On existence of equilibrium pair for constrained generalized games
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We obtain sufficient conditions for the existence of an equilibrium pair for a particular constrained generalized game as an application of a best proximity pair theorem. 1. Introduction Consider the following game involving n players. For the ith playera pair (Xi ,Yi ) of strategy sets is associated. Knowing the choice of strategies xi ∈ X i = nj=1, j =i X j of all other players, the ith-player choice is restricted to Ai (xi ) ⊆ Yi . Otherwise the choice will be made from Xi . According to these preferences, let fi : Yi × X i → R be the payoff function associated with the ith player for each i = 1,...,n. In this situation, it is natural to expect an optimal approximate solution which will fulfill the requirement to some exn tent. Therefore, it should be contemplated to find a pair (x, y) where x ∈ X = i=1 Xi and y ∈ Y = ni=1 Yi which will behave like an equilibrium point of a generalized game, that is, yi ∈ Ai (xi ) and maxz∈Ai (xi ) fi (z,xi ) = fi (yi ,xi ) for each i = 1,... ,n, and satisfy the optimization constraint, namely, the distance between x and y is minimum with respect to X and Y . In this case, the pair (x, y) is called an equilibrium pair and the game is termed as constrained generalized game. Indeed, in this paper, sufficient conditions for the existence of an equilibrium pair for this constrained generalized game are obtained as an application of a best proximity pair theorem. The entire edifice of game theory expounds with a mathematical search to strike an optimal balance between persons generally having conflicting interests. Each player has to select one from his fixed range of strategies so as to bring the best outcome according to his own preferences. Following the pioneering work of Debreu [1], the generalized game is one in which the choice of each player is restricted to a subset of strategies determined by the choice of other players. Mathematically, the situation is described as follows. compact convex sets in a normed Let there be n players. Let X1 ,...,Xn be nonempty linear space F. Let Xi be the strategy set and let fi : X = ni=1 Xi → R be the payoff function for the ith player, for each i = 1,...,n. Given the strategies xi of all other players, the choice Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:1 (2004) 21–29 2000 Mathematics Subject Classification: 47H10, 47H04, 54H25 URL: http://dx.doi.org/10.1155/S1687182004308132
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On best proximity pairs
of the ith player is restricted to the set Ai (xi ) ⊆ Xi . An equilibrium point in a generalized game is an element x ∈ X such that for each i = 1,...,n, xi ∈ Ai (xi ) and
maxi fi y,xi = fi xi ,xi = fi (x),
(1.1)
y ∈Ai (x )
where the following convenient notations are used. Notation 1.1. Denote X=
n
Xi =
Xi ,
i=1
n
Xj.
(1.2)
j =1 j =i
A point x of X whose ith coordinate is xi and xi ∈ X i is written as (xi ,xi ). The above definition of the equilibrium point is a natural extension of the Nash equilibrium point introduced by Nash in [6]. Since then a number of ge
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