Algorithms for Approximations of Nash Equilibrium ( 2003 ; Lipton , Markakis , Mehta , 2006

Nash [13] introduced the concept of Nash equilibria in non-cooperative games and proved that any game possesses at least one such equilibrium. A well-known algorithm for computing a Nash equilibrium of a 2-player game is the Lemke-Howson algorithm [11], however it has exponential worst-case running time in the number of available pure strategies [15]. Recently, Daskalakis et al [4] showed that the problem of computing a Nash equilibrium in a game with 4 or more players is PPAD-complete; this result was later extended to games with 3 players [7]. Eventually, Chen and Deng [2] proved that the problem is PPAD-complete for 2-player games as well. This fact emerged the computation of approximate Nash equilibria. There are several versions of approximate Nash equilibria that have been defined in the literature; however the focus of this entry is on the notions of -Nash equilibrium and -well-supported Nash equilibrium. An -Nash equilibrium is a strategy profile such that no deviating player could achieve a payoff higher than the one that the specific profile gives her, plus . A stronger notion of approximate Nash equilibria is the -well-supported Nash equilibria; these are strategy profiles such that each player plays only approximately best-response pure strategies with non-zero probability.