However, many of the commonly studied 2x2 games are at least ordinally symmetric.
The standard representations of chicken, the Prisoner's Dilemma, and the Stag hunt are all symmetric games.
Formally, in order for a 2x2 game to be symmetric, its payoff matrix must conform to the schema pictured to the right.
The requirements for a game to be ordinally symmetric are weaker, there it need only be the case that the ordinal ranking of the payoffs conform to the schema on the right.
Cheng et al. (2004) show that every two-strategy symmetric game has a (not necessarily symmetric) pure strategy Nash equilibrium.
Emmons et al. (2022) show that in every common-payoff game (a.k.a.
team game) (that is, every game in which all players receive the same payoff), every optimal strategy profile is also a Nash equilibrium.
These are in contrast to uncorrelated asymmetries which are purely informational and have no effect on payoffs (e.g. see Hawk-dove game).
, Partha Dasgupta and Eric Maskin give the following definition, which has been repeated since in the economics literature However, this is a stronger condition that implies the game is not only symmetric in the sense above, but is a common-interest game, in the sense that all players' payoffs are identical.