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Does a zero sum game always has a unique payoff, whatever the nash equilibrium selected is ? even with mixed strategies ?

If so, what is the proof ?

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up vote 2 down vote accepted

A unique average payoff, yes, for a zero-sum game with finitely many pure strategies. It is part of von Neumann's minimax theorem.

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To elaborate a little bit more. Let us say that G is a zero-sum game in which the pure action sets of the players are finite. To have an existence of a Nash equilibrium, we have to consider the mixed extension of G. In the context of zero-sum games, we call Nash equilibrium strategies as optimal strategies due to von Neumann - Morgenstern's seminal book. The proof is that every finite zero-sum game has a value, that is maxmin equals minmax. Besides, both players have optimal strategies. As Robert says, the average payoff is unique when the players play optimal strategies and it is equal to the value of the game.

p.s I am talking about two-player zero-sum games above. Regarding n-player games, even though players play an optimal strategy the strategy profile might not be a Nash equilibrium!

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