Fixed Points of Parameterized Perturbations

Let X be a convex subset of a locally convex topological vector space, let U ⊂ X be open with U compact, let F : U → X be an upper semicontinuous convex valued correspondence with no fixed points in U \U , let P be a compact absolute neighborhood retract, and let ρ : U → P be a continuous function. We show that if the fixed point index of F is not zero, then there is a neighborhood V of F in the (suitably topologized) space of upper semicontinuous convex valued correspondences from U to X such that for any continuous function g : P → V there is a p ∈ P and a fixed point x of g(p) such that ρ(x) = p. This implies that no normal form game satisfies the conditions specified in Section 4.6 of Levy (2013). Running Title: Parameterized Perturbations