Nash Equilibrium

A Nash Equilibrium is a Strategy Profile such that no player can improve their utility by unilaterally deviating. Every finite game has at least one mixed strategy Nash equilibrium. This is the most fundamental concept in Game Theory, and yields a “consistent prediction” for what players will do.

Formal Definition
A strategy profile is a Nash Equilibrium if:

where:

• is the Strategy Set for player ,
• is the Strategy Profile of all players except ,
• is the utility Function for player .

For examples, check out Nash Equilibrium Examples

Properties

• For a Mixed Strategy Nash, every pure action that is played with positive probability in (i.e., every action in the support of ) and be Individually Rational for all players.