Choosing a Strategy Based on the Maximin Quantile Criterion in a Game of a Company with Several Groups of Clients

The problem of choosing optimal strategies for the case when the second player is represented by a set of decision makers is considered within the framework of a hierarchical game with a random second player. It is assumed that the second player’s choice is described by a weighted sum of independent identically distributed random variables. The parametric quantile criterion (Value at Risk) is used as a criterion for the choice of the solution by the first player. The model can be used to justify for the choice of strategy of a large company (the first player) based on the analysis of customer response, represented by several heterogeneous groups of decision makers (the second player). The problem statement is close to a game with nature, but here the response of the second player is not represented by a single random variable, but by a weighted sum of independent random variables. When using the quantile criterion by the first player, in contrast to the mean value criterion, its solution is influenced not only by the parameters of the response distribution but also by the weights that reflect the number and heterogeneity in volume of consumers representing the second player. It is assumed that these weighting coefficients are not precisely specified, and the heterogeneity in volume is described by the following parameters: the maximum and minimum share of an individual decision maker in the total volume, as well as the total number of decision makers. An estimate of the greatest possible variance of a weighted sum of random variables, depending on these parameters, is obtained. Problems with a quantile criterion are usually solved with a fixed probability value, but the choice of probability is not typically justified. It is proposed to find a solution to the problem of parametric maximization of the worst quantile, which under the considered conditions reduces to choosing the maximum from linear functions. An algorithm for solving an optimization problem with a parametric quantile criterion is constructed, and model examples are considered.