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On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms

Srihari Govindan
Andrew McLennan

Abstract

Consider nonempty finite pure strategy sets S1,…,Sn, let S=S1×⋅⋅⋅×Sn, let Ω be a finite space of “outcomes,” let ?(Ω) be the set of probability distributions on Ω, and let θ: S→?(Ω) be a function. We study the conjecture that for any utility in a generic set of n-tuples of utilities on Ω there are finitely many distributions on Ω 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.