28. Mixed and Behavior Strategies in Infinite Extensive Games

We are concerned with infinite extensive games—not necessarily of perfect information—in which there may be a continuum of alternatives at some or all the moves; the games may also have unbounded or infinite play length. Our object is to define the notion of mixed strategy for such games, and to use this definition to prove the appropriate generalization of Kuhn’s theorem on optimal behavior strategies in games of perfect recall [K1]. Also, our methods give a solution to the conceptual problem raised by McKinsey under the heading ‘‘games played over function space’’ [Mc, pp. 355–357]. By-products are that our proof of Kuhn’s theorem makes no use of the rather cumbersome ‘‘tree’’ model for extensive games, that it explicitly uses conditional probabilities (which are implicitly used by Kuhn), and that it explicitly proves that in a game which is of perfect recall for one player, that player can restrict himself to behavior strategies (this also is implicit in Kuhn’s proof ). Our proof is longer and more complicated than Kuhn’s proof, but only because of the problems introduced by the non-denumerably infinite character of the game; the treatment of finite games by our methods would be considerably shorter.