Weak maximal elements and weak equilibria in ordinal games with applications to exchange economies

We study binary relations (preferences) and ordinal games when no continuity properties and its generalizations are assumed at all. We introduce weakening of the maximal element and the Nash equilibrium, called, respectively, the emph{weak maximal element} and the emph{weak equilibrium}, and prove the existence when the binary relations satisfy only convexity conditions. The weak maximal element (the weak equilibrium) is equivalent to the maximal element (the Nash equilibrium) if and only if a suitable generalization of the continuity is given. Moreover, we obtain the existence of emph{quasi-Pareto optimal} allocations in exchange economies.