Consider nonempty finite pure strategy sets ,…,, let =×⋅⋅⋅×, let Ω be a finite space of “outcomes,” let Δ(Ω) be the set of probability distributions on Ω, and let θ: →Δ(Ω) be a function. We study the conjecture that for any utility in a generic set of ‐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 . We give a counterexample refuting the conjecture for ≥3. Several special cases of the conjecture follow from well known theorems, and we provide some generalizations of these results.