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cult problem. Therefore, not
without reason the expression curse of dimensionality was introduced. Next we
shall present the solution of some speci
The solution of these functional equations is a dif
c cases. This task is solved in more detail
by Nitu et al. (2000a). For the solution of the proposed problem, we shall begin
with the case where in both systems the C- and R-elements are indistinguishable.
The diagram of interaction between systems H and N can then be represented in
such a form as shown in Fig.
10.2
. By taking into account the designations used in
this diagram, we obtain that the number of C
a
-and C
b
-elements that have reached
the C
b
-and C
a
-elements of the opposite system in a one-step operation will be
respectively: max{0, a
C
-N
Rb
} and max{0, b
C
-N
Ra
}. Therefore, the numbers of C
a
-
and C
b
-elements that are put out of operation on the average are max{0, b
C
-N
Ra
}
p
1
[N
Ra
] and max{0, a
C
-N
Rb
}p
2
[N
Rb
]. Consequently, after one step of the game the
following numbers of C
a
-and C
b
- elements remain in the system that are not
put out of operation: max[0, M
a
-max(0, b
C
-N
Ra
)p
1
[N
Ra
]]p
1
[N
Ra
], max[0, M
b
-max
(0, a
C
-N
Rb
)p
2
[N
Rb
]]p
2
[N
Rb
]. Going over to the multi-step situation, we introduce the
win function (
2.16
). The aim of system N is to destroy system H, or more precisely,
to put out of operation all of the latter
'
s a-elements. For this purpose the system
N cannot use all C
b
-elements, since it will then be left defenseless against the C
a
-
elements. An analogous situation holds true for the system H. Each system is
compelled to release after each successive step of the game a certain number of
working a-and b-elements. The magnitude of this payoff is proportional to the
difference M
b
−
b
C
(M
a
−
a
C
−
N
Ra
). Now the problems on both sides
become clear. The system N must strive to provide such a number of C
b
-elements in
all directions so as to maintain the maximum value of this difference. In other
−
N
Rb
−
Fig. 10.2 Schematic diagram of the case of mutual interchangeability
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