# The Guaranteed on Risks and Regrets Solution for a Hierarchical Model with Informed Uncertainty

Authors:

The paper formalizes a new model of solution-making under the conditions of uncontrolled (uncertain) factors in the form of a hierarchical game.
The problem of solution-making under uncertainty in the form of a hierarchical game with nature is considered
$$\Gamma = \left \langle U,\left \{ Y\left [ u \right ] | u \in U \right \}, f_{0} \left ( u,y\left ( u \right ) \right ) \right \rangle.$$
In this game $U$ is a set of top-level player strategies (center). Not an empty set $Y\left[ u \right]$ is a set of uncertainties (lower level player strategies, that is, nature). It that can be realized as a result of the chosen center strategy $u \in U$. The basic difference between the game and the known models $\left [ 3 \right ] - \left [ 5 \right ]$ is that nature "reacts" to the choice of the solution maker, changing the area of possible uncertainties.
Solution-making in the game $\Gamma$ is as follows. The first move is made by the top-level player using a certain strategy $u \in U$. The second move is made by nature, which realizes an any informed uncertainty $y\left(u \right)\in Y\left[ u \right]$. As a result of this procedure in the game $\Gamma$ there is a situation $\left(u,y\left(u \right) \right)$. In this situation the payoff function value of the center for equal to $f_{0}\left(u,y\left(u \right) \right)$.
In the game $\Gamma$ center, choosing a strategy $u \in U$ that seeks to maximize its payoff function $f_{0}\left(u,y\left(u \right) \right)$. A top-level player should consider the possibility of realization of any uncertainty $y\left(u \right)\in Y\left[ u \right]$. In this case, it can use different concepts of solution-making in problems under uncertainty.
The article discusses the approach to solution-making in this model, based on the concept of optimality Pareto and the principles of Wald and Savage.
A two-criterion problem is considered
$$P=\left \langle U,\left \{ R^{V}\left ( u \right ) , R^{S} \left ( u \right ) \right \} \right \rangle.$$
In this problem the function
$$R^{V}\left ( u \right ) = max_{u} min_{y\left ( u \right )} f_{0}\left ( u,y\left ( u \right ) \right ) - min_{y\left ( u \right )} f_{0}\left ( u, y\left ( u \right ) \right )$$
is a risk on Wald for the center, the function
$$R^{S}\left ( u \right ) = max_{y\left ( u \right )}\Phi _{0}\left ( u,y\left ( u \right ) \right ) - min_{u} max_{y\left ( u \right )} \Phi _{0} \left ( u,y\left ( u \right ) \right )$$
is a strategic regret of the center. The regret function is defined by the following equality
$$\Phi _{0} \left ( u,y\left ( u \right ) \right ) = max_{u\in U} f_{0}\left ( u, y\left ( u \right ) \right ) - f_{0}\left ( u, y\left ( u \right ) \right ).$$
The strategy of the center $u^{\ast } \in U$ will be called a guaranteed risk and regret solution for the game $\Gamma$, if it is the minimum Pareto solution to the problem $P$.
The article describes an algorithm for constructing a formalized optimal solution. The "performance" of the specified algorithm for finding the regret function and constructing a guaranteed risk and regret solution for the game on the example of a linear-quadratic optimization problem in terms of possible supply of imported products to the market is investigated.
Keywords: hierarchical game under uncertainty, Pareto minimum, risk function, regret function,
informed uncertainty

UDC:
519.83