On the existence of approximate equilibria and sharing rule solutions in discontinuous games

This paper studies the existence of equilibrium solution concepts in a large class of economic models with discontinuous payoff functions. The issue is well understood for Nash equilibria, thanks to Reny’s better-reply security con- dition ( Reny 1999) and its recent improvements (Barelli and Meneghel 2013, McLennan et al. 2011, Reny 2009, 2011). We propose new approaches, related to Reny’s work, and obtain tight conditions for the existence of approximate equi- libria and of sharing rule solutions in pure and mixed strategies ( Simon and Zame 1990 ). As byproducts, we prove that many auction games with correlated types admit an approximate equilibrium, and that many competition models have a sharing rule solution.