A coordination game is a game in which the players can obtain mutual gains by choosing compatible strategies. In a two-player coordination game with two strategies per player, the payoffs can be written
![[(A,a) (C,c); (B,b) (D,d)],](/images/equations/CoordinationGame/NumberedEquation1.svg) |
where 
,
, 
, and 
. The two diagonal outcomes are then pure-strategy Nash equilibria.
When the players rank the two equilibria differently, the result is the battle of the sexes. When both rank one equilibrium higher but strategic uncertainty can trap them at the other, the game exhibits a coordination failure.
