On Pseudo-Games

In the definition of a two-person zero-sum game given by Von Neumann and Morgenstern it is assumed that both players know the rules of the game (e. g. , the game tree, the information sets as well as the distributions of the ensuing payoffs for given strategy choices, etc. ). We use the term pseudo-game to denote the case where at least one player does not have complete information. In this paper we restrict our attention to those pseudo-games in which player I, say, is only aware of his set of pure strategy choices (assumed to contain m elements: 2 m 1, let () = (I₁ (), I₂ (), , Iₙ (), ) denote the partition on the set I of positive integers defined by the equations: equation*1. 3 Iₙ () = \ (^{n-1₊=₁ k^) + 1, (^n-1₊=₁ k^) + 2, , ⁿ₊=₁ k^\}; n = 1, 2, 3,. equation* For example, I₁ (2) = \1\, I₂ (2) = \2, 3, 4, 5\, I₃ (2) = \6, 7, , 14\; etc. We shall refer to Iₙ () as the nth interval of the partition (). Note that the cardinality of Iₙ () is n^. Let us suppose that player I is using some rule P that assigns, with probability 1, the same mixed strategy to the ith subgame as it does to the jth subgame whenever i and j belong to the same interval Iₙ (), n = 1, 2, 3,. In this case we say that P is constant on intervals. Thus if we say that player I is to play a certain strategy p during the nth interval of a partition (), we mean that he is to play p during every subgame whose index belongs to Iₙ (). The particular strategy that player I uses in the nth interval (a random variable depending on plays and losses occurring prior to the nth interval) will be denoted by pₙ. For j = 1, 2, 3, , N, let Xⱼ represent the loss incurred by player I during the jth subgame. Note that the sequence \Xₙ\ is a discrete stochastic process whose index set is the set I of positive integers and whose law of evolution is determined by the distributions P (₀, ₁) and by the rules P and Q that the players use. The first objective of this paper is to prove: THEOREM. Suppose players I and II are playing a sequence of identical pseudo-games G satisfying (i) and (ii): (i) Player I has m 2 pure strategy choices. (ii) The distributions P (₀, ₁) have uniformly bounded second moments and for each a A and every Borel set C, P (₀, ) (C) is B-measurable. Then there exists a class of rules \P\ₘ for player I such that for all rules Q that player II may use we have: P \P\ₘ Pr (₍ N^-1 N₉=₁ Xⱼ vG P, Q) = 1. We will show, that is, that the player with incomplete information can do as well asymptotically as he could if he had complete information. The members of \P\ₘ will all be constant on intervals. Our second objective will be to seek a strong convergence rate for N^-1 N₉=₁ Xⱼ. In the course of achieving this goal we will show that a good partition is obtained by setting equal to (m + 2) /m.