Uniqueness of Equilibria
In examining the existence conditions for equilibria in N-person games either the Banach or the Kakutani fixed point theorem was used. The Banach fixed point theorem guaranteed the existence of the unique equilibrium, and an iteration algorithm was also suggested to compute the equilibrium. However, the existence theorems based on the Kakutani fixed point theorem (Theorems 5.3 and 5.4) do not guarantee uniqueness. For example, by selecting constant payoff functions all strategies provide equilibria, and constant functions are continuous as well as concave. So the conditions of the Nikaido–Isoda theorem are satisfied if the strategy sets are nonempty, convex, closed, and bounded. It is well known from optimization theory that strictly concave functions cannot have multiple maximum points. Unfortunately for N-person games this result cannot be extended, since a simple example shows that there is the possibility of even infinitely many equilibria if all payoff functions are strictly concave.