Glicksberg's theorem explained

Glicksberg's theorem should not be confused with Glicksberg fixed-point theorem.

In the study of zero sum games, Glicksberg's theorem (also Glicksberg's existence theorem) is a result that shows certain games have a minimax value:^[1] .If A and B are Hausdorff compact spaces, and K is an upper semicontinuous or lower semicontinuous function on

A x B , then\sup _{f} inf _{g} \iint Kdfdg = inf _{g} \sup _{f} \iint Kdfdg

where f and g run over Borel probability measures on A and B.

The theorem is useful if f and g are interpreted as mixed strategies of two players in the context of a continuous game. If the payoff function K is upper semicontinuous, then the game has a value.

The continuity condition may not be dropped: see example of a game with no value.

Notes and References

1. Glicksberg, I. L. (1952). A Further Generalization of the Kakutani Fixed Point Theorem, with Application to Nash Equilibrium Points. Proceedings of the American Mathematical Society, 3(1), pp. 170-174, https://doi.org/10.2307/2032478