Symmetric, zero sum games are a special class of games that have distinct equilibrium qualities. These qualities make it substantially easier to solve them.
- A symmetric game is a game in which all players have the same actions and symmetric payoffs given each individual’s action.
- A zero sum game is a game in which the payoffs for all players in each outcome sum to zero.
- In the equilibrium of a two player, symmetric, zero sum game, each player must receive a payoff of 0.
- Two player, symmetric, zero sum games always have equilibria in symmetric strategies.
Note that the game must fulfill all of those conditions to guarantee the described properties of equilibrium. For example, symmetric, non-zero sum games can only have asymmetric equilibria (Fey 2012); symmetric, zero sum games with more than three players can have only asymmetric equilibria (Xefteris 2015).