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agent
i
takes the same action based on the same strategy both in
AG
and in
MG
. Thus,
we have
C
AG
(
t
)=
C
MG
(
t
)
at time
t
.
Next, we show that
C
AG
(
t
)
⊆
C
MG
(
t
)
at time
t
∈
T
. Notice that all the agents in
MG
increase the selling strategies for
h
m
(
t
m
. On the other hand, notice that
the agents in
AG
that have smaller mean asset values
v
(
t
)
than the price
p
(
t
)
increase
the selling strategies for
h
m
(
t
−
1) =
{
1
}
m
. Since every contrarian refers to the same
−
1) =
{
1
}
m
) of the strategy, he does not change his decision during the interval
T
.
If an agent increases the selling strategies in
AG
, it also increases the selling strategies
in
MG
. Thus we have
C
AG
(
t
)
part (i.e.,
{
1
}
C
MG
(
t
)
at time
t
∈
⊆
T
.
The similar argument holds for
C
MJ
(
t
)
C
AG
(
t
)
.
⊆
We call an agent a
bi-strategist
if he can take both buy and sell actions, that is, has
strategies
R
i,a
containing both actions, for
h
m
(
t−
m
.
The following lemma states that there is a time lag between the price rising and the
action of agent's payoff function.
m
or
h
m
(
t−
1) =
{
1
}
1) =
{−
1
}
Lemma 1.
In AG, suppose that a history
H
contains
h
m
(
t
m
. Even if a
bi-strategist keeps the opposite action of the payoff function for
H
p
steps, he takes the
same action as the payoff function after the
H
p
+1
-st step.
−
1) =
{
1
}
Proof.
Suppose that a bi-strategist
i
has a strategy
R
i,a
1
(resp. and a strategy
R
i,a
2
)
which takes the opposite action of (resp. the same action as) the payoff function. If
i
adopts the strategy
R
i,a
1
now, the score difference between
R
i,a
1
and
R
i,a
2
is at
most
2
H
p
. Since the difference decreases by 2 for a step, the scores of
R
i,a
1
and
R
i,a
2
becomes the same point at the
H
p
-th step. Then, after the
H
p
+1
-st step, he takes the
strategy
R
i,a
2
.
For simplicity, we assume that the size of
H
p
is greater than
m
enough.
Lemma 2.
In AG, suppose that a history
H
contains
h
m
(
t
m
. For any time
−
1) =
{
1
}
steps
t
1
,t
2
∈
T
=(
t,...,t
r
−
1)
,where
t
1
<t
2
, we have
C
AG
(
t
1
)
C
AG
(
t
2
)
.
⊆
Proof.
Suppose that agent
i
belongs to
C
AG
(
t
1
)
. We show that once the rising price
p
(
t
1
)
overtakes the mean asset value
v
i
(
t
1
)
of agent
i
,
v
i
(
t
1
)
will not overtake
p
(
t
1
)
as
long as
p
(
t
1
)
is rising. Since
v
i
(
t
)=
a
(
p
v
)
u
+
a
−
a
u
+
a
<
1
,
v
i
(
t
+1)
−
>
0
and
0
<
p>v
holds as long as
p
(
t
1
)
is rising. Thus, agent
i
is contrarian at time
t
1
+1
.We
have
C
AG
(
t
1
)
C
AG
(
t
1
+1)
, and can inductively show
C
AG
(
t
1
)
C
AG
(
t
2
)
.
⊆
⊆
We say that the bubble is
monotone
if
h
m
(
t
m
holds for any
t
−
1) =
{
1
}
∈
T
=
(
t,...,t
r
−
1)
.
Lemma 3.
In AG, as long as more than half population are bi-strategists, the price in
a monotone bubble will reach the upper bound.