Symmetric games

Symmetric games: all players are identical and indistinguishable. They have the same strategy sets, their utility functions are the same function of their own strategy and the other players' actions, and this function is symmetric in the other players' actions.

Examples: prinsoner's dilemma.

Nash proved that every symmetric game must have a symmetric equilibrium where all players play the same strategy.

J. F. Nash, Jr. Non-cooperative games. Annals of Mathematics, 54(2):286-295, 1951.