Nash Equilibrium (Pure and Mixed)

Abstract This article defines and examines the Nash equilibrium solution concept for noncooperative strategic form games. Examples of canonical (2 × 2) bimatrix form games are used to demonstrate that a noncooperative strategic form game may have a unique pure or mixed strategy Nash equilibrium (Prisoners' Dilemma and Matching Pennies, respectively), may have both pure and mixed strategy Nash equilibria (Battle of the Sexes), or an infinite number of Nash equilibria. Nash's application of the Kakutani fixed point theorem to prove the existence of equilibria is discussed and applied to specific games. Motivated by the possible multiplicity of Nash equilibria in noncooperative strategic form games, the number of equilibria, the construction of games with a unique, prespecified equilibrium, and the calculation of equilibria are considered. The article concludes with a short description of the attempts to refine the Nash equilibrium concept, including perfect and proper equilibria, as well as evolutionary stability.