A unified approach to the Nash equilibrium existence in large games from finitely many players to infinitely many players. (English) Zbl 1483.91011
The authors proposed a unified approach to the Nash equilibrium existence in games with infinitely players. They derived the unified approach coming from the case of finitely many players and reproving Nash equilibrium existence theorems. They derived the corresponding theorems for games in normal, qualitative, generalized, forms
Reviewer: Carlos Narciso Bouza Herrera (Habana)
MSC:
| 91A06 | \(n\)-person games, \(n>2\) |
| 91A10 | Noncooperative games |
| 91A07 | Games with infinitely many players |
References:
| [1] | Askoura, Y., The weak-core of a game in normal form with a continuum of players, J. Math. Econ., 47, 1, 43-47 (2011) · Zbl 1211.91053 · doi:10.1016/j.jmateco.2010.11.003 |
| [2] | Askoura, Y., On the core of normal form games with a continuum of players, Math. Soc. Sci., 89, 32-42 (2017) · Zbl 1415.91023 · doi:10.1016/j.mathsocsci.2017.06.001 |
| [3] | Aumann, RJ, Markets with a continuum of traders, Econometrica, 32, 39-50 (1964) · Zbl 0137.39003 · doi:10.2307/1913732 |
| [4] | Aumann, RJ, Existence of competitive equilibria in markets with a continuum of traders, Econometrica, 34, 1-17 (1966) · Zbl 0142.17201 · doi:10.2307/1909854 |
| [5] | Balder, EJ, A unifying approach to existence of Nash equilibria, Int. J. Game Theory, 24, 79-94 (1995) · Zbl 0837.90137 · doi:10.1007/BF01258205 |
| [6] | Deguire, P.; Tan, KK; Yuan, XZ, The study of maximal elements, fixed points for LS-majorized mappings and their applications to minimax and variational inequalities in product topological spaces, Nonlinear Anal., 37, 933-951 (1999) · Zbl 0930.47024 · doi:10.1016/S0362-546X(98)00084-4 |
| [7] | Ichiishi, I., Game Theory for Economic Analysis (1983), New York: Academic Press, New York · Zbl 0522.90104 |
| [8] | Khan, MA; Rath, KP; Sun, Y., On the existence of pure strategy equilibria in games with a continuum of players, J. Econ. Theory, 76, 13-46 (1997) · Zbl 0883.90137 · doi:10.1006/jeth.1997.2292 |
| [9] | Nash, J., Non-cooperative games, Ann. Math., 54, 286-295 (1951) · Zbl 0045.08202 · doi:10.2307/1969529 |
| [10] | Noguchi, M., Existence of Nash equilibria in large games, J. Math. Econ., 45, 168-184 (2009) · Zbl 1156.91418 · doi:10.1016/j.jmateco.2008.08.005 |
| [11] | Rath, KP, 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 · doi:10.1007/BF01211424 |
| [12] | Salonen, H., On the existence of Nash equilibria in large games, Int. J. Game Theory, 39, 351-357 (2010) · Zbl 1211.91054 · doi:10.1007/s00182-009-0180-7 |
| [13] | Schmeidler, D., Equilibrium points in nonatomic games, J. Stat. Phys., 7, 295-300 (1973) · Zbl 1255.91031 · doi:10.1007/BF01014905 |
| [14] | Shafer, W.; Sonnenschein, H., Equilibrium in abstract economies without ordered preferences, J. Math. Econ., 2, 3, 345-348 (1975) · Zbl 0312.90062 · doi:10.1016/0304-4068(75)90002-6 |
| [15] | Tarafdar, EU; Chowdhury, MSR, Topological Methods for Set-Valued Nonlinear Analysis (2008), Singapore: World Scientific Publishing Co. Pre. Ltd., Singapore · Zbl 1141.47033 · doi:10.1142/6347 |
| [16] | Weber, S., Some results on the weak core of a non-side-payment game with infinitely many players, J. Math. Econ., 8, 1, 101-111 (1981) · Zbl 0474.90093 · doi:10.1016/0304-4068(81)90015-X |
| [17] | Yang, Z., Some infinite-player generalizations of Scarf’s theorem: finite-coalition \(\alpha \)-cores and weak \(\alpha \)-cores, J. Math. Econ., 73, 81-85 (2017) · Zbl 1415.91034 · doi:10.1016/j.jmateco.2017.09.005 |
| [18] | Yang, Z., Some generalizations of Kajii’s theorem to games with infinitely many players, J. Math. Econ., 76, 131-135 (2018) · Zbl 1388.91039 · doi:10.1016/j.jmateco.2018.04.004 |
| [19] | Yang, Z., The weak \(\alpha \)-core of exchange economies with a continuum of players and pseudo-utilities, J. Math. Econ., 91, 43-50 (2020) · Zbl 1457.91050 · doi:10.1016/j.jmateco.2020.08.005 |
| [20] | Yang, Z.; Yuan, GX, Some generalizations of Zhao’s theorem: hybrid solutions and weak hybrid solutions for games with nonordered preferences, J. Math. Econ., 84, 94-100 (2019) · Zbl 1427.91066 · doi:10.1016/j.jmateco.2019.07.007 |
| [21] | Yang, Z.; Zhang, X., A weak \(\alpha \)-core existence theorem of games with nonordered preferences and a continuum of agents, J. Math. Econ., 94, 102464 (2021) · Zbl 1464.91012 · doi:10.1016/j.jmateco.2020.102464 |
| [22] | Yannelis, NC; Prabhakar, ND, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ., 12, 233-245 (1983) · Zbl 0536.90019 · doi:10.1016/0304-4068(83)90041-1 |
| [23] | Yu, J., Essential equilibria of n-person noncooperative games, J. Math. Econ., 31, 3, 361-372 (1999) · Zbl 0941.91006 · doi:10.1016/S0304-4068(97)00060-8 |
| [24] | Yuan, X.Z.: The study of equilibria for abstract economics in topological vector spaces-a unified approach. Nonlinear Anal., 37(4), 409-430 (1999) · Zbl 0943.47053 |
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.
