Geodesic
Mathematical foundations of the Golden Rule. I. Static variant
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 7 (2015) no. 3, pp. 16-47.

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The Berge equilibrium concept was suggested by Russian mathematician K. Vaisman in 1994. In the presented paper, we offer to use this concept as a mathematical model of the Golden Rule. The Berge–Pareto equilibrium is formalized, sufficient conditions for the existence of the equilibrium are found. For mixed strategies, the existence of the equilibrium is proved.
Keywords: non-cooperative game, Berge equilibrium
Mots-clés : Pareto maximum. 
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Vladislav I. Zhukovskiy; Konstantin N. Kudryavtsev. Mathematical foundations of the Golden Rule. I. Static variant. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 7 (2015) no. 3, pp. 16-47. http://geodesic.mathdoc.fr/item/MGTA_2015_7_3_a1/

[1] Vaisman K. S., Ravnovesie po Berzhu, Avtoreferat diss. ... kand. fiz.-mat. nauk, SPbGU, 1995

[2] Vaisman K. S., “Ravnovesie po Berzhu”: Zhukovskii V. I., Chikrii A. A., Lineino-kvadratichnye differentsialnye igry, Naukova dumka, Kiev, 1994, 119–143

[3] Vasin A. A., Krasnoschekov P. S., Morozov V. V., Issledovanie operatsii, Izdatelskii tsentr «Akademiya», M., 2008

[4] Danford N., Shvarts Dzh. T., Lineinye operatory, v. I, IL, M., 1962

[5] Dmitruk A. V., Vypuklyi analiz. Elementarnyi vvodnyi kurs, Maks-Press, M., 2012

[6] Zhukovskii V. I., Kooperativnye igry pri neopredelennosti i ikh prilozheniya, Izd. 2-e, Editorial URSS, M., 2010 | MR

[7] Zhukovskii V. I., Kudryavtsev K. N., “Uravnoveshivanie konfliktov pri neopredelennosti. I. Analog sedlovoi tochki”, Matematicheskaya teoriya igr i ee prilozheniya, 5:1 (2013), 27–44 | MR

[8] Zhukovskii V. I., Chikrii A. A., Lineino-kvadratichnye differentsialnye igry, Naukova dumka, Kiev, 1994

[9] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1985 | MR

[10] Lyusternik L. A., Sobolev V. I., Elementy funktsionalnogo analiza, Nauka, M., 1969

[11] Maschenko S. O., “Kontseptsiya ravnovesiya po Neshu i ee razvitie”, Zhurnal obchislovalnoi ta prikladnoi matematiki, 2012, no. 1(107), 40–65

[12] Morozov V. V., Sukharev A. G., Fedorov V. V., Issledovanie operatsii v zadachakh i uprazhneniyakh, Vysshaya shkola, M., 1986

[13] Podinovskii V. V., Nogin V. D., Pareto-optimalnye resheniya mnogokriterialnykh zadach, Fizmatlit, M., 2007

[14] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, IL, M., 1962

[15] Berge C., Théorie générale des jeux á $n$ personnes games, Gauthier Villars, Paris, 1957 ; Berzh K., Obschaya teoriya igr neskolkikh lits, Fizmatgiz, M., 1961 | MR

[16] Borel E., “Sur les systémes de formes linéaires a déterminant symétrique gauche et la théorie générale du jeu”, Comptes Rendus de l`Academie des Sciences, 184 (1927), 52–53

[17] Colman A. M., Körner T. W., Musy O., Tazdait T., “Mutual support in games: Some properties of Berge equilibria”, Journal of Mathematical Psychology, 55:2 (2011), 166–175 | DOI | MR | Zbl

[18] Glicksberg I. L., “A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points”, Proc. Amer. Math. Soc., 3:1 (1952), 170–174 | MR

[19] Nash J. F., “Equilibrium points in $N$-person games”, Proc. Nat. Academ. Sci. USA, 36 (1950), 48–49 | DOI | MR | Zbl

[20] Nash J. F., “Non-cooperative games”, Ann. Math., 54 (1951), 286–295 | DOI | MR | Zbl

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

[22] Von Neumann J., “Zur Theorie der Gesellschaftspiele”, Math. Ann., 100:1 (1928), 295–320 | DOI | MR | Zbl

[23] Radjef M. S., “Sur l`existenced`un èquilibre de Berge pour un jeu diffèrentiel anp orsonnes”, Cahiers Mathèmatiques de l`Universitè d`Oran, 1998, no. 1, 89–93 | MR

[24] Shubik M., “Review of C. Berge «General theory of $n$-person games»”, Econometrica, 29:4 (1961), 821 | DOI

[25] Vaisman K. S., “About differential game under uncertainty”, Nonsmooth and Discontinuous Problems of Control and Optimization, Abstr. of Third Intern. Workshop (St.-Petersburg, 1995), 45–48

[26] Vaisman K. S., “The Berge equilibrium for linear-quadratic differential game”, Multiple criteria problems under uncertainty, The 3-d Intern. Workshop, Abstracts (Orekhovo-Zuevo, Russia, 1994), 96

[27] Vaisman K. S., Zhukovskiy V. I., “The Berge equilibrium under uncertainty”, Multiple criteria problems under uncertainty, The 3-d Intern. Workshop, Abstracts (Orekhovo-Zuevo, Russia, 1994), 97–98

[28] Zhukovskiy V. I., Molostvov V. S., Vaisman K. S., “Non-cooperative games under uncertainty”, Game Theory and Application, 3 (1997), 189–222 | MR

[29] Zhukovskiy V. I., Salukvadze M. E., Vaisman K. S., The Berge equilibrium, Preprint, Institute of control systems, Tbilisi, 1994 | MR