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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.
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