Abstract
We present a brief review of the most important concepts and results concerning games in which the goal structure is formalized by binary relations called preference relations. The main part of the work is devoted to games with ordered outcomes, i.e., game-theoretic models in which preference relations of players are given by partial orders on the set of outcomes. We discuss both antagonistic games and n-person games with ordered outcomes. Optimal solutions in games with ordered outcomes are strategies of players, situations, or outcomes of the game. In the paper, we consider noncooperative and certain cooperative solutions. Special attention is paid to an extension of the order on the set of probabilistic measures since this question is substantial for constructing the mixed extension of the game with ordered outcomes. The review covers works published from 1953 until now.
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References
R. J. Aumann, “Utility theory without completeness axiom,” Econometrica, 30, No. 3, 445–462 (1962).
R. J. Aumann, “Utility theory without completeness axiom: a correction,” Econometrica, 32, No. 1–2, 210–212 (1964).
R. J. Aumann and B. Peleg, “Von Neumann–Morgenstern solutions to cooperative games without side payments,” Bull. Am. Math. Soc., 66, 173–179 (1960).
C. Berge, Théorie Générale des Jeux a n Personnes Games, Gauthier-Villar, Paris (1957).
G. Birkhoff, Lattice Theory, Am. Math. Soc., Providence, Rhode Island (1973).
D. Blackwell, “An analog of the minimax theorem for vector payoffs,” Pac. J. Math., 6, No. 1, 1–8 (1956).
H. Bohnenblust and S. Karlin, “On a theorem of Ville,” in: Contributions to the Theory of Games (H. W. Kuhn and A. W. Tucker, eds.), 1, Princeton Univ. Press (1950), pp. 155–160.
R. Farquharson, “Sur une generalization de la notion d’equilibrium,” C. R. Acad. Sci. Paris, 240, No. 1, 46–48 (1955).
G. Jentsch, “Some thoughts on the theory of cooperative games,” Ann. Math. Stud., 52, 407–442 (1964).
A. Y. Kiruta, M. Rubinov, and E. B. Yanovskaya, Optimal Choice of Distributions in Complex Socio-Economic Problems [in Russian], Nauka, Leningrad (1980).
O. I. Larichev, Science and Art in Decision Making [in Russian], Nauka, Moscow (1979).
R. Luce and H. Raiffa, Games and Decisions [in Russian], IL, Moscow (1961).
B. G. Mirkin, Problem of Group Choice [in Russian], Nauka, Moscow (1974).
V. V. Morozov, “Mixed strategies in game with vector payoffs,” Vestn. Mosk. Univ., 4, 44–49 (1978).
H. Moulin, Théorie des Jeux pour l’ Économie et la Politique, Hermann, Paris (1981).
J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton Univ. Press, Princeton (1953).
V. D. Nogin, Decision Making in Multicriteria Environment [in Russian], Fizmatlit, Moscow (2002).
B. Peleg, “The independence of game theory of utility theory,” Bull. Am. Math. Soc., 72, No. 6, 995–999 (1966).
V. V. Podinovski, “The principle of guaranteed result for partial preference relations,” Zh. Vychisl. Mat. Mat. Fiz., 19, No. 6, 1436–1450 (1979).
V. V. Podinovski, “General antagonistic games,” Zh. Vychisl. Mat. Mat. Fiz., 21, No. 5, 1140–1153 (1981).
V. V. Podinovski and V. D. Nogin, Pareto-Optimal Solutions of Multicriteria Problems [in Russian], Nauka, Moscow (1982).
V. V. Rozen, “Mixed extensions of games with ordered outcomes,” Zh. Vychisl. Mat. Mat. Fiz., 16, No. 6, 1436–1450 (1976).
V. V. Rozen, Goal—Optimality—Decision [in Russian], Radio i Svyaz, Moscow (1982).
V. V. Rozen, “Value for games with ordered outcomes,” in: Semigroup Theory and Its Applications [in Russian], Saratov (1987), pp. 59–70.
V. V. Rozen, “Cooperative games with quasi-ordered outcomes,” Kibernetika, 6, 77–89 (1988).
V. V. Rozen, “Equilibrium points in games with ordered outcomes,” Kibernetika, 5, 98–104 (1989).
V. V. Rozen, “Ordinal invariants and ‘environment’ problem for games with ordered outcomes,” Kibern. Sist. Anal., 2, 145–159 (2001).
V. V. Rozen, “An extension of ordering on the set of probabilistic measures,” Izv. Saratovsk. Univ. (N.S.), Ser. Mat. Mekh. Inform., 5, No. 1, 61–70 (2005).
V. V. Rozen, “Equilibrium in games with ordered outcomes,” Izv. Saratovsk. Univ. (N.S.), Ser. Mat. Mekh. Inf., 9, No. 3, 61–66 (2009).
V. V. Rozen, “Equilibrium points in games with ordered outcomes,” in: Contributions to Game Theory and Management, Vol. III, Saint Petersburg (2010), pp. 368–386.
V. V. Rozen, “Nash equilibrium in games with ordered outcomes,” in: Contributions to Game Theory and Management, Vol. IV, Saint Petersburg (2011), pp. 407–420.
V. Rozen, Decision Making under Quality Criteria. Mathematical Models [in Russian], Palmarium Academic Publ., Saarbrücken (2013).
V. Rozen and G. Zhitomirski, “A category approach to derived preference relations in some decision making problems,” Math. Social Sci., 51, 257–273 (2006).
T. F. Savina, “Homomorphisms and congruence relations for games with preference relations,” Contributions to Game Theory and Management, Vol. III, Saint Petersburg (2010), pp. 387–398.
L. S. Shapley, “Equilibrium points in games with vector payoffs,” Nav. Res. Logist. Quart. Washington, 6, No. 1, 57–61 (1959).
L. S. Shapley and M. Shubik, “Solutions of n-person games with ordinal utilities,” Econometrica, 21, No. 2, 348–349 (1953).
E. R. Smolyakov, Equilibrium Models under Divergent Interests of Parties [in Russian], Nauka, Moscow (1986).
E. Vilkas, “Optimality concepts in game theory,” in: Contemporary Directions of Game Theory [in Russian], Vilnus (1976), pp. 25–43.
E. Vilkas, Optimality in Games and Decisions [in Russian], Nauka, Moscow (1990).
N. N. Vorobiev, “The present state of game theory,” Usp. Mat. Nauk, 25, No. 2 (152), 81–140 (1970).
N. N. Vorobiev, Game Theory for Economists-Cyberneticists [in Russian], Nauka, Moscow (1985).
E. B. Yanovskaya, “Equilibrium points in games with non-archimedean utilities,” in: Mathematical Methods in Social Sciences, 4, Vilnus (1974), pp. 98–118.
E. B. Yanovskaya, “Antagonistic games,” in: Cybernetic Problems [in Russian], 34, Moscow (1978), pp. 221–246.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 136, Proceedings of the Seminar on Algebra and Geometry of Samara University, 2017.
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Rozen, V.V. Games with Ordered Outcomes. J Math Sci 235, 740–755 (2018). https://doi.org/10.1007/s10958-018-4091-7
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DOI: https://doi.org/10.1007/s10958-018-4091-7
Keywords and phrases
- game with ordered outcomes
- optimal strategy
- equilibrium point
- acceptable outcome
- extension of the order on the set of probabilistic measures