Ìàòåìàòèêà / 5. Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå

Romanuke V. V., c. t. s., associate professor

Khmelnytskyy National University, Ukraine

On all possible versions of relationships of relative uncertainties in four-dimensional problem of minimizing maximal imbalance with right-nonregular and left-nonregular two components of projector optimal strategy

A four-dimensional problem of minimizing maximal imbalance [1] lies in solving the antagonistic game with the kernel

(1)

as a model of uncertainties removing, defined on the Cartesian product of the parallelepiped

(2)

of pure strategies

(3)

of the first player and of the type (2) other parallelepiped of pure strategies

(4)

of the second player, where variables and in (1) due to

, (5)

are excluded. The known minimax procedure [1] for finding the optimal behavior

(6)

of the second player as the projector drives to solving the equation

, (7)

giving the optimal game value and regular components

, (8)

of (6) by the condition

. (9)

However, it may be occurred that for some specific cases of the ends and pre-evaluation [2], there may be violated one of the memberships (9). Will consider the violation with

, (10)

for by . This will fix right-nonregular and left-nonregular two components of projector optimal strategy (6) as the equation (7) of relative uncertainties becomes impracticable. Then there are the following versions of relationships of relative uncertainties, arising due to inequalities (10):

, (11)

, (12)

, (13)

, (14)

, (15)

where by . Obviously, the inequalities (11) — (15) should be equalized by the projector as even as possible. And it is clear that such equalization will produce incalculable number of its optimal strategies (6). So, right-nonregular and left-nonregular two components of projector optimal strategy (6) may be selected from a continuum of pure strategies by some subset of the set or . For instance, this concerns the right-nonregular -th component, being determined from (15), and the left-nonregular -th component, being determined from one of the inequalities (11) — (14). The only question that matters is that how to select them properly or, for more strictness, rationally.

References

1. Âîðîáü¸â Í. Í. Òåîðèÿ èãð äëÿ ýêîíîìèñòîâ-êèáåðíåòèêîâ / Âîðîáü¸â Í. Í. — Ì. : Íàóêà, Ãëàâíàÿ ðåäàêöèÿ ôèçèêî-ìàòåìàòè÷åñêîé ëèòåðàòóðû, 1985. — 272 ñ.

2. Romanuke V. V. Digression on the right off-bound projector optimal strategy in four props construction being pressed uncertainly / V. V. Romanuke // Ñèñòåìè îáðîáêè ³íôîðìàö³¿. — 2011. — Âèïóñê 2 (92). — Ñ. 129 — 132.