On finite strategy sets for finitely repeated zero-sum games

Abstract

We study finitely repeated two-person zero-sum games in which Player 1 is restricted to mixing over a fixed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 along with an optimal mixed strategy. We show that the entropy of this mixed strategy appears as a factor in an exact formula for the value of the game and thus is seen to have a direct numerical effect on the game's value. We develop upper and lower bounds on the value of these games that are within an additive constant and discuss how our results are related to the work of Neyman and Okada on strategic entropy (Neyman and Okada, 1999, Games Econ. Behav. 29, 191–223). Finally, we use these results to bound the value of repeated games in which one of the players uses a computer with a bounded memory and is further restricted to using a constant amount of time at each stage.