Equilibrium uniqueness in aggregative games: very practical conditions. (English) Zbl 1497.91011
Summary: Various Nash equilibrium results for a broad class of aggregative games are presented. The main ones concern equilibrium uniqueness. The setting presupposes that each player has \(\mathbb{R}_+\) as strategy set, makes smoothness assumptions but allows for a discontinuity of stand-alone payoff functions at 0; this possibility is especially important for various contest and oligopolistic games. Conditions are completely in terms of marginal reductions which may be considered as primitives of the game. For many games in the literature they can easily be checked. They automatically imply that conditional payoff functions are strictly quasi-concave. The results are proved by means of the Szidarovszky variant of the Selten-Szidarovszky technique. Their power is illustrated by reproducing quickly and improving upon various results for economic games.
MSC:
| 91A10 | Noncooperative games |
Keywords:
aggregative game; contest game; equilibrium (semi-)uniqueness; Nikaido-Isoda theorem; pseudo-concavity; Selten-Szidarovszky techniqueReferences:
| [1] | Acemoglu, D.; Jensen, MK, Aggregate comparative statics, Games Econ. Behav., 81, 27-49 (2013) · Zbl 1281.91011 · doi:10.1016/j.geb.2013.03.009 |
| [2] | Auslender, A., Optimisation (1976), Paris: Masson, Paris · Zbl 0326.90057 |
| [3] | Corchón, LC, Comparative statics for aggregative games, Math. Soc. Sci., 28, 151-165 (1994) · Zbl 0877.90091 · doi:10.1016/0165-4896(94)90001-9 |
| [4] | Corchón, LC, Theories of Imperfectly Competitive Markets (1996), Berlin: Springer-Verlag, Berlin · Zbl 0911.90044 · doi:10.1007/978-3-662-22531-8 |
| [5] | Cornes, R.; Hartley, R., Asymmetric contests with general technologies, Econ. Theory, 26, 923-46 (2005) · Zbl 1116.91009 · doi:10.1007/s00199-004-0566-5 |
| [6] | Cornes, R., Hartley, R.: Well-behaved aggregative games. Economic Discussion Paper May 24, School of Social Sciences. The University of Manchester (2011) · Zbl 1253.91007 |
| [7] | Cornes, R.; Sato, T.; von Mouche, PHM; Quartieri, F., Existence and uniqueness of Nash equilibrium in aggregative games: an expository treatment, Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi, 47-61 (2016), Cham: Springer, Cham · Zbl 1417.91018 · doi:10.1007/978-3-319-29254-0_5 |
| [8] | Finus, M., von Mouche, P.H.M., Rundshagen, B.: On uniqueness of coalitional equilibria. In: L.A. Petrosjan, N.A. Zenkevich (eds.) Contributions to Game Theory and Management, vol. VII, pp. 51-60. St. Petersburg State University (2014) · Zbl 1347.91028 |
| [9] | Folmer, H.; von Mouche, PHM, On a less known Nash equilibrium uniqueness result, J. Math. Sociol., 28, 67-80 (2004) · Zbl 1101.91003 · doi:10.1080/00222500490448190 |
| [10] | Forgó, F.: On the existence of Nash-equilibrium in n-person generalized concave games. In: S. Komlósi, T. Rapscák, S. Schaible (eds.) Generalized Convexity, Lecture Notes in Economics and Mathematical Systems, vol. 405, pp. 53-61. Springer-Verlag (1994) · Zbl 0802.90128 |
| [11] | Forgó, F.; Kánnai, Z.; Szidarovszky, F.; Bischi, GI, Necessary conditions for concave and Cournot oligopoly games, Games and Dynamics in Economics: Essays in Honour of Akio Matsumoto (2020), Singapore: Springer, Singapore · Zbl 1457.91240 |
| [12] | Gaudet, G.; Salant, SW, Uniqueness of Cournot equilibrium: new results from old methods, Rev. Econ. Stud., 58, 2, 399-404 (1991) · Zbl 0729.90010 · doi:10.2307/2297975 |
| [13] | Ginchev, I.; Ivanov, VI, Second-order characterizations of convex and pseudoconvex functions, J. Appl. Anal., 9, 2, 261-273 (2003) · Zbl 1040.49024 · doi:10.1515/JAA.2003.261 |
| [14] | Halkin, H., Implicit functions and optimization problems without differentiability of the data, SIAM J. Control, 12, 2, 229-236 (1974) · Zbl 0241.90057 · doi:10.1137/0312017 |
| [15] | Hefti, A., Equilibria in symmetric games: theory and applications, Theor. Econ., 12, 979-1002 (2017) · Zbl 1396.91030 · doi:10.3982/TE2151 |
| [16] | Hirai, S.; Szidarovszky, F., Existence and uniqueness of equilibrium in asymmetric contests with endogenous prizes, Int. Game Theory Rev., 15, 1, 1350005 (2013) · Zbl 1274.91099 · doi:10.1142/S0219198913500059 |
| [17] | Jensen, MK; Corchón, LC; Marini, MA, Chapter 4: Aggregative games, Handbook of Game Theory and Industrial Organization, 66-92 (2018), New York: Edward Elgar, New York · doi:10.4337/9781785363283.00010 |
| [18] | Kaneda, M., Matsui, A.: Do profit maximizers maximize profit?: Divergence of objective and result in oligopoly. Technical report, Mimeo, University of Tokyo (2003) |
| [19] | Kolstad, CD; Mathiesen, L., Necessary and sufficient conditions for uniqueness of a Cournot equilibrium, Rev. Econ. Stud., 54, 4, 681-690 (1987) · Zbl 0638.90014 · doi:10.2307/2297489 |
| [20] | Laffont, JJ; Tirole, J., Using cost observations to regulated firms, J. Polit. Econ., 94, 614-641 (1986) · doi:10.1086/261392 |
| [21] | Martimort, D.; Stole, L., Representing equilibrium aggregates in aggregate games with applications to common agency, Games Econ. Behav., 76, 753-772 (2012) · Zbl 1250.91060 · doi:10.1016/j.geb.2012.08.005 |
| [22] | McLennan, A.; Monteiro, PK; Tourky, R., Games with discontinuous payoffs; a strengthening of Reny’s existence theorem, Econometrica, 79, 5, 1643-1664 (2011) · Zbl 1272.91018 · doi:10.3982/ECTA8949 |
| [23] | McManus, M., Numbers and size in Cournot oligopoly, Yorkshire Bull., 14, 14-22 (1962) · doi:10.1111/j.1467-8586.1962.tb00310.x |
| [24] | Nikaido, H.; Isoda, K., Note on non-cooperative games, Pac. J. Math., 5, 807-815 (1955) · Zbl 0171.40903 · doi:10.2140/pjm.1955.5.807 |
| [25] | Okuguchi, K., The Cournot oligopoly and competitive equilibria as solutions to non-linear complementrity problems, Econ. Lett., 12, 127-133 (1983) · Zbl 1273.91221 · doi:10.1016/0165-1765(83)90123-4 |
| [26] | Okuguchi, K.; Suzumura, K., Uniqueness of the Cournot oligopoly equilibrium: a note, Econ. Stud. Q., 22, 81-83 (1971) |
| [27] | Okuguchi, K.; Yamazaki, T., Global stability of Nash equilibrium in aggregative games, Int. Game Theory Rev., 16, 4, 1450014 (2014) · Zbl 1310.91018 · doi:10.1142/S0219198914500145 |
| [28] | Quartieri, F.: Necessary and sufficient conditions for the existence of a unique Cournot equilibrium. Ph.D. thesis, Siena-Università di Siena, Italy (2008) |
| [29] | Ritz, R., Rand journal of economics, Int. J. Ind. Organ., 18, 452-498 (1987) |
| [30] | Ritz, R., Strategic incentives for market share, Int. J. Ind. Organ., 26, 586-597 (2008) · doi:10.1016/j.ijindorg.2007.04.006 |
| [31] | Selten, R., Preispolitik der Mehrproduktunternehmung in der Statischen Theorie (1970), Berlin: Springer-Verlag, Berlin · Zbl 0195.21801 · doi:10.1007/978-3-642-48888-7 |
| [32] | Stewart, G., Management objectives and strategic interactions among capitalist and labor-managed firms, J. Econ. Behav. Organ., 17, 423-431 (1992) · doi:10.1016/S0167-2681(95)90018-7 |
| [33] | Szidarovszky, F.: On the Oligopoly game. Technical report, Karl Marx University of Economics, Budapest (1970) · Zbl 0241.90075 |
| [34] | Szidarovszky, F.; Okuguchi, K., On the existence and uniqueness of pure Nash equilibrium in rent-seeking games, Games Econ. Behav., 18, 135-140 (1997) · Zbl 0889.90168 · doi:10.1006/game.1997.0517 |
| [35] | Szidarovszky, F.; Yakowitz, S., A new proof of the existence and uniqueness of the Cournot equilibrium, Int. Econ. Rev., 18, 787-789 (1977) · Zbl 0368.90020 · doi:10.2307/2525963 |
| [36] | Szidarovszky, F.; Yakowitz, S., Contributions to Cournot oligopoly theory, J. Econ. Theory, 28, 51-70 (1982) · Zbl 0486.90014 · doi:10.1016/0022-0531(82)90091-6 |
| [37] | Tan, K.; Yu, J.; Yuan, X., Existence theorems of Nash equilibria for non-cooperative \(n\)-person games, Int. J. Game Theory, 24, 217-222 (1995) · Zbl 0837.90126 · doi:10.1007/BF01243152 |
| [38] | Vasin, A.; Vasina, P.; Ruleva, T., On organization of markets of homogeneous goods, J. Comput. Syst. Sci. Int., 46, 93-106 (2007) · Zbl 1272.91058 · doi:10.1134/S106423070701011X |
| [39] | von Mouche, PHM; von Mouche, PHM; Quartieri, F., On the geometric structure of the Cournot equilibrium set: the case of concave industry revenue and convex costs, Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi, 63-88 (2016), Cham: Springer, Cham · Zbl 1417.91336 · doi:10.1007/978-3-319-29254-0_6 |
| [40] | von Mouche, PHM; Petrosyan, LA; Mazalov, VV, The Selten-Szidarovszky technique: the transformation part, Recent Advances in Game Theory and Applications, 147-164 (2016), Cham: Birkhäuser, Cham · Zbl 1397.91025 · doi:10.1007/978-3-319-43838-2_8 |
| [41] | von Mouche, P.H.M., Quartieri, F.: Existence of equilibria in Cournotian games with utility functions that are discontinuous at the origin. Technical report, SSRN 2528435 (2012) |
| [42] | von Mouche, PHM; Quartieri, F., On the uniqueness of Cournot equilibrium in case of concave integrated price flexibility, J. Glob. Optim., 57, 3, 707-718 (2013) · Zbl 1286.91084 · doi:10.1007/s10898-012-9926-z |
| [43] | von Mouche, PHM; Quartieri, F., Cournot equilibrium uniqueness via demi-concavity, Optimization, 67, 4, 441-455 (2017) · Zbl 1397.91374 · doi:10.1080/02331934.2017.1405954 |
| [44] | von Mouche, P.H.M., Quartieri, F., Szidarovszky, F.: On a fixed point problem transformation method. In: Proceedings of the 10th IC-FPTA, Cluj-Napoca, Romania, pp. 179-190 (2013) |
| [45] | von Mouche, P.H.M., Sato, T.: Cournot equilibrium uniqueness: at 0 discontinuous industry revenue and decreasing price flexibility. Int. Game Theory Rev. 21(2), 1940010 (2019) · Zbl 1417.91353 |
| [46] | von Mouche, PHM; Yamazaki, T., Sufficient and necessary conditions for equilibrium uniqueness in aggregative games, J. Nonlinear Convex Anal., 16, 2, 353-364 (2015) · Zbl 1311.47076 |
| [47] | Yamazaki, T., On the existence and uniqueness of pure-strategy Nash equilibrium in asymmetric rent-seeking contests, J. Public Econ. Theory, 10, 2, 317-327 (2008) · doi:10.1111/j.1467-9779.2008.00364.x |
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.
