Abstract
A constrained weak Nash-type equilibrium problem with multivalued payoff functions is introduced. By virtue of a nonlinear scalarization function, some existence results are established. The results extend the corresponding one of Fu (2003). In particular, if the payoff functions are singlevalued, our existence theorem extends the main results of Fu (2003) by relaxing the assumption of convexity.
Highlights
For a long time, real valued functions have played a central role in game theory
As Si(⋅) is upper semicontinuous and the set Si(x0) is compact, it follows that ui,0 ∈ Si(x0)
As Si(⋅) is lower semicontinuous, there exists wi,n ∈ Si(xn), such that wi,n → wi,0. It follows from compactness of Fi(wi,n, xni ) that there exists zi,n ∈ Fi(wi,n, xni ) such that ξei = max ξei (Fi)
Summary
Real valued functions have played a central role in game theory. More recently, motivated by applications to real-world situations, many authors have studied the existence of solutions of Pareto equilibria of multiobjective game with vector payoff functions; for example, see [1,2,3,4] and the references therein. Let E be a nonempty convex subset of X, let H : E → 2Z be a set-valued mapping, and let P ⊂ Z be a closed convex and pointed cone with int P ≠ 0. Let Xi be a nonempty, compact convex subset of Zi, respectively.
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