Economic and Operations Research Situations with Interval Data

6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011

Cooperative Game Theory. Operations ResearchGames. Applications to Interval Games

Lecture 5: Economic and Operations Research Situations withInterval Data

Sırma Zeynep Alparslan GokSuleyman Demirel UniversityFaculty of Arts and SciencesDepartment of Mathematics

Isparta, Turkeyemail:[email protected]

August 13-16, 2011


Outline

Introduction

Cooperative interval games

Classes of cooperative interval games

Economic situations with interval data

Operations Research situations with interval data

References


Introduction

Introduction

This lecture is based on Cooperative interval games by AlparslanGok which was the PhD Dissertation thesis from Middle EastTechnical University.

I The thesis is also published as a book by

Lambert Academic Publishing (LAP)

Cooperative Interval Games: Theory and Applications

I For more information please see:http://www.morebooks.de/store/gb/book/cooperative-interval-games/isbn/978-3-8383-3430-1


Introduction

Introduction

This lecture is also based on the papers

I Big boss interval games by Alparslan Gok, Branzei and Tijswhich was published in International Journal of Uncertainty,Fuzziness and Knowledge-Based Systems (IJUFKS),

I Airport interval games and their Shapley value by AlparslanGok, Branzei and Tijs which was published in OperationsResearch and Decisions,

I Bankruptcy problems with interval uncertainty by Branzei andAlparslan Gok which was published in Economics Bulletin and

I Sequencing interval situations and related games by AlparslanGok et al. which will appear in Central European Journal ofOperations Research (CEJOR).


Introduction

Motivation

Game theory:

I Mathematical theory dealing with models of conflict andcooperation.

I Many interactions with economics and with other areas suchas Operations Research (OR) and social sciences.

I Tries to come up with fair divisions.

I A young field of study:The start is considered to be the book Theory of Games andEconomic Behaviour by von Neumann and Morgernstern(1944).

I Two parts: non-cooperative and cooperative.


Introduction

Motivation continued...

I Cooperative game theory deals with coalitions whichcoordinate their actions and pool their winnings.

I The main problem: How to divide the rewards or costs amongthe members of the formed coalition?

I Generally, the situations are considered from a deterministicpoint of view.

I Basic models in which probability and stochastic theory play arole are: chance-constrained games and cooperative gameswith stochastic/random payoffs.


Introduction

Motivation continued...

Idea of interval approach:

I In most economic and OR situations rewards/costs are notprecise.

I Possible to estimate the intervals to which rewards/costsbelong.

Why cooperative interval games are important?

I Useful for modeling real-life situations.

Aim: generalize the classical theory to intervals and apply it toeconomic situations and OR situations.

I In this study, rewards/costs taken into account are notrandom variables, but just closed and bounded intervals ofreal numbers with no probability distribution attached.


Introduction

Interval calculus

I (R): the set of all closed and bounded intervals in RI , J ∈ I (R), I =

[I , I], J =

[J, J], |I | = I − I , α ∈ R+

I addition: I + J =[I + J, I + J

]I multiplication: αI =

[αI , αI

]I subtraction: defined only if |I | ≥ |J|

I − J =[I − J, I − J

]I weakly better than: I < J if and only if I ≥ J and I ≥ J

I I 4 J if and only if I ≤ J and I ≤ J

I better than: I � J if and only if I < J and I 6= J

I I ≺ J if and only if I 4 J and I 6= J



Classical cooperative games versus cooperative intervalgames

I < N, v >, N := {1, 2, ..., n}: set of players

I v : 2N → R: characteristic function, v(∅) = 0

I v(S): worth (or value) of coalition S

GN : the class of all coalitional games with player set N

I < N,w >, N: set of players

I w : 2N → I (R): characteristic function, w(∅) = [0, 0]

I w(S) = [w(S),w(S)]: worth (value) of S

IGN : the class of all interval games with player set NExample (LLR-game): Let < N,w > be an interval game withw({1, 3}) = w({2, 3}) = w(N) = J < [0, 0] and w(S) = [0, 0]otherwise.



Arithmetic of interval games

w1,w2 ∈ IGN , λ ∈ R+, for each S ∈ 2N

I w1 4 w2 if w1(S) 4 w2(S)

I < N,w1 + w2 > is defined by (w1 + w2)(S) = w1(S) + w2(S).

I < N, λw > is defined by (λw)(S) = λ · w(S).

I Let w1,w2 ∈ IGN such that |w1(S)| ≥ |w2(S)| for eachS ∈ 2N . Then < N,w1 − w2 > is defined by(w1 − w2)(S) = w1(S)− w2(S).

Classical cooperative games associated with < N,w >

I Border games: < N,w > and < N,w >

I Length game: < N, |w | >, where |w | (S) = w(S)− w(S) foreach S ∈ 2N .



Preliminaries on classical cooperative games< N, v > is called a balanced game if for each balanced mapλ : 2N \ {∅} → R+ we have∑

S∈2N\{∅}

λ(S)v(S) ≤ v(N).

The core (Gillies (1959)) C (v) of v ∈ GN is defined by

C (v) =

{x ∈ RN |

∑i∈N

xi = v(N);∑i∈S

xi ≥ v(S),∀S ∈ 2N

}.

Theorem (Bondareva (1963), Shapley (1967)): Let < N, v > be ann-person game. Then, the following two assertions are equivalent:

(i) C (v) 6= ∅.(ii) < N, v > is a balanced game.



Interval solution conceptsI (R)N : set of all n-dimensional vectors with elements in I (R).The interval imputation set:

I(w) =

{(I1, . . . , In) ∈ I (R)N |

∑i∈N

Ii = w(N), Ii < w(i), ∀i ∈ N

}.

The interval core:

C(w) =

{(I1, . . . , In) ∈ I(w)|

∑i∈S

Ii < w(S), ∀S ∈ 2N \ {∅}

}.

Example (LLR-game) continuation:

C(w) =

{(I1, I2, I3)|

∑i∈N

Ii = J,∑i∈S

Ii < w(S)

},

C(w) = {([0, 0], [0, 0], J)} .



Classical cooperative games (Part I in Branzei, Dimitrovand Tijs (2008))

< N, v > is convex if and only if the supermodularity condition

v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T )

for each S ,T ∈ 2N holds.< N, v > is concave if and only if the submodularity condition

v(S ∪ T ) + v(S ∩ T ) ≤ v(S) + v(T )

for each S ,T ∈ 2N holds.



Convex and concave interval games

I < N,w > is supermodular if

w(S) + w(T ) 4 w(S ∪ T ) + w(S ∩ T ) for all S ,T ∈ 2N .

I < N,w > is convex if w ∈ IGN is supermodular and|w | ∈ GN is supermodular (or convex).

I < N,w > is submodular if

w(S) + w(T ) < w(S ∪ T ) + w(S ∩ T ) for all S ,T ∈ 2N .

I < N,w > is concave if w ∈ IGN is submodular and |w | ∈ GN

is submodular (or concave).



Size monotonic interval games

I < N,w > is size monotonic if < N, |w | > is monotonic, i.e.,|w | (S) ≤ |w | (T ) for all S ,T ∈ 2N with S ⊂ T .

I SMIGN : the class of size monotonic interval games withplayer set N.

I For size monotonic games, w(T )− w(S) is defined for allS ,T ∈ 2N with S ⊂ T .

I CIGN : the class of convex interval games with player set N.

I CIGN ⊂ SMIGN because < N, |w | > is supermodular impliesthat < N, |w | > is monotonic.



I-balanced interval games

< N,w > is I-balanced if for each balanced map λ∑S∈2N\{∅}

λSw(S) 4 w(N).

IBIGN : class of interval balanced games with player set N.

CIGN ⊂ IBIGN

CIGN ⊂ (SMIGN ∩ IBIGN)

Theorem: Let w ∈ IGN . Then the following two assertions areequivalent:

(i) C(w) 6= ∅.(ii) The game w is I-balanced.



Solution concepts for cooperative interval games

Π(N): set of permutations, σ : N → N, of NPσ(i) =

{r ∈ N|σ−1(r) < σ−1(i)

}: set of predecessors of i in σ

The interval marginal vector mσ(w) of w ∈ SMIGN w.r.t. σ:

mσi (w) = w(Pσ(i) ∪ {i})− w(Pσ(i))

for each i ∈ N.

Interval Weber set W : SMIGN � I (R)N :

W(w) = conv {mσ(w)|σ ∈ Π(N)} .



The interval Shapley value

The interval Shapley value Φ : SMIGN → I (R)N :

Φ(w) =1

n!

∑σ∈Π(N)

mσ(w), for each w ∈ SMIGN .

Example: N = {1, 2}, w(1) = [0, 1],w(2) = [0, 2], w(1, 2) = [4, 8].

Φ(w) =1

2(m(12)(w) + m(21)(w));

Φ(w) =1

2((w(1),w(1, 2)− w(1)) + (w(1, 2)− w(2),w(2))) ;

Φ(w) =1

2(([0, 1], [4, 7]) + ([4, 6], [0, 2])) = ([2, 3

1

2], [2, 4

1

2]).



Classical big boss games versus big boss interval games

Classical big boss games (Muto et al. (1988), Tijs (1990)):< N, v > is a big boss game with n as big boss if

(i) v ∈ GN is monotonic, i.e.,v(S) ≤ v(T ) if for each S ,T ∈ 2N with S ⊂ T ;

(ii) v(S) = 0 if n /∈ S ;

(iii) v(N)− v(S) ≥∑

i∈N\S(v(N)− v(N \ {i}))for all S ,T with n ∈ S ⊂ N.

Big boss interval games:< N,w > is a big boss interval game if < N,w > and< N,w − w > are classical (total) big boss games.BBIGN : the class of big boss interval games.Marginal contribution of each player i ∈ N to the grand coalition:Mi (w) := w(N)− w(N \ {i}).



Properties of big boss interval games

Theorem: Let w ∈ SMIGN . Then, the following conditions areequivalent:

(i) w ∈ BBIGN .

(ii) < N,w > satisfies

(a) Veto power property:w(S) = [0, 0] for each S ∈ 2N with n /∈ S .

(b) Monotonicity property:w(S) 4 w(T ) for each S ,T ∈ 2N with n ∈ S ⊂ T .

(c) Union property:

w(N)− w(S) <∑

i∈N\S

(w(N)− w(N \ {i}))

for all S with n ∈ S ⊂ N.



T -value (inspired by Tijs(1981))

I the big boss interval point: B(w) := ([0, 0], . . . , [0, 0],w(N));

I the union interval point:

U(w) := (M1(w), . . . ,Mn−1(w),w(N)−n−1∑i=1

Mi (w)).

I The T -value T : BBIGN → I (R)N is defined by

T (w) :=1

2(U(w) + B(w)).



Holding situations with interval data

Holding situations: one agent has a storage capacity and otheragents have goods to store to generate benefits.In classical cooperative game theory, holding situations aremodeled by using big boss games (Tijs, Meca and Lopez (2005)).For a holding situation with interval data one can construct aholding interval game which turns out to be a big boss intervalgame.

Example: Player 3 is the owner of a holding house which hascapacity for one container. Players 1 and 2 have each onecontainer which they want to store. If player 1 is allowed to storehis/her container, then the benefit belongs to [10, 30] and if player2 is allowed to store his/her container, then the benefit belongs to[50, 70].



Example continues ...

The situation described corresponds to an interval game as follows:

I The interval game < N,w > with N = {1, 2, 3} andw(S) = [0, 0] if 3 /∈ S , w(∅) = w(3) = [0, 0],w(1, 3) = [10, 30] and w(N) = w(2, 3) = [50, 70] is a big bossinterval game with player 3 as big boss.

I B(w) = ([0, 0], [0, 0], [50, 70]) andU(w) = ([0, 0], [40, 40], [10, 30]) are the elements of theinterval core.

I T (w) = ([0, 0], [20, 20], [30, 50]) ∈ C(w).



Airport situations with interval dataIn airport situations, the costs of the coalitions are considered(Driessen (1988)):

I One runway and m types of planes (P1, . . . ,Pm pieces of therunway: P1 for type 1, P1 and P2 for type 2, etc.).

I Tj < [0, 0]: the interval cost of piece Pj .

I Nj : the set of players who own a plane of type j .

I nj : the number of (owners of) planes of type j .

I < N, d > is given byN = ∪m

j=1Nj : the set of all users of the runway;

d(∅) = [0, 0], d(S) =∑j

i=1 Ti

if S ∩ Nj 6= ∅, S ∩ Nk = ∅ for all j + 1 ≤ k ≤ m.

S needs the pieces P1, . . . ,Pj of the runway. The interval cost of

the used pieces of the runway is∑j

i=1 Ti .



Airport situations with interval data

Formally, d =∑m

k=1 Tku∗∪mr=kNr

, where

u∗K (S) :=

{1, K ∩ S 6= ∅0, otherwise

is the dual unanimity game.Interval Baker-Thompson allocation for a player i of type j :

γi :=

j∑k=1

(m∑

r=k

nr )−1Tk .

Proposition: Interval Baker-Thompson allocation agrees with theinterval Shapley value Φ(d).



Airport situations with interval data

Proposition: Let < N, d > be an airport interval game. Then,< N, d > is concave.

Proof: It is well known that non-negative multiples of classical dualunanimity games are concave (or submodular). By formaldefinition of d the classical games d =

∑mk=1 T ku∗k,m and

|d | =∑m

k=1 |Tk | u∗k,m are concave because T k ≥ 0 and |Tk | ≥ 0for each k , implying that < N, d > is concave.

Proposition: Let (N, (Tk)k=1,...,m) be an airport situation withinterval data and < N, d > be the related airport interval game.Then, the interval Baker-Thompson rule applied to this airportsituation provides an allocation which belongs to C(d).



Example:< N, d > airport interval game interval costs: T1 = [4, 6],T2 = [1, 8],d(∅) = [0, 0], d(1) = [4, 6], d(2) = d(1, 2) = [4, 6] + [1, 8] = [5, 14],d = [4, 6]u∗{1,2} + [1, 8]u∗{2},

Φ(d) = ( 12 ([4, 6] + [0, 0]), 1

2 ([1, 8] + [5, 14])) = ([2, 3], [3, 11]),γ = ( 1

2 [4, 6], 12 [4, 6] + [1, 8]) = ([2, 3], [3, 11]) ∈ C(d).

Figure:



Sequencing situations with interval data

Sequencing situations with one queue of players, each with onejob, in front of a machine order. Each player must have his/her jobprocessed on this machine, and for each player there is a costaccording to the time he/she spent in the system (Curiel, Pederzoliand Tijs (1989)).

A one-machine sequencing interval situation is described as a4-tuple (N, σ0, α, p),σ0: a permutation defining the initial order of the jobsα = ([αi , αi ])i∈N ∈ I (R+)N , p = ([p

i, pi ])i∈N ∈ I (R+)N : vectors

of intervals with αi , αi representing the minimal and maximalunitary cost of the job of i , respectively, p

i, pi being the minimal

and maximal processing time of the job of i , respectively.




I To handle such sequencing situations, we propose to use eitherthe approach based on urgency indices or the approach basedon relaxation indices. This requires to be able to compute

either ui =[αip

i

, αipi

](for each i ∈ N) or ri =

[p

iαi, piαi

](for each

i ∈ N), and such intervals should be pair-wise disjoint.

Interval calculus: Let I , J ∈ I (R+).We define · : I (R+)× I (R+)→ I (R+) by I · J := [I J, I J].Let Q :=

{(I , J) ∈ I (R+)× I (R+ \ {0}) | I J ≤ I J

}.

We define ÷ : Q → I (R+) by IJ := [ I

J ,IJ

] for all (I , J) ∈ Q.



Sequencing situations with interval dataExample (a): Consider the two-agent situation withp1 = [1, 4], p2 = [6, 8], α1 = [5, 25], α2 = [10, 30]. We can computeu1 =

[5, 25

4

], u2 =

[53 ,

154

]and use them to reorder the jobs as the

intervals are disjoint.

Example (b): Consider the two-agent situation withp1 = [1, 3], p2 = [4, 6], α1 = [5, 6], α2 = [11, 12]. Here, we cancompute r1 =

[15 ,

12

], r2 =

[4

11 ,12

], but we cannot reorder the jobs

as the intervals are not disjoint.

Example (c): Consider the two-agent situation withp1 = [1, 3], p2 = [5, 8], α1 = [5, 6], α2 = [10, 30]. Now, r1 is definedbut r2 is undefined. On the other hand, u1 is undefined and u2 isdefined, so no comparison is possible; consequently, the reorderingcannot take place.



Sequencing situations with interval dataLet i , j ∈ N. We define the interval gain of the switch of jobs i andj by

Gij :=

{αjpi − αipj , if jobs i and j switch[0,0], otherwise.

The sequencing interval game:

w :=∑

i ,j∈N:i<j

Giju[i ,j].

Gij ∈ I (R) for all switching jobs i , j ∈ N andu[i ,j] is the unanimity game defined as:

u[i ,j](S) :=

{1, if {i , i + 1, ..., j − 1, j} ⊂ S0, otherwise.




I The interval equal gain splitting rule is defined byIEGSi (N, σ0, α, p) = 1

2

∑j∈N:i<j

Gij + 12

∑j∈N:i>j

Gij , for each

i ∈ N.

Proposition: Let < N,w > be a sequencing interval game. Then,i) IEGS(N, σ0, α, p) = 1

2 (m(1,2...,n)(w) + m(n,n−1,...,1)(w)).ii) IEGS(N, σ0, α, p) ∈ C(w).

Proposition: Let < N,w > be a sequencing interval game. Then,< N,w > is convex.Example: Consider the interval situation with N = {1, 2},σ0 = {1, 2}, p = (2, 3) and α = ([2, 4], [12, 21]).The urgency indices are u1 = [1, 2] and u2 = [4, 7], so that the twojobs may be switched.We have:G12 = [18, 30], IEGS(N, σ0, α, p) = ([9, 15], [9, 15]) ∈ C(w).



Bankruptcy situations with interval dataIn a classical bankruptcy situation, a certain amount of money Ehas to be divided among some people, N = {1, . . . , n}, who haveindividual claims di , i ∈ N on the estate, and the total claim isweakly larger than the estate.The corresponding bankruptcy game vE ,d :vE ,d(S) = (E −

∑i∈N\S di )+ for each S ∈ 2N , where

x+ = max {0, x} (Aumann and Maschler (1985)).

I A bankruptcy interval situation with a fixed set of claimantsN = {1, 2, . . . , n} is a pair (E , d) ∈ I (R)× I (R)N , whereE = [E ,E ] < [0, 0] is the estate to be divided and d is thevector of interval claims with the i-th coordinate di = [d i , d i ](i ∈ N), such that [0, 0] 4 d1 4 d2 4 . . . 4 dn andE <

∑ni=1 d i .

BRIN : the family of bankruptcy interval situations with set ofclaimants N.



Bankruptcy situations with interval dataWe define a subclass of BRIN , denoted by SBRIN , consisting of allbankruptcy interval situations such that

|d(N \ S)| ≤ |E | for each S ∈ 2N with d(N \ S) ≤ E .(∗)

I We call a bankruptcy interval situation in SBRIN a strongbankruptcy interval situation. With each (E , d) ∈ SBRIN weassociate a cooperative interval game < N,wE ,d >, defined by

wE ,d(S) := [vE ,d(S), vE ,d(S)] for each S ⊂ N.

Note that (∗) implies vE ,d(S) ≤ vE ,d(S) for each S ∈ 2N .

SBRIGN : the family of all bankruptcy interval games wE ,d

with (E , d) ∈ SBRIN .



Bankruptcy situations with interval dataWe notice that wE ,d ∈ SBRIGN is supermodular because vE ,d andvE ,d ∈ GN are convex. The following example illustrates that

wE ,d ∈ SBRIGN is supermodular but not necessarily convex.Example: Let (E , d) be a two-person bankruptcy situation.We suppose that the claims of the players are closed intervalsd1 = [70, 70] and d2 = [80, 80], respectively,and the estate is E = [100, 140].Then, the corresponding game < N,wE ,d > is given by

wE ,d(∅) = [0, 0],wE ,d(1) = [20, 60],wE ,d(2) = [30, 70]

and wE ,d(1, 2) = [100, 140].

I This game is supermodular, but is not convex because|wE ,d | ∈ GN is not convex.



Minimum cost spanning tree situations with interval data

There are also interesting results for interval extension of minimimspanning tree situations. For further details please see the paper

I Connection situations under uncertainty and cost monotonicsolutions by Moretti et al. which was published in Computersand Operations Research.


References

References

[1]Alparslan Gok S.Z., Cooperative interval games, PhDDissertation Thesis, Institute of Applied Mathematics, Middle EastTechnical University, Ankara-Turkey (2009).[2]Alparslan Gok S.Z., Cooperative Interval Games: Theory andApplications, Lambert Academic Publishing (LAP), Germany(2010) ISBN:978-3-8383-3430-1.[3]Alparslan Gok S.Z., Branzei R., Fragnelli V. and Tijs S.,Sequencing interval situations and related games, to appear inCentral European Journal of Operations Research (CEJOR).[4]Alparslan Gok S.Z., Branzei R. and Tijs S., Airport intervalgames and their Shapley value, Operations Research and Decisions,Issue 2 (2009).


References

References

[5]Alparslan Gok S.Z., Branzei R. and Tijs S., Big boss intervalgames, International Journal of Uncertainty, Fuzziness andKnowledge-Based Systems (IJUFKS), Vol. 19, no:1 (2011)pp.135-149.[6]Aumann R. and Maschler M., Game theoretic analysis of abankruptcy problem from the Talmud, Journal of EconomicTheory 36 (1985) 195-213.[7]Bondareva O.N., Certain applications of the methods of linearprogramming to the theory of cooperative games, ProblemlyKibernetiki 10 (1963) 119-139 (in Russian).[8] Branzei R. and Alparslan Gok S.Z., Bankruptcy problems withinterval uncertainty, Economics Bulletin, Vol. 3, no. 56 (2008) pp.1-10.


References

References[9]Branzei R., Dimitrov D. and Tijs S., Models in CooperativeGame Theory, Springer, Game Theory and Mathematical Methods(2008).[10]Curiel I., Pederzoli G. and Tijs S., Sequencing games,European Journal of Operational Research 40 (1989) 344-351.[11]Driessen T., Cooperative Games, Solutions and Applications,Kluwer Academic Publishers (1988).[12]Gillies D. B., Solutions to general non-zero-sum games. In:Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theoryof games IV, Annals of Mathematical Studies 40. PrincetonUniversity Press, Princeton (1959) pp. 47-85.[13] Moretti S., Alparslan Gok S.Z., Branzei R. and Tijs S.,Connection situations under uncertainty and cost monotonicsolutions, Computers and Operations Research, Vol.38, Issue 11(2011) pp.1638-1645.


References

References

[14] Muto S., Nakayama M., Potters J. and Tijs S., On big bossgames, The Economic Studies Quarterly Vol.39, No. 4 (1988)303-321.[15]Shapley L.S., On balanced sets and cores, Naval ResearchLogistics Quarterly 14 (1967) 453-460.[16] Tijs S., Bounds for the core and the τ -value, In: MoeschlinO., Pallaschke D. (eds.), Game Theory and MathematicalEconomics, North Holland, Amsterdam (1981) pp. 123-132.[17] Tijs S., Big boss games, clan games and information marketgames. In:Ichiishi T., Neyman A., Tauman Y. (eds.), Game Theoryand Applications. Academic Press, San Diego (1990) pp.410-412.


References

References

[18]Tijs S., Meca A. and Lopez M.A., Benefit sharing in holdingsituations, European Journal of Operational Research 162 (1)(2005) 251-269.[19] von Neumann, J. and Morgernstern, O., Theory of Games andEconomic Behaviour, Princeton: Princeton University Press(1944).

Economic and Operations Research Situations with Interval Data

Education

airport interval games

interval uncertainty

interval gameslecture

interval datareferences

cooperative intervalgames

papersbig boss interval

player set n n

interval game withw