Same strategy, same payoff?

Question

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.

Answer 1

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.