Outcome and Risk in a Multi-step Positional Problem under Uncertainty.

In single-criteria problem under strategical uncertainty from the point of view of DM tasks of decision making are examined. DM tries to increase the guaranteed outcome with possible smaller guaranteed risk. We are based on the principle of minimax regret (SavageNichans) with the help of mathematical apparatus of the method of dynamic programming for discrete problems.
First, we examine single-criteria problem of two forms which differs by pairs: contrstrategy — pure uncertainty and pure strategy — strategical uncertainty.
In the problem of the first type, a hierarchical procedure of formation the counterstrategies of DM is used. The discrimination of uncertainty appears: the decision-maker learns the uncertainty which has realized and only then he chooses his strategy, namely for example for single-criteria problem under uncertainty

\[Γ = \langleΣ, U_Z, Z, J(U_Z, Z, x0)\rangle\]

\[Σ ÷ x(k + 1) = f(k, x(k), u(k, x(k), z(k, x(k))), x(0) = x_0~ k = 0, ..., K,\]

\[U_Z = (U_Z(0), U_Z(1), ..., U_Z(K − 1)) ÷ (u(0, x, z), u(1, x, z), ..., u(K − 1, x, z)),\]

\[Z = (Z(0), Z(1), ..., Z(K − 1)) ÷ (z(0, x), ..., z(K − 1, x)),\]

\[U_Z = \{U_Z\}, Z = \{Z\}.\]

\[J(U_Z, Z, x_0) = Φ(x(K)) + \sum_{k=0}^{K - 1} F_i(k, x(k), u[k], z[k]),\]

the functions of risk are defined

\[R(U_Z, Z, x_0) = \max_{U_Z ∈U_Z} J(U_Z, Z, x_0) − J(U_Z, Z, x_0).\]

In the problem of the second type strategic uncertainties are used. They are formed on the assumption of discrimination DM, who transmits them to the researcher selected by him pure strategy for the formation of strategic uncertainty.
In the problem about uncertainties only boundaries of changes are known (any probabilistic characteristics are absent).
In the first case the regret function is constructed, in the second case — guarantee of outcome and risk. Secondly, it is offered the initial single -criteria problem to set in accordance two-criteria discrete positional problem, where the first criteria — the guaranteed outcomes, the second one — “minus” guaranteed risk. For this two-criteria problem we construct the Pareto maximal pure strategy, which defines the value of guaranteed outcomes and guaranteed risk accompanying the realization of guaranteed outcome.
As the example, the explicit form of suggested solution for linear-quadratic one step variant of single-criteria problem is obtained.

Keywords: multi-step problem, management problem strategy, multicriteria problem, guarantees, Pareto maximum.