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).