The guaranteed on risks and regrets solution for a hierarchical model with informed uncertainty
A. E. Bardin,
Yu. N. Zhiteneva Moscow State Regional Institute of Humanities, Orekhovo-Zuevo, Moskovskaya obl.
Abstract:
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.
Keywords:
hierarchical game under uncertainty, Pareto minimum, risk function, regret function, informed uncertainty.
UDC:
519.83
MSC: 91A65