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In the context of game theory, I wonder if the following statement is true for any game, if so, how do we prove it.

If every player plays the same strategy in a given game, then the payoff must be the same for everyone.

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Usually, payoffs are defined to be functions of the strategies, so the result is trivially true. – Michael Greinecker Feb 18 '12 at 1:02
If everyone plays the strategy of marrying the first person who proposes, the payoffs will definitely not be the same for everyone. – Gerry Myerson Feb 18 '12 at 2:19
I guess the question cannot be answered without a notion of sameness of strategies. – Michael Greinecker Feb 18 '12 at 11:24
@MichaelGreinecker, i mean pure strategies in my question. – Simon Hughes Feb 18 '12 at 12:44
No, not only for symmetric games. $S_1=S_2=\{a,b\}$. Payoffs are given by $u_1(a,a)=u_1(bb)=u_2(a,a)=u_2(bb)=u_2(a,b)=u_2(b,a)=1$, and $u_1(a,b)=u_1(b,a)=2$. This game is not symmetric, but satisfies the criterion. – Michael Greinecker Feb 19 '12 at 0:44

By definition, this is true iff the game is symmetric. From Wikipedia:

A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them.

Note that here "strategy" is used in the sense of "pure strategy"; players playing the same mixed strategy in a symmetric game will generally get different payoffs because of different random decisions.

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