Using Statistical Estimates in the Game with Nature as an Investment Model

Purpose of the study. The aim of the research is to develop new principles of decision making (principles of optimality) in games with nature and their application to analyze statistical data and choose strategies for stock investment.Materials and methods. We analyze Russian and foreign bibliography on the research problem. A model of decision making in a game with nature with known state probabilities is proposed. The mathematical expectation of the player's payoff is taken as an assessment of efficiency, and the standard deviation or variance is taken as a risk assessment. This two-criterion task is formalized by transferring the efficiency assessment into a constraint. As a result, for the case of mixed strategies, a nonlinear (quadratic) task of mathematical programming arises. To solve it, an approach based on the Lagrange function and the Karush-Kuhn-Tucker optimality conditions is used. As an application of the methods obtained, the problems of stock investment are considered.Results. Analytical methods for solving the indicated optimization problem and an algorithm for finding optimal mixed strategies are obtained. Practical examples of application of the proposed approach on real statistical data are given. As the initial data in this study, we used stock quotes of Russian companies in the electric power industry for the period from 01.07.2020 to 01.10.2020, taken from the website of the FINAM Investment Company. The developed method allows one to find the optimal strategy and the corresponding values of profitability and risk based on only the initial data (statistical characteristics of financial instruments and the threshold value of profitability), i.e. provides, in our opinion, a convenient analysis tool for the investor.Conclusion. The concept of the principle of optimality in decision making problems under conditions of incomplete information is very ambiguous. The decision maker should be able to choose from a range of decision making models that reflect the dependence of the type of rational behavior on the available information and the attitude to risk. The paper proposes a model of this type for the case of probabilistic uncertainty, which leads to the problem of minimizing variance as a risk assessment with a lower bound on the mathematical expectation as an assessment of efficiency.