Overdetermined Systems of Linear Equations
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the decisions (Xj)j¢ i of the other players. In this framework one t h e n has:
An oligopoly consists of a finite number (usually few) firms involved in the production of a good. Oligopolies are a basic economic market structure, with examples ranging from large firms in automobile, computer, chemical, or mineral extraction industries to small firms with local markets• Oligopolies are examples of imperfect competition in that the producers or firms are sufficiently large t h a t they affect the prices of the goods• A monopoly, on the other hand, consists of a single firm which has full control of the market• Oligopoly theory dates to A. Cournot [1], who considered competition between two producers, the so-called duopoly problem, and is credited with being the first to study noncooperative behavior, in which the agents act in their own self-interest. In his study, the decisions made by the producers or firms are said to be in equilibrium if no producer can increase his income by unilateral action, given that the other producer does not alter his decision.
where F ( x ) = ( - V x l U l ( X ) , . . . , - V x m U m ( X ) ) , and is assumed to be a row vector, where Vx, ui(x) denotes the gradient of ui with respect to xi. []
J.F. Nasa [18], [19] generalized Cournot's concept of an equilibrium for a behavioral model consisting of several agents or players, each acting in his own self-interest, which has come to be called a noncooperative game. Specifically, consider m players, each player i having at his disposal a strategy vector xi = { X i l , . . . , xin} selected from a closed, convex set K i C R n, with a utility function ui: K --+ R 1, where K = K 1 x ... x K m C R rnn. T h e rationality postulate is t h a t each player i selects a strategy vector xi E K i that maximizes his utility level u i ( x l , . . . , xi-1, xi, x i + l , . . . , xm) given
If the feasible set K is compact, then existence is guaranteed under the assumption that each utility function ui is continuously differentiable. J.B. Rosen [22] proved existence under similar conditions. S. K a r a m a r d i a n [12] relaxed the assumption of compactness of K and provided a proof of existence and uniqueness of Nasa equilibria under the strong monotonicity condition• As shown by D. Gabay and H. Moulin [6], the imposition of a coercivity condition on F ( x ) will guarantee existence of a Nasa equilibrium x* even if the feasible set is no longer compact. Moreover, if F ( x ) satisfies
DEFINITION 1 (Nasa equilibrium) A Nash equilibrium is a strategy vector x* = ( x ~ , . . . , X m ) C K, such that > A
e K
S
where z i = (x~,
•
, Xi_l,Xi+l,..., * * z*).
. .
vi,
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It has been shown (cf. [6], [10]) that Nasa equilibria satisfy variational inequalities• In the present context, under the a s s u m p t i o n that each ui is continuously differentiable on K and concave with respect to xi, one has THEOREM 2 (variational inequality formulation) Under the previous assumptions, x* is a Nasa equilibrium if and only if x* E K is a solution of the variational
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