Existence of Equilibria for Discontinuous Games in General Topological Spaces with Binary Relations

We provide several results on the existence of equilibria for discontinuous games in general topological spaces without any convexity structure. All of the theorems yielding existence of equilibria here are stated in terms of the player’s preference relations over joint strategies.

A New Fixed Point Theorem and Applications

A new fixed point theorem is established under the setting of a generalized finitely continuous topological space (GFC-space) without the convexity structure. As applications, a weak KKM theorem and a minimax inequalities of Ky Fan type are also obtained under suitable conditions. Our results are different from known results in the literature.

An Internal Solution to the Problem of Linearization of a Convexity Space

Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.