Variational Inequalities In Cournot Oligopoly

ConsiderG = (X1,…,XM,g1,…,gM)anM-player game in strategic form, where the setXiis an interval of real numbers and the payoff functionsgiare differentiable with respect to the related variablexi∈ Xi. If they are also concave, with respect to the related variable, then it is possible to associate to the gameGa 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 anyi = 1,…,M, we haveXi= [0,+∞); the market price function is inC1and convex; the cost production function of the playeriis linear and the functionxi→ 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 anyi = 1,…,M, we haveXi= [0,+∞); the market price function is inC2and concave and the cost production function of thei-player is inC2and 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 gameGis well posed with respect to its variational inequality.