Hierarchical model of competition under uncertainty

Oligopoly is a basic concept in the theory of competition. This structure is the central object of research in the economics of markets. There are many mathematical models of the market that are formalized in the form of an oligopoly in economic theory.

The Cournot oligopoly is an elementary mathematical model of competition. The principle of equilibrium formalizes the non-cooperative nature of the conflict. Each player chooses the equilibrium strategy of behavior that provides the greatest profit, provided that the other competitors adhere to their equilibrium strategies.

The Stackelberg model describes a two-level hierarchical model of firm competition. The top-level player (center, leader) chooses his strategy, assuming reasonable (optimal) decision-making by the lower-level players. Lower-level players (agents, followers) recognize the leadership of the center. They consider the center's strategies known. These players choose their strategies, wanting to maximize their payoff functions. This hierarchical structure is from a game point of view a case of a hierarchical game $\\Gamma_1$.

The indefinite uncontrolled factors (uncertainties) are the values for which only the range of possible values is known in this paper. Recently, studies of game models under uncertainty have been actively conducted. In particular, non-coalitional games under uncertainty are investigated.

The concepts of risk and regret are formalized in various ways in the theory of problems with uncertainty. At the same time, the decision-maker takes into account both the expected losses and the possibility of favorable actions of factors beyond his control.\nThis article examines the two-level hierarchical structure of decision-making in the problem of firm competition. A linear-quadratic model with two levels of hierarchy is considered. This model uses the concepts of Cournot and Stackelberg under uncertainty. Uncontrolled factors (uncertainties) are identified with the actions of the importing company.

The Wald and Savage principles are used to formalize the solution. According to Wald's maximin criterion, game with nature is seen as a conflict with a player who wants to harm the decision-maker as much as possible.\n\nSavage's minimax regret criterion, when choosing the optimal strategy, focuses not on winning, but on regret. As an optimal strategy, the strategy is chosen in which the amount of regret in the worst conditions is minimal.

A new approach to decision-making in the game with nature is formalized. It allows you to combine the positive features of both principles and weaken their negative properties. The concept of $U$-optimal solution of the problem in terms of risks and regrets is considered.\nThe problems of formalization of some types of optimal solutions for a specific linear-quadratic problem with two levels of hierarchy are solved.