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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.2012-02-18
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    If everyone plays the strategy of marrying the first person who proposes, the payoffs will definitely not be the same for everyone.2012-02-18
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    I guess the question cannot be answered without a notion of sameness of strategies.2012-02-18
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    @MichaelGreinecker, i mean pure strategies in my question.2012-02-18
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    Take a two player game with strategy sapces $S_1=S_2=\{a\}$. Let $u_1(a,a)=1$ and $u_2(a,a)=0$. Then the payoffs differ even though both choose the same strategy.2012-02-18
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    @MichaelGreinecker, so i guess the above statement is true only if the game is symmetric as joriki pointed out below in his answer.2012-02-19
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    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.2012-02-19

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