“Glicksberg's theorem”的意思、由来-开放百科全书

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Glicksberg's theorem

释义

References

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 , then

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. ^{{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}}

1 : Game theory

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