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.
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.
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.