Handbook >> Game Theory >> Useful Concepts in Game Theory >> The concept of Equilibrium and some solution concepts >> Nash Equilibrium in Pure StrategiesNash equilibrium is one of the central solution concepts for games. The basic idea of a Nash equilibrium is that if each player chooses their part of the Nash equilbrium strategy, then no other player has a reason to deviate to another strategy. A simple example is a coordination game, such as the one in the figure below.
Nash Equilibrium in Pure StrategiesIn this game, both (L, l) and (R, r) are Nash equilibria. If Player 1 chooses L then Player 2 gets 1 by playing l and 0 by playing r; if Player 1 chooses R then Player 2 gets 2 by playing r and 0 by playing l. The two stratgies L and R for Player 1 and the two strategies l and r for Player 2 are called "pure strategies" and the strategy pairs (L, l) and (R, r) are called "pure strategy equilibria." Nash Equilibrium in Mixed StrategiesSome games, such as RockPaperScissors, don't have a pure strategy equilibrium. In this game, if Player 1 chooses R, Player 2 should choose p, but if Player 2 chooses p, Player 1 should choose S. This continues with Player 2 choosing r in response to the choice S by Player 1, and so forth.
In games like RockPaperScissors, a player will want to randomize over several actions. If a player randomly chooses a pure strategy, we say that the player is using a "mixed strategy." In a pure strategy a player chooses an action for sure, whereas in a mixed strategy, he chooses a probability distribution over the set of actions available to him.  
