Mathematical models of the battle between parties for electorate or between companies for markets
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MATHEMATICAL MODELS OF THE BATTLE BETWEEN PARTIES FOR ELECTORATE OR BETWEEN COMPANIES FOR MARKETS V. V. Ostapenko,a O. S. Ostapenko,b E. N. Belyaeva,a† and Y. V. Stupnitskaya c
UDC 518.9
Abstract. A model of optimization of strategies for any number of political parties (companies) in election (advertising) campaigns is proposed. We formulated the problem of choosing an optimal strategy for allocating scarce resources among regions (markets) in which political parties (companies) have different influences. The optimization model is analyzed and the existence of a unique solution is proved for different functions of dependencies between investments and winning. An analytical solution to the model is found for the case of two parties (companies). A direction for further development is the study of the sensitivity of an optimal solution to model parameters and formation of a stochastic model. Keywords: sociological process, conflict situation, noncooperative game, equilibrium situation, condition for the existence of a solution to a game. INTRODUCTION Many mathematical models of sociological processes are based on probability theory and mathematical statistics [1–5]. However, the complexity and variety of problems arising in sociology require the introduction of an extensive mathematical apparatus that includes graph theory, cluster analysis, analysis of expert judgments, pattern recognition, numerical optimization methods, and other systems analysis methods [4]. Conflict situations arise in the majority of sociological processes, for example, battles between parties for their electorate or between companies for markets or optimizing (for example, allocation of capital) under uncertainty. Such processes can be naturally modelled within the framework of noncooperative game theory [6]. Antagonistic games are most investigated; in such games, one player battles against the others and considers that they are allied with one another against him. However, even in a 2 player game, this does not confirmed since each of them pursues his own objectives. This article develops methods from [7–9] and constructs models for processes of allocation of funds for advertising in different regions (for parties) or in markets (for companies) with a view to maximizing their electorate or profit. Issues of mutual influence of parties (companies) in a concrete region (market) are considered, the concept of an optimal solution is given for each player, and issues of existence of such a solution and numerical methods for finding it are investigated. MODELLING THE MUTUAL INFLUENCE AMONG PLAYERS Let us analyze the process of the battle between two parties for their electorate in one region (the battle between two companies for a concrete market can similarly be described). a
Educational-Scientific Complex “Institute for Applied Systems Analysis” of NTTU “KPI,” Ministry of Education and Sciences, Youth and Sports of Ukraine and National Academy of Sciences of Ukraine, Kyiv, Ukraine; † [email protected]. bV. M. Glushkov Institute of Cybern
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