Existence of equilibria in countable games: an algebraic approach

Details Existence of equilibria in countable games: an algebraic approach Existence of equilibria in countable games: an algebraic approach

2012-03-11

arxiv.org/abs/1203.2301v1

Computer Science

Computer Science and Game Theory

25 pages. A variational principle for a certain class of amenable groups is proved in Sec.3, Theorem 3.5

Scientific paper

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are not applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group and the payoffs are functions of the group operation. In order to obtain the existence of equilibria, finitely additive mixed strategies have to be allowed. This creates a problem of selection of a product measure of mixed strategies. We propose a family of such selections and prove existence of an equilibrium that does not depend on the selection. As a byproduct we show that if finitely additive mixed strategies are allowed, then Wald's game admits an equilibrium.

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