Zero-Sum Games: The Finite Case

Abstract

This chapter studies zero-sum games, where the sets of strategies are finite. We prove the minmax theorem of von Neumann, which states that if a game is played with mixed strategies (probability distribution on strategies) then the value exists, as well as an optimal mixed strategy for each player. We then consider extensions such as Loomis’ theorem and Ville’s theorem. Finally, we introduce and study the convergence of the learning process “Fictitious Play”, where the initial game is repeated and each player plays at every period a best response to the average of the strategies played by his opponent in the past.