Glicksberg's theorem

Not to 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 ${\displaystyle A\times B}$, then

${\displaystyle \sup _{f}\inf _{g}\iint K\,df\,dg=\inf _{g}\sup _{f}\iint K\,df\,dg}$

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.^[2]