chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2023-24-4-33-47cheb-1593Research ArticleСтатьиArticleМетоды нахождения оптимальных смешанных стратегий в матричных играх с коррелированными случайными выигрышамиMethods for Determining Optimal Mixed Strategies in Matrix Games with Correlated Random PayoffsГореликВиктор АлександровичGorelikVictor Alexandrovich

доктор физико-математических наук, профессор

doctor of physical and mathematical sciences, professor

vgor16@mail.ruЗолотоваТатьяна ВалерьяновнаZolotovaTatiana Valerianovna

доктор физико-математических наук, профессор

doctor of physical and mathematical sciences, professor

tgold11@mail.ruФедеральный исследовательский центр «Информатика и управление» Российской академии наук (ФИЦ ИУ РАН); Московский педагогический государственный университетРоссияFederal Research Center “Informatics and Management” of the Russian Academy of Sciences; Moscow State Pedagogical UniversityRussian FederationФинансовый университет при Правительстве РФРоссияFinancial University under the Government of the Russian FederationRussian Federation2023250120242443347Copyright © Горелик В.А., Золотова Т.В., 20232023Горелик В.А., Золотова Т.В.Gorelik V.A., Zolotova T.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/1593

Рассматривается игра с природой при известных вероятностях состояний. Предлагается принцип оптимальности для принятия решений для игр с природой, основанный на оценках эффективности и риска. В отличие от традиционного подхода к определению смешанной стратегии в теории игр, в данной работе рассматривается возможность корреляционной зависимости случайных значений выигрышей для начальных альтернатив.Предлагаются два варианта реализации двухкритериального подхода к определению прин-ципа оптимальности. Первый вариант — минимизировать дисперсию как оценку риска сболее низким порогом математического ожидания выигрыша. Второй вариант — максимизировать математическое ожидание выигрыша с верхним порогом дисперсии. Получены аналитические решения обеих задач. Рассмотрено применение полученных результатов на примере процесса инвестирования на фондовом рынке. Инвестор, как правило, формирует портфель не сразу, а в виде последовательного процесса приобретения того или иного финансового актива. В этом случае смешанная стратегия может быть реализована в ее имманентном смысле, т.е. покупки осуществляются случайным образом с распределением, определяемым ранее найденным оптимальным решением. Если этот процесс достаточно длительный, то структура портфеля будет примерно соответствовать типу смешанной стратегии. Такой подход использования игры с природой с учетом корреляционной зависимости случайного выигрыша чистых стратегий может быть применен и к задачампринятия решений в других областях управления рисками.

A game with nature for known state probabilities is considered. An optimality principle is proposed for decision-making for games with nature, based on efficiency and risk estimates.In contrast to the traditional approach to the definition of a mixed strategy in game theory, this paper considers the possibility of correlation dependence of random payoff values for initial alternatives. Two variants of the implementation of the two-criteria approach to the definition of the optimality principle are suggested. The first option is to minimize the variance as a riskestimate with a lower threshold on the mathematical expectation of the payoff. The second option is to maximize the mathematical expectation of the payoff with an upper threshold on the variance. Analytical solutions of both problems are obtained. The application of the obtained results on the example of the process of investing in the stock market is considered.An investor, as a rule, does not form a portfolio all at once, but as a sequential process of purchasing one or another financial asset. In this case, the mixed strategy can be implemented in its immanent sense, i.e. purchases are made randomly with a distribution determined by the previously found optimal solution. If this process is long enough, then the portfolio structurewill approximately correspond to the type of mixed strategy. This approach of using the game with nature, taking into account the correlation dependence of random payoff of pure strategies, can also be applied to decision-making problems in other areas of risk management.

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The authors declare that there are no conflicts of interest present.