operations research situations and games

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 2: Operations Research Situations and Games

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

Isparta, Turkeyemail:[email protected]

August 13-16, 2011


Outline

Introduction

Cooperative games

Economic and Operations Research situations and games

Market situations and big boss games

Bankruptcy situations and games

Sequencing situations and games

Airport situations and games

Final Remark

References


Introduction

IntroductionI Operations research (OR) is an interdisciplinary mathematical

science that studies the effective use of technology byorganizations.

I Operations research use techniques of other sciences such asstatistics, optimization, probability theory, game theory,mathematical modeling and simulation to solve the problems.

I We pay much attention to the modelling part; that is how togo from an economic or Operations Research situation togame theory.

I Economic and OR games: Cooperative games associated withseveral types of economic and OR problems in which variousdecision makers (players) are involved, who face a jointoptimization problem (in trying to minimize/maximize totaljoint costs/rewards) and an allocation problem in how todistribute the joint costs/rewards among them.


Cooperative games

Preliminaries on cooperative gamesA cooperative game (Transferable utility (TU-game)) is a pair< N, v >, where

I N = {1, 2, ..., n} is the set of playersI v : 2N → R is the characteristic function of the game with

v(∅) = 0.I v(S) - value of the coalition, S ⊂ N.I GN - the family of coalitional games with player set N.

We denote the size of a coalition S ⊂ N by |S |.GN is a (2|N| − 1) dimensional linear space for which unanimitygames form a basis.Let S ∈ 2N \ {∅}. The unanimity game based on S , uS : 2N → Ris defined by

uS(T ) =

{1, S ⊂ T0, otherwise.


Cooperative games

Core

How to distribute the profit generated by the cooperating players?An important role is played by the allocations in the core of thegame.The core (Gillies (1959)) is defined by

C (v) =

{x ∈ RN |

∑i∈N

xi = v(N),∑i∈S

xi ≥ v(S)for each S ∈ 2N

},

for each v ∈ GN .

I∑

i∈N xi = v(N): Efficiency condition

I∑

i∈S xi ≥ v(S), S ⊂ N: Coalitional rationality condition


Cooperative games

The Shapley value

Π(N): the set of all permutations σ : N → N of N.Pσ(i) :=

{r ∈ N|σ−1(r) < σ−1(i)

}: the set of all predecessors of i

with respect to the permutation σ.Let v ∈ GN and σ ∈ Π(N).The marginal contribution vector mσ(v) ∈ Rn with respect to σand v has the i-th coordinate the valuemσ

i (v) := v(Pσ(i) ∪ {i})− v(Pσ(i)) for each i ∈ N.The Shapley value (Shapley (1953)) φ(v) of a game v ∈ GN is theaverage of the marginal vectors of the game:

φ(v) :=1

n!

∑σ∈Π(N)

mσ(v).


Cooperative games

Preliminaries on cooperative games

A game < N, v > is calledsuperadditive if v(S ∪ T ) ≥ v(S) + v(T ) for all S ,T ⊂ N withS ∩ T = ∅;subadditive if v(S ∪ T ) ≤ v(S) + v(T ) for all S ,T ⊂ N withS ∩ T = ∅;convex (or supermodular) if v(S ∪ T )+v(S ∩ T ) ≥ v(S) + v(T )for all S ,T ⊂ N;concave (or submodular) if v(S ∪T )+v(S ∩T ) ≤ v(S) + v(T ) forall S ,T ⊂ N.Each convex (concave) game is also superadditive (subadditive).For a game v ∈ GN and a coalition T ∈ 2N \ {∅}, the subgamewith player set T , (T , vT ), is the game vT defined byvT (S) := v(S) for all S ∈ 2T .For details see Branzei, Dimitrov and Tijs (2008).




We study some types of Economic and Operations Researchsituations and their relation with game theory.

I Market situations and big boss games

I Bankruptcy situations and games

I Sequencing situations and games

I Airport situations and games




A class of cooperative games suitable to model market situationswith two corners regarding the outcome of cooperation is big bossgames.In one corner there is a powerful player called the big boss; theother corner contains players that need the big boss to benefitfrom cooperation.In a big boss game, the big boss has veto power (i.e. the worth ofeach coalition which does not include the big boss is zero) and thecharacteristic function of the game has a monotonicity property(i.e. joining the big boss is more beneficial when the size ofcoalitions grows larger) and a union property (expressed in terms ofmarginal contributions to the grand coalition of coalitions andindividuals).



Example (Find the treasure)

There is a hotel with 6 rooms. In one of the rooms a treasure ishidden. The value of the treasure is 600 units. Each room’sprobability of having the treasure is same, i.e. 1

6 .

I Agent O is allowed to look only in one room and to keep thetreasure if found.

I Agents A and B know something about the position of thetreasure:

1. If the treasure is in room 3 or 6, then agent A knows it.2. If the treasure is in room 4,5 or 6 then agent B knows it.

The big boss game with O as a big boss. v({O}) = 16 600 = 100,

v({A}) = v({B}) = 0, v({O,A}) = 36 600 = 300,

v({O,B}) = 46 600 = 400, v({A,B}) = v(∅) = 0,

v({O,A,B}) = 56 600 = 500.



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.

I Property (ii) expresses the veto power of the big boss: theworth of each coalition which does not include the big boss iszero.

I Property (i) says that joining the big boss is more beneficialwhen the size of coalitions grows larger, while (iii) expressesthe fact that the force of non big boss players is in their union.

We denote the set of all big boss games with n as a big boss byBBGN .



Big boss gamesLet v ∈ GN . For each i ∈ N, the marginal contribution of player ito the grand coalition N is Mi (v) := v(N)− v(N \ {i}).The core C (v) of a big boss game is always nonempty and equal to

{x ∈ I (v)|0 ≤ xi ≤ Mi (v) for each i ∈ N \ {n}} .

For a big boss game < N, v > (with n as a big boss) of v ∈ GN

two particular elements of its core are the big boss point B(v)defined by

Bj(v) :=

{0, if j ∈ N \ {n}v(N), if j = n,

and the union point U(v) defined by

Uj(v) :=

{Mj(N, v), if j ∈ T \ {n}v(N)−

∑i∈N\{n}Mi (N, v), if j = n.



τ -value of a big boss game

For big boss games τ -value, introduced by Tijs (1981), is theelement in the center of the core.

τ -value of a big boss game < N, v > is defined by

τ(v) =B(v) + U(v)

2.

Note that if the game is a convex big boss game thenτ(v) = φ(v).

For details see Tijs (1981, 1990).



ExampleLet < N, v > be a three-person game with v(N) = 5,v({1, 3}) = 3, v({2, 3}) = 4, v(S) = 0 otherwise.This game is a big boss game, with 3 as a big boss (fromdefinition). Now,

M1(v) = v(N)− v(N \ {1}) = 5− 4 = 1,

M2(v) = v(N)− v(N \ {2}) = 5− 3 = 2.

C (v) ={x ∈ R3 : x(N) = v(N), 0 ≤ x1 ≤ M1(v), 0 ≤ x2 ≤ M2(v)

}C (v) =

{x ∈ R3 :

3∑i=1

xi = 5, 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 2

}C (v) = conv {(0, 0, 5), (1, 0, 4), (0, 2, 3), (1, 2, 2)}, which is aparalellogram.B(v) = (0, 0, 5),U(v) = (1, 2, 2) and τ(v) = ( 1

2 , 1,72 ) (core

elements).



Bi-monotonic allocation schemes (bi-mas)We denote by Pn the set {S ⊂ N|n ∈ S} of all coalitionscontaining the big boss.Let v ∈ GN be a big boss game with n as a big boss. We call ascheme a = (aiS)i∈S ,S∈Pn a bi-monotonic allocation scheme(bi-mas) (Branzei, Tijs and Timmer (2001)) if

(i) (aiS)i∈S is a core element of the subgame < S , v > for eachcoalition S ∈ Pn,

(ii) aiS ≥ aiT for all i ∈ S \ {n} with S ⊂ T and anS ≤ anT for allS ,T ∈ Pn with S ⊂ T .

I Interpretation: In a bi-mas the big boss is weakly better off inlarger coalitions, while the other players are weakly worse off.

I For (total) big boss games the τ -value generates a bi-mas.Further, each core element generates a bi-mas for thesegames.



Example

Let < N, v > be a big boss game (with 3 as big boss) andv({i}) = 0 for i ∈ N, v({1, 2}) = 0,v({1, 3}) = 6, v({2, 3}) = 5, v(N) = 9.The (total) τ -value is a bi-monotonic allocation scheme:

(1, 2, 3)(1, 3)(2, 3)(3)

1 2 32 1 1

2 5 12

3 ∗ 3∗ 2 1

2 2 12

∗ ∗ 0

.In larger coalitions player 3 is better off, other players worse off.




Bankruptcy situations and bankruptcy games have been intensivelystudied in literature (O’Neill (1982), Aumann and Maschler(1985)).The story is based on a certain amount of money (estate) whichhas to be divided among some people (claimants) who haveindividual claims on the estate, and the total claim is weakly largerthan the estate.

A bankruptcy situation with set of claimants N is a pair (E , d),where E ≥ 0 is the estate to be divided and d ∈ RN

+ is the vectorof claims such that

∑i∈N di ≥ E .




We assume that 0 ≤ d1 ≤ d2 ≤ . . . ≤ dn and denote by BRN theset of bankruptcy situations with player set N.The total claim is denoted by D =

∑i∈N di .

A bankruptcy rule is a function f : BRN → RN which assigns toeach bankruptcy situation (E , d) ∈ BRN a payoff vectorf (E , d) ∈ RN such that 0 ≤ f (E , d) ≤ d (reasonability) and∑

i∈N fi (E , d) = E (efficiency).




The proportional rule (PROP) is one of the most often used in reallife, defined by

PROPi (E , d) =di∑

j∈N djE

for each bankruptcy problem (E , d) and all i ∈ N.

Another interesting rule the rights-egalitarian (f RE ) rule is definedby f RE

i (E , d) = di + 1n (E −

∑i∈N di ), for each division problem

(E , d) and all i ∈ N.




To each bankruptcy situation (E , d) ∈ BRN one can associate abankruptcy game vE ,d defined by vE ,d(S) = (E −

∑i∈N\S di )+ for

each S ∈ 2N , where x+ = max {0, x}.The game vE ,d is convex and the bankruptcy rules PROP and f RE

provide allocations in the core of the game (Aumann and Maschler(1985)).



Example

A bankruptcy situation (E , d) is given by E = 500 andd = (100, 200, 300). The associated bankruptcy game (bydefinition) vE ,d is as follows

S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} N

vE ,d(S) 0 0 100 200 200 300 400 500.

For example vE ,d({1, 2}) = max{

0,E −∑

i∈N\{1,2} di

}=

max {0, 500− 300} = 200.

I (vE ,d) = conv {(0, 100, 400), (0, 300, 200), (200, 100, 200)} ,

C (vE ,d) = conv {(0, 200, 300), (100, 200, 200), (100, 100, 300)} .



Example continues...

Marginal vectors of the bankruptcy game can be observed from thefollowing table.

σ(123)(132)(213)(231)(312)(321)

mσ1 (vE ,d) mσ

2 (vE ,d) mσ3 (vE ,d)

0 200 3000 200 300

100 100 300100 100 300100 200 200100 200 200

.

φ(vE ,d) = (200

3,

500

3,

800

3) ∈ C (vE ,d)

(bankruptcy games are convex).




Let us calculate PROP and f RE

PROP1(E , d) = d1∑j∈N dj

E = 100600 500 = 250

3 ,


E = 200600 500 = 500

3 ,


E = 300600 500 = 250.

f RE1 (E , d) = d1 + 1

3 (E −∑

i∈N di ) = 100 + 13 (500− 600) = 200

3 ,f RE2 (E , d) = d2 + 1

3 (E −∑

i∈N di ) = 200 + 13 (500− 600) = 500

3f RE3 (E , d) = d3 + 1

3 (E −∑

i∈N di ) = 300 + 13 (500− 600) = 800

3Note that PROP(E , d) = ( 250

3 , 5003 , 250) and

f RE (E , d) = ( 2003 , 500

3 , 8003 ) are also core elements, because the

game is convex.Note that f RE (E , d) = φ(vE ,d).




Waiting lines are part of everyday life (people standing in line,service jobs, manufacturing jobs, machines to be repaired,telecommunication transmissions etc..).

We consider the sequencing situations with one queue of players,each with one job, in front of a machine order.

Each player must have his/her job processed on this machine, andfor each player there is a cost according to the time he/she spentsin the system.




First problem is to find an optimal order of the jobs taking intoaccount the individual processing times and the costs per unit oftime because it is useful for reducing the cost connected with thetime spent in the system.After an optimal order is found, second problem is to motivate theagents to switch their positions according to the new order sinceagents are basically interested in their individual benefit .

I First problem: An optimal order may be obtained simplyreordering the jobs for weakly decreasing urgency indices(Smith (1956)).

I Second problem: Cooperative games arising from suchsequencing situations is useful to solve this problem which isdone by offering fair shares of the gain generated bycooperation(Curiel, Pederzoli and Tijs (1989)).




Formally, a one-machine sequencing situation is a 4-tuple(N, σ0, α, p) where:

I N = {1, ..., n} is the set of jobs;

I σ0 : N → {1, ..., n} is a permutation that defines the initialorder of the jobs;

I α = (αi )i∈N ∈ Rn+ is a non-negative real vector, where αi is

the cost per unit of time of job i ;

I p = (pi )i∈N ∈ Rn+ is a positive real vector, where pi is the

processing time of job i .




Given a sequencing situation and an ordering σ of the jobs, foreach i ∈ N we denote by P(σ, i) the set of jobs preceding i ,according to the order σ.

The time spent in the system by job i is the sum between thewaiting time that jobs in P(σ, i) need to be processed and theprocessing time of job i yielding the related cost

αi

(∑j∈P(σ,i) pj + pi

).

The (total) cost associated with σ:

Cσ, is given by Cσ =∑

i∈N αi

(∑j∈P(σ,i) pj + pi

).




The optimal order of the jobs σ∗ produces the minimum cost

Cσ∗ =∑

i∈N αi

(∑j∈P(σ∗,i) pj + pi

)or the maximum cost saving

Cσ0 − Cσ∗ .Smith (1956) proved that an optimal order can be obtainedreordering the jobs according to decreasing urgency indices, wherethe urgency index of job i ∈ N is defined as ui =

αi

pi.

The following question arises: Is it possible to share this costsavings Cσ0 − Cσ∗ among the agents in such a way that no agentwill protest? This question can be answered by using cooperativegames.



Example

Let us find the optimal order of the three agents, whereα1 = 20, α2 = 60, α3 = 100 and p1 = 2, p2 = 3, p3 = 4.

The urgency indices areu1 = α1

p1= 20

2 = 10, u2 = α2p2

= 603 = 20 and u3 = α3

p3= 100

4 = 25.

Optimal order of service is σ∗ = (3, 2, 1).




A sequencing game is a pair < N, v > where N is the set ofplayers, that coincides with the set of jobs, and the characteristicfunction v assigns to each coalition S the maximal cost savingsthat the members of S can obtain by reordering only their jobs.

We say that a set of jobs T is connected according to an order σ iffor all i , j ∈ T and k ∈ N, σ(i) < σ(k) < σ(j) implies k ∈ T .A switch of two connected jobs i and j , where i precedes j ,generates a change in cost equal to αjpi − αipj .We denote the gain of the switch as

gij = (αjpi − αipj)+ = max{0, αjpi − αipj}.




For a connected coalition T according to an order σ isv(T ) =

∑j∈T

∑i∈P(σ,j)∩T

gij .

If S is not a connected coalition, the order σ induces a partition inconnected components, denoted by S/σ. For a nonconnectedcoalition S , v(S) =

∑T∈S/σ

v(T ) for each S ⊂ N.

The characteristic function v of the sequencing game can bedefined as v =

∑i ,j∈N:i<j

giju[i ,j], where u[i ,j] is the unanimity game

defined as:

u[i ,j](S) =

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

.




Curiel, Pederzoli and Tijs (1989) show that sequencing games areconvex games and, consequently, their core is nonempty.Moreover, it is possible to determine a core allocation withoutcomputing the characteristic function of the game.They propose to share equally between the players i , j the gain gij

produced by the switch and call this rule the Equal Gain Splittingrule (EGS-rule).EGSi (N, σ0, α, p) = 1

2

∑k∈P(σ,i) gki + 1

2

∑j :i∈P(σ,j) gij for each

i ∈ N.



Example

Let us find the optimal order of the three agents whereα1 = 20, α2 = 10, α3 = 30 and p1 = 1, p2 = 1, p3 = 1.The urgency indices areu1 = α1

p1= 20, u2 = α2

p2= 10 and u3 = α3

p3= 30.

Optimal order of service is σ∗ = (3, 1, 2).Solution: Go from σ0 to optimal order by neigbour switches andshare equally the gain. This can be done by two neigbour services.First 2 and 3 switch, gain is g23 = α3p2 − α2p3 = 30− 10 = 20.Second 1 and 3 switch, gain is g13 = α3p1 − α1p3 = 30− 20 = 10.EGS(N, σ0, α, p) = (∗, 10, 10) + (5, ∗, 5) = (5, 10, 15) ∈ C (v).




The sequencing game < N, v > (convex) with N = {1, 2, 3} canbe constructed as follows: v({1}) = v({2}) = v({3}) = v(∅) = 0v({1, 2}) = 0 (1 is more urgent than 2), v({1, 3}) = 0 (switch isnot allowed because 2 is in between), v({2, 3}) = 20,v(N) = 20 + 10 = 30.Note that v = 20u[2,3] + 10u[1,3].For examplev({2, 3}) = 20u[2,3]({2, 3}) + 10u[1,3]({2, 3}) = 20 + 0 = 20,v(N) = 20u[2,3]({1, 2, 3}) + 10u[1,3]({1, 2, 3}) = 20 + 10 = 30.

I (v) = conv {(0, 0, 30), (0, 30, 0), (30, 0, 0)} ,

W (v) = C (v) = conv {(0, 0, 30), (0, 30, 0), (10, 0, 20), (10, 20, 0)} .




Marginal vectors of the sequencing game can be observed from thefollowing table.

σ(123)(132)(213)(231)(312)(321)

mσ1 (v) mσ

2 (v) mσ3 (v)

0 0 300 30 00 0 30

10 0 200 30 0

10 20 0

.

φ(v) = ( 103 ,

403 ,

403 ) ∈ C (v) (sequencing games are convex).




In airport situations costs of the coalitions are considered (Driessen(1988)).Airport situations lead to concave games and the Shapley valuebelongs to the core of the game.Baker(1965)-Thompson(1971): only users of a piece of the runwaypay for that piece and they share the cost of it equally.Littlechild and Owen (1973) showed that the Baker-Thompson rulecorresponds to the Shapley value.




Consider the aircraft fee problem of an airport with one runway.

I Suppose that the planes which are to land are classified intom types.

I For each 1 ≤ j ≤ m, denote the set of landings of planes oftype j by Nj and its cardinality by nj .

I Then N = ∪mj=1Nj represents the set of all landings.

I Let cj represent the cost of a runway adequate for planes oftype j . We assume that the types are ordered such that0 = c0 < c1 < . . . < cm.




I We consider the runway divided into m consecutive pieces Pj ,1 ≤ j ≤ m, where P1 is adequate for landings of planes oftype 1; P1 and P2 together for landings of planes of type 2,and so on. The cost of piece Pj , 1 ≤ j ≤ m, is the marginalcost cj − cj−1.

I The Baker-Thompson rule is given byβi =

∑jk=1[

∑mr=k nr ]−1(ck − ck−1) whenever i ∈ Nj . That is,

every landing of planes of type j contributes to the cost of thepieces Pk , 1 ≤ k ≤ j , equally allocated among its users∪m

r=kNr .

I The airport TU game < N, c > is given byc(S) = max {ck |1 ≤ k ≤ m,S ∩ Nk 6= ∅} for all S ⊂ N.



Example

The players 1, 2 and 3 own planes which need landing strip oflength |AD|. The strip |AD| is divided into 3 pieces |AB|, |BC |and |CD|. Player 1 needs to use the strip of length |AB| with thecost k1, player 2 needs to use the strip of length |AC | with the costk1 + k2, player 3 needs to use the strip of length |AD| with thecost k1 + k2 + k3.In cooperation they can share strips leading to the cost game< N, c > with N = {1, 2, 3},c(∅) = 0, c({1}) = k1, c({2}) = c({1, 2}) = k1 + k2,c(N) = c({1, 3}) = c({2, 3}) = c({3}) = k1 + k2 + k3.The Baker Thompson rule isβ = (β1, β2, β3) = (k1

3 ,k13 + k2

2 ,k13 + k2

2 + k3).We know that Baker-Thompson rule is equal to the Shapley value.




Marginal vectors of the airport game can be observed from thefollowing table.

σ(123)(132)(213)(231)(312)(321)

mσ1 (c) mσ

2 (c) mσ3 (c)

k1 k2 k3

k1 0 k2 + k3

0 k1 + k2 k3

0 k1 + k2 k3

0 0 k1 + k2 + k3

0 0 k1 + k2 + k3

.

φ(c) = (k1

3,

1

6(2k1 + 3k2),

1

6(6k3 + 3k2 + 2k1)).

φ(c) ∈ C (c) (airport games are concave).


Final Remark

Final Remark

For other interesting Operations research games see

Operations Research Games: A Survey

by Borm, Hamers and Hendrickx published in TOP (theOperational Research journal of SEIO (Spanish Statistics andOperations Research Society)).


References

References

[1]Aumann R. and Maschler M., “Game theoretic analysis of abankruptcy problem from the Talmud”, Journal of EconomicTheory 36 (1985) 195-213.[2]Baker J.Jr., “Airport runway cost impact study”, Reportsubmitted to the Association of Local Transport Airlines, Jackson,Mississippi (1965).[3]Borm P., Hamers H. Hendrickx R., (2001)Operations ResearchGames: A Survey, TOP 9, 139-216.[4]Branzei R., Dimitrov D. and Tijs S., “Models in CooperativeGame Theory”, Game Theory and Mathematical Methods,Springer (2008).[5]Branzei R., Tijs S. and Timmer J., “Information collectingsituations and bi-monotonic allocation schemes”, MathematicalMethods of Operations Research 54 (2001) 303-313.


References

References

[6] Curiel I., Pederzoli G. and Tijs S., “Sequencing games”,European Journal of Operational Research 40 (1989) 344-351.[7] Driessen T., “Cooperative Games, Solutions and Applications”,Kluwer Academic Publishers (1988).[8] 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.[9] Littlechild S.C. and Owen G., “A simple expression for theShapley value in a special case”, Management Science 20 (1973)370-372.[10]Muto S., Nakayama M., Potters J. and Tijs S., “On big bossgames”, The Economic Studies Quarterly Vol. 39, no. 4 (1988)303-321.


References

References[11] O’Neill B., “A problem of rights arbitration from the Talmud”,Mathematical Social Sciences 2 (1982) 345-371.[12] Shapley L.S., “A value for n-person games”, Annals ofMathematics Studies 28 (1953) 307-317.[13] Smith W., “Various optimizers for single-stage production”,Naval Research Logistics Quarterly 3 (1956) 59-66.[14] Thompson G.F., “Airport Costs and Pricing”, UnpublishedPhD. Dissertation, University of Birmingham (1971).[15] Tijs S., “Big boss games, clan games and information marketgames.” In: Ichiishi T., Neyman A., Tauman Y. (Eds.), GameTheory and Applications. Academic Press, San Diego (1990) pp.410-412.[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.

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