The "Value at Risk" Principle in Hierarchical Game

Abstract

A hierarchical game of two persons with random factors is considered. It is assumed that the top-level player has the right of the first move. It is believed that the lower level player at the time of decision making knows exactly the realization of the random factor and the choice of partner. And the top-level player at the time of decision making knows only a probabilistic measure on the set of values ​​of an uncertain factor. The principle of optimality is new: it is believed that a top-level player is ready to neglect some of the "unpleasant" events, the total probability of which is given, but otherwise he is careful. Under these assumptions, the maximum guaranteed result of the top-level player is calculated. The structure of strategies providing such a result is clarified. Two cases were investigated: a game with and without feedback. To solve the problem, an original definition of the maximum guaranteed result is proposed. It is equivalent to the classical definition, but is simpler. Using this technique, solving of the problem reduces to identical transformations of the formulas for predicate calculus. As a result of the solution, the optimal strategy and the set of "unpleasant" cases which are excluded from consideration search task is reduced to calculating multiple maximins on finite-dimensional spaces. In this case, the operation of calculating the expected value with respect to given probabilistic measure is considered to be "elementary". Models of this type can have different interpretations. One can use them for methodological justification of the principle of maximum guaranteed result. One can use them when solving risk management tasks. One can consider them as models for managing the "customer base" of the service company. The proposed method allows to study such models at a qualitative level, and in some cases to obtain quantitative results.