We study binary relations (preferences) and ordinal games in the case where no continuity-like properties are assumed at all. We introduce generalizations of the maximal element and Nash equilibrium, called, respectively, the weak maximal element and weak equilibrium, and give existence results when binary relations satisfy only convexity conditions. The weak maximal element (weak equilibrium) is equivalent to the maximal element (Nash equilibrium) if and only if a generalization of continuity is given. Moreover, we obtain the existence of quasi-Pareto optimal allocations in exchange economies.
Binary relations Preference relations Weak maximal element Ordinal games Weak equilibrium Exchange economies Quasi-Pareto optimality
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Aliprantis, C.D., Brown, D.J., Burkinshaw, O.: Existence and optimality of competitive equilibria. Springer-Verlag, Berlin (1990)CrossRefGoogle Scholar
Baye, M.R., Tian, G., Zhou, J.: Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Rev. Econ. Stud. 60, 935–948 (1993)CrossRefGoogle Scholar
Bich, P., Laraki, R.: On the existence of approximate equilibria and sharing rule solutions in discontinuous games, Mimeo (2012)Google Scholar
Carmona, G., Podczeck, K.: Existence of Nash equilibrium in ordinal games with discontinuous preferences. Econ. Theory 61(3), 457–478 (2016)CrossRefGoogle Scholar