C. A. PENSAVALLE () (Department of Mathematics and Physics, University of Sassari, Via Vienna 2, 07100 Sassari, Italy, IT, Italy) G. PIERI () (D.S.A., University of Genoa, Stradone S. Agostino 37, 16136 Genova, Italy, IT, Italy)
Abstract
Consider G = (X1,â¦,XM,g1,â¦,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 â 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,â¦,M, we have Xi = [0,+â); the market price function is in C1 and convex; the cost production function of the player i is linear and the function xi â gi(â¦,xi,â¦) 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,â¦,M, we have Xi = [0,+â); 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.
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