Glicksberg's theorem

{{distinguish|Glicksberg fixed-point theorem}}

{{technical|date=May 2025}}

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.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

If A and B are Hausdorff compact spaces, and K is an upper semicontinuous or lower semicontinuous function on $A\backslash times\; B$, then

:$$

\sup_{f}\inf_{g}\iint K\,df\,dg = \inf_{g}\sup_{f}\iint K\,df\,dg

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.{{citation | chapter = On a game without a value | first1 = Maurice | last1 = Sion | first2 = Phillip | last2 = Wolfe | pages = 299–306 | title = Contributions to the Theory of Games III | editor1-first = M. | editor1-last = Dresher | editor2-first = A. W. | editor2-last = Tucker | editor3-first = P. | editor3-last = Wolfe | year = 1957 | isbn = 9780691079363 | publisher = Princeton University Press | series = Annals of Mathematics Studies 39}}