyce John Dickhaut and Kevin McCabe lsquolsquoTrust Reciprocity and SocialHistoryrsquorsquo Games and Economic Behavior X (1995) 122ndash142

Bewley Truman lsquolsquoA Depressed Labor Market as Explained by ParticipantsrsquorsquoAmerican Economic Review Papers and Proceedings LXXXV (1995) 250ndash254 lsquolsquoWhy not Cut Payrsquorsquo European Economic Review XLII (1998) 459ndash490


Blinder Alan S and Don H Choi lsquolsquoA Shred of Evidence on Theories of WageSticknessrsquorsquo Quarterly Journal of Economics CV (1990) 1003ndash1016

Blount Sally lsquolsquoWhen Social Outcomes Arenrsquot Fair The Effect of Causal Attribu-tions on Preferencesrsquorsquo Organizational Behavior and Human Decision Proces-ess LXIII (1995) 131ndash144

Bolle Friedel and Alexander Kritikos lsquolsquoSelf-Centered Inequality Aversion versusReciprocity and Altruismrsquorsquo Discussion Paper Europa-Universitat ViadrinaFrankfurt (Oder) 1998

Bolton Gary E and Axel Ockenfels lsquolsquoA Theory of Equity Reciprocity andCompetitionrsquorsquo Discussion Paper Pennsylvania State University 1997

Bolton Gary E Jordi Brandts and Elena Katok lsquolsquoA Simple Test of Explanationsfor Contributions in Dilemma Gamesrsquorsquo Discussion Paper Pennsylvania StateUniversity 1997

Bolton Gary E Jordi Brandts and Axel Ockenfels lsquolsquoMeasuring Motivations forReciprocal Responses Observed in Simple Dilemma Gamesrsquorsquo DiscussionPaper Universitat Magdeburg 1997

Burlando Roberto and John D Hey lsquolsquoDo Anglo-Saxons Free-Ride Morersquorsquo Journalof Public Economics LXIV (1997) 41ndash60

Camerer Colin and Richard Thaler lsquolsquoUltimatums Dictators and MannersrsquorsquoJournal of Economic Perspectives IX (1995) 209ndash219

Cameron Lisa lsquolsquoRaising the Stakes in the Ultimatum Game ExperimentalEvidence from Indonesiarsquorsquo Discussion Paper Princeton University 1995

Campbell Carl M and Kunal Kamlani lsquolsquoThe Reasons for Wage Rigidity Evidencefrom a Survey of Firmsrsquorsquo Quarterly Journal of Economics CXII (1997)759ndash789

Charness Gary lsquolsquoAttribution and Reciprocity in a Labor Market An ExperimentalInvestigationrsquorsquo Games and Economic Behavior forthcoming

Clark Andrew E and Andrew J Oswald lsquolsquoSatisfaction and Comparison IncomersquorsquoJournal of Public Economics LXI (1996) 359ndash381

Croson Rachel T A lsquolsquoExpectations in Voluntary Contributions MechanismsrsquorsquoDiscussion Paper Wharton School University of Pennsylvania 1995 lsquolsquoPartners and Strangers Revisitedrsquorsquo Economics Letters LIII (1996) 25ndash32

Davis Douglas and Charles Holt Experimental Economics (Princeton NJPrinceton University Press 1993)

Davis J A lsquolsquoA Formal Interpretation of the Theory of Relative DeprivationrsquorsquoSociometry XXII (1959) 280ndash296

Dawes Robyn M and Richard Thaler lsquolsquoCooperationrsquorsquo Journal of EconomicPerspectives II (1988) 187ndash197

Dufwenberg Martin and Georg Kirchsteiger lsquolsquoA Theory of Sequential Reciproc-ityrsquorsquo Discussion Paper CentER Tilburg University 1998

Falk Armin and Urs Fischbacher lsquolsquoA Theory of Reciprocityrsquorsquo Discussion PaperUniversity of Zurich 1998

Falkinger Josef Ernst Fehr Simon Gachter and Rudolf Winter-Ebmer lsquolsquoA SimpleMechanism for the Efficient Provision of Public GoodsmdashExperimental Evi-dencersquorsquo American Economic Review forthcoming

Fehr Ernst and Armin Falk lsquolsquoWage Rigidity in a Competitive Incomplete ContractMarketrsquorsquo Journal of Political Economy CVII (1999) 106ndash134

Fehr Ernst and Simon Gachter lsquolsquoCooperation and PunishmentmdashAn Experimen-tal Analysis of Norm Formation and Norm Enforcementrsquorsquo Discussion PaperInstitute for Empirical Research in Economics University of Zurich 1996

Fehr Ernst Simon Gachter and Georg Kirchsteiger lsquolsquoReciprocity as a ContractEnforcement Devicersquorsquo Econometrica LXV No 4 (1996) 833ndash860

Fehr Ernst Georg Kirchsteiger and Arno Riedl lsquolsquoDoes Fairness Prevent MarketClearing An Experimental Investigationrsquorsquo Quarterly Journal of EconomicsCVIII (1993) 437ndash460

Festinger L lsquolsquoA Theory of Social Comparison Processesrsquorsquo Human Relations VII(1954) 117ndash140

Forsythe Robert N E Horowitz L Hoel Savin and Martin Sefton lsquolsquoFairness inSimple Bargaining Gamesrsquorsquo Games and Economic Behavior VI (1988) 347ndash369

Franciosi Robert Praveen Kujal Roland Michelitsch Vernon Smith and GangDeng lsquolsquoFairness Effect on Temporary and Equilibrium Prices in Posted-OfferMarketsrsquorsquo Economic Journal CV (1995) 938ndash950


Frank Robert H Choosing the Right PondmdashHuman Behavior and the Quest forStatus (Oxford Oxford University Press 1985)

Friedman Daniel and John Rust The Double Auction MarketmdashInstitutionsTheories and Evidence (Reading MA Addison-Wesley Publishing Company1993)

Guth Werner Nadelsquo ge Marchand and Jean-Louis Rulliere lsquolsquoOn the Reliability ofReciprocal FairnessmdashAn Experimental Studyrsquorsquo Discussion Paper HumboldtUniversity Berlin 1997

Guth Werner Rolf Schmittberger and Bernd Schwarze lsquolsquoAn ExperimentalAnalysis of Ultimatum Bargainingrsquorsquo Journal of Economic Behavior andOrganization III (1982) 367ndash388

Guth Werner Rolf Schmittberger and Reinhard Tietz lsquolsquoUltimatum BargainingBehaviormdashA Survey and Comparison of Experimental Resultsrsquorsquo Journal ofEconomic Psychology XI (1990) 417ndash449

Haltiwanger John and Michael Waldman lsquolsquoRational Expectations and the Limitsof Rationalityrsquorsquo American Economic Review LXXV (1985) 326ndash340

Hayashi Nahoko Elinor Ostrom James Walker and Toshio Yamagishi lsquolsquoReciproc-ity Trust and the Sense of Control A Cross-Societal Studyrsquorsquo Discussion PaperIndiana University Bloomington 1998

Hoffman Elizabeth Kevin McCabe and Vernon Smith lsquolsquoOn Expectations andMonetary Stakes in Ultimatum Gamesrsquorsquo International Journal of GameTheory XXV (1996) 289ndash301

Homans G C Social Behavior Its Elementary Forms (New York Harcourt Braceamp World 1961)

Isaac Mark R and James M Walker lsquolsquoGroup Size Effects in Public GoodsProvision The Voluntary Contribution Mechanismrsquorsquo Quarterly Journal ofEconomics CIII (1988) 179ndash199

Isaac Mark R and James M Walker lsquolsquoCostly CommunicationAn Experiment ina Nested Public Goods Problemrsquorsquo in Thomas R Palfrey ed LaboratoryResearch in Political Economy (Ann Arbor University of Michigan Press1991)

Isaac Mark R James M Walker and Arlington M Williams lsquolsquoGroup Size and theVoluntary Provision of Public Goods Experimental Evidence Utilizing LargeGroupsrsquorsquo Journal of Public Economics LIV (1994) 1ndash36

Kachelmeier Steven J and Mohamed Shehata lsquolsquoCulture and Competition ALaboratory Market Comparison between China and the Westrsquorsquo Journal ofEconomic Organization and Behavior XIX (1992) 145ndash168

Kahneman Daniel Jack L Knetsch and Richard Thaler lsquolsquoFairness as a Con-straint on Prot Seeking Entitlements in the Marketrsquorsquo American EconomicReview LXXVI (1986) 728ndash741

Keser Claudia and Frans van Winden lsquolsquoPartners Contribute More to PublicGoods than Strangers Conditional Cooperationrsquorsquo Discussion Paper Univer-sity of Karlsruhe 1996

Ledyard John lsquolsquoPublic Goods A Survey of Experimental Researchrsquorsquo in J KagelandA Roth eds Handbook of Experimental Economics (Princeton PrincetonUniversity Press 1995)

Levine David K lsquolsquoModelingAltruism and Spitefulness in Experimentsrsquorsquo Review ofEconomic Dynamics forthcoming (1997)

Loewenstein George F Leigh Thompson and Max H Bazerman lsquolsquoSocial Utilityand Decision Making in Interpersonal Contextsrsquorsquo Journal of Personality andSocial Psychology LVII (1989) 426ndash441

McKelvey Richard D and Thomas R Palfrey lsquolsquoQuantal Response Equilibria forNormal Form Gamesrsquorsquo Games and Economic Behavior X (1995) 6ndash38

Mueller Denis Public Choice II (Cambridge Cambridge University Press 1989)Ockenfels Axel and Joachim Weimann lsquolsquoTypes and PatternsmdashAn Experimental

East-West Comparison of Cooperation and Solidarityrsquorsquo Discussion PaperDepartment of Economics University of Magdeburg 1996

Ostrom Elinor and James M Walker lsquolsquoCooperation without External Enforce-mentrsquorsquo in Thomas R Palfrey ed Laboratory Research in Political Economy(Ann Arbor University of Michigan Press 1991)

Pollis N P lsquolsquoReference Groups Re-examinedrsquorsquo British Journal of Sociology XIX(1968) 300ndash307


Rabin Matthew lsquolsquoIncorporating Fairness into Game Theory and EconomicsrsquorsquoAmerican Economic Review LXXXIII (1993) 1281ndash1302

Rehder Robert lsquolsquoJapanese Transplants After the Honeymoonrsquorsquo Business Horizons(1990) 87ndash98

Roth Alvin E lsquolsquoBargaining Experimentsrsquorsquo in J Kagel and A Roth eds Handbookof Experimental Economics (Princeton Princeton University Press 1995)

Roth Alvin E and Ido Erev lsquolsquoLearning in Extensive-Form Games ExperimentalData and Simple Dynamic Models in the Intermediate Termrsquorsquo Games andEconomic Behavior VIII (1995) 164ndash212

Roth Alvin E Vesna Prasnikar Masahiro Okuno-Fujiwara and Shmuel ZamirlsquolsquoBargaining and Market Behavior in Jerusalem Ljubljana Pittsburgh andTokyo An Experimental Studyrsquorsquo American Economic Review LXXXI (1991)1068ndash1095

Runciman Walter G Relative Deprivation and Social Justice (New York Penguin1966)

Russell Thomas and Richard Thaler lsquolsquoThe Relevance of Quasi Rationality inCompetitive Marketsrsquorsquo American Economic Review LXXV (1985) 1071ndash1082

Sadrieh Abdolkarim The Alternating Double Auction Market (Berlin Springer1998)

Skinner Jonathan and Joel Slemrod lsquolsquoAn Economic Perspective on Tax EvasionrsquorsquoNational Tax Journal XXXVIII (1985) 345ndash353

Slonim Robert and Alvin E Roth lsquolsquoFinancial Incentives and Learning inUltimatum and Market Games An Experiment in the Slovak RepublicrsquorsquoEconometrica LXVI (1997) 569ndash596

Smith Vernon L lsquolsquoAn Experimental Study of Competitive Market BehaviorrsquorsquoJournal of Political Economy LXX (1962) 111ndash137 lsquolsquoMicroeconomic Systems as an Experimental Sciencersquorsquo American EconomicReview LXXII (1982) 923ndash955

Smith Vernon L and Arlington W Williams lsquolsquoThe Boundaries of CompetitivePrice Theory Convergence Expectations and Transaction Costsrsquorsquo in L Greenand J H Kagel eds Advances in Behavioural Economics Vol 2 (NorwoodNJ Ablex Publishing Corporation 1990)

Stouffer Samuel A The American Soldier (Princeton Princeton UniversityPress 1949)

Thaler Richard H lsquolsquoThe Ultimatum Gamersquorsquo Journal of Economic Perspectives II(1988) 195ndash206

Tversky A and D Kahneman lsquolsquoLoss Aversion in Riskless Choice A Reference-Dependent Modelrsquorsquo Quarterly Journal of Economics CVI (1991) 1039ndash1062

Watabe M S Terai N Hayashi and T Yamagishi lsquolsquoCooperation in the One-ShotPrisonerrsquos Dilemma Based on Expectations of Reciprocityrsquorsquo Japanese Journalof Experimental Social Psychology XXXVI (1996) 183ndash196

Whyte William F Money and Motivation (New York Harper and Brothers 1955)


subjects do not care solely about material payoffs [Guth and Tietz1990 Roth 1995 Camerer and Thaler 1995] However there isalso evidence that seems to suggest that fairness considerationsare rather unimportant For example in competitive experimen-tal markets with complete contracts in which a well-denedhomogeneous good is traded almost all subjects behave as if theyare only interested in their material payoff Even if the competi-tive equilibrium implies an extremely uneven distribution of thegains from trade equilibrium is reached within a few periods[Smith and Williams 1990 Roth Prasnikar Okuno-Fujiwara andZamir 1991 Kachelmeier and Shehata 1992 Guth Marchandand Rulliere 1997]

There is similarly conicting evidence with regard to coopera-tion Reality provides many examples indicating that people aremore cooperative than is assumed in the standard self-interestmodel Well-known examples are that many people vote pay theirtaxes honestly participate in unions and protest movements orwork hard in teams even when the pecuniary incentives go in theopposite direction1 This is also shown in laboratory experiments[Dawes and Thaler 1988 Ledyard 1995] Under some conditions ithas even been shown that subjects achieve nearly full cooperationalthough the self-interest model predicts complete defection [Isaacand Walker 1988 1991 Ostrom and Walker 1991 Fehr andGachter 1996]2 However as we will see in more detail in SectionIV there are also those conditions under which a vast majority ofsubjects completely defect as predicted by the self-interest model

There is thus a bewildering variety of evidence Some piecesof evidence suggest that many people are driven by fairnessconsiderations other pieces indicate that virtually all peoplebehave as if completely selsh and still other types of evidencesuggest that cooperation motives are crucial In this paper we askwhether this conicting evidence can be explained by a singlesimple model Our answer to this question is affirmative if one iswilling to assume that in addition to purely self-interestedpeople there are a fraction of people who are also motivated byfairness considerations No other deviations from the standard

1 On voting see Mueller [1989] Skinner and Slemroad [1985] argue that thestandard self-interest model substantially underpredicts the number of honesttaxpayers Successful team production in eg Japanese-managed auto factoriesin North America is described in Rehder [1990] Whyte [1955] discusses howworkers establish lsquolsquoproduction normsrsquorsquo under piece-rate systems

2 Isaac and Walker and Ostrom and Walker allow for cheap talk while inFehr and Gachter subjects could punish each other at some cost




















(1) Ui(x) 5 xi 2 a i

1

n 2 1 ojTHORN i

max xj 2 xi0

2 b i

1

n 2 1 ojTHORN i

max xi 2 xj0



















AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


































































































(1) Ui(x) 5 xi 2 a i

1

n 2 1 ojTHORN i

max xj 2 xi0

2 b i

1

n 2 1 ojTHORN i

max xi 2 xj0



















AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES



























































































(1) Ui(x) 5 xi 2 a i

1

n 2 1 ojTHORN i

max xj 2 xi0

2 b i

1

n 2 1 ojTHORN i

max xi 2 xj0



















AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES






















































































(1) Ui(x) 5 xi 2 a i

1

n 2 1 ojTHORN i

max xj 2 xi0

2 b i

1

n 2 1 ojTHORN i

max xi 2 xj0



















AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































(1) Ui(x) 5 xi 2 a i

1

n 2 1 ojTHORN i

max xj 2 xi0

2 b i

1

n 2 1 ojTHORN i

max xi 2 xj0



















AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES





























































































AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES
























































































AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































AND MARKET GAMES










s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES



















































































s s8( a 2) a 2(1 1 2 a 2) 05


(3) s

5 05 if b 1 05

[ [s8( a 2)05] if b 1 5 05

5 s8(a 2) if b 1 05


(4) p 5

1 if s $ s8( a )

F(s(1 2 2s)) [ (01) if s8( a ) s s8( a ))

0 if s s8( a )



(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

















































































(5) s

5 05 if b 1 05

[ [s8( a ) 05] if b 1 5 05

[ (s8( a ) s8( a )] if b 1 05





Stake size(country)


s 02


04 s 05



0 66



5 57


67 $5 and $10(USA)

0 82



8 61


24 $10(USA)

0 83


27 $100(USA)

4 74


115 $10(USA)

75a



3 70



04d 75



8d 69


875 38 71




s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

















































































s8( a 2) a 2(1 1 2 a 2) 05




























(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES






































































































(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES
































































































(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


























































































(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































(8) s 5 mini [ 2n

a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i

1

2

Proof See Appendix







(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































(9) s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is 0


(10) s s8i a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i



















(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES































































































(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

























































































(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES




















































































(11) xi( g1 gn) 5 y 2 gi 1 a oj 5 1

n

gj 1n a 1









(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES



















































































(12) xi( g1 gn p1 pn) 5 y 2 gi 1 a oj5 1

n

gj 2 oj5 1

n

pji 2 c oj 5 1

n

pij











Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES






















































































Totalnumber

of subjects

























(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES































































































(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES






















































































(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































(13) c a i

(n 2 1)(1 1 a i) 2 (n8 2 1)( a i 1 b i)

for all i [ 1 n8



Proof See Appendix















(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES




























































































(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES
























































































(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































(14) s(b ) 5

05 if b i 05

04 if 0235 b i 05

03 if b i 0235






OF SELLERS


































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES















































































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES









































































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


































































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES




























































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES























































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES



















































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES













































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES






































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES































































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

























































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































IX SUMMARY






APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































APPENDIX




05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES
















































































05 because

(A1) s 21

n 2 1b i(s 2 1 1 s) 2

n 2 2

n 2 1b i(s 2 0) $ 0


(A2) (n 2 1) s $ b i(ns 2 1)


(A3) (n 2 1)s $ ns 2 1



(A4) Ui(si) 5 2a i

n 2 1s 2

a i

n 2 1(1 2 s) 5 2

a i

n 2 1


(A5) 1 2 si 2a i

n 2 1(2si 2 1) 2

n 2 2

n 2 1b i(1 2 si)

(1 2 si)[1 2n 2 2

n 2 1b i] 2

a i

n 2 1 2

a i

n 2 1





QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

















































































QED





(A6) s 21

n 2 1a i(1 2 s 2 s) 2

n 2 2

n 2 1b i(s 2 0)

$ 0 21

n 2 1a i(1 2 s) 2

1

n 2 1a is


(A7) (1 2 b i)(n 2 1) 1 2 a i 1 b i $ 0



(A8) U1(s) 5 1 2 s 21

n 2 1b 1(1 2 s 2 s) 2

n 2 2

n 2 1b 1(1 2 s)



(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

















































































(A9)dU1

ds5 2 1 1

2

n 2 1b 1 1

n 2 2

n 2 1b 1


(A10) b 1 (n 2 1)n



(A11) s a i

(1 2 b i)(n 2 1) 1 2 a i 1 b i i [ 2 n


(A12) 0 $ s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is



(A13)

s 2a i

n 2 1(1 2 2s) 2

n 2 2

n 2 1b is $ 0 2

a i

n 2 1(1 2 s) 2

a i

n 2 1s





QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

















































































QED



(A14) U1( g1 5 0) 5 y 1 a oj 5 2

n

gj 2b

n 2 1 oj5 2

n

gj


U1(g1 0)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 2a 1

n 2 1 oj5 2

k

(g1 2 gj)

y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 k1 1

n

(gj 2 g1) 1b 1

n 2 1 oj5 2

k

(g1 2 gj)

5 y 2 g1 1 ag1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

1b 1

n 2 1(n 2 1)g1

5 y 2 (1 2 a 2 b 1)g1 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj

y 1 a oj5 2

n

gj 2b 1

n 2 1 oj5 2

n

gj 5 U1(g1 5 0)





n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES


















































































n

gj 2b l

n 2 1 oj 5 l 1 1

n

( gj 2 gl)

2a l

n 2 1 oj5 1

l 2 1

gl 5 y 1 a oj 5 l 1 1

n

gj 2b l

n 2 1 oj 5 l 1 1

n

gj

2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl

5 Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 a l

l 2 1

n 2 1gl


(A16) Ul( gl) Ul(0) 2 (1 2 a)gl 1 b l

n 2 l

n 2 1gl 2 b l

l 2 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1 b l

n 2 2(k 1 1) 1 1

n 2 1gl

Ul(0) 2 (1 2 a)gl 1n 2 2k 2 1

n 2 1gl

5 Ul(0) 2(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1gl

Thus if

(A17)(1 2 a)(n 2 1) 2 (n 2 2k 2 1)

n 2 1$ 0




(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

















































































(A18) (1 2 a)(n 2 1) $ n 2 2k 2 1

Ucirc a 1 2n 2 2k 2 1

n 2 1

Ucirc a n 2 1 2 n 1 2k 1 1

n 2 15

2k

n 2 1

Ucirck

n 2 1$

a

2





Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k)ag 2 D a

2a j

n 2 1k( g 2 D ) 2

b j

n 2 1(n 2 k 2 1) D

5 y 2 g 2 (n 2 k)ag 2a j

n 2 1kg

1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)

5 Uj( g) 1 D 1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1)


1 2 a 1a j

n 2 1k 2

b j

n 2 1(n 2 k 2 1) 0


(A20) k(n 2 1) (a 1 b j 2 1)(a j 1 b j)



a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES

















































































a 1 b j 2 1

2 b j

a







n8 2 c

(A22) xj 5 y 2 g 1 a[(n 2 1)g 1 gi] 2 cg 2 gi

n8 2 c2

n8 2 c

n8 2 c( gi 2 gi)

5 y 2 gj 1 a[(n 2 1)g 1 gi] 2 (n8 2 c 1 c)g 2 gi

n8 2 c5 xi



than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES
















































































than the gain if

(A23)

1


c e

n 2 1



QED












REFERENCES





















































































REFERENCES

















































































































































































































A THEORY OF FAIRNESS, COMPETITION, AND ...web.stanford.edu/~niederle/Fehr.Schmidt.1999.QJE.pdfA THEORY OF FAIRNESS, COMPETITION, AND COOPERATION

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