O’Neill’s Theorem for Games
将奥尼尔定理推广到有限博弈，证明可通过添加重复策略和扰动，使新博弈的均衡集与给定指标和完全混合策略配置对应。
We present the following analog of O’Neill’s theorem (O’Neill B (1953) Essential sets and fixed points. Amer. J. Math. 75(3):497–509 (theorem 5.2)) for finite games. Let [Formula: see text] be the components of Nash equilibria of a finite normal-form game G. For each i, let [Formula: see text] be the index of [Formula: see text]. For each [Formula: see text], there exist pairwise disjoint neighborhoods [Formula: see text] of the components such that for any choice of finitely many distinct completely mixed strategy profiles [Formula: see text] for each [Formula: see text] and numbers [Formula: see text] such that [Formula: see text], there exists a normal-form game [Formula: see text] obtained from G by adding duplicate strategies and an [Formula: see text]-perturbation [Formula: see text] of [Formula: see text] such that the set of equilibria of [Formula: see text] is [Formula: see text], where for each i, j, (1) [Formula: see text] is equivalent to the profile [Formula: see text] and (2) the index [Formula: see text] equals [Formula: see text]. Funding: L. Pahl acknowledges financial support from the Hausdorff Center for Mathematics [Deutsche Forschungsgemeinschaft Project 390685813].