Establishing the existence of equilibria is different from finding them. Section 5.3's proof of the existence of equilibria assumes the existence of the incentives and responses that determine equilibria but does not rest on a specification of them. Hence it does not show how to spot equilibria. It establishes the existence of equilibria in ideal normal-form games but does not provide a way of identifying them. This chapter formulates a procedure for finding equilibria in ideal normal-form games.

Let us begin with a few points about ideal games, our search procedure's subject. Although we have defined equilibrium in terms of incentives rather than payoff increases, in ideal games incentives follow payoff increases. Wherever an agent has an opportunity to increase his payoff, he has an incentive to switch strategy, and vice versa. This greatly facilitates identification of incentives and therefore equilibria. Also, in an ideal game agents are jointly rational if possible. Now that we have shown the existence of equilibria in ideal normal-form games, we also assume the existence of solutions. Therefore in the ideal games for which our search procedure is designed, we take the idealization about rationality to entail that agents are jointly rational. We also assume that their joint rationality carries over to their pursuit of incentives, which as a result obeys rules presented in Sections 6.2 and 6.3.

Our procedure for identifying equilibria applies to a restricted class of ideal normal-form games. The restrictions, related to the structure of incentives, are relatively mild. They are presented in the course of the chapter.