Value of Infinite Matrix Games

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On the Value of Some Infinite Matrix Games

Author(s): Luciano Mndez-NayaSource: Mathematics of Operations Research, Vol. 26, No. 1 (Feb., 2001), pp. 82-88Published by: INFORMSStable URL: http://www.jstor.org/stable/3690436

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MATHEMATICSOF OPERATIONSRESEARCHVol. 26, No. 1, February2001, pp. 82-88Printed in U.S.A.

ON THE VALUE OF SOME INFINITE MATRIX GAMESLUCIANO MENDEZ-NAYA

It is shown that a zero-sum two-personnoncooperativegame A defined by a bounded infinitematrixin which each row convergesto the same real number 3 and each column to the same realnumbera has a value V(A) if and only if a < /3, in which case a < V(A) < 3. For any gamedefined by a bounded infinite matrix A = (aj), a necessary condition for V(A) to exist is thatinfj liminfi ao OVieN ,i=l

and let A = (aij)iEN;jEN, be an infinite matrix with upper bound M = sui,jEN laijl. Alsodenoted by A is the zero-sum two-person game in which S is the set of strategies of bothplayers and K, the payoff function of the first player, is defined by

K(x, y) = xAy V(x, y) E S x S.The superior and inferior values of A, V(A) and V(A) respectively, are defined by

V(A) = inf sup K(x, y),yES xeSV(A) = supinf K(x, y),xeS yES

ReceivedDecember 17, 1999; revisedJuly 10, 2000.AMS2000 subjectclassification. Primary:91A05; secondary:91A10.OR/MSsubjectclassification.Primary:Games/groupdecisions; secondary:Noncooperative.Keywords. Two-persongames, infinite matrixgames, value of a game.82

0364-765X/01/2601/0082/$05.001526-5471 electronicISSN, ? 2001, INFORMS


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INFINITEMATRIXGAMES

and a strategyx E S (y e S) is said to be optimalfor the first (second) playerifinfK(x, y) = V(A) supK(x, ) = V(A)).yES XES

If V(A) = V(A), A has a value V(A) that is the commonvalue of V(A) and V(A).Analogous definitions can be given when A is a boundedfinite or semi-infinitematrix,in which cases a playerwith a finite numbern of pure strategieshas mixed strategysetnSn= xe Rn/ xi = 1,xi >0 Vi= 1,...,n n.i=l

If A = (aij)iEN;jENs a bounded nfinitematrix,then for each i', j' E N the matricesAi,j,,A, and Ai are definedbyAi,j, = (aij)l 0 such that

supxAy' < oy' -E.xeSHence if ei E S is such that its ith component s unity and the otherszero,

eiAy'


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where n(j) is a numbersuch thatVn> n(j) a - 3(-aflj- 3(j1)

(the existence of n(j) is guaranteed,Vj, by the definition of w). For each i greaterthan n'andj less thanj', Ei- 3(j'- 1)

Summingover j < j',j'-1E(aij-)j)>-3 Vi>nj=1

Hence Equation(2) implies thatVi > n',-E > (eiA - o)y' = (aij - oij)yj=1

j'-l oo= E(aij - wj)yj+ E (aij - oj)yjj=1 j=j'e oo00> -- laij- jlY5 >'-3- 2MyJEJ= 2-E e 2>---2M =--e >-E- 3 6M 3

which is a contradiction.This proves (1), and hence thatV(A) = inf supxAy > inf coy= inf ot.yeS S yS jEN

Similarly,V(A) < sup i.iEN

ThusV(A) < sup 8i < inf co < V(A),ieN jEN

and game A does not have a value. -EXAMPLE 3. Waldgave the following exampleof an infinitematrixgame with no value:Aw = (aij), where -1 if i < j,

a=j0= if i = j,1 if i > j.Given Theorem2, the fact that Aw has no value is an immediateresult, since it is obviousthatfor this matrix8i = -1 for all i EN and aoj= 1 for all j EN.

THEOREM4. If A is a bounded infinitematrixsuch that all its rows converge to thesame real number 3 and all its columnsto the same real numbera, then(1) if a > 3, A has no value;(2) if a = ,, A has a value and V(A) = a = 8;(3) if a


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INFINITEMATRIX AMESLEMMA 5. If A is a bounded infinite matrix such that all its rows converge to thesame real number13and all its columns to the same real numbera, and a < f, thena < V(A) < V(A) < 13.LEMMA 6. If A is a boundedinfinitematrix,and (il, jl) EN x N, then(1) if V(i, j) such that j < jl and i > il, aij < V(A), then V(A) < V(Ailj);(2) if V(i, j)such that i < il and j > jl, aij > V(A), then V(A) > V(Ai1j).PROOF OF LEMMA 5. Given e > 0, let s E N be such that M/s < e/2. Since the rows ofA convergeto i3and its columns to a, we can define the sequence il, i2, .. is, js asfollows:

i1 eN/Vi >il, ail jl, i < i', aj < +ai2o-} 2ss

E

is E N/V(i, j) such that i > is, j < js, aij < a+ sj E N/V(i, j) such that j > j,, i i, ai < 13+

We now definey' E Sj byy -_ ^ ifj E {1, jl, J2 .. s.lj 0 otherwise,

and we calculatee1Aisy'Vi eN. If 1 < i < il, thenEaij < 13+s VJ jl2s

sois s

eiAsy'=

aijyj= aily' + E aij,yjj=l t=l

M e s< -t+ +1s+l 2(s+1) s+lE e< +- + =8+ .2 2

If i i< i2, then E Eail j2,aij


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Similararguments how that for all i < is, e,AJy' < 3 + E.If i > is,E Eaij < a+ - < + - j < s-1'2s 2s

soJs s-IeiAj,y' = Laijyj =ailYl + aij,, +aijyjj=1 t=l

s ) ( s s ) < E.(+12s s+l +Ij44Thus

eiAJAy'< 3+ e Vi EN,and thereforesupxAJs' _< +E,xES

whenceV(Ai')= V(AJ') = inf supxAJiy< f+E.

yESjs, xESHence

VE> 0 3j, EN such that V(AJi)< j3+E,and since V(A) = limn V(An) (Tijs 1977), then

V(A) = lim V(AJ')< .E-*)OSimilarlyit can be provedthat a < V(A), so

a < V(A)< V(A)5 il,

eiA _< V(A)(since aij < V(A) for all j < jl). Thus

eiA5 < max{V(Aij,), V(A)},and

V(A) = infsupxAy < supxA5 = supeiA5 < max{V(Ai,j,), V(A)}.yES XES xES iENBut max{V(Ai,j,), V(A)1 = V(Ai,h, so V(A) < V(Ai,j,).

(2) The proof is analogousto that of part(1). 03

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PROOF OF THEOREM 4. Since the only accumulationpointof each row and each columnis its limit, assertion(1) follows from Theorem2.Assertion(2) is a direct consequenceof Lemma 5.Supposenow that a V(A) + E . In view of (3), for all s E N we can define thesequence il, ji.. . is, j, as follows:

il E N/Vi > il ail I /V(i, ) such that j > jl, i il, aij > V(A)- E,

is > is- I V(i, j) such that i > is, j < j, 1, a ij < V(A) + E,js > 1s51V(i, j) such thatj 5 jsh' < is, a1j ? V(A) - E.

By Lemma6 and (3), V(A) < V(Ai211) nd V(A) > V(A,1).Now let 3 = .. . , 33),where -, = (- jk ..... Yj) for k = 1,... , s, be an optimalstrategyfor the second player in the matrixgame Ai,li (Jo= 0), and let i be an optimalstrategyfor the firstplayerin the matrixgame Ai21. Then

V(A) > V(AiSjS) supxA3jSj3 'Ajsj.~Hence, since (i) ii = 0 Vi> i2, (ii) a1j> -M for (i, j) such thati < i2, il V(A) - E for (i, j) such thati 22, 1 > j12

V(A)> -Aisj3 > V(A2j)1)31I I M 121II (V(A))(I - 13II-1Y21)and as V(A 2il) V(A),V A) ? 'V(A)j 11 - M112III+V(A) e) (I - 11 dj1 3 111)

- lYiIIiEI1133211,1(M(A)- E) +V(A)- E,so

11Y2ii V(A) - (V(A)+ E)+ IY11lE V(A) -(V(A)?E)M + V(A) - E M + V(A) - EBy similararguments,

V(A) - (V(A) + E) (2 t s- 1).M + V(A)-,EHence

+ 13A1>2)(A) - (V A)+ E)Ilull=y il ?+ + Il1 s > (s- 2) M+ V(A)-ETherefore, since V(A) - (V(A) + E) > 0 and the above inequalityis true for all naturalnumberss, Es such that Ij33I11, which is impossible.Hence V(A) = V(A) and the matrixgame A has a value. E

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L. MENDEZ-NAYAREMARK. In the case of part (3) of the theorem, the value can be any real numberS E [a, /]. For example, if the matrix A = (aij)iEN;jEN is defined by

8 if i= 1 and j= 1,ai= a if i > 1 and i > j,/8 if j > 1 and j > i,

then e, is a dominantstrategyfor both playersand V(A) = 8.The following exampleillustrates he use of Theorem4 to provethe existence of a valuefor a certain class of recursivegames (for more on recursivegames see Owen 1995).EXAMPLE. Consider a zero-sumgame G in which, at each stage, Player i has strate-gies mi and ni. The outcomes (mi, m2), (m,, n2) and (n1, m2) are terminal, with payoffsK(m,, m2)= 5, K(m,, n2) = 3, and K(n,, m2)= a for Player 1, but if (nl, n2) is playedthen G is played again. Since this game terminatesas soon as eitherof the playerschooseshis firststrategy, t can be treatedas an infinite matrixgame:the strategyset of each playeris N (the set of naturalnumbers),Player i's choice of strategyk E N is equivalentto hischoosingto play ni in the firstk- 1 stages andmi in the kth(if the otherplayer,j, does notterminate the game sooner by playing mj), and the payoff matrix is A = (aij)iEN; jN, where

S if i=j,aii = a if i> j,3 if j >i.

This matrixclearly satisfies the conditions of Theorem4, accordingto which it thereforehas a value if and only if / > a.Acknowledgments. The author thanksI. GarciaJurado,Ian-CharlesColeman and X.L. Quiiioa for helpful comments and suggestions. Financialsupportfor this research wasgrantedby the Xunta de Galicia underprojectPGIDTOOPXI20104PR.

ReferencesCegielski, A. 1991. Approximationof some zero-sumnoncontinuousgames by a matrixgame. Comment.Math.2 261-267.Gomez, E. 1988. Games with convex payoff function in the first variable.Internat.J. Game Theory3 201-204.Marchi,E. 1967. On the minimaxtheoremof the theoryof games. Ann. Mat. Pura Appl. 77 207-282.Mendez-Naya, L. 1996. Zero-sum continuous games with no compact support.Internat. J. Game Theory 2593-111.

.1998. Weaktopology and infinite matrixgames. Internat.J. Game Theory27 219-229.Owen, G. 1995. Game Theory.AcademicPress, San Diego, CA.Tijs, S. 1977. Semi-infiniteand infinitematrix and bimatrixgames. Ph.D. thesis, Universityof Nijmegen.Wald,A. 1950. Note on zero sum two persongames. Ann. Math.52 739-742.

L. Mendez-Naya:Departamento e MetodosCuantitativos araa Economfa,Facultadede Ciencias Econ6micase Empresariais,Avda. X6an XXIII, s/n, 15704-Santiagode Compostela, Spain;e-mail:[email protected]

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Value of Infinite Matrix Games