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.