O’Neill’s Theorem For Games

We present the following analog of O’Neill’s Theorem (Theorem 5.2 in [17]) for finite games. Let C1, . . . , Ck be the components of Nash equilibria of a finite normal-form game G. For each i, let ci be the index of Ci. For each ε > 0, there exist pairwise disjoint neighborhoods V1, ..., Vk of the components such that for any choice of finitely many distinct completely mixed strategy profiles {σij}ij, σij ∈ Vi for each i = 1, . . . , k and numbers rij ∈ {−1, 1} such that j rij = ci, there exists a normal-form game G¯ obtained from G by adding duplicate strategies and an ε-perturbation G¯ε of G¯ such that the set of equilibria of G¯ε is {σ¯ij}ij , where for each i, j:(1) σ¯ij is equivalent to the profile σij; (2) the index σ¯ij equals rij.