Geoscience Reference
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takes place, and as a result of the k-step process a total win of system N can be
described by the equation:
Q
k
¼ Q
k
u
0
;
½
u
1
;
...
;
u
k
1
;
m
0
; m
1
;
...
; m
k
1
;
M
a
ð
0
Þ;
M
b
ð
0
Þ
¼
X
k
1
f
m
a
ð
n
Þ
p
1
½N
Ra
ð
n
Þ
m
b
ð
n
Þ
p
2
½N
Rb
ð
n
Þ
g
ð
10
:
20
Þ
n¼0
There are several methods with the aid of which an analysis can be made of this
k-step process. One can consider this k-step game as a one-step game, in which case
system H must select simultaneously a plurality of the vector {u
0
,u
1
,
…
,u
k-1
}, and
system N, a plurality of the vector {v
0
,v
1
,
,v
k-1
}, where the selection of u
k
and v
k
depends on the previous values obtained according to Eq. (
10.14
).
For a solution of this quite complex problem, it is suggested to substitute for it a
similar problem of multi-step optimization. The values of this game can be
expressed as:
…
Z
Q
k
dF
ð
v
0
;
V
k
¼ max
F
mi
G
f
v
1
;
...
;
v
k
1
Þ
dG
ð
u
0
;
u
1
;
...
;
u
k
1
Þg
¼ mi
G
ma
F
f
...
g
ð
10
:
21
Þ
where, the distribution functions F and G are determined on the boundaries of the
complex form:
0
v
0
M
b
ð
0
Þ
0
u
0
M
a
ð
0
Þ
0
v
1
M
b
ð
1
Þ
0
u
1
M
a
ð
1
Þ
...
...
0
v
k
1
M
b
ð
k
1
Þ
0
u
k
1
M
a
ð
k
1
Þ
ð
10
:
22
Þ
By utilizing the optimality principle and taking into account the dependence
V
k
=V
k
[M
a
(0), M
b
(0)], we obtain the following functional equation:
V
n
þ
1
½M
a
ð
0
Þ;
M
b
ð
0
Þ
ZZ
f
R
ð
u
;
v
Þþ
n
½M
a
ð
n
1
Þ;
M
b
ð
n
1
Þg
dF
ðmÞ
dG
ð
u
Þ
¼ mi
G
ma
F
½
:::
¼ max
F
mi
G
½
0
u
M
a
ð
n
1
Þ
0
v
M
b
ð
n
1
Þ
ð
10
:
23
Þ
where
ZZ
V
1
½M
a
ð
0
Þ;
M
b
ð
0
Þ
¼ max
F
mi
G
½
R
ð
u
;
v
Þ
dF
ð
v
Þ
dG
ð
u
Þ
¼ mi
G
ma
F
½
...
0
u
M
a
ð
0
Þ
0
v
M
b
ð
0
Þ
ð
10
:
24
Þ
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