The discussion of strategies and solution concepts in Chapter 3 largely ignored issues of computation. We start by asking the most basic question: How hard is it to compute the Nash equilibria of a game? The answer turns out to be quite subtle, and to depend on the class of games being considered.

We have already seen how to compute the Nash equilibria of simple games. These calculations were deceptively easy, partly because there were only two players and partly because each player had only two actions. In this chapter we discuss several different classes of games, starting with the simple two-player, zero-sum normal-form game. Dropping only the zero-sum restriction yields a problem of different complexity—while it is generally believed that any algorithm that guarantees a solution must have an exponential worst case complexity, it is also believed that a proof to this effect may not emerge for some time. We also consider procedures for n-player games. In each case, we describe how to formulate the problem, the algorithm (or algorithms) commonly used to solve them, and the complexity of the problem. While we focus on the problem of finding a sample Nash equilibrium, we will briefly discuss the problem of finding all Nash equilibria and finding equilibria with specific properties. Along the way we also discuss the computation of other game-theoretic solution concepts: maxmin and minmax strategies, strategies that survive iterated removal of dominated strategies, and correlated equilibria.