In the study of

zero sum Zero-sum game is a mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is e ...

games A game is a structured form of play, usually undertaken for entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator sports or games) or art (such ...

, Glicksberg's theorem (also Glicksberg's existence theorem) is a result that shows certain games have a

minimax Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When de ...

value:Glicksberg, I. L. (1952). A Further Generalization of the

Kakutani Fixed Point Theorem In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed poi ...

, 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 space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

s, and ''K'' is an

upper semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, rou ...

lower semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, rou ...

function on

A\times B

, then :

\sup_\inf_\iint K\,df\,dg = \inf_\sup_\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 In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...

of two players in the context of a

continuous game A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe (noughts and crosses) or checkers (draughts). In other words, it extends the notion of a discrete game, where the playe ...

. 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

Game theory {{gametheory-stub