Weak maximal elements and weak equilibria in ordinal games with applications to exchange economies
Tóm tắt
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.
Tài liệu tham khảo
Aliprantis, C.D., Brown, D.J., Burkinshaw, O.: Existence and optimality of competitive equilibria. Springer-Verlag, Berlin (1990)
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)
Bich, P., Laraki, R.: On the existence of approximate equilibria and sharing rule solutions in discontinuous games, Mimeo (2012)
Carmona, G., Podczeck, K.: Existence of Nash equilibrium in ordinal games with discontinuous preferences. Econ. Theory 61(3), 457–478 (2016)
Corson, H.H., Lindenstrauss, J.: Continuous selections with nonmetrizable range. Trans. Am. Math. Soc. 121, 492–504 (1966)
He, W., Yannelis, N.C.: Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Econ. Theory 61(3), 497–513 (2016)
Prokopovych, P.: The single deviation property in games with discontinuous payoffs. Econ. Theory 53, 383–402 (2013)
Prokopovych, P.: Majorized correspondences and equilibrium existence in discontinuous games. Econ. Theory 61(3), 541–552 (2016)
Reny, P.: On the existence of pure and mixed strategy equilibria in discontinuous games. Econometrica 67, 1029–1056 (1999)
Reny, P.: Further results on the existence of Nash equilibria in discontinuous games. University of Chicago, Mimeo (2009)
Reny, P.: Nash equilibrium in discontinuous games. Econ. Theory 61(3), 553–569 (2016)
Scalzo, V.: On the existence of maximal elements, fixed points and equilibria of generalized games in a fuzzy environment. Fuzzy Sets Syst. 272, 126–133 (2015)
Scalzo, V.: Remarks on the existence and stability of some relaxed Nash equilibrium in strategic form games. Econ. Theory 61(3), 571–586 (2016)
Shafer, W., Sonnenschein, H.: Equilibrium in abstract economies without ordered preferences. J. Math. Econ. 2, 345–348 (1975)
Tarafdar, E.: On nonlinear variational inequalities. Proc. Am. Math. Soc. 67, 95–98 (1977)
Wu, X., Shen, S.: A further generalization of Yannelis-Prabhakar’s continuous selection theorem and its applications. J. Math. Anal. Appl. 197(1), 61–74 (1996)
Yannelis, N.C., Prabhakar, N.: Existence of maximal elements and equilibria in linear topological spaces. J. Math. Econ. 12, 233–245 (1983)