Variational Inequalities in Cournot Oligopoly

Abstract

Consider G = (X1,…,XM,g1,…,gM) an M-player game in strategic form, where the set Xi is an interval of real numbers and the payoff functions gi are differentiable with respect to the related variable xi ∈ Xi. If they are also concave, with respect to the related variable, then it is possible to associate to the game G a variational inequality which characterizes its Nash equilibrium points. In this paper it is considered the variational inequality for two sets of Cournot oligopoly games. In the first case, for any i = 1,…,M, we have Xi = [0,+∞); the market price function is in C1 and convex; the cost production function of the player i is linear and the function xi → gi(…,xi,…) is strictly concave. We prove the existence and uniqueness of the Nash equilibrium point and illustrate, with an example, an algorithm which calculates its components. In the second case, for any i = 1,…,M, we have Xi = [0,+∞); the market price function is in C2 and concave and the cost production function of the i-player is in C2 and convex. In these circumstances, as a consequence of well known facts, the existence and uniqueness of the Nash equilibrium point are guaranteed and also the Tykhonov and Hadamard well-posedness of the game. We prove that the game G is well posed with respect to its variational inequality.