Glicksberg's theorem
From HandWiki
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 [math]\displaystyle{ A\times B }[/math], then
- [math]\displaystyle{ \sup_{f}\inf_{g}\iint K\,df\,dg = \inf_{g}\sup_{f}\iint K\,df\,dg }[/math]
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.[2]
References
- ↑ 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
- ↑ Sion, Maurice; Wolfe, Phillip (1957), "On a game without a value", in Dresher, M.; Tucker, A. W.; Wolfe, P., Contributions to the Theory of Games III, Annals of Mathematics Studies 39, Princeton University Press, pp. 299–306, ISBN 9780691079363
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