Two-level hierarchical model of competition between three firms under uncertainty.

The paper considers a problem of competition between three manufacturing firms in the
market of homogeneous infinitely divisible products. It is assumed that the nature of the
interaction of manufacturing firms in the market has a hierarchical structure. Namely, one of
the companies, the leader company, is the leading manufacturer and is the first to decide on the
volume of product deliveries to the market. While the other two firms decide on the number of
products delivered after the leading firm and must take into account the volume of its deliveries.
Taking into account such a hierarchy in the interaction of firms leads to the need to formalize
the problem in the form of a two-level hierarchical game. In this case, the leader firm is identified
with the top-level player, and the other two firms are identified with the lower-level players. In
addition, it is assumed that the cooperation of lower-level players is impossible. As a result, when
formalizing the optimal solution for lower-level players, the concept of Nash equilibrium from
the game theory is used.
In addition to the above, the problem under consideration assumes the presence of
uncontrolled uncertain factors, about which only a set of possible values is known, and there are
no probabilistic characteristics. The presence of uncertainty in the framework of this problem
is interpreted as the presence of an importing company on the market, the volume of products
supplied by which is not known in advance by any of the manufacturing companies. However,
it is possible for them to estimate the limits of the estimated volume of imports. The presence
of uncertainty obliges all manufacturing firms to take this fact into account and, as a result, in
their choice of the optimal solution, use one of the principles of the theory of decision-making
under uncertainty. The paper considers the case when one of the lower-level players uses the
Wald principle (maximin principle, the principle of guaranteed results), and the second one is
guided by the Savage principle (the principle of minimax regret) when choosing his decision. For
a top-level player --- a leading firm --- two cases are considered in this paper. The first is when
the player uses the Wald principle, and the second is the Savage principle.
Thus, the problem described in this paper is formalized as a two-level hierarchical game with
uncertainty. For the game in this setting, the algorithm for constructing the proposed optimal
solution is described and its explicit form is found for a specific type of payoff functions of all
participants in the game. In addition, coefficient criteria for the existence of an optimal solution
are obtained.

Keywords: hierarchical game, uncertainty, Nash equilibrium, Wald's principle, Savage's principle.