Information Technology Reference
In-Depth Information
16.5.2 The Game with an Arbitrary Number of Periods
For an undefined but finite
L
, the payoff matrix has the following appearance:
s
1
s
2
s
3
s
L
...
⎛
⎝
⎞
⎠
h
1
1
+
T
1
1
...
1
h
2
12
+
T
2
...
2
h
3
A
L
,
1
1
2
3
+
T
...
3
=
.
.
.
.
.
.
.
.
h
L
1
2
3
...
L
+
T
We claim that in equilibrium every hider and searcher strategy must have some
positive weight. If this is the case then every hider strategy must provide the same
expected payoff, and this payoff will be the value of the game. Denoting a generic
searcher strategy
s
and a generic hider strategy
h
,where
q
i
again denotes the proba-
bility of the searcher playing strategy
s
i
,and
p
i
=
h
i
p
(
)
the probability of the hider
playing
h
i
:
L
i
=
2
q
i
=
T
,
h
1
T
(
,
s
)=
q
1
(
1
+
T
)+
(16.14)
Since:
L
i
=
2
q
i
=
1
−
q
1
,
(16.15)
we conclude:
1
T
,
q
1
=
1
−
(16.16)
Similarly, from the second row we obtain:
T
−
2
+
q
1
1
T
−
1
T
2
,
q
2
=
=
1
−
(16.17)
T
These two “boundary conditions” on
q
1
and
q
2
will come in handy shortly. The
reader will note that clearly it would be possible to continue in this vein to obtain
every
q
i
as a function of
T
; we could then find a formula for
T
by substituting
into
L
i
=
1
q
i
=
1. One relatively tidy way of doing this is to note that by taking
the difference-in-differences between rows, we obtain a homogeneous recurrence
relation, which must equal zero in equilibrium:
∑
(
T
(
h
x
+
1
,
s
)
−
T
(
h
x
,
s
))
−
(
T
(
h
x
,
s
)
−
T
(
h
x
−
1
,
s
)) =
0
,
(16.18)
1
+
2
T
T
⇒
q
x
−
1
−
q
x
+
q
x
+
1
=
0
,
(16.19)
Search WWH ::
Custom Search