A survey on computation methods for Nash equilibrium

Abstract

A dominant solution concept of non-cooperative game theory is the concept of Nash equilibrium. A Nash equilibrium is a strategy profile from where unilateral deviations do not pay. A nice property of this concept is the well known fact that every finite game has at least one Nash equilibrium. The proof given by Nash (1950) is based on Brouwer fixed point theorem which is very non-constructive. A natural question to ask is whether Nash equilibrium can be computed efficiently. This is still unknown in terms of complexity. Very recently (Daskalakis et al., 2006), it has been shown that the computation of Nash equilibrium is PPAD-complete; which is a new complexity class introduced by Papadimitriou (1994) mainly to capture the complexity of Nash equilibrium. This note tries to give a brief survey of the various algorithms that exist in the literature for computing the Nash equilibrium in finite games. Also, it will briefly touch upon the various existential conditions for Nash equilibrium for infinite games.