Recent Developments in Equilibria Algorithms

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Abstract

Nash proved in 1951 that every game has a mixed Nash equilibrium [6]; whether such an equilibrium can be found in polynomial time has been open since that time. We review here certain recent results which shed some light to this old problem.

Even though a mixed Nash equilibrium is a continuous object, the problem is essentially combinatorial, since it suffices to identify the support of a mixed strategy for each player; however, no algorithm better than exponential for doing so is known. For the case of two players we have a simplex-like pivoting algorithm due to Lemke and Howson that is guaranteed to converge to a Nash equilibrium; this algorithm has an exponential worst case [9]. But even such algorithms seem unlikely for three or more players: in his original paper Nash supplied an example of a 3-player game, an abstraction of poker, with only irrational Nash equilibria.

Research supported by an NSF ITR grant, and by a grant from Microsoft Research.