Affine minimax variational inequalities and matrix two-person games

Abstract

The concept of minimax variational inequality was proposed by Huy and Yen (Acta Math Vietnam 36, 265–281, 2011). This paper establishes some properties of monotone affine minimax variational inequalities and gives sufficient conditions for their solution stability. Then, by transforming a two-person zero sum game in matrix form (Barron in Game Theory. An Introduction, 2nd edn, Wiley, New Jersey, 2013) to a monotone affine minimax variational inequality, we prove that the saddle point set in mixed strategies of the matrix game is a nonempty compact polyhedral convex set and it is locally upper Lipschitz everywhere when the game matrix is perturbed. The rate of convergence of the extragradient method of Korpelevich applied to the matrix game is also discussed.