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## Archive of Issues

Russia Moscow; Orekhovo-Zuevo
Year
2013
Issue
3
Pages
28-33
 Section Mathematics Title Method of settlement of conflicts under uncertainty Author(-s) Zhukovskii V.I.a, Soldatova N.G.b Affiliations Lomonosov Moscow State Universitya, Moscow State Regional Institute of Humanitiesb Abstract As a mathematical model of conflict the non-cooperation game $\Gamma$ of two players under uncertainty is considered. About uncertainty only the limits of change are known. Any characteristics of probability are absent. To estimate risk in $\Gamma$ we use Savage functions of risk (from principle of minimax regret). The quality of functioning of conflict's participants is estimated according to two criteria: outcomes and risks, at that each of the participants tries to increase the outcome and simultaneously to decrease the risk. On the basis of synthesis of principles of minimax regret and guaranteed result, Nash equilibrium and Slater optimality as well as solution of the two-level hierarchical Stackelberg game, the notion of guaranteed equilibrium in $\Gamma$ (outcomes (prize) and risks) is formalized. We give the example. Then the existence of such a solution in mixed strategies at usual limits in mathematical game theory is established. Keywords strategy, situations, uncertainty, non-cooperative game, Nash equilibrium, Slater maximum and minimum UDC 519.833 MSC 91A10 DOI 10.20537/vm130303 Received 5 July 2013 Language Russian Citation Zhukovskii V.I., Soldatova N.G. Method of settlement of conflicts under uncertainty, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 3, pp. 28-33. References Krasovskii N.N., Subbotin A.I. Game-theoretical control problems, New York-Berlin: Springer-Verlag, 1988, 517 p. Venttsel' E.S. Issledovanie operatsii (Operations research), Moscow: Znanie, 1976, 63 p. Tsvetkova E.V., Arlyukova I.O. Riski v ekonomicheskoi deyatel'nosti (Risks in economic activity), Saint-Petersburg: IVESEP, 2002, 64 p. Savage L.Y. The theory of statistical decision, J. Am. Stat. Assoc., 1951, no. 46, pp. 55-67. Cheremnykh Yu.N. Mikroekonomika. Prodvinutyi uroven' (Microeconomics. The advanced level), Moscow: INFRA, 2008, 843 p. Von Neumann J. Zur Theorie der Gesellschaftspiele, Math. Ann., 1928, vol. 100, pp. 295-320. Nash J.F. Equilibrium points in N-person games, Proc. Nat. Acad. Sci. USA, 1950, vol. 36, pp. 48-49. Zhukovskii V.I., Kudryavtsev K.N. Equilibrating conflicts under uncertainty. I. Analogue of a saddle-point, Mat. Teor. Igr Pril., 2013, vol. 5, no. 1, pp. 27-44. Full text