Nash Equilibrium in Discontinuous Games: A Weakening of Reny’s Robust Better-Reply Correspondence Property

This paper weakens the notion of robust better-reply correspondence property introduced in [Reny, P. J. [2020] Nash equilibrium in discontinuous games, Annu. Rev. Econ. 12, 439–470] to prove the existence of pure strategy Nash equilibrium in compact convex discontinuous and possibly nonquasiconcave games. Our weakening of this property is satisfied by a large class of these games and our equilibrium existence results strictly generalize the most important ones in the literature, namely those obtained in [Reny, P. J. [2020] Nash equilibrium in discontinuous games, Annu. Rev. Econ. 12, 439–470; Reny, P. [2016b] Introduction to the symposium on discontinuous games, Econ. Theory 61, 423–429; Carmona, G. and Podczeck, K. [2016] Existence of Nash equilibrium in ordinal games with discontinuous preferences, Econ. Theory 61, 457–478] (in a special case), [Reny, P. [1999] On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056; McLennan, A., Monteiro, P. K. and Tourky, R. [2011] Games with discontinuous payoffs: A strengthening of Reny’s existence theorem, Econometrica 79, 1643–1664; Barelli, P. and Meneghel, I. [2013] A note on the equilibrium existence problem in discontinuous games, Econometrica 81, 813–824; Nessah, R. [2011] Generalized weak transfer continuity and Nash equilibrium, J. Math. Econ. 47, 659–662; Nessah, R. and Tian, G. [2016] On the existence of Nash equilibrium in discontinuous games, Econ. Theory 61, 515–540].