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C
∈
X
x
(
C
)=
1. Similarly, a mixed strategy for
such that
x
(
C
)
≥
0forall
C
∈
X
and
player II will be given by a function
y
:
Y
→
R
C
∈
Y
y
(
C
)=
1. When the players use their mixed
such that
y
(
C
)
≥
0forall
C
∈
Y
and
strategies
x
and
y
, the payoff
M
(
x
,
y
)
is the expected value of
M
(
A
,
B
)
.
n
,
m
and
A very simple lattice game
G
=(
X
,
Y
,
M
)
is the following,
X
=
Y
⊂ F
1if
A
=
B
,
M
(
A
,
B
)=
(7.2)
0if
A
=
B
.
It is easy to see that an optimal strategy for both players is the uniform distribution
on the set of their pure strategies and the value of the game
v
is given by
1
v
=
|
,
|
X
but, to completely solve this game we have to know the cardinality of the set
X
.
Given
A
∈ F
n
,
m
we denote its increments by
Δ
A
(
i
)
,
Δ
A
(
i
)=
A
(
i
+
1
)
−
A
(
i
)
for
i
=
1
,
2
,...,
n
−
1and
Δ
A
(
n
)=
A
(
n
)
−
A
(
1
)
. We consider the following sets:
0
n
,
m
F
=
{
A
∈ F
n
,
m
:
A
(
i
)
∈{
0
,
1
,−
1
},
i
=
1
,...,
n
−
1
},
(7.3)
1
n
,
m
F
=
{
A
∈ F
m
:
A
(
i
)
∈{
0
,
1
,−
1
,
m
−
1
,
1
−
m
},
n
,
i
=
1
,...,
n
−
1
},
(7.4)
n
F
=
{
A
∈ F
n
,
m
:
A
(
i
)
∈{
0
,
1
,−
1
},
i
=
1
,...,
n
},
(7.5)
,
m
n
F
=
{
A
∈ F
n
,
m
:
A
(
i
)
∈{
0
,
1
,−
1
,
m
−
1
,
1
−
m
},
i
=
1
,...,
n
}
(7.6)
,
m
2
n
,
m
0
n
,
m
3
n
,
m
1
n
,
m
Clearly
F
⊂ F
⊂ F
n
,
m
and
F
⊂ F
⊂ F
n
,
m
. The elements of the
n
set
F
m
can be interpreted as paths on the cylinder, that is paths on the lattice
,
{
1
,
2
,...,
n
+
1
}×{
1
,
2
,...,
m
}
where the points
(
1
,
j
)
and
(
n
+
1
,
j
)
are considered
n
to be the same point. In a similar way the elements of the set
m
can be interpreted
as paths which can surround a cylinder one or more times and the elements of
F
,
n
F
m
as paths which can surround a cylinder or as paths on a torus. Figure
7.1
shows a
,
representation of the element
(
1
,
6
)
,
(
2
,
7
)
,
(
3
,
8
)
,
(
4
,
9
)
,
(
5
,
10
)
,
(
6
,
1
)
,
(
7
,
2
)(
8
,
2
)
,
3
1
(
9
,
3
)
,
(
10
,
4
)
,
(
11
,
3
)
,
(
12
,
4
)
,
(
13
,
5
)
}∈F
13
,
10
⊂ F
13
,
10
⊂ F
13
,
10
on the lattice
n
n
n
L
=
{
1
,
2
,...,
13
}×{
1
,
2
,...,
10
}
and on the torus. The sets
F
m
,
F
m
,
F
m
and
,
,
,
n
,
m
appear in many situations in which the path of a person needs to be described.
F
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