Nash Equilibria: Where We Stand
 Christos H. Papadimitriou
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Abstract
In the Fall of 2005 it was shown that finding an εapproximate mixed Nash equilibrium in a normalform game, even with two players, is PPADcomplete for small enough (additive) ε — and hence, presumably, an intractable problem. This solved a longstanding open problem in Algorithmic Game Theory, but created many open questions. For example, it is known that inverse polynomial ε is enough to make the problem intractable, while, for two player games, relatively simple polynomial algorithms are known to achieve ε near \(1\over 3\); bridging this gap is an important open problem.
When the number of strategies per player is small, a different set of algorithmic techniques comes into play; it had been known, for example, that symmetric games of this sort can be solved in polynomial time, via a reduction to the existential theory of the reals. In ongoing joint work with Costis Daskalakis we have shown that a simple exhaustive approach works in a broader, and more useful in practice, class of games known as anonymous games, in which the payoff of each player and strategy is a symmetric function of the strategies chosen by the other players; that is, a player’s utility depends on how many other players have chosen each of the strategies, and not on precisely which players have. In fact, a variant of the same algorithmic technique gives a pseudopolynomialtime approximation scheme for general nplayer games, as long as the number of strategies is kept a constant. Improving this to polynomial seems a challenging problem.
A third important front in this research project is exploring equilibrium concepts that are more attractive computationally than the mixed Nash equilibrium, and possibly more natural, yet no less universal (guaranteed to exist under quite general assumptions). A number of such alternatives have been explored recently, some of them in joint work with Alex Fabrikant. For example, we show that twoplayer games with random entries of the utility matrices are likely to have a natural generalization of a pure Nash equilibrium called unit recall equilibrium.
Finally, it had long been believed that Nash equilibria of repeated games are much easier to find, due to a cluster of results known in Game Theory as the Folk Theorem. We shall discuss how recent algorithmic insights cast doubt even to this reassuring fact.
 Title
 Nash Equilibria: Where We Stand
 Book Title
 Algorithms – ESA 2007
 Book Subtitle
 15th Annual European Symposium, Eilat, Israel, October 810, 2007. Proceedings
 Pages
 p 1
 Copyright
 2007
 DOI
 10.1007/9783540755203_1
 Print ISBN
 9783540755197
 Online ISBN
 9783540755203
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 4698
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors
 Authors

 Christos H. Papadimitriou ^{(1)}
 Author Affiliations

 1. Computer Science Division, UC Berkeley,
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