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Proof. First, the mean asset values that has been overtaken by the price will not exceed
the price again from the proof of Lemma 2.
Second, any bi-strategist i with v i ( t ) >p ( t ) will take a buying action in the H p +1
steps from Lemma 1. Since v i ( t +1)
1) / ( u + a ) < 0 , the mean asset
value decreases. Thus, the rising price will eventually reach the greatest mean asset
value in the set of contrarians.
Third, since all the bi-strategists increase the selling strategies, they will take selling
actions in H p +1 steps. After that, A/N < 0 holds and the price falls down.
v i ( t )= a ( p
From Lemma 3, the following theorem is straightforward.
Theorem 2. In AG, as long as more than half population are bi-strategists, the mono-
tone bubble will terminate.
4
Simulation
Here we present simulation results by using the basic constants in Table 1 2 .
Ta b l e 1 . Basic constants
Symbol
Meaning
Va l u e
N
Number of agents
501
S
Number of strategies
4
m
Memory size
4
H p
Score memory
4
T
Number of steps
5000
Initial agent's money 10000
Initial agent's assets
100
r
Investment rate
0.01
Our first question with respect to ExAG is :
1. What values are suitable for the constant λ and the threshold K in ExAG ?
Our next question with respect to AG is :
2. How does the inequality of wealth distribution vary in AG ?
Then, our further questions with respect to several games are as follows.
3. How widely do the Pareto indices of games differ from practical data ?
4. How widely do the skewness / kurtosis of games differ from practical data ?
5. How widely do the volatilities differ in several games ?
6. How widely do the volatility autocorrelations differ from practical data ?
For the first issue, Figure 3 shows the patterns of price behavior for three kinds of λ
values. From the definition of the direct action probability (see (11)), the smaller the λ
2
We repeated the experiments up to 30 times and obtained averaged results.
 
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