Toward an Integration of Social Learning and Individual Learning in Agent-Based Computational Stock Markets: The Approach Based on Population Genetic Programming

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Toward an Integration of Social Learning and Individual

Learning in Agent-Based Computational Stock

Markets: The Approach Based on Population Genetic

Programming

∗

Chia-Hsuan Yeh

AI-ECON Research Center

Department of Information Management

I-Shou University

Kaohsiung County, Taiwan 84008

Tel&Fax: 886-7-3552758

E-mail: spockyeh@ms38.hinet.net

WWW: http://econo.nccu.edu.tw/

∼spock/

Shu-Heng Chen

AI-ECON Research Center

Department of Economics

National Chengchi University

Taipei, Taiwan 11623

Tel: 886-2-29387308

Fax: 886-2-27386874

E-mail: chchen@nccu.edu.tw

Abstract

Artificial stock market is a growing field in the past few years. The essence of this issue is the interaction between many heterogeneous agents. In order to model thiscomplex adaptive system, the techniques of

evolutionary computation have been employed. Chen and Yeh (2000) proposed a new architecture to con-struct the artificial stock market. This framework is composed of a single-population genetic programming (SGP) based adaptive agent with a SA (Simulated Annealing) learning process and abusiness school.

However, one of the drawbacks of SGP-based framework is that the traders can’t work out new ideas by themselves. The only way is to consult researchers in the business school. In order to make the traders more intelligent, we employ multi-population GP (MGP) based framework with the mechanism ofschool.

This extension is not only reasonable, but also has the economic implications. How do the more intelligent agents influence the economy? Are the econometric properties of the simulation results based on MGP more like the phenomena found in the real stock market? In this paper, the comparison between SGP and MGP is studied from two sides. One is related to the micro-structure, traders’ behavior and believe. The other is macro-properties, the properties of time series. The line of research is helpful in understanding the foundation of economics and finance, and constructing more realistic economic models.

Keywords: Evolutionary Computation, Genetic Programming, Agent-Based Modeling, Artiﬁcial Stock

Mar-ket, Simulated Annealing

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1 Background and Motivation

Artificial stock market has been a hot topic in the fields of agent based computational economics and finance. Genetic algorithms, artificial neural net and genetic programming have been used to model this framework. The main difference between these approaches is twofold.

• representation

• social learning vs. individual learning

Different representation constitutes different strategy space. Similarly, different style of learning explains different kind of human behavior. Both of them may induce different phenomena. Therefore, in order to obtain meaningful and reasonable results, employing the appropriate representation and learning behavior are the most important steps in studying agent-based computational economics and finance. In Lucas (1986),

In general terms, we view or model an individual as a collection of decision rules (rules that dictate the action to be taken in given situations) and a set of preferences used to evaluate the outcomes arising from particular situation-action combinations. These decision rules are continuously under review and revision; new decision rules are tried and tested against experience, and rules that produce desirable outcomes supplant those that do not. (pp. 217)

From the viewpoint of representation, if a decision rule can hopefully be written and implemented as a computer program, and since every program in terms of its input-output structure can be understood as a function. Then, based on the language of LISP program, every function can be represented as a LISP S-Expression, and hence a parse tree. This representation of decision rule is exactly what genetic programming does. Consequently, the Lucasian adaptive economic agent can be modeled as

• evolving a population of decision rules • evolving a population of functions • evolving a population of programs

• evolving a population of LISP S-Expression • evolving a population of parse trees

Moreover, from the perspective of genetic programming, these decision rules can be reviewed and revised under the genetic operators (including reproduction, crossover and mutation). The performance of new decision rules are validated based on the ﬁtness function. Selection is then conducted under the survival-of-the-ﬁttest principle which approximates the concept of rules producing desirable outcomes supplant those that do not. Therefore, we can restate the Lucasian economic agent in terms of the language of genetic programming:

In general terms, we view or model an individual as a population of LISP programs (which is generated by a set of function and terminals) and a set of ﬁtness functions used to evaluate the performance arising from particular LISP programs. These programs are continuously under review and revision based on the genetic operators, new programs are tried and tested against experience, and selection is conducted according to the survival-of-the-ﬁttest principle

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However, when we extend this idea to model a society of economic agents, a population of genetic programming is then employed. Each agent in this society is formed by a genetic programming. The action is determined by his own decision rules (strategies) and ﬁtness function(s). The social and economic activities are the aggregate phenomena generated from these agents’ interaction and coordination. This is the concept of multi-population genetic programming (MGP) which is distinguished from single-population genetic programming (SGP). In Vriend (2000), the implications of SGP/SGAs and MGP/MGAs are distinguished from social and individual learning. In other words, in social learning, agents learn from other agents’ experience, whereas in individual learning, agents learn from their experience and thinking. Therefore, what Lucas (1986) mentioned focused

on individual learning. Moreover, due to the criticisms given by Harrald (1998),1it further demonstrated the

importance of multi-population GP/GAs.2 However, there also exists problems in the MGP/MGAs modeling.

The important phenomena found in economic activities, such as following the herd and rumors dissemination, or the style of social learning, are totally missing in the architecture of MGP/MGAs. In the past few years, MGAs is widely employed in the economic modeling. Although it passes the Harrald’s criticisms, whereas the concept of Lucasian economic agent is not well captured in the representation of GAs. Also, the traditional representation of GAs is the ﬁxed-length bit string, the important economic activity, such as innovation and creation, can’t be modeled by GAs. Therefore, genetic programming seems to be the best tool to model the Lucasian economic agent.

The problem mentioned above is twofold. Firstly, SGP ignores individual learning. Moreover, the tradi-tional approach of SGP fails to pass the Harrald’s criticism. Secondly, the architecture of MGP merely focuses on individual learning. Therefore, we need a new architecture to integrate both of the key feature of social learning and individual learning. In Chen and Yeh (2000), a new architecture was proposed to solve Harrald’s criticism. The mechanism of “school” is introduced into SGP framework. However, it is still a type of social learning. The concept of Lucasian adaptive economic agent is not modeled. In this paper, we extend the pre-vious research to a new architecture which is a multi-population GP framework with the mechanism of school. This framework passes Harrald’s criticism, it also integrates both of social learning and individual learning. This extension is very reasonable, while the economic implications of this extension are more important.

In Lucas (1986), he didn’t mention how many decision rules are necessary (or enough) to model an indi-vidual. To some extent, this is related to the memory or intelligence. In other words, the more intelligent the traders are, the more decision rules (ideas) they have. From this perspective, the traders in our previous research (Chen and Yeh, 2000) are very naive. Each trader has only one idea in his mind. They are unable to work out new ideas by themselves. The only source of knowledge is the business school. On the other hand, the traders in this paper are more intelligent. Each trader has k (for example, 20) ideas. They have the ability of reasoning and imagination to create useful strategies. Therefore, we are interested in how the level of traders’ intelligence inﬂuences the economy. Is the economy more stable (or unstable) if the traders are more intelligent? Is the interaction between these more intelligent agents (or inherent complexity) more com-plicated (or simpler)? Consider a market composed of many traders. Each trader is modeled by a computer. The traders in our previous framework (Chen and Yeh, 2000) behave like basic computers with the ability of validation only. They have no way to reason. That means, the traders have only the basic intelligence. On the other hand, the traders in this paper behave like supercomputers (very intelligent traders). They could

1_{He mentioned that the traditional distinction between the phenotype and genotype in biology and doubted whether the}

adaptation can be directly operated on the genotype via the phenotype in social processes. In other words, it is not easy to justify why we can learn or know other agents’ strategies (phenotype) by means of their actions (phenotype).

2_{Arifovic (1995a, 1996), Miller (1996), Vila (1997), Arifovic, Bullard and Duﬀy (1997), Bullard and Duﬀy (1998a, 1998b, 1999),}

Staudinger (1998) are examples of SGA, while Andrews and Prager (1994), Chen and Yeh (1996, 1997a, 1997b, 1998), and Chen, Duﬀy and Yeh (1996) are examples of SGP. Examples of MGA can be found in Palmer et al. (1994), Tayler (1995), Arthur et al. (1997), Price (1997), Heymann, Pearzzo and Schuschny (1998).

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be the econometricians at school, or the technicians in the stock market. They have the powerful abilities in computing (in computation complexity and speed) and reasoning. The interesting questions is, how are these supercomputers (more intelligent traders) diﬀerent from those basic computers (less intelligent traders)? Of course, they may quickly discover the useful knowledge. Does it imply more eﬃcient market just what the textbooks told us? If so, what is the reason behind it? Is it because they quickly realize the fundamental of market? Or the interaction between these intelligent traders makes the market more complicated such that they can’t predict others’ behavior and market dynamics well. In principle, the knowledge base of these intelligent traders is larger, so they have higher probabilities to get useful information. In other words, their adaptability is much higher than those who are less intelligent. Therefore, the survivability is also higher. However, they may cause the market more complicated beyond control. Their survivability is then reduced. Which one is the most possible outcome needed to be studied.

As to the macro-phenomena, we also care about whether the econometric properties of time series are aﬀected by the level of traders’ intelligence. Does it still have the stylized facts found in our previous simula-tions? More importantly, are the emergent properties more richer? If the answer is yes, what is the driving force behind it? If no, how does it happen?

The psychological activities of each trader are also important. When does the trader intend to look for new strategies? When does he decide to do it by himself? However, researchers and traders may have different focuses. Researchers care about the accuracy of prediction (prediction accuracy oriented), while traders care about how to make money (profit oriented). Both of these criteria don’t necessarily lead to the same goal. It may cause traders to make wrong decisions when they consult with researchers. This further impacts the market dynamics. In the real world, there are many different types of traders. Sometimes, they could be “econometricians”, “technicians” or none of them. The diversity of traders’ behavior and believe is the key phenomenon in economics and financial market, it is what the conventional approaches lack. In the more general framework proposed in this paper, we can study the dynamics of interaction between these different types of traders and their impact on the market dynamics. Of course, in a more general point of view, according to Lucas (1986), each trader has a set of preferences to evaluate outcomes arising from particular situation-action combinations. Therefore, the performance of traders’ situation-action may be influenced by several criteria. They could be profit, prediction accuracy, or others. Basically, these criteria could also evolve over time. However, it is not easy to model this idea. Also, the architecture used in this paper is more complicated than that used in Chen and Yeh (2000). Therefore, we have to be very prudent when we employ more complicated model.

The questions proposed above are not easy to answer. Of course, we can’t have the answers by means of deductive processes. The architecture used in this paper provides us the chance to understand these issues. We can trace the simulations step by step, record the information about the dynamics of traders’ behavior, including the prediction about the stock price in the next period, the amount of stock holding, the strategies they use, how sophisticated the strategies are...., and so on.

In the past few years, the new field of artificial stock market has been emphasized by several researchers. The reason of this field growing fast is that it opens a broader view, so we can study basic problems in the financial stock market. For example, why are the herd behavior, volatility clustering (autoregressive conditional heteroskedasticity, ARCH), excess kurtosis (fat-tail distribution), bubbles and crashes, chaos, and unit roots usually found in the financial markets? And, how do they happen?3

The stock market is known as a complex adaptive system, the traditional techniques which are top-down perspective can’t serve this purpose. Furthermore, the technique tends toward the agent-based modeling which

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is a bottom-up approach. Such idea is more appropriate and reasonable to model social or economic activities.

Genetic programming serves this task better than other evolutionary techniques.4 It is not only a new approach,

but also a methodological innovation to economics.

In sum, the advantages of this new framework is as follows: • It captures the concept of Lucasian adaptive economic agent.

• It allows us to discuss the inﬂuence of the level of traders’ intelligence.

• It allows more rooms to discuss the psychological activities which determine when the traders visit “school” to improve their strategies and when they would like to modify their thought by themselves. In this primary study, we focus on the characteristics of multi-population GP with the mechanism of school. Besides replicating the stylized facts, the issues related to the comparison between SGP-based and MGP-based simulations are discussed. In Section 2, the analytical model of our artiﬁcial market is described. The experimental design is provided in Section 3. In Section 4, we analyze the simulation results and the concluding remarks is given in Section 5.

2 The Framework of Artificial Stock Market

The basic framework of the artiﬁcial stock market considered in this paper is the standard asset pricing model employed in Grossman and Stiglitz (1980). The dynamics of market is determined by an interaction of many heterogeneous agents. Each of them, based on his forecast of the future, maximize his expected utility.

2.1 Description about Traders

For simplicity, we assume that all traders share the same constant absolute risk aversion (CARA) utility function,

U (W_i,t) =−exp(−λW_i,t) (1)

where Wi,t is the wealth of trader i at time period t, and λ is the degree of relative risk aversion. Traders can

accumulate their wealth by making investments. There are two assets available for traders to invest. One is the riskless interest-bearing asset called money, and the other is the risky asset known as the stock. In other words, at each period, each trader has two ways to keep his wealth, i.e.,

Wi,t= Mi,t+ Pthi,t (2)

where Mi,tand hi,t denotes the money and shares of the stock held by trader i at time t. Given this portfolio

(M_i,t,h_i,t), a trader’s total wealth W_i,t+1 is thus

Wi,t+1= (1 + r)Mi,t+ hi,t(Pt+1+ Dt+1) (3)

where P_t is the price of the stock at time period t and D_tis per-share cash dividends paid by the companies

issuing the stocks. D_tcan follow a stochastic process not known to traders. Given this wealth dynamics, the

goal of each trader is to myopically maximize the one-period expected utility function,

E_i,t(U (W_i,t+1)) = E(−exp(−λW_i,t+1)| I_i,t) (4)

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subject to

W_i,t+1= (1 + r)M_i,t+ h_i,t(P_t+1+ D_t+1), (5)

where E_i,t(.) is trader i’s conditional expectations of W_t+1given his information up to t (the information set

Ii,t), and r is the riskless interest rate.

It is well known that under CARA utility and Gaussian distribution for forecasts, trader i’s desire demand,

h∗_i,t+1 for holding shares of risky asset is linear in the expected excess return:

h∗_i,t= Ei,t(Pt+1+ Dt+1)− (1 + r)Pt

λσ_i,t2 , (6)

where σ2_i,t is the conditional variance of (P_t+1+ D_t+1) given I_i,t.

The key point in the agent-based artiﬁcial stock market is the formation of Ei,t(.). In this paper, the

expectation is modeled by genetic programming. The details is described in next section.

2.2 The Mechanism of Price Determination

Given h∗_i,t, the market mechanism is described as follows. Let b_i,t be the number of shares trader i would like

to submit a bid to buy at period t, and let o_i,t be the number trader i would like to oﬀer to sell at period t.

It is clear that

b_i,t=

h∗_i,t− hi,t−1, h∗_i,t≥ hi,t−1,

0, otherwise. (7)

and

o_i,t =

hi,t−1− h∗i,t, h∗i,t < hi,t−1,

0, otherwise. (8) Furthermore, let B_t= N i=1 b_i,t (9) and Ot= N i=1 oi,t (10)

be the totals of the bids and oﬀers for the stock at time t, where N is the number of traders. Following Palmer

et al. (1994), we use the following simple rationing scheme:5

h_i,t=   

h_i,t−1+ b_i,t− o_i,t, if B_t= O_t,

hi,t−1+Ot

Btbi,t− oi,t, if Bt> Ot,

hi,t−1+ bi,t−_OB_ttoi,t, if Bt< Ot.

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All these cases can be subsumed into

h_i,t = h_i,t−1+ Vt Btbi,t−

V_t

Otoi,t (12)

where V_t≡ min(B_t, O_t) is the volume of trade in the stock.

Based on Palmer et al.’s rationing scheme, we can have a very simple price adjustment scheme, based solely

on the excess demand B_t− O_t:

Pt+1= Pt(1 + β(Bt− Ot)) (13)

5_{This simple rationing scheme is chosen mainly to ease the burden of intensive computation. An realistic alternative is to}

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where β is a function of the diﬀerence between B_t and O_t. β can be interpreted as speed of adjustment of prices. One of the β functions we consider is:

β(Bt− Ot) =

tanh(β₁(B_t− O_t)) if B_t≥ O_t,

tanh(β₂(B_t− O_t)) if B_t< O_t (14)

where tanh is the hyperbolic tangent function:

tanh(x)≡ e

x_{− e}−x

ex+ e−x (15)

The price adjustment process introduced above implicitly assumes that the total number of shares of the stock circulated in the market is ﬁxed, i.e.,

H_t=

i

h_i,t= H. (16)

In addition, we assume that dividends and interests are all paid by cash, so M_t+1=

i

M_i,t+1= M_t(1 + r) + H_tD_t+1. (17)

2.3 Formation of Adaptive Traders

As to the formation of traders’ expectations, Ei,t(Pt+1+ Dt+1), we assume the following functional form for

E_i,t(.). E_i,t(P_t+1+ D_t+1) =    (Pt+ Dt)(1 + θ1fi,t∗ 10−4), if − 10000.0 ≤ fi,t≤ 10000.0, (Pt+ Dt)(1 + θ1), if fi,t > 10000.0, (Pt+ Dt)(1− θ1), if fi,t <−10000.0 (18)

The population of f_i,t (i=1,...,N ) is formed by genetic programming. That means, the value of f_i,t is decode

from its GP tree gp_i,t. According to the martingale hypothesis, the trader holds martingale belief if E_i,t(P_t+1+

D_t+1) = P_t+ D_t. Therefore, the cardinality of set {i | E_i,t(P_t+1+ D_t+1)− (P_t+ D_t) = 0}, denoted by N_1,t,

gives us the information how well the eﬃcient market hypothesis is accepted among traders.

As to the subjective risk equation, we modiﬁed the equation originally used by Arthur et al. (1997).

1 is simply the historical volatility based on the past n1observations.

2.4 Single-Population Based Business School

Before introducing the diﬀerent architecture of multi-population GP, we have to review the mechanism of “business school” once again. The business school serves as a faculty of researchers. Traders can consult with them when they face the peer pressure or losing lot of money. However, the researchers and traders may have diﬀerent focus. Traders care about the models or strategies which are helpful for making money. While the

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researchers put attention on the accuracy of forecasting, for example, mean absolute percentage error (MAPE). Therefore, the business school considered here can be viewed as a collection of forecasting models. Then, the single-population can be applied to model its evolution.

Each researcher (forecasting model) is represented by a tree (GP parse tree). The school will be evaluated

with a prespeciﬁed schedule, say once for every m₁trading days. The procedure proceeds as follows.

At the evaluation date t, the business school will generate a group of new forecasting models in order to

ﬁt (or survive in) the new situation. Each forecasting model gp_i,t−1 at period t− 1 will be examined by a

new model which is generated from the same business school at period t− 1 by one of the following three

genetic operators, reproduction, crossover and mutation, each with probability pr, pc and pm(Table 1). The

tournament selection is applied in the procedures of three genetic operators as follows: • Reproduction:

Two forecasting models (GP trees) are randomly selected from GP_i,t−1. The one with lower MAPE over

the last m2 days’ forecasts is chosen as the new model.

• Mutation:

Two forecasting models are randomly selected from GP_i,t−1. The one with lower MAPE over the last

m₂ days’ forecasts is chosen as the candidate with the probability of p_M (In Table 1, the probability is

0.3) being mutated. No matter the candidate is mutated or not, the one (the new one if it is mutated) is chosen as the new model.

• Crossover:

Two pairs of forecasting models are randomly chosen, say (gp_j₁_,t−1,gp_j₂_,t−1) and (gp_k₁_,t−1,gp_k₂_,t−1). The

one with lower MAPE in each pairs is chosen as parent. One of the two children which is born by the crossover of their parents is randomly chosen as the new model.

Therefore, each forecasting model at period t− 1 is compared with the new model generated by one of

the three genetic operators based on the criterion of MAPE. The lower one is selected as the new forecasting model in next period (generation). The following is a pseudo program of the procedure of Business School (Also see Flowchart 1). Table 1 is an example of the speciﬁcation of the control parameters to evolve the business school.

Procedure [Business School]

0. begin

1. Calculate M AP E(gp_i,t)

2. A = Random(R,C,M) with (p_r,p_c,p_m)

3. If A = C, go to step (11).

4. (gp₁, gp₂) = (Random(GP_t−1),Random(GP_t−1))

5. Calculate M AP E(gp1) and M AP E(gp2).

6. gp_new= Tournament Selection (M AP E(gp₁),M AP E(gp₂))

7. If A = R, go to step (17).

8. gp_new← Mutation (gp_new)

9. Calculate M AP E(gp_new)

10. Go to step (17)

11. Randomly select two pairs of trees from GP_t−1

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13. gp₁ = Tournament Selection (Pair 1)

14. gp2 = Tournament Selection (Pair 2)

15. (gp₁, gp₂)← Crossover (gp₁,gp₂)

16. gpnew = Random(gp1, gp2)

17. gp_i,t = Tournament Selection (M AP E(gp_i,t−1),M AP E(gp_new))

18. end

2.5 The Interaction between Traders’ Behavior and Business School

The main distinction between SGP and MGP is on the formation of traders’ behavior. In the architecture of MGP, we allow the traders to think about how to react the environment by themselves. Therefore, at the evaluation date t, each trader has to make a decision. Should he change his mind (the strategy used in the previous period)? If the answer is yes, where should he consult? the business school or himself.

The way we use to model this psychological activities can be summarized as the following procedure. First,

whether each trader change his mind or not depends on his net change of wealth over the last n2days compared

with other traders. Let R_i,t b e his rank and ∆Wn2

i,t be this net change of wealth of trader i at time period t,

i.e.,

∆Wn2

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Table 1: Parameters of the Stock Market (I)

The Stock Market

Shares of the stock (H) 100

Initial money supply (M1) 100

Interest rate (r) 0.1

Stochastic process (D_t) Uniform distribution, U(5.01,14.99)

Price adjustment function tanh

Price adjustment (β₁) 0.2× 10−4

Price adjustment (β₂) 0.2× 10−4

Parameters of Genetic Programming

Function set {+, −, ×, %, Sin, Cos, Exp, Rlog, Abs, Sqrt}

Terminal set {Pt, Pt−1,· · · , Pt−10, Pt+ Dt,· · · , Pt−10+ Dt−10}

Selection scheme Tournament selection

Tournament size 2

Probability of creating a tree by reproduction 0.10

Probability of creating a tree by immigration 0.20

Probability of creating a tree by crossover 0.35

Probability of creating a tree by mutation 0.35

Probability of mutation 0.3

Probability of leaf selection under crossover 0.5

Mutation scheme Tree mutation

Replacement scheme (1+1) Strategy

Maximum depth of tree 17

Maximum number in the domain of Exp 1700

Number of generations 10000

Then, the probability for the trader i changing his mind at period t is determined by p_i,t=Ri,t

N . (23)

Equation (23) means that the trader with higher rank faces higher peer pressure. Hence, he has higher motivation to change mind.

Second, in addition to the peer pressure, each trader also cares about his own satisfaction. That means,

traders intend to improve their growth rate of income. Let the growth of income over the last n₂ days be

δn2 i,t = ∆Wn2 i,t − ∆Wi,t−nn2 2 | ∆Wn2 i,t−n2 | , (24)

and let qi,t be the probability that trader i will look for new strategies at the end of the tth period, assume

that it is determined by

q_i,t= 1

1 + expδi,tn2

. (25)

Therefore, the traders make great (less) progress have lower (higher) probability to change mind. lim

δi,tn2→∞

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Table 2: Parameters of the Stock Market (II)

Business School

Number of faculty members (F ) 500

Criterion of ﬁtness (Faculty members) MAPE

Evaluation cycle (m₁) 20

Sample Size (MAPE) (m₂) 10

Search intensity in Business School (I_s∗) 5

Traders

Number of traders (N ) 100

Number of ideas for each trader 20

Degree of RRA (λ) 0.5

Criterion of ﬁtness (Traders) Increments in wealth (Income)

Sample size of σ_t|n2 ₁ (n1) 10

Evaluation cycle(n₂) 1

Sample size (n₃) 10

Initial probability of consulting business school (psm_i,t) 0.5

Search intensity by Trader itself (I_h∗) 5

θ₁ 0.5

θ2 0.0133

and

lim

δn2_i,t→−∞qi,t= 1. (27)

Based on the description above, we know the probability (r_i,t) that trader i decides to change mind.

ri,t= pi,t+ (1− pi,t)qi,t= Ri,t

N +

N− R_i,t N

1

1 + expδki,t (28)

However, we have not yet mentioned how a trader come up with a new idea. In order to model this process,

we introduce a probability measure to describe this psychological activity. Let psm_i,t be the probability that

trader i would like to look for new ideas from business school. On the other hand, the probability that trader i decide to work out new ideas by himself is 1− psm_i,t. This probability is determined by

psm_i,t =           

psm_i,t−1− (ri,t− ri,t−1)psm_i,t−1, if ri,t− ri,t−1≥ 0, Case1,

psm_i,t−1− (r_i,t− r_i,t−1)(1− psm_i,t−1), if r_i,t− r_i,t−1< 0, Case1, psm_i,t−1+ (ri,t− ri,t−1)(1− psm_i,t−1), if ri,t− ri,t−1≥ 0, Case2,

psm_i,t−1+ (r_i,t− r_i,t−1)psm_i,t−1, if r_i,t− r_i,t−1< 0, Case2,

psm_i,t−1, Case3.

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where Case1 means that trader i looked for new idea from business school at period t− 1, Case2 means that

trader i made new idea by himself at period t− 1, and Case3 means that trader i didn’t change mind at period

t− 1.

The idea of Equation (29) is very straightforward. If a trader has high motivation to change mind, he will think about whether the result is due to the wrong decision, for example, consulted with researchers in the business school, made in the previous period or not. Therefore, he is prone to reduce the conﬁdence about business school.

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Once a trader decides to go to business school, he has to consult with one researcher at school randomly (or pick up one forecasting model at school randomly). Then, he compares the new idea with his old one used in the previous period based on the criterion of MAPE by means of calculating the stock price and dividends

over the last n3 trading days. If the new idea outperforms his old idea, he will adopt the new one. Otherwise,

he will look for new one at school once again until either he succeeds or he fails for I_s∗ times. Of course, it

is very possible that the trader decides to work out a new idea by himself. The new idea is also generated by three genetic operators which is happened in his mind. He has to compare the new idea with the old one

based on the net change of wealth over the last n₂trading days.6 If the new one outperforms the old one, he

will adopt it. Otherwise, he will think about it once again until he succeeds or he fails for I_h∗ times. These

procedures are summarized as a pseudo program listed below (Also see Flowchart 2).

Procedure 1 [Come up with a new idea]

0. begin

1. If go the business school, go to Procedure 2 2. Go to Procedure 3

3. end

Procedure 2 [Visiting Business School]

0. begin

1. Calculate M AP E(f_i,t)

2. I_s← 1

3. Randomly select a gp_j,t (∼ U[1, 500])

4. Calculate M AP E(gpj,t)

5. If M AP E(gp_j,t) < M AP E(f_i,t), go to Step (10)

6. Is← Is+ 1 7. If Is≤ Is∗, go to step (3) 8. f_i,t+1= f_i,t 9. Go to Step (11) 10. f_i,t+1 = gp_j,t 11. end

Procedure 3 [Work out new ideas by himself ]

0. begin

1. Calculate ∆Wn2

i,t (Wi,t− Wi,t−n2)

2. Ih ← 1

3. Generate a new gp_i,t,j based on one of three genetic operators

4. Calculate ∆Wn2

i,t,j

5. If ∆Wn2

i,t < ∆Wi,t,jn2 , go to Step (10)

6. I_h ← I_h + 1

7. If Ih≤ Ih∗, go to step (3)

8. f_i,t+1= f_i,t 9. Go to Step (11)

6_{Of course, the new idea doesn’t really be used. We can assume that the trader used the new idea since}_n2_{trading days before,}

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3 Experimental Designs

In order to facilitate us to understand the influence of “intelligent” agents, we consider three different scenarios, Market A, B and C. The difference between these markets are shown in Table 3. In Market A, the SGP based market is simulated. The results are compared with that in the Market B where traders work out new ideas by themselves rather that consult with researchers. The difference between these two simulations provides the effects of prediction accuracy oriented and profit oriented agents. Market C is a more realistic one. The agents can adapt themselves to modify the confidence about the school (or themselves). This design coincides with part of human’s psychological activity.

Based on these diﬀerent designs, simulations are conducted according to the parameters shown in Table 1 and 2. Each trader in our simulations has twenty ideas in his mind. These ideas also evolve from generation to generation. Therefore, the traders are very adaptive. In Table 4, the important variables related to the traders and market are summarized. These are helpful for us to go one step further to analyze our simulation results.

For example, the number of martingale believers (N1,t) tell us how many traders hold martingale belief at

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Table 3: The Market Structure

Market Architecture Probability of consulting business school

A SGP with business school 1.0

B MGP with business school 0.0

C MGP with business school Adaptive adjustment, Equation (29)

Table 4: Time Series Generated from the Artiﬁcial Stock Market:

Aggregate Variables

Stock price Pt

Trading volumes V_t

Totals of the bids B_t

Totals of the oﬀers O_t

# of martingale believers N_1,t

# of traders registered to business school N_2,t

# of traders with successful search in business school N3,t

# of traders registered to himself N4,t

# of traders with successful thinking N5,t

Individual Trader

Forecasts f_i,t

Subjective risks σ_i,t

Bid to buy b_i,t

Oﬀer to sell o_i,t

Wealth Wi,t

Income ∆W_i,t1

Rank of proﬁt-earning performance Ri,t

Complexity (depth of fi,t) ki,t

Complexity (# of nodes of f_i,t) κ_i,t

traders’ believe. We are also interested in how well the traders “live” in the market. Do they change mind

usually? Do they beneﬁt from business school or their mind? These subjects can be refereed to (N_2,t,N_4,t) and

(N_3,t,N_5,t). As to the k_i,t and κ_i,t, they are complexity measures of traders’ strategies in terms of GP-tree.

How the complexity of traders’ strategies coevolve with market dynamics is also an important issue. From the viewpoint of econometrics, the typical phenomena found in the ﬁnancial markets are analyzed in the paper.

For example, is{Pt} normal or stationary? is return series ({Rt}) independently and identically distributed?

Or, is{R_t} nonlinearly dependent? Does {R_t} have the property of GARCH...., and so on.

4 Simulation Results

Based on the experimental design given in Table 1 and 2, a single run with 10000 generations (periods) is conducted for Market A, B and C respectively. The time series data is further divided into ten subperiods. The basic statistics, econometric properties and the important variables shown in Table 4 for each subperiod are

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then calculated. In this paper, the simulation results are analyzed from two parts. One is the macro-properties, the econometric properties of time series. The other is micro-structure, traders’ behavior and believes. This approach provides us the information about the relation between the traders’ believes and market dynamics. Therefore, we can understand how these intelligent traders inﬂuence the market.

4.1 Macro-properties

As to the properties of time series, whether this artificial stock market can replicate the stylized facts found in the financial markets or not is the first question for the researchers working on this field will face.

1. Are stock prices and stock return normally distributed? 2. Does the price series have a unit root?

3. Are stock return independently and identically distributed?

The time series of the stock price in the last 9000 periods for each market is drawn in Figure 1, 2 and 3

respectively.7 In these ﬁgures, the range of price ﬂuctuation in Market B is higher than that in Market A and

C. However, the homogeneous rational expectations equilibrium price under full information is Pf = 1

r(d− λσ

2_h) ₍₃₀₎

where r is interest rate, d is the average of dividends, σ2is the variance of dividends series and h is the average

of shares of the stock for each trader. Therefore, the fundamental price (P_f) in these markets is 58.375. It

implies that the market composed of proﬁt oriented traders tends to overestimate the intrinsic value of the stock.

Stock return is derived by

rt= ln(Pt)− ln(Pt−1) (31)

Figure 4, 5, and 6 are the time series plots of the stock return, and the basic statistics of the time series of stock price and return are given in Table 5 and 6. According to the Jarqu-Bera normality test, neither the stock price nor return follows normal distribution. Moreover, the leptokurtosis of the stock return also confirms the fat tail phenomenon usually found in the financial data. In Table 7, the result of Dickey-Fuller test shows that there exists a unit root in the price series for each market except the first subperiod of Market A and the second subperiod of Market C based on 99% significance level.

As to the third question, it is related to the classical version of efficient market hypothesis. Technically speaking, the market is efficient if there exists no linear and nonlinear structure in the return series. Here, we employ the procedure proposed by Chen, Lux and Marchesi (1999). First, the Rissanen’s predictive stochastic complexity (PSC) is used to filter the linear signal. Once the linear signal is filtered, if there is any structure left in the residual, it must be nonlinear. Therefore, the most frequently used nonlinear test, BDS test, is then applied to the residual series. However, there are two parameters needed to be chosen. One is the distance parameter (0 standard deviations), and the other is the embedding dimension (DIM). Here, the result of BDS test is performed under 0 = 1 and DIM=2,3,4,5. In Table 8, we found that there exists linear structure in the

three markets, while the R2 is very low. Moreover, most periods fail to reject the nonexistence of nonlinear

signal. However, it is well known in econometrics that nonlinearity could be found in the second moment. The

7_{In the beginning of our simulation, trader are unaware of the characteristics of the market.} _{Therefore, price is highly}

overestimated. However, as time goes on, traders gradually modify their expectations. Price then goes down below $90. In order to focus on the major range of price ﬂuctuation, we only report the last 9000 periods.

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Table 5: Basic Statistics of the Artiﬁcial Stock Price Series

Market A

Periods P σ Skewness Kurtosis Jarqu-Bera p-value

1-1000 78.32788 13.62220 0.692494 2.200712 106.5439 0.000000 1001-2000 63.39534 1.357275 -0.354404 2.312132 40.64887 0.000000 2001-3000 61.91586 1.180842 0.917530 3.636987 157.2165 0.000000 3001-4000 61.86186 0.888477 -0.331832 2.725438 21.49304 0.000022 4001-5000 60.10845 1.181903 0.927996 3.232536 145.7824 0.000000 5001-6000 61.60884 0.896830 0.132566 2.951264 3.027935 0.220035 6001-7000 61.78188 1.633962 0.356041 2.035466 59.89105 0.000000 7001-8000 60.45598 1.156648 -0.521288 2.493351 55.98573 0.000000 8001-9000 61.87491 1.345820 0.902358 3.066857 135.8945 0.000000 9001-10000 60.10790 1.285679 0.461578 2.999811 35.50906 0.000000 Market B

1-1000 108.0933 12.68692 -0.451207 1.650351 109.8293 0.000000 1001-2000 82.78370 3.343650 0.232044 2.110596 41.93407 0.000000 2001-3000 73.32885 2.196288 1.075537 3.077568 193.0472 0.000000 3001-4000 76.48211 2.449680 -0.572735 2.382626 70.55220 0.000000 4001-5000 72.99718 1.330041 1.273381 5.006009 437.9195 0.000000 5001-6000 74.01037 1.124712 -0.481808 2.604814 45.19695 0.000000 6001-7000 76.87204 2.054136 0.659705 2.813349 73.98679 0.000000 7001-8000 76.97926 1.258317 0.429422 2.765607 33.02311 0.000000 8001-9000 74.16528 1.437827 0.134599 2.422469 16.91711 0.000212 9001-10000 76.18609 1.726372 0.278032 2.009311 53.77799 0.000000 Market C

1-1000 95.37813 12.08421 -0.435627 1.738648 97.92054 0.000000 1001-2000 68.12425 2.558067 1.192156 3.467249 245.9694 0.000000 2001-3000 68.60801 2.415664 -0.676687 2.251292 99.67445 0.000000 3001-4000 68.13919 2.498183 0.592067 2.058576 95.35216 0.000000 4001-5000 67.38430 2.232205 0.929403 2.813418 145.4156 0.000000 5001-6000 67.99319 1.176889 0.671658 2.598026 81.92002 0.000000 6001-7000 68.13972 0.931234 0.497421 3.292627 44.80586 0.000000 7001-8000 71.38652 1.850401 0.856242 3.030693 122.2310 0.000000 8001-9000 73.89949 1.524528 0.394129 2.855530 26.75926 0.000002 9001-10000 72.78556 2.353734 -0.239208 2.479806 20.81177 0.000030

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Table 6: Basic Statistics of the Artiﬁcial Stock Return Series

Market A

Periods r σ Skewness Kurtosis Jarqu-Bera p-value

1-1000 -0.000334 0.002317 3.285575 18.63701 11987.34 0.000000 1001-2000 -0.000059 0.001681 1.118201 5.572807 484.2014 0.000000 2001-3000 0.000012 0.001765 1.167794 5.453934 478.1986 0.000000 3001-4000 0.000031 0.001692 0.891895 4.381115 212.0578 0.000000 4001-5000 -0.000022 0.001652 1.028622 4.881467 323.8405 0.000000 5001-6000 -0.000014 0.001635 0.728591 3.365408 94.03773 0.000000 6001-7000 -0.000039 0.001653 1.495067 10.35198 2624.689 0.000000 7001-8000 0.000019 0.001627 0.890006 4.498190 225.5423 0.000000 8001-9000 -0.000006 0.001846 2.006383 12.64014 4543.108 0.000000 9001-10000 0.000056 0.001567 0.418674 2.545671 37.77750 0.000000 Market B

1-1000 -0.000020 0.002738 2.283185 8.642649 2195.468 0.000000 1001-2000 -0.000115 0.001555 1.308220 7.179182 1012.972 0.000000 2001-3000 -0.000070 0.001479 0.868873 3.959215 164.1606 0.000000 3001-4000 0.000063 0.001601 1.019151 5.090089 355.1311 0.000000 4001-5000 -0.000061 0.001404 0.556026 2.728130 54.60715 0.000000 5001-6000 0.000005 0.001504 0.844090 3.664948 137.1712 0.000000 6001-7000 0.000032 0.001638 1.074103 4.932855 347.9466 0.000000 7001-8000 0.000005 0.001541 0.814424 3.891260 143.6455 0.000000 8001-9000 -0.000025 0.001470 0.950137 5.776774 471.7296 0.000000 9001-10000 -0.000008 0.001642 2.076164 15.74482 7486.346 0.000000 Market C

1-1000 -0.000176 0.002498 2.356092 10.60410 3334.458 0.000000 1001-2000 -0.000148 0.001514 1.115679 6.842539 822.6695 0.000000 2001-3000 0.000105 0.001710 1.116244 5.578395 484.6717 0.000000 3001-4000 -0.000087 0.001489 0.635668 3.135544 68.11116 0.000000 4001-5000 0.000071 0.001570 0.669616 3.427464 82.34453 0.000000 5001-6000 -0.000018 0.001497 0.730715 3.544839 101.3596 0.000000 6001-7000 0.000039 0.001636 1.990691 15.69916 7379.999 0.000000 7001-8000 0.000016 0.001587 1.045142 5.452793 432.7285 0.000000 8001-9000 -0.000001 0.001622 1.370530 7.217321 1054.134 0.000000 9001-10000 0.000042 0.001677 1.284784 6.277304 722.6417 0.000000

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Table 7: Unit Root Test

Market A Market B Market C

Periods DF of P_t DF of P_t DF of P_t 1-1000 -3.743852 -0.205775 -1.629092 1001-2000 -1.102648 -2.378888 -3.287183 2001-3000 0.139497 -1.641350 1.907363 3001-4000 0.556529 1.176659 -1.993159 4001-5000 -0.452132 -1.490206 1.412270 5001-6000 -0.070920 0.119053 -0.387298 6001-7000 -0.775960 0.592499 0.773165 7001-8000 0.399396 -0.019152 0.194885 8001-9000 -0.163501 -0.550615 -0.163876 9001-10000 1.145653 -0.205708 0.787532

The MacKinnon critical values for rejection of hypothesis of a unit root at 99% (95%) significance level is -2.5668 (-1.9395).

(G)ARCH family of time series are designed to capture the regularities in the behavior of volatility which is the phenomenon of volatility clustering. Therefore, we carried out the Lagrange multiplier test for the presence of ARCH effects. If the null hypothesis of ARCH effect is rejected, we will further identify the GARCH structure by using the Bayesian Information Criterion (BIC). The results are exhibited in Table 9. Clearly, the presence of GARCH effects seems to be very robust compared with the BDS test.

4.2 Micro-structure

Based on the result described above, the diﬀerence between the three markets is not large. However, it doesn’t imply anything about the micro-structure. We are interested in how the traders behave in the three markets. The basic questions proposed in the previous research (Chen and Yeh, 2000) would help us to examine this issue. Are they martingale believers? Do they search new ideas intensively? What kind of strategies do the traders employ?...., and so on.

In Table 10, it evidenced that, on the average, the martingale belief doesn’t survive in the traders’ mind. The time series plot of the number of martingale believers for each market is also given in Figure 7, 8 and 9 respectively. In the 10000 periods, there is no more than eight traders holding martingale believes in each period. Now, we may interested in what the traders actually do if they don’t believe martingale hypothesis. In Table 11 and 12, we can get an impression about the traders’ search and thinking activities. In each market, there is about 90% traders trying to change their ideas which means the price dynamics is not easily captured. However, the inside information in each market is different. In Market A, the traders follow the ideas from business school which is prediction accuracy oriented. There is about 50% traders who registered to business school benefit from their search. Clearly, search is useful. It also implies that the useful forecasting models change over time. There is no robust forecasting model in this environment. On the other hand, the business school updates knowledge every 20 periods. As time goes on, the updated models have been out-of-date before it updates once again. In this situation, even though the traders is very adaptive in terms of they modify strategies at each period, but they can only reuse the old ideas. Therefore, the chance they benefit from business school is getting lower. It is also exhibited in the decrease of the average of traders with successful

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Table 8: PSC Filtering and BDS Test

Market A

Periods (p,q), (R2) DIM=2 DIM=3 DIM=4 DIM=5 Reject

1-1000 (1,0) (0.084586) 2.763187 3.355127 3.573923 3.801602 Yes 1001-2000 (1,2) (0.095338) 1.170001 1.536866 1.657808 1.732317 No 2001-3000 (1,2) (0.130978) 1.446862 1.503426 1.545025 1.537045 No 3001-4000 (1,0) (0.046633) 0.983322 1.267603 1.522006 1.634546 No 4001-5000 (2,0) (0.069238) 1.314665 1.588530 1.696342 1.709995 No 5001-6000 (0,3) (0.051332) 0.876030 1.318596 1.522116 1.618994 No 6001-7000 (1,0) (0.066648) 0.804488 0.863171 0.959267 1.000311 No 7001-8000 (0,2) (0.072331) 1.254773 1.393243 1.425195 1.392453 No 8001-9000 (1,2) (0.148256) 1.156910 1.209763 1.267034 1.342426 No 9001-10000 (1,0) (0.017866) 0.822629 0.955026 1.098682 1.245252 No Market B

1-1000 (1,0) (0.120571) 2.631589 2.950451 3.164891 3.472194 Yes 1001-2000 (3,3) (0.086179) 0.910340 1.215087 1.292173 1.361617 No 2001-3000 (1,0) (0.029251) 1.436759 1.382518 1.465204 1.517484 No 3001-4000 (1,2) (0.079725) 0.992550 1.185630 1.262048 1.248313 No 4001-5000 (0,0) (0.000000) 1.028910 1.154727 1.229629 1.275487 No 5001-6000 (2,2) (0.048942) 1.465323 1.476508 1.532547 1.544393 No 6001-7000 (2,2) (0.086259) 1.217710 1.568579 1.670191 1.674020 No 7001-8000 (1,0) (0.036988) 0.993627 1.141134 1.259636 1.346128 No 8001-9000 (1,0) (0.048028) 0.777911 0.794150 0.835682 0.798633 No 9001-10000 (2,2) (0.083901) 1.003050 1.238080 1.438938 1.546275 No Market C

1-1000 (1,2) (0.172231) 3.198414 3.851904 4.487734 5.187971 Yes 1001-2000 (1,0) (0.043393) 0.831786 1.056206 1.160091 1.175626 No 2001-3000 (0,2) (0.081779) 1.679190 2.057802 2.273284 2.336968 No 3001-4000 (2,2) (0.037457) 0.993473 1.261842 1.242837 1.188526 No 4001-5000 (3,2) (0.053997) 1.157223 1.210224 1.183651 1.155643 No 5001-6000 (4,0) (0.033742) 0.850240 0.984305 1.084103 1.078252 No 6001-7000 (1,0) (0.056553) 0.767305 0.884078 1.002873 1.017948 No 7001-8000 (1,0) (0.060121) 1.452094 1.545296 1.572037 1.598763 No 8001-9000 (1,0) (0.056991) 1.591326 1.694601 1.693094 1.603306 No 9001-10000 (3,3) (0.108706) 0.928695 1.148883 1.323939 1.417919 No

The BDS test statistic is asymptotically normal with mean 0 and standard deviation 1. The significance level of the test is set at 0.95.

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Table 9: GARCH Modeling

Periods Market A Market B Market C

1-1000 (1,1) (2,2) (1,2) 1001-2000 (1,1) (1,2) (1,1) 2001-3000 (0,1) (1,1) (1,1) 3001-4000 (1,1) (1,1) (1,1) 4001-5000 (1,1) (1,1) (1,1) 5001-6000 (1,1) (0,1) (0,1) 6001-7000 (1,1) (1,1) (1,1) 7001-8000 (1,1) (1,1) (1,1) 8001-9000 (1,1) (0,1) (0,1) 9001-10000 (0,1) (1,1) (1,1)

search on the h day after the business school has updated the information (See Table 13).

In Market B, the traders’ action follows profit and they renew their ideas at each period. It induces a different phenomenon. On the average, there is about 17 traders (except first subperiod) who benefit from their thinking. The average of the ratio of traders with successful thinking is also lower. The reasons are as follows. First, the traders’ action is to myopically maximize the one-period expected utility and they evaluate the ideas too frequently (at each period). It makes the strategies have lower chance to survive. Second, the traders’ ideas in their mind also evolve at each period. Therefore, even though each trader has 20 ideas, these

ideas easily tend to evolve similar structures.8 It explains the low ratio of traders with successful thinking.

In the beginning of the simulation of Market C, there exists both types of traders, prediction accuracy and profit oriented traders. Due to both criteria coevolving in this market, it makes the traders more difficult to capture the price dynamics and make profit. Therefore, more traders intend to change their strategies (See the final column in Table 11), and the number of traders getting useful ideas gradually decreases (See the final column in Table 12). The interesting thing is that the traders’ behavior tends to be profit oriented. The market dynamics is getting dominated by them, so the number of profit oriented traders with successful thinking increases. This makes the prediction accuracy oriented traders more difficult to survive. There are two possible reasons to explain this phenomenon.

• Proﬁt oriented traders are more easier to survive.

• Proﬁt oriented traders are more adaptive compared with business school.

In order to test the hypotheses, we can set the equal evaluation cycle for the traders and business school

(m₁ = n₂), for example, 20. The second conjecture is related to the inﬂuence of speculators who care about

short-term proﬁts and investors who focus on long-term proﬁts. These problems will be discussed in the future research.

The information about the complexity of evolving strategies also conﬁrms the diﬀerences between these markets. The results are exhibited in Table 14. Figure 10-15 are the time series plots of the complexity of

8_{How to design the evolution of traders’ mind is an important issue. It will inﬂuence the traders’ adaptability. We may only}

evolve the realized strategies and keep the other strategies unchanged, or the synthesis of both methods. Of course, this problem is not easy to solve. It is left to the future research.

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Table 10: Average of the Number of Martingale Believers (N1)

Periods Market A Market B Market C

1-1000 0.323 0.812 0.501 1001-2000 0.191 0.547 0.437 2001-3000 0.133 0.625 0.552 3001-4000 0.229 0.617 0.411 4001-5000 0.224 0.715 0.460 5001-6000 0.148 0.519 0.443 6001-7000 0.191 0.583 0.523 7001-8000 0.206 0.487 0.535 8001-9000 0.112 0.562 0.493 9001-10000 0.090 0.653 0.616

Table 11: Average of the Number of Traders Registered to Business School and Himself

Periods N₂ N₄ N₂ N₄ N₂+ N₄ 1-1000 88.762 87.196 47.280 45.452 92.732 1001-2000 88.956 87.400 40.054 53.035 93.089 2001-3000 88.877 87.282 36.092 56.763 92.855 3001-4000 88.756 87.310 30.314 62.666 92.980 4001-5000 88.689 87.258 26.297 66.627 92.924 5001-6000 88.814 87.247 22.560 70.390 92.950 6001-7000 88.859 87.088 20.106 72.809 92.915 7001-8000 88.982 87.432 16.052 76.799 92.851 8001-9000 88.915 87.465 14.264 78.799 93.063 9001-10000 88.884 87.312 10.546 82.429 92.975

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Table 12: Average of the Number of Traders with Successful Search and thinking

Periods N₃ N₅ N₃ N₅ N₃+ N₅ 1-1000 43.048 (0.48417) 28.285 (0.32433) 28.987 (0.61098) 15.677 (0.35173) 44.664 (0.48132) 1001-2000 44.066 (0.49585) 17.455 (0.19970) 23.501 (0.58711) 10.656 (0.20121) 34.157 (0.36690) 2001-3000 42.588 (0.47887) 17.826 (0.20426) 21.140 (0.58583) 11.417 (0.20107) 32.557 (0.35061) 3001-4000 44.776 (0.50463) 17.470 (0.20007) 18.874 (0.62134) 12.598 (0.20096) 31.472 (0.33852) 4001-5000 43.284 (0.48778) 17.357 (0.19890) 15.657 (0.59712) 13.361 (0.20058) 29.018 (0.31234) 5001-6000 42.664 (0.47976) 17.322 (0.19853) 13.511 (0.59780) 14.234 (0.20217) 27.745 (0.29837) 6001-7000 45.340 (0.51040) 17.428 (0.20005) 12.052 (0.59938) 14.574 (0.20011) 26.626 (0.28653) 7001-8000 42.439 (0.47678) 17.497 (0.20014) 9.733 (0.60698) 15.305 (0.19935) 25.038 (0.26970) 8001-9000 42.650 (0.47960) 17.460 (0.19960) 8.571 (0.60027) 15.887 (0.20171) 24.458 (0.26289) 9001-10000 45.490 (0.51168) 17.337 (0.19860) 6.112 (0.58299) 16.589 (0.20128) 22.701 (0.24428)

The value shown in the parentheses is the average of the ratio of traders with successful search and thinking.

evolving strategies. In the business school, the strategies try to trace the price dynamics in the past 10 periods

(m₂) over time. Therefore, they tend to become more complex in order to ﬁt the nonlinear structure. However,

as mentioned above, the traders’ action in Market B is to myopically maximize the one-period expected utility. Therefore, it is not necessary to evolve complex structure. Moreover, the ideas is renewed at each period, it further makes the strategies have no chance getting complicated. On the other hand, in Market C, due to the increasing of the proportion of proﬁt oriented traders, the complexity of strategies decreases gradually.

5 Concluding Remarks

In the primary research, we built an environment composed of multi-population genetic programming based traders. Besides replicating the stylized facts, the comparison between SGP-based and MGP-based simulations are also discussed. From the marco-phenomena point of view, we don’t get too much diﬀerence, while the micro-structure does. The diﬀerence may come from:

• The different oriented traders, profit and prediction accuracy. • The different evaluation cycle.

• The evolution of traders’ mind.

Of course, the inﬂuence of these points will be discussed more detail in the future research. Moreover, the eﬀect of the level of the traders’ intelligence (the number of ideas for each trader) is also an important issue. However, due to highly computation-intensive, this problem can’t be done easily at this moment.

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Table 13: Average of the Number of Traders with Successful Search on the h day after Business School Has Updated the Information

h N_3,h N_5,h N_3,h N_5,h 1 49.622 (0.56006) 18.76 (0.21546) 17.450 (0.66427) 13.798 (0.21455) 2 46.322 (0.52227) 18.78 (0.21613) 17.202 (0.65034) 13.988 (0.21654) 3 45.516 (0.51218) 19.07 (0.21768) 16.802 (0.63761) 14.108 (0.21862) 4 44.876 (0.50410) 18.78 (0.21525) 16.348 (0.62345) 13.856 (0.21362) 5 44.314 (0.49816) 18.64 (0.21367) 15.918 (0.60160) 14.072 (0.21569) 6 43.464 (0.48949) 18.39 (0.21079) 16.070 (0.61127) 13.912 (0.21334) 7 43.646 (0.49112) 18.36 (0.20980) 15.990 (0.60924) 14.160 (0.21701) 8 44.214 (0.49616) 18.28 (0.20994) 15.812 (0.60298) 13.940 (0.21304) 9 42.672 (0.48105) 18.08 (0.20739) 15.312 (0.57831) 14.276 (0.22227) 10 43.152 (0.48583) 18.44 (0.21139) 15.436 (0.58023) 14.098 (0.21655) 11 41.934 (0.47137) 18.50 (0.21193) 15.248 (0.58407) 14.166 (0.21728) 12 41.820 (0.47076) 18.49 (0.21151) 15.382 (0.57872) 13.904 (0.21358) 13 41.948 (0.47298) 18.27 (0.20932) 15.038 (0.56905) 14.230 (0.21954) 14 42.562 (0.47924) 18.42 (0.21081) 15.210 (0.57724) 13.718 (0.21103) 15 43.636 (0.48890) 18.73 (0.21448) 15.522 (0.58260) 13.830 (0.21356) 16 42.996 (0.48294) 18.47 (0.21094) 14.942 (0.57212) 14.244 (0.21901) 17 42.656 (0.48038) 18.58 (0.21294) 15.398 (0.58360) 13.918 (0.21322) 18 43.118 (0.48457) 19.21 (0.22044) 15.546 (0.58831) 13.944 (0.21273) 19 42.304 (0.47602) 18.45 (0.21136) 15.764 (0.59703) 14.210 (0.21941) 20 41.918 (0.47145) 18.09 (0.20715) 15.886 (0.58758) 14.224 (0.21972)

The values shown in the parentheses are the ratios of traders with successful search on theh day after business school has updated the information.

References

[1] Andrews, M. and R. Prager (1994), “Genetic Programming for the Acquisition of Double Auction Market Strategies,” in K. E. Kinnear (1994) (eds.), Advances in Genetic Programming, Vol. 1, MIT Press. pp. 355-368.

[2] Arifovic, J. (1995a), “Genetic Algorithms and Inﬂationary Economies,” Journal of Monetary Economics, Vol. 36, No. 1, pp. 219-243.

[3] Arifovic, J. (1996), “The Behavior of the Exchange Rate in the Genetic Algorithm and Experimental Economies,” Journal of Political Economy, Vol. 104, No. 3, pp. 510-541.

[4] Arifovic, J., J. Bullard and J. Duﬀy (1997), “The Transition From Stagnation to Growth: An Adaptive Learning Approach,” Journal of Economic Growth, Vol. 2, No. 2, pp. 185-209.

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Table 14: Complexity of Evolving Strategies

Periods k κ k κ k κ 1-1000 6.49306 15.45485 2.05669 3.07381 6.72513 18.38097 1001-2000 10.63684 45.34048 1.50741 1.74106 6.92334 16.87815 2001-3000 12.92895 76.73194 1.49331 1.72647 6.40866 15.28082 3001-4000 11.83218 60.33760 1.50069 1.72664 5.74555 14.30512 4001-5000 11.37810 43.41115 1.49665 1.73112 4.99250 14.59051 5001-6000 11.25262 40.83188 1.49975 1.73287 4.39418 11.63130 6001-7000 12.01427 66.41879 1.48403 1.69795 4.14494 12.61487 7001-8000 11.73566 65.89867 1.49715 1.72324 3.70228 10.19008 8001-9000 11.75256 74.46129 1.49465 1.71892 3.99040 15.93009 9001-10000 13.85391 100.87724 1.49338 1.73230 3.25615 10.64821

k and κ are the average of ktandκt taken over each period.

[5] Arthur, W. B., J. H. Holland, B. LeBaron, R. Palmer, and P. Taylor (1997), “Asset pricing Under Endogenous Expectations in an Artiﬁcial Stock Market,” in The Economy as an Evolving Complex System, Vol. II, W. B. Arthur, S. Durlauf, and D. Lane (eds.), Santa Fe Institute Studies in the Sciences of Complexity, Proceedings Volume XXVII, Reading, MA: Addison-Wesley. pp. 15-44.

[6] Bullard, J. and J. Duﬀy (1998a), “A Model of Learning and Emulation with Artiﬁcial Adaptive Agents,” Journal of Economic Dynamics and Control, Vol. 22, No. 2, pp. 179-207.

[7] Bullard, J. and J. Duﬀy (1998b), “Learning and the Stability of Cycles,” Macroeconomic Dynamics, Vol. 2, pp. 22-48.

[8] Bullard, J. and J. Duﬀy (1999), “Using Genetic Algorithms to Model the Evolution of Heterogeneous Beliefs,” Computational Economics, Vol. 13, pp. 41-60.

[9] Chen, S.-H., and C.-H. Yeh (1996), “Genetic Programming Learning and the Cobweb Model,” in P. Angeline (ed.) Advances in Genetic Programming, Vol. 2, Chapter 22, MIT Press, Cambridge, MA. 1996, pp. 443-466.

[10] Chen, S.-H., J. Duﬀy, and C.-H. Yeh (1996), “Genetic Programming in the Coordination Game with a Chaotic Best-Response Function,” in P. Angeline, T. Back, and D. Fogel (eds.) Evolutionary Programming V: Proceedings of the Fifth Annual Conference on Evolutionary Programming, MIT Press, Cambridge, MA, 1996, pp. 277-286.

[11] Chen, S.-H., and C.-H. Yeh (1997a), “Modeling Speculators with Genetic Programming,” in P. Angeline, R. G. Reynolds, J. R. McDonnell, and R. Eberhart (eds.), Evolutionary Programming VI, Lecture Notes

in Computer Science, Vol. 1213, Berlin: Springer-Verlag. 1997. pp. 137-147.

[12] Chen, S.-H., and C.-H. Yeh (1997b), “Speculative Trades and Financial Regulations: Simulations Based on Genetic Programming,” Proceedings of the IEEE/IAFE 1997 Computational Intelligence for Financial Engineering (CIFEr’97), New York City, U.S.A., March 24-25, 1997. IEEE Press, pp. 123-129.