Sofronidis, Nikolaos Efstathiou Mathematical economics and descriptive set theory. (English) Zbl 1098.91515 J. Math. Anal. Appl. 264, No. 1, 182-205 (2001). Summary: The purpose of this paper is first to show that for any integer \(n\geq 2\), there exist no Borel measurable necessary and sufficient conditions on \(n\) lower semi-continuous payoff functions defining a non-cooperative \(n\)-person game in strategic form which can assert the existence of non-cooperative Nash equilibria (in either pure or mixed strategies). Second we show that there exist no Borel measurable necessary and sufficient conditions on an exchange economy with preference relations that are represented by lower semi-continuous utility functions which can assert the existence of Walrasian equilibria. And third we show that there exist no Borel measurable necessary and sufficient conditions on a triple of a lower semi-continuous one-period return function, a compact-valued continuous constraint correspondence, and a discount factor defining a deterministic discrete infinite horizon macroeconomic model which can assert the existence of optimal plans starting at some point. Cited in 1 Document MSC: 91A44 Games involving topology, set theory, or logic 91A10 Noncooperative games 91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general) × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis: A Hitchhiker’s Guide (1999), Springer-Verlag: Springer-Verlag Berlin · Zbl 0938.46001 [2] Aliprantis, C. D.; Burkinshaw, O., Principles of Real Analysis (1998), Academic Press: Academic Press San Diego · Zbl 0436.46009 [3] Aliprantis, C. D.; Chakrabarti, S. K., Games and Decision Making (2000), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0949.91008 [4] Aliprantis, C. D.; Brown, D. J.; Burkinshaw, O., Existence and Optimality of Competitive Equilibria (1990), Springer-Verlag: Springer-Verlag Berlin [5] Dal Maso, G., An Introduction to Γ-convergence, Progr. Nonlinear Differential Equations Appl., 8 (1993) · Zbl 0816.49001 [6] Dasgupta, P.; Maskin, E., The existence of equilibrium in discontinuous economic games. I. Theory, Rev. Econom. Stud., 53 (1986) · Zbl 0697.90092 [7] Dasgupta, P.; Maskin, E., The existence of equilibrium in discontinuous economic games. II. Applications, Rev. Econom. Stud., 53, 27-41 (1986) · Zbl 0578.90099 [8] De Giorgi, E., \(G\)-operators and Γ-convergence, Proceedings of the International Congress of Mathematicians (1982) [9] Jost, J.; Li-Jost, X., Calculus of Variations (1998), Cambridge Univ. Press · Zbl 0913.49001 [10] Kechris, A. S., Classical Descriptive Set Theory (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.04002 [11] Nash, J. F., Non-cooperative games, Ann. Math., 54, 286-295 (1951) · Zbl 0045.08202 [12] Osborne, M. J.; Rubinstein, A., A Course in Game Theory (1998), MIT Press: MIT Press Cambridge [13] Rath, K. P., Existence and upper hemicontinuity of equilibrium distributions of anonymous games with discontinuous payoffs, J. Math. Econom., 26, 305-324 (1996) · Zbl 0871.90131 [14] Reny, P. J., On the existence of pure mixed strategy Nash equilibria in discontinuous games, Econometrica, 67, 1029-1056 (1999) · Zbl 1023.91501 [15] Royden, H. L., Real Analysis (1988), Macmillan: Macmillan New York · Zbl 0704.26006 [16] N. E. Sofronidis, Topics in Descriptive Set Theory Related to Equivalence Relations, Complex Borel, and Analytic Sets, Ph.D. Thesis, California Institute of Technology, 1999.; N. E. Sofronidis, Topics in Descriptive Set Theory Related to Equivalence Relations, Complex Borel, and Analytic Sets, Ph.D. Thesis, California Institute of Technology, 1999. [17] Sofronidis, N. E., Analytic non-Borel sets and vertices of differentiable curves in the plane, Real Anal. Exchange, 26, 735-748 (2000/2001) · Zbl 1019.51013 [18] Stokey, N. L.; Lucas, R. E.; Prescott, E. C., Recursive Methods in Economic Dynamics (1989), Harvard Univ. Press: Harvard Univ. Press Cambridge · Zbl 0774.90018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.