Mathematical models of decision making under risk with ordered outcomes

We consider mathematical models of decision making under risk as follows. Realization structure of the model consists of a set of strategies $X$, a set of states of an environment $Y,$ set of outcomes $A$, realization function $F:X\times Y\to A$ and some distribution of probabilities on set $Y$. Goal structure of the model by partial order relation $\omega $ on the set of outcomes $A$ is given. Fix a strategy $x\in X$ we can consider some probabilistic vector of standard simplex $S(A)$ and the order relation $\omega $ may be extended on $S(A)$. An optimal solution we mean as a strategy $x^* \in X$ that the corresponding probabilistic vector is a maximal element under extension of the order $\omega $. The main problem of the article is development of method for finding of optimal solutions in considered model. In the case the sets $X$, $Y$, $A$ are finite, we propose some algorithm for solving of this problem. Every step of the algorithm is based on the check of solvability some finite system of linear inequalities.