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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An approximation theorem for infinite games


Author: Michael Orkin
Journal: Proc. Amer. Math. Soc. 36 (1972), 212-216
MSC: Primary 90D05
DOI: https://doi.org/10.1090/S0002-9939-1972-0319583-8
MathSciNet review: 0319583
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Abstract: We consider infinite, two person zero sum games played as follows: On the nth move, players A, B select privately from fixed finite sets, $ {A_n},{B_n}$, the result of their selections being made known before the next selection is made. A point in the associated sequence space $ \Omega = \prod\nolimits_{n = 1}^\infty {({A_n} \times {B_n})} $ is thus produced upon which B pays A an amount determined by a payoff function defined on $ \Omega $. We show that if the payoff functions of games $ \{ {G_n}\} $ are upper semicontinuous and decrease pointwise to a function which is the payoff for a game, G, then $ {\text{Val}}({G_n}) \downarrow {\text{Val}}(G)$. This shows that a certain class of games can be approximated by finite games. We then give a counterexample to possibly more general approximation theorems by displaying a sequence of games $ \{ {G_n}\} $ with upper semicontinuous payoff functions which increase to the payoff of a game G, and where $ {\text{Val}}({G_n}) = 0$ for all n but $ {\text{Val}}(G) = 1$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0319583-8
Keywords: Imperfect information, zero sum two person game, infinite game, random strategy
Article copyright: © Copyright 1972 American Mathematical Society