AISB Journal 1(1), c 2001 http://www.aisb.org.uk
Market Diversity and Market Efficiency: The
Approach Based on Genetic Programming
Chia-Hsuan Yeh∗and Shu-Heng Chen†
∗Department of Information Management, Yuan Ze University, Chungli, Taoyuan
320, Taiwan, imcyeh@saturn.yzu.edu.tw
†Department of Economics, National Chengchi University, Taipei 11623, Taiwan, chchen@nccu.edu.tw
Abstract
The relation between market diversity and market efficiency has been studied. Economic heterogeneity is a fundamental driving force and an essential property in the economic systems. People who have different perspectives, technologies, or en-dowments may benefit from their trading behavior which constitutes economic ac-tivities. In this paper, economic simulation based on the growing field of artificial
stock market is employed to study this issue. Market size and different learning styles
are used to discuss the influence of heterogeneity. Simulation results have demon-strated that more participants and individual learning cause higher degree of traders’ diversity, which, in turn, enhances market efficiency.
1
Introduction
The relationship between competitiveness and market performance has been discussed for a long time. In a competitive economic environment, each firm or individual is un-able to influence the market. It has been mentioned in the economics courses that the competitive market is more efficient and has higher social welfare. Therefore, it is the de-sirable picture that economists intend to draw. The concept of competitiveness is related to market size, i.e., the number of market participants. The idea here is that the larger economy contributes to microeconomic heterogeneity, for example, behavior and strate-gies, profitability and market shares, production technology and efficiency. People who have different perspectives about the future implies that there exits room for the economic activity and they may benefit from their trading behavior. In other words, the higher de-gree of heterogeneity may provide more opportunities for trading. It is also an important seed of innovation.
However, few papers addressed the influence of the number of agents in the economic simulation literature. Usually, the number of agents in the simulated economies is de-termined arbitrarily or under the consideration of computational load. In (Arifovic et al. (1997)), they mentioned that the minimal number of strings (agents) for effective search is usually taken to be 30 according to the artificial intelligence literature. However, the point of view from optimization is quite different from the agent-based simulated economies (and/or the real world). (Aoki (1999)) pointed out that there exists different statistical properties when the number of agents (N ) goes to infinity. (Egenter et al. (1999)) have shown that the behavior of the simulated market became quite smooth or periodic and thus predictable if N → ∞. (Den Haan (2001)) also described the importance of the number
of different agents. In the framework of asset-pricing model, several properties depend on the number of types of agents. For example, the dynamics of interest rate, investment behavior, and agent’s welfare. In this paper, we try to restudy its relation to the market efficiency. This issue is very important in the financial market because it is well known that the thin and thick markets may have different financial properties and phenomena.
Economic simulations have been widely used in the study of economics. In order to model the interaction between many heterogeneous agents, the techniques of evolutionary computation were employed, for example, artificial neural nets, genetic algorithms (GAs) and genetic programming (GP). Different representation about agents’ behavior will in-fluence what they can learn, which may explain different scenarios. Genetic programming serves this purpose better than other techniques. Therefore, GP is applied to modeling the evolution of traders’ behavior in this paper.
However, there are two ways to implement GP, namely, single-population GP and multi-population GP. The former is a way to model social learning, whereas the latter focuses on individual learning ((Chen and Yeh (2001))). In this paper, we also attempt to figure out how differently the relation between market size and efficiency may emerge under these two different versions of GP. All these issues are studied within the context of agent-based artificial stock markets, which is probably the most powerful tool to study behavioral finance.
The organization of this paper is as follows. In Section 2, we present the framework of the agent-based artificial stock market. The implementation of social learning and individual learning is described in Section 3. Experimental designs and simulation results are shown in Section 4 and 5 respectively. Section 5 concludes.
2
Model Description
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, maximizes his expected utility.
2.1
Traders
For simplicity, we assume that all traders share the same constant absolute risk aversion (CARA) utility function,
U(Wi,t) = −exp(−λWi,t), (1) where Wi,tis the wealth of trader i at period t, and λ is the degree of relative risk aver-sion. 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,tdenote the money and shares of the stock held by trader i at period t respectively, and Ptis the price of the stock at period t. Given this portfolio (Mi,t,hi,t), a trader’s total wealth Wi,t+1is thus
where Dtis per-share cash dividends paid by the companies issuing the stocks and r is the riskless interest rate. Dtcan 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,
Ei,t(U (Wi,t+1)) = E(−exp(−λWi,t+1) | Ii,t) (4) subject to Equation (3), where Ei,t(.) is trader i’s conditional expectations of Wt+1given his information up to t (the information set Ii,t).
It is well known that under CARA utility and Gaussian distribution for forecasts, trader i’s desire demand, h∗i,t+1for holding shares of risky asset is linear in the expected excess return: h∗ i,t= Ei,t(Pt+1+ Dt+1) − (1 + r)Pt λσ2 i,t , (5)
where σ2i,tis the conditional variance of (Pt+1+ Dt+1) given Ii,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 detail is described in the next section.
2.2
Price Determination
Given h∗i,t, the market mechanism is described as follows. Let bi,tbe the number of shares trader i would like to submit a bid to buy at period t, and let oi,tbe the number of shares trader i would like to offer to sell at period t. It is clear that
bi,t= h∗
i,t− hi,t−1, h∗i,t ≥ hi,t−1,
0, otherwise. (6)
and
oi,t=
hi,t−1− h∗
i,t, h∗i,t< hi,t−1,
0, otherwise. (7) Furthermore, let Bt=N i=1 bi,t, and Ot= N i=1 oi,t (8)
be the totals of the bids and offers for the stock at period t, where N is the number of traders. Following (Palmer et al. (1994)), we use the following simple rationing scheme:
hi,t=
hi,t−1+ bi,t− oi,t, if Bt= Ot,
hi,t−1+Ot
Btbi,t− oi,t, if Bt> Ot, hi,t−1+ bi,t−Bt
Otoi,t, if Bt< Ot.
(9)
All these cases can be subsumed into
hi,t= hi,t−1+BVt tbi,t−
Vt
Otoi,t (10)
where Vt≡ min(Bt, Ot) is the volume of trade in the stock.
According to Palmer et al.’s rationing scheme, we can have a very simple price ad-justment scheme, based solely on the excess demand Bt− Ot:
where β is a function of the difference between Btand Ot. β can be interpreted as speed of adjustment of prices. The β function we consider is:
β(Bt− Ot) =
tanh(β1(Bt− Ot)) if Bt≥ Ot,
tanh(β2(Bt− Ot)) if Bt< Ot (12)
wheretanhis the hyperbolic tangent function:
tanh(x) ≡ eexx− e−x
+ e−x (13)
The price adjustment process introduced above implicitly assumes that the total num-ber of shares of the stock circulated in the market is fixed, i.e.,
Ht=
i
hi,t= H. (14)
In addition, we assume that dividends and interests are all paid by cash, so Mt+1=
i
Mi,t+1= Mt(1 + r) + HtDt+1. (15)
2.3
Formation of Expectations
As to the formation of traders’ expectations, Ei,t(Pt+1+Dt+1), we assume the following
functional form for Ei,t(.).1
Ei,t(Pt+1+ Dt+1) = (Pt+ Dt)(1 + θ1fi,t∗ 10−4), if − 104≤ fi,t≤ 104, (Pt+ Dt)(1 + θ1), if fi,t > 104, (Pt+ Dt)(1 − θ1), if fi,t < −104 (16) The population of fi,t (i=1,...,N ) is formed by genetic programming. That means, the value of fi,tis decoded from its GP tree gpi,t.
As to the subjective risk equation, we modified the equation originally used by (Arthur et al. (1997)), σ2 i,t= (1 − θ2)σ2t−1|n1+ θ2[(Pt+ Dt− Ei,t−1(Pt+ Dt))2]. (17) where σ2 t−1|n1= n1−1 j=0 (Pt−j− Pt|n1)2 n1− 1 (18) and Pt|n1 = n1−1 j=0 Pt−j n1 (19) In other words, σ2t−1|n
1 is simply the historical volatility based on the past n1
observa-tions.
Given each trader’ expectations, Ei,t(Pt+1+ Dt+1), according to Equation (5) and his own subjective risk equation, we can obtain each trader’s desire demand, h∗i,t+1shares of the stock, and then how many shares of stock each trader intends to bid or offer based on Equation (6) or (7).
1There are several alternatives to model traders’ expectations. The interested reader is referred to (Chen et al. (2001)).
3
The Framework of Social and Individual Learning
In this paper, genetic programming is employed to model the formation of traders’ ex-pectations. In agent-based computational economics, there are two different styles to implement GP, namely, single-population GP and multi-population GP. While, from the optimization point of view, the choice of the two styles (architectures) may have an ef-fect on the search efficiency, it brings different implications for the agent-based economic modeling. In the framework of single-population GP (SGP), each agent is represented by a tree (forecasting model, computer program). The evolution of this population of agents is performed through natural selection and genetic operations (reproduction, crossover, and mutation) on all of these trees (agents). In this case, agents can learn from other agents’ experiences, and hence is called social learning. On the other hand, in the frame-work of multi-population GP (MGP), each agent is endowed with a population of trees. Evolution is respectively manipulated in terms of the population of trees for each agent. In this case, agents learn only from their own experiences and reasoning, and hence is called individual learning.
According to (Vriend (2000)) and (Vriend (2001)), there is an essential difference between individual and social learning, and the underlying cause for this is the so-called spite effect. The spite effect may occur in a social learning GA, but can never occur in an individual learning GA. To see how the spite effect can influence the outcome of the evolutionary process, Vriend uses the two different GAs to simulate the learning process of an oligopoly game. The simulation results show that while the individual learning GA moves close to the Cournot-Nash output level, the social learning GA converges to the competitive Walrasian output level.
Vriend’s finding pushes us to think harder of the choice of the two styles of learn-ing. But, SGP (social learning) has another issue which is generally overlooked. (Harrald (1998)) criticized the traditional implementation of social learning. He mentioned the tra-ditional 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 pro-cesses. In particular, it is not easy to justify why we can learn or imitate other agents’ strategies (genotype) by means of their actions (phenotype).
Motivated by this criticism, (Chen and Yeh (2001)) proposed a modified version of social learning. The idea is to include a mechanism, called the “business school”. The business school serves as a faculty of researchers or a library. Knowledge in the business school or library is open for everyone. Traders can consult the researchers when they are under great peer pressure or economic loss. More precisely, the business school can be viewed as a collection of forecasting models. SGP (social learning) can then be applied to model the evolution of the business school without inviting Haralld’s criticism.2
Each researcher (forecasting model) is represented by a GP parse tree. Initially, the forecasting models (GPt=1) are randomly generated based on the function set and termi-nal set given in Table 1. For example,
✒✑ ✏ × ✖✕ ✗✔ Pt−1 ✒✑ ✏ – ✖✕ ✗✔ Pt−3 ✖✕ ✗✔ Pt−6 ✑ ✑ ✑ ✑ ◗◗ ◗◗
Each tree in the initial population of GP trees (forecasting models) is randomly generated
by either the growth method or the full method based on a toss of fair coin.3 In the
initialization stage, the maximum depth of a tree is restricted to 6. The performance of each forecasting model is determined by the pre-assigned fitness function, i.e., mean absolute percentage error (MAPE).
In the business school, the forecasting models will be evaluated with a pre-specified schedule, say once for every m1 trading days. The evaluation procedure proceeds as follows. At the evaluation date t, each old model (model at period t− 1, GPt−1) shall compete with a new model which is generated from GPt−1 itself by using one of the following four genetic operators: reproduction, mutation, crossover, and immigration, each with a probability pr, pm, pc, and pI (Table 1). The winner, the one with lower MAPE, will be selected as a model of next generation, GPt. When the evaluation is done over all the old models, a new generation of forecasting models (GPt) is then born. The four genetic operators are detailed as below.
• Reproduction:
Two forecasting models are randomly selected from GPt−1. The one with lower MAPE over the last m2days’ forecasts is chosen as the new model.
• Mutation:
Two forecasting models are randomly selected from GPt−1. The one with lower MAPE over the last m2days is chosen as a candidate, and has a probability of pM (Table 1) being mutated. There are several ways to implement mutation. What implemented here is known as tree mutation. If a tree is chosen to be mutated, one of its nodes is randomly selected, and the subtree originating from the node is replaced by a new subtree which is randomly generated. For example, consider the forecasting model A, whose tree structure is shown below. Suppose that the point M is chosen as the mutation point. Then the subtree originated from point M is removed and is replaced by a subtree B, which is randomly generated. And the result is the new model C.4
Tree A M Tree B N Tree C • Crossover:
Two pairs of forecasting models are randomly chosen, (gpj1,t−1, gpj2,t−1) and (gpk1,t−1, gpk2,t−1). The one with lower MAPE in each pair is chosen as a parent. The crossover operation is a sexual operation that starts with two parental models. One of the nodes for each parent is selected randomly, and the subtrees originating
3For more details, please refer to (Koza (1992)). 4But, if Tree A is not mutated (with a probability1 − P
from the nodes are exchanged. Two offspring are then produced. One of them is randomly chosen as the new model. For example, consider Tree A and B shown above as the parents, and points M and N are the crossover points. After crossover by exchanging the subtrees originating from M and N, the two offspring are Tree Aand B shown below.
Tree A
N
Tree B
M
• Immigration:
A forecasting model is randomly created as the new model. This operator is used to approximate the concept of imagination.
As to the evolution of traders’ behavior, each trader is endowed with NI forecasting models. These forecasting models are used to predict Ei,t(U (Wi,t+1)). In the beginning, they are also randomly generated. The performance of these forecasting models is mea-sured by profits, not MAPE, under a validation process. Traders’ adaptation proceeds as follows. At the evaluation date t, each trader has to make a decision. Should he change his forecasting model used in the previous period? This decision is affected by two psycho-logical factors, namely, peer pressure and a sense of self-realization. Generally speaking, if a trader’s profits earned in the last period is inferior to many other traders, or his prof-its comes to a historical low, then he will have a higher motivation, and hence a higher probability, to change.
The probabilities that traders consider to change a model can be mathematically de-scribed as follows. First, suppose that traders are ranked by the net change of wealth over the last n2trading days. Let∆Wn2
i,t be this net change of wealth of trader i at time period
t, i.e.,
∆Wn2
i,t ≡ Wi,t− Wi,t−n2, (20)
and, let Ri,tbe her rank. Then, the probability that trader i will change a model at the end of period t is assumed to be determined by
pi,t= Ri,t
N . (21)
The choice of the function pi,tis quite intuitive. It simply means that
pi,t< pj,t, if Ri,t< Rj,t. (22)
In words, the traders who come out top shall suffer less peer pressure, and hence have less motivation to change models than those who are ranked at the bottom.
In addition to peer pressure, a trader may also decide to change a model out of a sense of self-realization. Let the growth rate of wealth over the last n2days be
δn2
i,t =
Wi,t− Wi,t−n2
| Wi,t−n2 |
Table 1: Parameters of the Stock Market
The Stock Market
Shares of the stock (H) 100 (500) Initial money supply (M1) 100 (500)
Interest rate (r) 0.1
Stochastic process(Dt) 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 (pr) 0.10 Probability of creating a tree by immigration (pI) 0.20 Probability of creating a tree by crossover (pc) 0.35 Probability of creating a tree by mutation (pm) 0.35 Probability of mutation (PM) 0.30 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 4000
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 (Is∗) 5
Traders
Number of traders (N) 100
Number of ideas for each trader (NI) 1 (A), 10 (B), 25 (C)
Degree of RRA (λ) 0.5
Criterion of fitness (Traders) Increments in wealth (Income) Evaluation cycle (n2) 1
Search intensity by trader himself (Ih∗) 5
θ1 0.5
and let qi,tbe the probability that trader i will change a model at the end of the tth trading day, then it is assumed that
qi,t= 1
1 + expδi,tn2
. (24)
The choice of this density function is also straightforward. Notice that
lim
δn2i,t→∞qi,t = 0, (25)
and
lim
δi,tn2→−∞qi,t= 1. (26)
Therefore, the traders who have made great progress will naturally be more confident and hence have little need for changing ideas, whereas those who suffer devastating regression will have a strong desire for changing ideas.
In sum, for trader i, the decision to change a model can be considered as a result of a two-stage independent Bernoulli experiments. The success probability of the first experiment is pi,t. If the outcome of the first experiment is success, the trader will change a model. If, however, the outcome of the first experiment is failure, the trader will continue to carry out the second experiment with the success probability qi,t. If the outcome of the second experiment is success, then the trader will also change a model. Otherwise, the trader will keep her model unchanged. Based on the description above, the probability of changing the idea (ri,t) for trader i at period t is
ri,t= pi,t+ (1 − pi,t)qi,t =Ri,t N +
N − Ri,t
N 1 + exp1 δk i,t
(27)
Once the trader decides to change, how he changes depends on the two styles of learn-ing. In the case of social learning, he will register to the business school, and randomly catch a model there. In the case of individual learning, he will search a new model on his own by running genetic programming on the population of his NIexisting models (think tank).
Once he gets a new model, he will test the new model with the historical data and see whether the new model will bring him more profits than the old one. If it does, the old model will be replaced by the new model. Otherwise, he will keep on searching in the business school (in the case of social learning) or his think tank (in the case of individual learning) until either he succeeds or he fails for a pre-specified times, Is∗or Ih∗(Table 1). The process of traders’ evolution is also shown in Flowchart 1.
4
Experimental Designs
In order to examine the effect of market size on market efficiency, we consider exper-iments associated with two different market size (numbers of participants). The small market, M1, has 100 traders, whereas the large market, M2has 500 traders. Furthermore, to see the effect of social and individual learning on market efficiency, three scenarios corresponding to different styles of learning are proposed. Traders in Market A follow the social learning scheme (driven by SGP), and Traders in Markets B and C follow the individual learning scheme (driven by MGP).
Markets B and C are distinguished by the number of models (NI) (the size of think tank) assigned to each trader. In SGP, market size (number of traders) and population size
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(number of GP trees or models) refer to the same thing, because each trader is represented by one model in SGP. However, in MGP, market size and population size refer to two different things. Therefore, it is important to distinguish these two different size effects on market efficiency. In this paper, traders in Market B are endowed with 10 models, whereas traders in Market C are equipped with 25 models.
Therefore, totally, we conduct six distinct experiments, namely,(M1, A), (M1, B),
(M1, C), (M2, A), (M2, B), (M2, C). Each experiment consists of 10 trials of 4000
trading periods. In terms of genetic programming, the number of generations is not nec-essary the same as the number of trading periods. In fact, the former is the latter divided by evaluation cycle. In the case of Markets B and C, since GP is applied to traders, and the evaluation cycle (n2) is 1 (Table 1), the number of generations is 4000, i.e., the same as the number of trading periods. Nevertheless, in Market A, GP is not applied to traders but to the business school, and the evaluation cycle (m1) for the business school is 20 (Table 1); therefore the number of generations is only 200.
Each simulation will generate a time series data with 4000 observations of price. To avoid the effect of initialization, we shall drop the first 1000 observations, and base our analysis only on the last 3000 observations.
5
Simulation Results
5.1
The Efficiency Measure
The purpose of our experiments is to examine the effect of market size and learning styles on market efficiency. Before proceeding further, we have to give market efficiency a tech-nical notion so that our our numerical results can be presented and analyzed accordingly. In financial econometrics, the market is said to be efficient if the return series{rt} is
unpredictable, where
rt= ln(Pt) − ln(Pt−1). (28) Technically speaking, a series is unpredictable if there exists no linear and nonlinear structures in it, or the series is independent. To test whether a series is independent, we followed the procedure of (Chen et al. (2001)). This procedure is composed of two steps, namely, the PSC filtering and the BDS testing. We first applied the Rissanen’s predictive stochastic complexity (PSC) to filter the linear process. The PSC criterion is frequently used as a model selection tool in time series analysis. By using the criterion, we can determine the best linear structure, i.e., the best ARMA (AutoRegressive and Moving Average), of the time series in question.5 Once the linear signals are filtered, any signals
left must be nonlinear. The next step is then to apply the BDS test ((Brock et al. (1996))), one of the most frequently used tests for nonlinearity, to the residual series, i.e., the series after PSC filtering. The null hypothesis of the BDS test is that the series in question is identically and independently distributed. If we fail to reject the null hypothesis, then the series is said to be independent. Since the BDS test statistic under a large sample follows a standardized normal distribution, it is quite easy to have an eyeball check on the results.
5.2
Results: Price and Market Efficiency
The two-stage econometric procedure as outlined above gives three statistics for each market experiment, namely, PSC, R2, and BDS. They are all shown in Table 2. The
(p, q) under the column “PSC” is the orders p and q selected based on the Rissanen’s PSC
criterion. The column “R2” reports the coefficient of determination derived by running the PSC-selected ARM A(p, q) regression. The column “BDS” gives the BDS test statistic. If the return series is independent, there shall be completely no structure, be it linear or nonlinear, found in the series. In our statistical language, p, q, and R2should all be zero; the BDS test should be as low as possible, but not greater than a critical value, say, 1.96.6
Using these reference numbers, we can have a quick grasp of our simulation results. First of all, we would like to draw readers’ attention to the experiment(M2, C). As detailed in the previous section, the market in the experiment has three characteristics, large number of traders, large population size, and individual learning. Based on the reference numbers given above, one can easily see that the return series generated by this experiment are all independent. The only exception is Run 2, which has a linear structure being detected (p= 1). But the associated R2is so small (0.009) that one can hardly be interested in this linear signal. So, one can generally conclude that markets
5The ARMA process is a canonical linear time series model. Given a time series{X
t}, we say that {Xt}
follows a AR(p) process if
Xt= a0+ Σpi=1aiXt−i+ εt, (29)
and{Xt} follows a MA(q) process if
Xt= b0+ Σqj=1bjεt−j+ εt, (30)
whereεtis a Gaussian white noise, i.e.,
εt∼ N(0, σ2), and E(εs, εt) = 0 if s = t. (31)
{Xt} is said to follow a ARMA(p,q) process, if
Xt= c0+ Σpi=1aiXt−i+ Σqj=1bjεt−j+ εt (32)
Table 2: PSC Filtering and BDS Test
Market Size:M1
Market A Market B Market C
Run PSC R2 BDS PSC R2 BDS PSC R2 BDS 1 (2,2) 0.104 7.860 (2,3) 0.057 5.479 (2,3) 0.045 6.349 2 (3,3) 0.089 7.749 (0,2) 0.049 7.104 (3,0) 0.046 6.561 3 (2,2) 0.072 6.845 (2,2) 0.047 6.369 (0,0) None 4.948 4 (1,2) 0.086 8.462 (2,2) 0.066 6.988 (2,2) 0.085 8.153 5 (3,3) 0.082 7.750 (0,2) 0.029 6.802 (4,5) 0.086 6.274 6 (2,2) 0.072 6.909 (0,2) 0.033 7.052 (2,2) 0.056 6.099 7 (2,3) 0.102 6.958 (1,2) 0.091 10.831 (0,2) 0.047 7.726 8 (1,0) 0.058 6.890 (1,0) 0.051 7.012 (3,4) 0.085 6.413 9 (4,5) 0.133 7.431 (0,2) 0.061 7.777 (1,2) 0.044 6.073 10 (0,3) 0.058 6.956 (2,2) 0.057 5.727 (0,2) 0.031 5.833 Average 0.086 7.381 0.054 7.114 0.058 6.443 Market Size:M2
Market A Market B Market C
Run PSC R2 BDS PSC R2 BDS PSC R2 BDS 1 (1,0) 0.012 3.947 (1,0) 0.006 0.385 (0,0) None -0.009 2 (1,0) 0.011 3.096 (1,0) 0.007 1.479 (1,0) 0.009 0.788 3 (1,0) 0.018 2.971 (0,0) None 0.801 (0,0) None 0.455 4 (1,0) 0.015 2.076 (1,0) 0.007 0.373 (0,0) None -0.163 5 (1,0) 0.009 3.452 (0,0) None 0.319 (0,0) None 0.591 6 (1,0) 0.011 3.221 (1,0) 0.005 2.107 (0,0) None 0.180 7 (1,0) 0.029 2.635 (0,0) None 0.003 (0,0) None 0.678 8 (2,3) 0.027 3.529 (1,0) 0.009 1.292 (0,0) None 1.565 9 (1,0) 0.011 3.866 (1,0) 0.010 0.945 (0,0) None 1.154 10 (0,2) 0.025 3.944 (1,0) 0.013 0.034 (0,0) None 1.318 Average 0.0168 3.274 0.008 0.774 0.009 0.656
The BDS test statistic is asymptotically normal with mean 0 and standard deviation 1. The signifi-cance level of the test is set at 0.95. In BDS test, the distance parameter (standard deviations) is set to be 1, and the embedding dimension is set to be 5.
with large number of participants, each learn with the evolution of his own large number of forecasting models, is highly efficient in the sense that the efficient market hypothesis is sustained (return series is unpredictable).
From(M2, C), we have three directions to move. We would first like to know what will happen when the number of forecasting models assigned to each trader is reduced. To answer this question, we move from(M2, C) to (M2, B). In contrast, markets under
(M2, B) has a stronger linear structure: the AR(1) signal is detected in seven out of the
ten markets, while their associated R2s are generally very low. Furthermore, in one case (Run 6), nonlinear structure is also found (BDS =2.107). All these evidences suggest that the number of forecasting models can have a positive effect on the market efficiency. The larger the number, the higher the efficiency.
The second direction to move is to consider the effect of the number of market par-ticipants (market size), i.e., to move from(M2, C) to (M1, C). From PSC, we can see that the linear structure is even stronger than the case(M2, C). Linear signals are perva-sively found in all markets with Run 3 as the only exception. Some of the linear signal are even highly structured, say, Run 5. Turn to BDS, none of these markets fail to reject the null hypothesis. As a result, all markets are not efficient. The striking fact is that in
experiment(M2, C), we have all market efficient, but now all inefficient in (M1, C). The only change is the number of participants. Therefore, the market size can have a dramatic impact on the market efficiency. The more the participants, the higher the efficiency.
The third direction to move is to examine the effect of social learning. Here, we move from(M2, C) to (M2, A). From all the three statistics, one can easily see that all markets under(M2, A) are relatively inefficient. Hence, the social learning scheme represented by SGP with the business school can have a negative impact upon market efficiency.
5.3
Results: Trading Volume and Market Diversity
In addition to price dynamics, another important observation in stock market is trading volume. The trading volume serves as an important indicator of the diversity of traders’ expectations. Consider an extreme case where all traders hold the same expectation (fore-cast). With the same expectation, traders always stand in the same position, either all to buy or all to sell at any given price. In this case, no trade can possibly happen in the market, and this is corresponding to the famous no-trade theorem in neo-classical economics ((Tirole (1982))). On the other hand, if traders’ expectations are diversified (heterogeneous), then optimistic traders can easily be matched to pessimistic traders at many different prices, and one can expect a large trading volume appearing in the market. To have a slice of the idea that the trading volume is very different among the experi-ments, Figures 1-3 plots the time series of the trading volume observed in one typical run of the three market experiments(M1, A), (M1, B), and (M1, C). By presenting these fig-ures together, one can immediately see that the market(M1, A) is rather quiet as opposed to the other two. Its trading volume is almost nil in many of the trading days. Therefore, except for a few cases, traders in market(M1, A) share very similar forecasts.
Of course, three individual cases may tell us little what may cause the difference in the diversity of traders’ expectations. To be more systematic, Table 3 gives us two sum-mary statistics of all experiments. One is the average daily trading volume (the “Mean” column), and the other is the volatility of the daily trading volume (the “Std. Dev.” col-umn). Based on this table, one can well articulate the main factors which contribute to the diversity of traders’ expectations, and there are two factors, learning styles and market size.
First, learning styles. To see how learning styles can change the diversity of traders’ expectations, let us compare the experiment of social learning ((M1, A)) to the experiment of individual learning ((M1, B), (M1, C)). The ten markets in the experiment (M1, A) are all very quiet. On average, the mean trading volume over these ten markets is only 0.15 units, whereas the same figure in the other two experiments are 15 ((M1, B) and 14
((M1, C) units respectively, i.e. almost 100 times higher. This sharp contrast also appears
in the comparison between the experiment(M2, A) and the experiments (M2, B) and
(M2, C). For the former, the mean trading volume is only 1 unit, but for the latter, the
figure jumps to 81 and 79.
These figures clearly show that social learning and individual learning can have dra-matic impact upon the diversity on traders’ expectations. This result may not come to us as a surprise. If traders learn and adapt via a pool of common knowledge (business school, library,...), then the diversity of them is constrained by the diversity of the pool. Since in the business school direct imitation (reproduction) is feasible, competitive forecasting models can be disseminated to a large number of faculty members, which reduces the diversity of the pool, and the diversity of traders’ expectations. On the other hand, traders who learn and adapt on their own would not allow other traders to imitate their best fore-casting models kept as a business secret. Therefore, dissemination of knowledge is more
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difficult to proceed in the context of individual learning, and hence it tends to maintain a greater diversity of traders’ expectation.
Second, market size. To see the effect of market size on market diversity, the ex-periment(M1, A) is compared to the experiment (M2, A), i.e., the one with one hundred traders to the one with five hundred traders. From Table 3, one can see that when the num-ber of traders increases by 5 times, the mean daily trading volume also increase about 9 to 10 times, from 0.15 units to 1 units. But, caution should be exercised on interpreting these numbers. The thing is that when market size increases by 5 times, the total num-ber of shares also increases by that amount (so that share per capita remains unchanged). Therefore, it would not surprise us if trading volume only increases in proportion to the increase in market size. However, here we see that the trading volume increases by more than 5 times, and also double that figure; therefore, the effect of market size on market diversity is not superficial. One can confirm this finding by the other two sets of exper-iments under individual learning, i.e., {(M1, B), (M2, B)}, and {(M1, C), (M2, C)}. By comparing the same figure pairwisely, one can see once again that when market size increases by five times, the trading volume increases by more than five times.
There is a simple explanation for the impact of market size upon market diversity. In the case of social learning, even though there is only a single pool of common knowledge, a larger number of traders (students) implies a larger sampling of the pool, and hence a greater diversity. In the case of individual learning, since there is no channel for direct dissemination of knowledge, a larger number of traders implies more secrets kept by each individuals, and hence also a greater diversity.
But, this explanation is only one-way and may oversimplify the situation. The mar-kets which we study are typical examples of co-evolving systems. What has not been mentioned in the explanation above is the interaction among different components of the market. For example, in the case of social learning, a greater diversity of traders’ expecta-tions may result in more complex aggregate (price) phenomena, which in turn also nurse the diversity of the business school. And a pool with a greater diversity may make traders’ expectations even more diversified. This reinforcing mechanism can go on and on, which eventually increases the trading volume by far more than 5 times.
As we have discussed earlier, in the context of individual learning, there is another key parameter which has an effect on market efficiency, i.e., population size. However, it is interesting to note that population size does not have an positive effect on trading volume. If we compare the experiment(M1, B) to the experiment (M1, C), we see the mean daily trading volume decreases from 15 units to 14 units when population size increases from 10 to 25. A similar result is also observed in the other pair of experiments,{(M2, B),
(M2, C)}.
Why does population size have a negative effect on the diversity of traders’ expecta-tions? The reason is also intuitive. In the case of social learning, each trader can base their final decision only on one model no matter how many models, be it 10 or 25, they are able to process. As a result, the diversity of traders’ expectations is not directly affected by population size. Nonetheless, a larger population size means to equip each trader with a better capability to process information. Hence, if there is a best model at a point in time, the chance of discovering this model improves for all traders. Even though they do not come up with the same model, it is still likely that they find similar ones. That explains why the diversity of traders’ expectations and the trading volume may actually drop with the presence of a larger population size.
Table 3: Trading Volume
Market Size:M1
Market A Market B Market C Run Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.
1 0.150 1.384 16.216 13.034 15.189 12.805 2 0.277 1.328 17.665 14.517 15.363 13.067 3 0.410 1.998 15.373 13.785 15.613 13.035 4 0.150 0.928 16.713 13.898 15.843 13.323 5 0.049 0.285 15.122 13.405 13.702 12.439 6 0.080 0.480 14.342 12.345 14.613 12.380 7 0.044 0.328 13.955 11.980 12.892 12.168 8 0.172 1.242 15.174 13.116 12.841 11.371 9 0.059 0.329 14.365 12.535 14.820 12.369 10 0.105 1.134 14.153 12.447 14.838 12.503 Average 0.150 0.944 15.308 13.106 14.571 12.546 Market Size:M2
Market A Market B Market C Run Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.
1 0.728 5.172 73.330 47.525 71.622 49.105 2 0.935 4.042 82.657 53.155 82.150 53.602 3 0.718 7.004 84.541 50.787 82.813 51.563 4 0.778 5.195 72.865 50.061 76.502 49.392 5 1.112 8.127 95.009 60.465 86.131 57.656 6 1.319 6.277 85.942 52.902 79.125 52.582 7 0.409 1.807 74.807 48.631 90.215 55.193 8 0.877 7.791 78.962 50.327 70.133 45.129 9 1.944 10.873 81.123 53.567 78.114 49.935 10 1.453 7.110 83.132 51.648 78.295 52.167 Average 1.0273 6.340 81.237 51.907 79.510 51.632
5.4
Results: Portfolio and Market Diversity
In the previous section, market diversity is studied from the perspective of the trading volume. There is another way to observe market diversity, i.e, trader’s portfolios. A portfolio is the distribution of traders’ wealth into money and shares of the stock. If any point in time traders’ are homogeneous in their portfolio decision, then they must hold the same shares of stock, and the variance (diversity) of shares hold among all trader is zero. Therefore, the variance of shares hold provides us another measure to examine market diversity.
Corresponding to Figures 1-3, Figures 4-6 present the time series plots of the variance of shares hold in the three different market experiments. A large degree of homogeneity of portfolios is observed in the experiment(M1, A). In many of the trading days, the variance is almost zero, which suggests that traders hold the same amount shares. But
(M1, B) and (M1, C) give quite different pictures.
Table 4 gives the mean variance of all experiments. We shall not give detail accounts of the results here, because the findings are the same as those in Table 3. Briefly speaking,
Table 4: The Mean Variance of Stock Shares between Traders
Market Size:M1 Market Size:M2
Run Market A Market B Market C Market A Market B Market C
1 0.004 1.743 1.652 0.003 1.733 1.722 2 0.008 2.034 1.757 0.004 1.961 1.937 3 0.012 1.837 1.768 0.005 1.916 1.870 4 0.007 1.957 1.780 0.005 1.732 1.866 5 0.001 1.690 1.601 0.005 2.299 2.047 6 0.002 1.556 1.688 0.006 1.967 1.966 7 0.001 1.584 1.451 0.002 1.782 2.110 8 0.006 1.700 1.433 0.007 1.826 1.612 9 0.001 1.642 1.618 0.017 1.982 1.794 10 0.003 1.576 1.679 0.010 1.825 1.918 Average 0.005 1.732 1.643 0.006 1.902 1.884
a greater diversity of portfolios is observed under a larger market size and individual learning. However, a larger population size slightly reduce the diversity.
5.5
Market Diversity and Market Efficiency
The experimental results obtained above can be summarized as two effects on market efficiency (price predictability), namely, the size effect and the learning effect. The size effect says that the market will become efficient when the number of traders (market size) and/or the number of models (GP trees) processed by each traders (population size) increases. The learning effect says that the price will become more efficient if traders’ adaptive behavior become more independent and private. Coming to market diversity, we observe very similar effects except population size: market diversity does not go up with population size.
These findings motivate us to search for a linkage between market diversity and market efficiency. A suggested argument goes like follows: a larger market size, and a more isolated learning style will increase the diversity of traders’ expectations, which in turn make the market become more active (high trading volumes), and hence more efficient (less predictable). While the statement is somewhat plausible, we cannot give it a formal proof. In fact, in complex adaptive systems, the route from cause to effect are sometimes so complicated that no one can follow every step of it. Nevertheless, since the argument is empirically relevant, it can be taken as a hypothesis to test with real data.
Consider the Taiwan stock market as an example. A large proportion of market par-ticipants are individual investors who has little control of the market. Based on our “the-orem”, a large market size implies efficiency, Taiwan stock market should be efficient. However, empirical evidences have shown that this market is not that efficient ((Chen and Tan (1996))). What is missing here? By our “theorem”, the answer rests on social learning. Actually, in this market, individual traders usually consult the professionals from companies, institutions and even mass medium., which is exactly a kind of social learning. Also, take the United States as another example. In the U.S. stock market, a
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large proportion of market participants are institutional investors, each have their own research department. This is more similar individual learning. By our “theorem”, this market should be very efficient. Empirical studies have shown that it is indeed the case ((Chen and Tan (1996))).
6
Concluding Remarks
In this paper, the relation between market diversity and market efficiency is investigated. The simulation results reveal that the important driving force which makes market effi-ciently is market diversity. The increase of market size contributes to the market efficiency by means of introducing greater diversity into the market. Moreover, individual learning further reinforces the effect, whereas population size plays a mixing role. These three factors co-influence the market dynamics.
Of course, there exists some other determinants for market efficiency. For example, the degree of traders’ prudence. This behavioral parameter concerns the number of peri-ods (time horizon) which the traders look back while making their forecasts. A prudent trader cares long-term profits, which, in our framework, corresponds to a long evaluation cycle (n2). Also, in a more flexible design, traders should be able to adaptively switch between individual learning and social learning, rather than getting stuck to only one learning style. The effects of these determinants are left for further research.
Acknowledgements
Research support from NSC grants No. 89-2415-H-214-002 is gratefully acknowledged. This paper was revised from its original version by taking the advice and comments of four anonymous referees. The authors are very grateful for their painstaking review of this paper. Of course, all remaining errors are the authors’ sole responsibility.
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