On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms.

Consider nonempty finite pure strategy sets S[subscript 1], . . . , S[subscript n], let S = S[subscript 1] times . . . times S[subscript n], let Omega be a finite space of "outcomes," let Delta(Omega) be the set of probability distributions on Omega, and let theta: S approaches Delta(Omega) be a function. We study the conjecture that for any utility in a generic set of n-tuples of utilities on Omega there are finitely many distributions on Omega induced by the Nash equilibria of the game given by the induced utilities on S. We give a counterexample refuting the conjecture for n >= 3. Several special cases of the conjecture follow from well-known theorems, and we provide some generalizations of these results.