Doubly Strong Equilibrium

Scalzo, Vincenzo (2020): Doubly Strong Equilibrium.

Abstract

We present a new concept for (generalized) strategic form games, called \emph{doubly strong equilibrium}, and give an existence result when the players have non-ordered and discontinuous preferences. Since a doubly strong equilibrium is a strong equilibrium in the sense of Aumann, we get the existence of strong equilibria in discontinuous games. The result has been obtained by using the \emph{quasi-Ky Fan minimax inequality}. Applications to exchange economies are given. We prove the existence of \emph{doubly strong allocations}, which maximize consumers' preferences on the set of feasible allocations. The doubly strong allocations belong to the core of the economy. When consumers' preferences are selfish, we have that the doubly strong allocations are fair in the sense of Schmeidler and Yaari. So, we get the existence of fair allocations in the setting of non-ordered and discontinuous preferences.\

• Item Type: MPRA Paper
• Language: English
• Keywords: Generalized games.Discontinuous and non-ordered preferences. Doubly strong equilibrium. Quasi-Ky Fan minimax inequality. Exchange economies. Core allocations. Fair allocations.
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C70 - General; D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D51 - Exchange and Production Economies; D - Microeconomics > D6 - Welfare Economics > D63 - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

References

\bibitem{ali1} Aliprantis, C.D., Brown, D.J., Burkinshaw, O.: Existence and Optimality of Competitive Equilibria. Springer-Verlag, Berlin, Heidelberg (1989)

\bibitem{ali2} Aliprantis, C.D., Border, K.: Infinite Dimensional Analysis. Springer-Verlag, Berlin, Heidelberg, New York (1999)

\bibitem{aum1} Aumann, R.J.: Acceptable points in general cooperative n-person games, Contributions to the Theory of Games, vol. IV, Ann. of Math.\ Stud., vol. 40, Princeton University Press, Princeton, NJ (1959)

\bibitem{aum2} Aumann, R.J.: The core of a cooperative game without side payments. Transactions of the American Mathematical Society 98, 539-552 (1961)

\bibitem{bag1} Bagh, A.: Pure strategy equilibria in Bertrand games with discontinuous demand and asymmetric tie-breaking rules. Economics Letters 108 (3), 277-279 (2010)

\bibitem{bag2} Bagh, A., Jofre, A.: Reciprocal upper semicontinuity and better reply secure games: a comment. Econometrica 74, 1715-1721 (2006)

\bibitem{bm} Barelli, P., Meneghel, I.: A note on the equilibrium existence problem in discontinuous games. Econometrica 81, 813-824 (2013)

\bibitem{ba} Basile, A., Scalzo, V.: Non-emptiness of the alpha-core: sufficient and necessary conditions. mimeo (2019)

\bibitem{bay} Baye, M.R., Tian, G., Zhou, J.: Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Review of Economic Studies 60, 935-948 (1993)

\bibitem{ber} Bertrand, J.: Theorie mathematique de la richesse sociale. Journal des Savants, 499-508 (1883)

\bibitem{bor} Border, K.C.: A core existence theorem for games without ordered preferences. Econometrica 52 (6), 1537-1542 (1984)

\bibitem{car1} Carmona, G.: An existence result for discontinuous games. Journal of Economic Theory 144, 1333-1340 (2009)

\bibitem{car2} Carmona, G., Podczeck, K.: Existence of Nash equilibrium in games with a measure space of players and discontinuous payoff functions. Journal of Economic Theory 152 (1), 130-178 (2014)

\bibitem{car3} Carmona, G., Podczeck, K.: Existence of Nash equilibrium in ordinal games with discontinuous preferences. Economic Theory 61 (3), 457-478 (2016)

\bibitem{car4} Carmona, G., Podczeck, K.: The conditions in the existence results for discontinuous games by Reny and by Simon and Zame are incomparable. Games and Economic Behavior 111, 16-19 (2018)

\bibitem{cast} Castellani, M., Giuli, M., Pappalardo, M.: A Ky Fan minimax inequality for quasi-equilibria on finite-dimensional spaces. Journal of Optimization Theory and Applications 179, 53-64 (2018)

\bibitem{cors} Corson, H.H., Lindenstrauss, J.: Continuous selections with nonmetrizable range, Transactions of the American Mathematical Society 121, 492-504 (1966)

\bibitem{das} Dasgupta, P., Maskin, E.: The existence of equilibrium in discontinuous economic games, I: theory. Review of Economic Studies 53, 1-26 (1986)

\bibitem{fan} Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities, vol. 3. Academic Press, New York (1972)

\bibitem{fol} Foley, D.: Resource allocation and the public sector. Yale Economic Essays 7 (1967)

\bibitem{he} He, W., Yannelis, N.C.: Discontinuous games with asymmetric information: An extension of Reny's existence theorem. Games and Economic Behavior 91, 26-35 (2015)

\bibitem{he2} He, W., Yannelis, N.C.: Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Economic Theory 61 (3), 497-513 (2016)

\bibitem{hol} Holy, C.: An exchange economy can have an empty alpha-core. Economic Theory 4, 453-461 (1994)

\bibitem{hot} Hotelling, H.: The stability of competition. Economic Journal 39, 41-57 (1929)

\bibitem{ich} Ichiishi, T.: A social coalitional equilibrium existence lemma, Econometrica 49, 369-377 (1981)

\bibitem{kaj} Kajii, A.: A generalization of Scarf’s theorem: an $\alpha$-core existence theorem without transitivity or completeness. Journal of Economic Theory 56, 194-205 (1992)

\bibitem{mic1} Michael, E.: A note on paracompact spaces. Proceedings of the American Mathematical Society 4, 831-838 (1953)

\bibitem{mic2} Michael, E.: Continuous Selections I. Annals of Mathematics 63 (2), 361-382 (1956)

\bibitem{mcl} Mc Lennan, A., Monteiro, P.K., Tourky, R.: Games with discontinuous payoffs: a strengthening of Reny’s existence theorem. Econometrica 79, 1643-1664 (2011)

\bibitem{nes1} Nassah, R., Tian, G.: The existence of equilibria in discontinuous games. IESEG School of Management, mimeo (2008)

\bibitem{nes1} Nassah, R., Tian, G.: On the existence of strong Nash equilibria. Journal of Mathematical Analysis and Applications 414 (2), 871-885 (2014)

\bibitem{nes2} Nassah, R., Tian, G.: On the existence of Nash equilibrium in discontinuous games. Economic Theory 61 (3), 515-540 (2016)

\bibitem{pr1} Prokopovych, P.: The single deviation property in games with discontinuous payoffs. Economic Theory 53, 383-402 (2013)

\bibitem{pr2} Prokopovych, P.: Majorized correspondences and equilibrium existence in discontinuous games. Economic Theory 61 (3), 541-552 (2016)

\bibitem{ren1} Reny, P.J.: On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029-1056 (1999)

\bibitem{ren2} Reny, P.J.: Further results on the existence of Nash equilibria in discontinuous games. University of Chicago, Mimeo (2009)

\bibitem{ren3}Reny, P.J.: Strategic approximations of discontinuous games. Economic Theory 48 (1), 17-19 (2011)

\bibitem{ren5} Reny, P.J.: Nash equilibrium in discontinuous games, Economic Theory 61 (3), 553- 569 (2016)

\bibitem{sc1} Scalzo, V.: Essential equilibria of discontinuous games, Economic Theory 54 (1), 27-44 (2013)

\bibitem{sc2} Scalzo, V.: On the existence of maximal elements, fixed points and equilibria of generalized games in a fuzzy environment, Fuzzy Sets and Systems 272, 126-133 (2015)

\bibitem{sc3} Scalzo, V.: Equilibrium existence in games: slight single deviation property and Ky Fan minimax inequality, Journal of Mathematical Economics 82, 197-201 (2019a)

\bibitem{sc4} Scalzo, V.: Continuity properties of the Nash equilibrium correspondence in a discontinuous setting, Journal of Mathematical Analysis and Applications 473, 1270-1279 (2019b)

\bibitem{sc5} Scalzo, V.: Existence of strong equilibrium in discontinuous games. mimeo (2019c)

\bibitem{scr2} Scarf, H.E.: On the existence of a cooperative solution for a general class of n-person games. Journal of Econnomic Theory 3, 169-181 (1971)

\bibitem{schm} Schmeidler, D., M. Yaari, M.: Fair allocations. mimeo (1971)

\bibitem{sim} Simon, L.K., Zame, W.R.: Discontinuous games and endogenous sharing rules. Econometrica 58 (4), 861-872 (1990)

\bibitem{sve} Svensson, L.G.: On the existence of fair allocations. Zeitschrift f\"{u}r National Oekonomie 43, 301-308 (1983)

\bibitem{tho} Thomson, W.: Fair Allocation Rules. Rochester Center for Economic Research, Working Paper No.\ 539 (2007)

\bibitem{tia} Tian, G., Zhou, J.:Quasi-varitional inequalities with non-compact sets. Journal of Mathematical Analysis and Applications 160, 583-595 (1991)

\bibitem{uya} Uyanik, M.: On the nonemptiness of the α-core of discontinuous games: Transferable and nontransferable utilities, Journal of Econnomic Theory 158 , 213-231 (2015)

\bibitem{var} Varian, H.: Equity, envy, and efficiency. Journal of Economic Theory 9, 63-91 (1974)

\bibitem{nic} Yannelis, N.C.: The core of an economy without ordered preferences. BEBR Reprint Number 91-025, Springer-Verlag, 102-123 (1991)

\bibitem{zho} Zhou, J., Chen, G.: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. Journal of Mathematical Analysis and Applications 132, 213-225 (1988)