3. Zero-Sum Games: The General Case

This chapter considers general zero-sum games and proves various minmax theorems. We start with Sion’s theorem with geometrical and topological assumptions on the game: convex compact strategy sets and payoff functions quasi-concave upper semi-continuous in the first variable and quasi-convex lower semi-continuous in the second variable. Then we prove the standard minmax theorem in mixed strategies for measurable bounded payoff functions, extending von Neumann’s theorem, under topological assumptions: compact Hausdorff strategy sets and payoff functions upper semi-continuous in the first variable and lower semi-continuous in the second variable. We conclude the chapter with the introduction of the value operator and the notion of a derived game, which play an important role in the operator approach to repeated games.