Strong coalitional equilibria in games under uncertainty

V. I. Zhukovskiy   ; L. V. Zhukovskaya   ; K. N. Kudryavtsev   ; M. Larbani

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 2, pp. 189-207

Résumé

The Strong Coalitional Equilibrium (SCE) is introduced for normal form games under uncertainty. This concept is based on the synthesis of the notions of individual rationality, collective rationality in normal form games without side payments, and a proposed coalitional rationality. For presentation simplicity, SCE is presented for 4-person games under uncertainty. Sufficient conditions for the existence of SCE in pure strategies are established via the saddle point of the Germeir's convolution function. Finally, following the approach of Borel, von Neumann and Nash, a theorem of existence of SCE in mixed strategies is proved under common minimal mathematical conditions for normal form games (compactness and convexity of players' strategy sets, compactness of uncertainty set and continuity of payoff functions).

Détail

Keywords: normal form game, uncertainty, guarantee, mixed strategies, saddle point, equilibrium.
Mots-clés : Germeier convolution

Bibliographie

[1] Nash J., “Equillibrium points in $n$-person games”, Proc. Nat. Academ. Sci. USA, 36:1 (1950), 48–49 | DOI | MR | Zbl

[2] Nash J., “Non-cooperative games”, Annals of Mathematics, 54:2 (1951), 286–295 | DOI | MR | Zbl

[3] Gillies D. B., “Solutions to general non-zero-sum games”, Contributions to the Theory of Games (AM-40), 4, Princeton University Press, Princeton, 1959, 47–86 | DOI | MR

[4] Chwe M. S.-Y., “Farsighted coalitional stability”, Journal of Economic Theory, 63:2 (1994), 299–325 | DOI | MR | Zbl

[5] Mariotti M., “A model of agreement in strategic form games”, Journal of Economic Theory, 74:1 (1997), 196–217 | DOI | MR | Zbl

[6] Kornishi H., Ray D., “Coalition formation as a dynamic process”, Journal of Economic Theory, 110:1 (2003), 1–41 | DOI | MR

[7] Ray D., Vohra R., “Equilibrium binding agreements”, Journal of Economic Theory, 73:1 (1997), 30–78 | DOI | MR | Zbl

[8] Aumann R. J., “Acceptable points in general cooperative $n$-person games”, Contributions to the Theory of Games (AM-40), 4, Princeton University Press, Princeton, 1959, 287–324 | DOI | MR

[9] Aumann R. J., “The core of a cooperative game without side payments”, Transactions of the American Mathematical Society, 98:3 (1961), 539–552 | DOI | MR | Zbl

[10] Berge C., Théorie générale des jeux à $n$ personnes, Gauthier-Villar, Paris, 1957 http://eudml.org/doc/192658 | MR

[11] Zhukovskiy V. I., “Some problems of non-antagonistic differential games”, Mathematical Methods in Operations Research, Bulgarian Academy of Sciences, Sofia, 1985, 103–195

[12] Zhukovskiy V. I., Larbani M., “Alliance in three person games”, Researches in Mathematics and Mechanics, 22:1(29) (2017), 115–129 | DOI

[13] Sally D., “Conversation and cooperation in social dilemmas: A meta-analysis of experiments from 1958 to 1992”, Rationality and Society, 7:1 (1995), 58–92 | DOI

[14] Kahneman D., Knetsch J. L., Thaler R. H., “Fairness and the assumptions of economics”, The Journal of Business, 59:4-2 (1986), 285–300 | DOI

[15] Fehr E., Schmidt K. M., “The economics of fairness, reciprocity and altruism: Experimental evidence and new theories”, Handbook of the Economics of Giving, Altruism and Reciprocity, v. 1, 2006, 615–691 | DOI

[16] Engel C., “Dictator games: A meta study”, Experimental Economics, 14:4 (2011), 583–610 | DOI

[17] Bernheim B. D., Peleg B., Whinston M. D., “Coalition-proof Nash equilibria I. Concepts”, Journal of Economic Theory, 42:1 (1987), 1–12 | DOI | MR | Zbl

[18] Zhao J., “The hybrid solutions of an $N$-person game”, Games and Economic Behavior, 4:1 (1992), 145–160 | DOI | MR | Zbl

[19] Larbani M., Nessah R., “Sur l'équilibre fort selon Berge”, RAIRO - Operations Research, 35:4 (2001), 439–451 | DOI | MR | Zbl

[20] Nessah R., Larbani M., Tazdait T., “A note on Berge equilibrium”, Applied Mathematics Letters, 20:8 (2007), 926–932 | DOI | MR | Zbl

[21] Zhao J., “A cooperative analysis of covert collusion in oligopolistic industries”, International Journal of Game Theory, 26:2 (1997), 249–266 | DOI | MR | Zbl

[22] Scarf H. E., “On the existence of a cooperative solution for a general $n$-person game”, Journal of Economc Theory, 3:2 (1971), 169–181 | DOI | MR

[23] Luce R. D., Raiffa H., Games and decisions, John Wiley, New York, 1957 | MR | Zbl

[24] Wald A., Statistical decision functions, John Wiley, New York, 1950 | MR | Zbl

[25] Savage L. J., The foundations of statistics, John Wiley, New York, 1954 | MR | Zbl

[26] Zhukovskiy V. I., Kudryavtsev K. N., “Pareto-optimal Nash equilibrium: Sufficient conditions and existence in mixed strategies”, Automation and Remote Control, 77:8 (2016), 1500–1510 | DOI | MR

[27] Germeier Yu. B., Non-antagonistic games, Springer, Boston, 1986 | MR

[28] Borel E., “La théorie du jeu et les equations intégrales à noyau symétrique”, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 173 (1921), 1304–1308 | Zbl

[29] von Neumann J., “Zur theorie der gesellschaftsspiele”, Mathematische Annalen, 100:1 (1928), 295–320 | DOI | MR | Zbl

[30] Berge C., Espace topologiques, Dunod, Paris, 1963 | MR

[31] Morozov V. V., Sukharev A. G., Fedorov V. V., Operational research in problems and exercises, URSS, M., 2016

[32] Glicksberg I. L., “A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points”, Proceedings of the American Mathematical Society, 3:1 (1952), 170–174 | DOI | MR

[33] Zhukovskiy V. I., Kudryavtsev K. N., “Mathematical foundations of the Golden Rule. I. Static case”, Automation and Remote Control, 78:10 (2017), 1920–1940 | DOI | MR | Zbl

[34] Larbani M., Zhukovskiy V. I., “Berge equilibrium in normal form static games: a literature review”, Izvestia Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 49 (2017), 80–110 | DOI | MR | Zbl