# Nash Equilibria: Where We Stand

## Abstract

In the Fall of 2005 it was shown that finding an *ε*-approximate mixed Nash equilibrium in a normal-form game, even with two players, is PPAD-complete for small enough (additive) *ε* — and hence, presumably, an intractable problem. This solved a long-standing 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 on-going 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 pseudopolynomial-time approximation scheme for general *n*-player 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 two-player 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.