Geodesic
Methods for determining optimal mixed strategies in matrix games with correlated random payoffs
Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 33-47.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 risk estimate 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 structure will 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.
Keywords: risk management, optimality principle, two-criteria approach, mathematical expectation, standard deviation. 
@article{CHEB_2023_24_4_a3,
     author = {V. A. Gorelik and T. V. Zolotova},
     title = {Methods for determining optimal mixed strategies in matrix games with correlated random payoffs},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {33--47},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a3/}
}
TY  - JOUR
AU  - V. A. Gorelik
AU  - T. V. Zolotova
TI  - Methods for determining optimal mixed strategies in matrix games with correlated random payoffs
JO  - Čebyševskij sbornik
PY  - 2023
SP  - 33
EP  - 47
VL  - 24
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a3/
LA  - en
ID  - CHEB_2023_24_4_a3
ER  - 
%0 Journal Article
%A V. A. Gorelik
%A T. V. Zolotova
%T Methods for determining optimal mixed strategies in matrix games with correlated random payoffs
%J Čebyševskij sbornik
%D 2023
%P 33-47
%V 24
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a3/
%G en
%F CHEB_2023_24_4_a3
V. A. Gorelik; T. V. Zolotova. Methods for determining optimal mixed strategies in matrix games with correlated random payoffs. Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 33-47. http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a3/

[1] Simon H. A., Administrative behavior, Simon and Schuster, United States, 2013

[2] Wald A., Sequential analysis, Courier Corporation, United States, 2004

[3] Samuelson L., “Game theory in economics and beyond”, Voprosy Ekonomiki, 13 (2017), 89–115 | DOI

[4] Allen F., Morris S. E., “Game theory models in finance”, International Series in Operations Research and Management Science, Springer, New York, 2014, 17–41 | MR

[5] Breton M., “Dynamic Games in Finance”, Handbook of Dynamic Game Theory, eds. Başar T., Zaccour G., Springer, Cham, 2018 | MR

[6] Askari G., Gordji M. E., Park C., “The behavioral model and game theory”, Palgrave Communications, 5:15 (2019), 1–8 | DOI

[7] Fox W. P., Burks R., “Game Theory”, Applications of Operations Research and Management Science for Military Decision Making, International Series in Operations Research and Management Science, 283, Springer, Cham, 2019

[8] Allayarov S., “Game Theory And Its Optimum Application For Solving Economic Problems”, Turkish Journal of Computer and Mathematics Education, 12:9 (2021), 3432–3441

[9] Jiang Y., Zhou K., Lu X., Yang S., “Electricity trading pricing among prosumers with game theory-based model in energy blockchain environment”, Applied Energy, 271 (2020), 115239 | DOI

[10] Lee S., Kim S., Choi K., Shon T., “Game theory-based security vulnerability quantification for social internet of things”, Future Generation Computer Systems, 82 (2018), 752–760 | DOI

[11] Hafezalkotob A., Mahmoudi R., Hajisami E., Wee H. M., “Wholesale-retail pricing strategies under market risk and uncertain demand in supply chain using evolutionary game theory”, Kybernetes, 47:8 (2018), 1178–1201 | DOI

[12] Piraveenan M., “Applications of game theory in project management: a structured review and analysis”, Mathematics, 7:13 (2019), 858 | DOI

[13] Shafer G., Vovk V., Game-Theoretic Foundations for Probability and Finance, Wiley Series in Probability and Statistics, 455, John Wiley and Sons, United States, 2019 | DOI | MR

[14] Wald A., Statistical Decision Functions, Wiley, Oxford UK, 1951 | DOI | MR

[15] Hurwicz L., Optimality Criteria for Decision Making Under Ignorance, Statistics, 370, 1951

[16] Savage L. J., “The Theory of Statistical Decision”, J. Am. Stat. Assoc., 46 (1951), 55–67 | DOI | Zbl

[17] Harman R., Prus M., “Computing optimal experimental designs with respect to a compound Bayes risk criterion”, Statistics and Probability Letters, 137 (2018), 135–141 | DOI | MR | Zbl

[18] Kuzmics C., “Abraham Wald's complete class theorem and Knightian uncertainty”, Games and Economic Behavior, 104 (2017), 666–673 | DOI | MR | Zbl

[19] Radner R., “Decision and choice: bounded rationality”, International Encyclopedia of the Social and Behavioral Sciences, Second Edition, Elsevier, Florida United States, 2015, 879–885 | DOI

[20] Gorelik V. A., Zolotova T. V., “The optimality principle “mathematical expectation-var” and its application in the Russian stock market”, Proceedings 12th International Conference Management of Large-Scale System Development (IEEE Conference Publications) (Moscow), 2019, 1–4 | DOI

[21] Gorelik V. A., Zolotova T. V., “Risk management in stochastic problems of stock investment and its application in the Russian Stock Market”, Proceedings 13th International Conference Management of Large-Scale System Development (IEEE Conference Publications) (Moscow), 2020, 1–5 | DOI

[22] Zhukovsky V. I., Kirichenko M. M., “Risks and outcomes in a multicriteria problem with uncertainty”, Risk Management, 2 (2016), 17–25

[23] Labsker L. G, “The property of synthesizing the Wald-Savage criterion and its economic application”, Economics and Mathematical Methods, 55:4 (2019), 89–103 | DOI

[24] García F., González-Bueno J. A., Oliver J., “Mean-variance investment strategy applied in emerging financial markets: evidence from the Colombian stock market”, Intellectual Economics, 9:15 (2015), 22–29 | DOI

[25] Xu Y., Xiao J., Zhang L., “Global predictive power of the upside and downside variances of the U.S. equity market”, Economic Modelling, 93 (2020), 605–619 | DOI

[26] Sharpe W. F., Alexander G. J., Bailey J. V., Investments, Prentice-Hall, Englewood Cliffs, 1999 | DOI

[27] Investment company “FINAM”, (accessed 5 March 2021) https://www.finam.ru/