The «value at risk» principle in hierarchical game

The two-player hierarchical game is considered. The bottom level player’s payoff is supposed to depend on random factor. The top level player is supposed to know the set of possible values of uncertain factors and probability measure on this set. And he assumes that the bottom level player knows the realized value of random factor when he chooses his control. The optimality principle is new. It is supposed that top level player wishes to obtain maximal possible result with prescribed probability. In such a way the model permits to take into account the inclination to risk of the top level player. Open loop and closed loop models are investigated. In both cases the original setting of the problem contains non elementary operation of choice of a set of negligible values of uncertain factors. The obtained results permit to replace this operation by the operation of calculating of mathematical expectation of random value. In both models the problem of calculating of maximal guaranteed result of top level player and search of his optimal strategy is reduced to calculating a minimax on “finite-dimensional” sets.