Approximations of Nash equilibria

摘要

Inspired by previous works on approximations of optimization problems and recent papers on the approximation of Walrasian and Nash equilibria and on stochastic variational inequalities, the present paper investigates the approximation of Nash equilibria and clarifies the conditions required for the convergence of the approximate equilibria via a direct approach, a variational approach, and an optimization approach. Besides directly addressing the issue of convergence of Nash equilibria via approximation, our investigation leads to a deeper understanding of various notions of functional convergence and their interconnections; more importantly, the investigation yields improved conditions for convergence of the approximate Nash equilibria via the variational approach. An illustrative application of our results to the approximation of a Nash equilibrium in a competitive capacity expansion model under uncertainty is presented. Keywords Variational inequalities - Stochastic - Equilibrium models - Epiconvergence - Competitive capacity expansion under uncertainty Mathematics Subject Classification (2000) 90C15 - 90C31 - 90C33 - 91A06 - 91A10 - 91A15 It is with great pleasure that we dedicate this paper to our esteemed colleague and mentor, Professor Steve Robinson, on the occasion of his 65th birthday. Steve’s many seminal works have deeply influenced our research; the present paper is an example of his influence.