Andrew McLennan,  School of Economics Discussion Paper No. 526 July 2014, School of Economics, The University of Queensland.

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The effect of perturbing a parameter—comparative statics—is, of course, a familiar and important issue in economic analysis. Perfection of a single Nash equilibrium (Selten (1975)) is defined by requiring that at least some perturbations in a given class give rise to perturbed games that have nearby equilibria. Roughly, Kohlberg and Mertens (1986) define strategic stability of a set of equilibria by requiring that for all sufficiently small perturbations, the perturbed games have equilibria near the set. This note presents a topological result concerning the behavior of such nearby equilibria when there is a function from a neighborhood of the relevant set of equilibria to the space of perturbations.