Acceptable points in games of perfect information.

This is the second of a series of papers on the theory of acceptable points in ^-person games. The first was [1]; in it the notion of acceptable points was defined for cooperative games, and a fundamental theorem was proved relating the acceptable expected payoffs for a single play of a game to probable average payoffs for ' 'strong equilibrium points" in its supergame. The chief result of the current paper, Theorem 5.4, is a generalization of von Neumann's classical Theorem on two-person zero-sum games of perfect information (see [11]). Roughly, it states that strong equilibrium points in the supergame of a stable game of perfect information can be achieved in pure supergame strategies. An example shows that not all games possess this property; and in fact, it is conjectured that the property is characteristic of game structures of perfect information. The theorem stated above holds whether G is interpreted as a cooperative or as a non-cooperative game. To lend meaning to this statement, we will have to extend the theory introduced in [1] to non-cooperative games. We plan to do this in full in a subsequent paper. Here just enough definitions and theorems will be used to enable us to state and prove the chief result for non-cooperative games of perfect information. The paper is divided into two parts, the first centering around the proof of the chief result for cooperative games, the second dealing with the extension to non-cooperative games. Section 1, the introduction, serves mainly to supply background from [1] and from the literature. In § 2, we show that the naive approach to generalizing von Neumann's theorem on games of perfect information fails; that is, we bring an example of a stable game of perfect information that has no acceptable point in pure strategies. It is then shown intuitively that an appropriate generalization of the von Neumann Theorem should involve the supergame. Sections 3 and 4 are devoted to the proof of preliminary theorems, dealing with supergame pure strategies and supergames of perfect information, respectively. In § 5 we establish the chief result. Section 6, which completes the first part of the paper, is devoted to the example and conjecture mentioned in connection with the chief result.