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

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

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

1. 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