Glicksberg's 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 compact sets, and K is an upper semicontinuous or lower semicontinuous function on $A\times B$ , 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.

References

Sion, Maurice; Wolfe, Phillip (1957), "On a game without a value", in Dresher, M.; Tucker, A. W.; Wolfe, P. (eds.), Contributions to the Theory of Games III, Annals of Mathematics Studies 39, Princeton University Press, pp. 299–306, ISBN 9780691079363

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[1] Sion, Maurice; Wolfe, Phillip (1957), "On a game without a value", in Dresher, M.; Tucker, A. W.; Wolfe, P. (eds.), Contributions to the Theory of Games III, Annals of Mathematics Studies 39, Princeton University Press, pp. 299–306, ISBN 9780691079363