The guaranteed on risks and regrets solution for a hierarchical model with informed uncertainty

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
\begin{equation*} \Gamma=\langle U, \{ Y[u] \mid u\in U \}, f_0(u,y(u))\rangle. \end{equation*}
In this game $U$ is a set of top-level player strategies (center). Not an empty set $Y[u]$ 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 [3]–[5] 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 $y(u)\in Y[u]$. As a result of this procedure in the game $\Gamma$ there is a situation $(u,y(u))$. In this situation the payoff function value of the center for equal to $f_0(u,y(u))$.

In the game $\Gamma$ center, choosing a strategy $u\in U$ that seeks to maximize its payoff function $f_0(u,y(u))$. A top-level player should consider the possibility of realization of any uncertainty $y(u)\in Y[u]$. 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
\begin{equation*} P=\langle U, \{ R^V(u), R^S(u) \} \rangle. \end{equation*}
In this problem the function
\begin{equation*} R^V(u)=\max\limits_u \min\limits_{y(u)}f_0(u,y(u))-\min\limits_{y(u)}f_0(u,y(u)) \end{equation*}
is a risk on Wald for the center, the function
\begin{equation*} R^S(u)=\max\limits_{y(u)} \Phi_0(u,y(u))-\min\limits_u\max\limits_{y(u)} \Phi_0(u,y(u)) \end{equation*}
is a strategic regret of the center. The regret function is defined by the following equality
\begin{equation*} \Phi_0(u,y(u))=\max\limits_{u\in U} f_0(u,y(u))-f_0(u,y(u)). \end{equation*}
The strategy of the center $u^*\in U$ 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.