A Model for Managing Activity Constraints

This paper considers a model of a hierarchical system of the Principal–agent type, in which the Principal manages the set of agent’s choices. The model of such a system is a hierarchical two-player game with forbidden situations. In this game, the Principal chooses some subset of a fixed set, whereas the agent chooses his control from this subset. The agent’s payoff explicitly depends on his own choice only, whereas the Principal’s payoff depends both on the agent’s control and on its own choice. The dependence of the Principal’s payoff on his choice is assumed to be monotonic with respect to the inclusion relation on the set of his strategies. The problems formulated below are to calculate the Principal’s maximum guaranteed result, under the assumption of the agent’s benevolence and without this assumption. A new definition of the Principal’s maximum guaranteed result in the game with a benevolent agent is proposed, which remains correct also in the case when the agent’s payoff achieves no maximum for some of the Principal’s strategies. The equivalence of this definition to the classical Stackelberg’s definition is proved for the cases in which the latter is correct. In the general case, the problems formulated below require calculating the maximin with bound constraints on complex infinite-dimensional spaces. Some methods for simplifying these problems are suggested. In the case of a finite basic set, algorithms for solving the problem in a polynomial time with respect to the number of elements in this set are developed. In the case of infinite basic set, the problem is reduced to a sequence of ordinary optimization problems. The methods proposed below can be used to construct and investigate many applications-relevant models of this type.