×

On the existence of pure strategy equilibria in games with a continuum of players. (English) Zbl 0883.90137

Summary: We present results on the existence of pure strategy Nash equilibria in nonatomic games. We also show by means of counterexamples that the stringent conditions on the cardinality of action sets cannot be relaxed, and thus resolve questions which have remained open since D. Schmeidler’s paper [J. Stat. Phys. 7, No. 4, 295–300 (1973; Zbl 1255.91031)].

MSC:

91A07 Games with infinitely many players

Citations:

Zbl 1255.91031
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Arrow, K. J.; Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290 (1954) · Zbl 0055.38007
[2] Aumann, R. J., Integrals of set valued functions, J. Math. Anal. Appl., 12, 1-12 (1965) · Zbl 0163.06301
[3] Aumann, R. J., An elementary proof that integration preserves upper semicontinuity, J. Math. Econ., 3, 15-18 (1976) · Zbl 0352.28003
[4] Aumann, R. J.; Katznelson, Y.; Radner, R.; Rosenthal, R. W.; Weiss, B., Approximate purification of mixed strategies, Math. Oper. Res., 8, 327-341 (1983) · Zbl 0532.90101
[5] Berge, C., Topological Spaces (1963), Oliver & Boyd: Oliver & Boyd London · Zbl 0114.38602
[6] Bergin, J.; Bernhardt, D., Anonymous sequential games with aggregate uncertainty, J. Math. Econ., 21, 543-562 (1992) · Zbl 0779.90079
[7] Bewley, T. F., The equality of the core and the set of equilibria in economies with infinitely many commodities and a continuum of agents, Int. Econ. Rev., 14, 383-393 (1973) · Zbl 0275.90007
[8] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[9] Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions. Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, 580 (1977), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0346.46038
[10] Constantinides, G. M.; Rosenthal, R. W., Strategic analysis of the competitive exercise of certain finacial options, J. Econ. Theory, 32, 128-138 (1984) · Zbl 0552.90008
[11] Dubey, P.; Mas-Colell, A.; Shubik, M., Efficiency properties of strategic market games: An axiomatic approach, J. Econ. Theory, 22, 339-362 (1980) · Zbl 0443.90013
[12] Diestel, J.; Uhl, J. J., Vector Measures (1977), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0369.46039
[13] Fan, K., Fixed points and minimax theorems in locally convex linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38, 121-126 (1952) · Zbl 0047.35103
[14] Glicksberg, I., A further generalization of Kakutani’s fixed point theorem with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3, 170-174 (1952) · Zbl 0046.12103
[15] Jovanovic, B.; Rosenthal, R. W., Anonymous sequential games, J. Math. Econ., 17, 77-87 (1988) · Zbl 0643.90107
[16] Karni, E.; Levin, D., Social attributes and strategic equilibrium: A restaurant pricing game, J. Pol. Econ., 102, 822-840 (1994)
[17] Karni, E.; Schmeidler, D., Fixed preferences and changing tastes, Amer. Econ. Rev., 80, 262, .267 (1990)
[18] Khan, M. Ali, On the integration of set-valued mappings in a non-reflexive Banach space, Simon Stevin, 59, 257-267 (1985) · Zbl 0606.28007
[19] Khan, M. Ali, Equilibrium points of nonatomic games over a Banach space, Trans. Amer. Math. Soc., 293, 737-749 (1986) · Zbl 0594.90104
[20] Khan, M. Ali; Sun, Y. N., On large games with finite actions: A synthetic treatment, Mita J. Econ. (Mita Gakkai Zasshi), 87, 73-84 (1994)
[21] Khan, M. Ali; Sun, Y. N., Pure strategies in games with private information, J. Math. Econ., 24, 633-653 (1995) · Zbl 0840.90137
[22] Khan, M. Ali; Sun, Y. N., Integrals of set-valued functions with a countable range, Math. Oper. Res., 21, 946-954 (1996) · Zbl 0868.28006
[23] Khan, M. Ali; Yannelis, N. C., Equilibrium Theory in Infinite Dimensional Spaces (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0734.00037
[24] Mas-Colell, A., Non-cooperative approaches to the theory of perfect competition, J. Econ. Theory, 22 (1980) · Zbl 0476.90011
[25] Mas-Colell, A., On a theorem of Schmeidler, J. Math. Econ., 13, 201-206 (1984) · Zbl 0563.90106
[26] Massó, J., Undiscounted equilibrium payoffs of repeated games with a continuum of players, J. Math. Econ., 22, 243-264 (1993) · Zbl 0793.90105
[27] Massó, J.; Rosenthal, R. W., More on the “anti-folk theorem,”, J. Math. Econ., 18, 281-290 (1989) · Zbl 0684.90107
[28] Nash, J. F., Noncooperative games, Ann. Math., 54, 286-295 (1951) · Zbl 0045.08202
[29] Ostroy, J.; Zame, W. R., Non-atomic economies and the boundaries of perfect competition, Econometrica, 62, 593-633 (1994) · Zbl 0798.90011
[30] Parthasarathy, K. R., Probability Measures on Metric Spaces (1967), Academic Press: Academic Press New York · Zbl 0153.19101
[31] Pascoa, M. R., Noncooperative equilibrium and Chamberlinian monopolistic competition, J. Econ. Theory, 60, 335-353 (1993) · Zbl 0796.90008
[32] Pascoa, M. R., Approximate equilibrium in pure strategies for non-atomic games, J. Math. Econ., 22, 223-241 (1993) · Zbl 0806.90133
[33] Peleg, B.; Yaari, M., Markets with countably many commodities, Int. Econ. Rev., 11, 369-377 (1970) · Zbl 0211.23004
[34] Radner, R.; Rosenthal, R. W., Private information and pure-strategy equilibria, Math. Oper. Res., 7, 401-409 (1982) · Zbl 0512.90096
[35] Rath, K. P., A direct proof of the existence of pure strategy equilibria in games with a continuum of players, Econ. Theory, 2, 427-433 (1992) · Zbl 0809.90139
[36] Rath, K. P.; Sun, Y. N.; Yamashige, S., The nonexistence of symmetric equilibria in anonymous games with compact action spaces, J. Math. Econ., 24, 331-346 (1995) · Zbl 0834.90145
[37] Rényi, A., Foundations of Probability (1970), Holden-Day: Holden-Day San Francisco · Zbl 0203.49801
[38] Rob, R., Entry, fixed costs and the aggregation of private information, Rev. Econ. Stud., 54, 619-630 (1987) · Zbl 0626.90008
[39] Rustichini, A.; Yannelis, N. C., Edgeworth’s conjecture in economies with a continuum of agents and commodities, J. Math. Econ., 20, 307-326 (1991) · Zbl 0736.90012
[40] Rudin, W., Functional Analysis (1973), McGraw-Hill: McGraw-Hill New York · Zbl 0253.46001
[41] Rudin, W., Real and Complex Analysis (1974), McGraw-Hill: McGraw-Hill New York
[42] Schmeidler, D., Equilibrium points of non-atomic games, J. Statist. Phys., 7, 295-300 (1973) · Zbl 1255.91031
[43] Sun, Y. N., Integration of correspondences on Loeb spaces, Trans. Amer. Math. Soc., 349, 129-153 (1997) · Zbl 0867.28013
[44] Valadier, M., Young measures, Methods of Nonconvex Analysis. Methods of Nonconvex Analysis, Lecture Notes in Mathematics, 1446 (1993), Springer-Verlag: Springer-Verlag Berlin/New York
[45] Yannelis, N. C., Integration of Banach-valued correspondences, (Khan, M. Ali; Yannelis, N. C., Equilibrium Theory in Infinite Dimensional Spaces (1991), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0747.28005
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.