View source for Uniform honeycombs in hyperbolic space

{{distinguish2|[[Glicksberg theorem]]}}

In the study of [[zero sum]] [[game theory|games]], '''Glicksberg's theorem''' (also '''Glicksberg's existence theorem''') is a result that shows certain games have a [[minimax]] value
.<ref>{{citation | chapter = On a game without a value | first1 = Maurice | last1 = Sion | first2 = Phillip | last2 = Wolfe | pages = 299-306 | title = Contributions to the Theory of Games III | editor1-first = M. | editor1-last = Dresher | editor2-first = A. W. | editor2-last = Tucker | editor3-first = P. | editor3-last = Wolfe | year = 1957 | isbn = 9780691079363 | publisher = Princeton University Press | series = Annals of Mathematics Studies 39}}</ref>

If ''A'' and ''B'' are [[compact set]]s, and ''K'' is an [[upper semicontinuous]] or [[lower semicontinuous]] function on <math>A\times B</math>, then

:<math>
\sup_{f}\inf_{g}\int\int K\,df\,dg = \inf_{g}\sup_{f}\int\int K\,df\,dg
</math>

where ''f'' and ''g'' run over [[Borel probability measure]]s on ''A'' and ''B''.

The theorem is useful if ''f'' and ''g'' are interpreted as [[mixed strategy|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 ==
{{reflist}}

{{DEFAULTSORT:Glicksberg's Theorem}}
[[Category:Game theory]]


{{gametheory-stub}}