minority game as a model for the artificial financial markets

8/2/2019 Minority Game as a Model for the Artificial Financial Markets

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mieko

Abstract — What are the conditions required for a good

artificial market? Simplicity, expandability, reality, good

learning capability, sustainability are the elements definitely to

be included into the model. Minority Game is one of those

candidate models that depict necessary features for an artificial

market. It is an evolutionary game designed for modeling the

financial market made of heterogeneous agents with learning

ability. Those agents are equipped with limited intelligence in

space and time and they have access to public information, but

independent otherwise. However, the rule of the original form

of the minority game has an essential defect. The winners

receive the same reward no matter how small the number of the

minority agents is. This is unnatural, and this is the very reason

why the conventional form of the minority game fails toreproduce the realistic wealth distribution among agents,

namely, Gini’s coefficient being much too small compared to

the observed value. To remedy this, winners with larger risk

ought to be rewarded better. We show how this modification

saves the model and reproduce the realistic wealth distribution

( 0.5G ≅ ) known from the statistics.

I. I NTRODUCTION

HE minority game (MG) is an evolutionary game

designed for modeling the financial market made of

heterogeneous agents with learning ability [1, 2]. The major

function of those agents is to decide one out of two possible

actions, say 0 or 1, corresponding to buy or sell, buy or not to buy, etc. and all the agents who have made the minority

choice win the game. Which choice has won the game

becomes the public information open to all the agents. Each

agent decides 0 or 1 following the strategy with the highest

score in their strategy table and they are bound to choose the

best strategy out of the individual strategy table at each time.

Each strategy is a choice of 0 or 1 corresponding to the M-step

history. There arek

2 possible strategies corresponding toM

2k = possible histories, and the size S of the strategy table

is supposed to be much smaller than the maximum

This work was supported in part by the Japan Foundation of

Promoting Science for the Grant-In-Aid Scientific Research

(C)(2) 14580385.

M. Tanaka-Yamawaki is with the Department of Information

and Knowledge Engineering, Tottori University, Tottori

680-8552 JAPAN (phone: +81-857-31-5223; fax:

+81-857-31-0879; e-mail: [email protected]).

S. Tokuoka., is with the Department of Information and

Knowledge Engineering, Tottori University, Tottori

680-8552 JAPAN

number k

2 . This implies the agent’s intelligence is highly

limited in terms of space, i.e., each agent has a capacity of

having S strategies in his strategy table. A strategy set is

produced for each agent randomly at the beginning of the

game and is kept unchanged throughout the game. Learning

ability is limited to the choice of the best strategy according to

the point given to the strategy. If an agent wins/loses the

game by using the chosen strategy, it gains/loses a point in the

strategy table. Agents do not communicate each other; and

they do not know other agents’ strategy table. Their

intelligence is also limited in terms of time. The memory

length M is not very long. All the agents have the same S andM. An example of a strategy table is shown in Table 1 for the

case of M=2 and S=3.

Though MG is invented as a model of an artificial market,simulation results of MG-oriented artificial society indicate

that the numbers of winning games for most agents remain

almost equal throughout a series of games. In other words, the

society made of MG agents is a highly equal society. Since

individual agents are given different sets of strategies, this

fact tells us that the used strategies are also equal and there is

no outstanding one.

In order to make the model more meaningful, we consider

two ways of modifying the standard MG. One is to change the

rewarding scheme (RMG) and the other is to introduce

agents’ assets explicitly (IMG), and its further modification

with tax effect (TMG). We will show that this new modelTMG derives a realistic wealth distribution among agents in

terms of Gini’s coefficient as well as Pareto’s index.

The structure of the paper is as follows. Section II describes

our interpretation for the meaning of MG. Section III is

devoted to explain the problem of equal agents in the standard

MG (SMG) and to introduce the first way (RMG) of solving

the equality problem by changing the rewarding scheme.

Section IV presents the result of RMG in terms of Gini’s

coefficient, which is compared with the result of IMG and

Minority Game as a Model for the Artificial Financial Markets

Mieko Tanaka-Yamawaki and Seiji Tokuoka, Department of Information and Knowledge Engineering,

Faculty of Engineering, Tottori University, Tottori 680-8550 Japan

T

TABLE IEXAMPLE OF THE STRATEGY TABLE (M=2,S=3)

Winning History STRATEGY1 STRATEGY2 STRATEGY3

00 1 1 101 0 0 110 0 1 1

11 0

0

0

0-7803-9487-9/06/$20.00/©2006 IEEE

2006 IEEE Congress on Evolutionary ComputationSheraton Vancouver Wall Centre Hotel, Vancouver, BC, CanadaJuly 16-21, 2006

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TMG, the second way of solving the equality problem by

introducing assets explicitly. We then conclude that TMG

can solve the problem nicely when the tax rate and the

investment rate are about the same. Section V examines

Pareto’s law by looking for the linearity in log-log plot of

accumulated wealth distributions and show that TMG is in

good agreement to the observed Pareto’s index. Section VI

concludes the paper.

II. IMPLICATION OF THE MINORITY GAME (MG)

The basic assumption of the minority game sounds a little odd.

First, the traders seem to follow the majority in order to gain

profit in the trades. Indeed many day-traders behave as trend-

followers and take small profits by making frequent small

trades. On the other hand, investors or long-term traders do

not chase small profits. They invest before the stock price

rises and sell when many others start following the trend. In

other words, they are minorities.

Traders always try to outwit others. They would think of

buying while others are selling, and selling while others are

buying. They are content if such a choice was a success.

Seemingly a result of much deliberation, a trader’s

satisfaction simply comes from the action he took was indeed

the right one. Traders also know the history is useful if it is

not too long. Contrary to other games such as chess, a long

process of thinking is meaningless for the game of trading.

What is important is when the current trend ends and the

market enter the reverse trend cycle. Those who can judge the

moment of turnaround would win by outwitting the others.

Traders usually do not exchange ideas each other. They are

independent agents. They all have a common knowledge of

price history of the market in the near past, e.g., whether

many traders sold last weekend or not, and use it to maketheir own decisions. The market trend is formed when the

bulk of traders make the same choice. This occurs as a

stochastic process. In other words no one can predict based on

any deterministic logic exactly when it occurs and how. The

asset price is another stochastic process in the market. Those

two processes are correlated each other. One drastic

assumption is to regard those two processes as a single

stochastic process of the system. MG is thus derived

following such assumptions.

a v e r a g e n u m b e r o f w i n n e r s

86

88

90

92

94

96

98

100

0 2 4 6 8 10 12

memory

S = 2S = 4S = 8

S = 16S = 32random

Fig. 1 Number of winning agents is shown as a function of memory M for various S values, from 2 (largest for M>1) to 32 (lowest for M>1), for agentnumber N=201 in the case of original MG. The results are the averaged numbers

over 100 trials, where each trial has 10,000 repeated games of MG.

MG exhibits self-organized behavior [1]. An MG-based

market has consistently larger winners than the case of

random strategy as shown in Fig.1.

Since the agents do not communicate each other and theyare not given any motivation to collaborate, this spontaneous

emergence of self-organization is a nontrivial result. Fig.1

shows that significantly many agents win for S < 10 in most

range of memory size M, since the maximum number of

winners is 100 for the case of N=201.

It is a little odd to see that the performance of MG for S >

10 for M < 5 is much worse than the horizontal line at 94.8

representing the random strategy, since we would expect

better performance for more intelligent agents having large

memory length M and memory size S. From now on, we focus

on the parameter range S < 10 and M<10, in which the

strategy makes sense.

III. PROBLEMS OF SMG AND ITS SOLUTION BY RMG

Minority game in its standard form is unsatisfactory

considering the traders’ nature, or even at the stage of El

Farol problem concerning a chance of visiting the pub at a

good time or crowded time [3]. The more profit is promised to

each winner for the less number of winners. This factor is

missing in the standard minority game. Giving the same

reward to the owners of the winning strategy, the standard

minority game successfully maintains a long-lived society for

a very long time, with a penalty of unnaturally even

distribution of wealth among agents. As shown in Fig.2, the

agents have almost equal winning chances throughout thegames. Assuming that the agents receive rewards

proportional to the number of winning games, the wealth

distribution among agents is almost even. In order to remedy

this defect, we attempt to alter the basic rule of MG in various

manners.

3200

3400

3600

3800

4000

4200

4400

4600

4800

5000

5200

0 50 100 150 200

w

i n T i m e s

player

M = 1M = 3M = 5M = 7M = 9

Fig. 2 Average numbers of winning games out of 10,000 at each trial are shownin the descending order for N=201 players, for the Standard MG with S=4,M=1,3,5,7,9. Note that the numbers of winning games are located within a very

narrow range except for M=1, indicating that the wealth distribution is highlyeven over the entire society.

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First, we try to change the rewarding scheme by giving

+2 for winning strategy and -1 for each losing strategy. By

encouraging the winners, we expect to promote selfish

competitions among agents. We call this model the

reward-driven Minority Game (RMG) [4]. The result

shown in Fig. 3 surprisingly tells us that this model

promotes more cooperation among agents, and less

dependence on the size of intelligence S. Also for small

memory length (M=1-3) the average winners are close to

the maximum (100 for N=201), for which the society

members enjoy the maximum profits as a whole. The only

disadvantage of our reward-driven MG of (+2,-1)

rewarding system compared to the standard model seems

to be the biased distribution of wealth.

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200

w i n T i m e s

player

M = 1M = 3M = 5M = 7M = 9

Fig.4 Average numbers of winning games out of 10,000 at each trial are shownin the descending order for N=201 players, for the Reward-driven MG of (+2,-1)

rewarding scheme with S=4, M=1,3,5,7,9. The numbers of winning games aredistributed over wider range compared to the standard MG, indicating that thewealth distribution is more biased compared to the standard model.

In order to see what is happening more closely, we

compare the number of winning games for each agent for the

standard MG (SMG) and our reward-driven MG (RMG).

Fig.2 shows the number of winning games of all the 201

agents in the descending order for the case of SMG. We can

see that, except for the case of M=1, most agents win

approximately 4,800 times out of total 10,000 games, which

indicates that the society of SMG is highly even. Fig.4 shows

the corresponding result for RMG for S=4, M=1,3,5,7,9,

N=201, which indicates that the numbers of winning games

for 201 agents distribute over wider range from 3,000 to

7,000 for the case of RMG (+2,-1). The society of RMG is

uneven compared to that of SMG.

We have also tried other choices of rewarding scheme such

as (+3,-1), (+4,-1), (+10,-1), and (+1,-2) for N=201, S=8. The

first three choices (+3,-1), (+4,-1), and (+10,-1) do not make

any essential difference from (+2, -1). The choice of opposite

direction (+1,-2), on the other hand, makes the numbers of

winners much smaller (ranging from 80 to 92) than the

random strategy (slightly bellow 95) thus discarded. The

reason why this rewarding scheme fails to produce the

spontaneous collaboration is supposed to be as follows. Under

such a situation, the agents keep changing the strategies and

loose a chance to stick to a certain selected strategy while

wondering around the strategy apace. In this regard, the

reward-driven scheme tends to stick to a very few strategies at

the early stage of the simulation [4].

IV. GINI’S COEFFICIENT AND TMG

Gini’s coefficient is frequently used to represent the evenness

of wealth distribution. The range of this coefficient is

N/)1 N(G0 −≤≤

where the minimum value G=0 occurs when the wealth is

equally distributed to all the agents, while the maximum

G=(N-1)/N is achieved when the whole wealth is

concentrated to a single agent. The more biased wealth

distribution corresponds to the larger value of G.The statistical value of G is reported to be around G=0.5 in

the case of Japanese household in 2002 [5] and around G=0.7

for the world average.

Gini’s coefficient for the SMG reproduced for the case of

N=201, S=4, M=8 is shown in Fig.5 as a time series of a

simulation. The curve quickly falls to a very small value close

to 0.01, which is far from the reality. This situation reflects

the fact that all the agents are equally rich in this artificial

society.

Our model RMG saves the situation considerably. The

biased winning chances in RMG as shown in Fig. 4 results in

much larger Gini’s coefficient as shown in Fig. 6, where the

curve saturates around G=0.2. This amounts to about one half

of the statistical value in Ref. 5.

Another way of having a biased wealth distribution is to

introduce a direct exchange of asset to reward the winners. By

doing this, we can avoid the defect of MG where the degree of

minority is neglected entirely and the agent is rewarded by

the equal amount no matter how small the minority group is.

Introduction of direct asset exchange among agents requires a

constitutional change of the model.

Fig. 3 Number of winning agents as a function of M for thereward-driven model (RMG). Note that the winning rate isconsistently higher than the random strategy represented by a

horizontal line at 94.8.

94

95

96

97

98

99

100

0 2 4 6 8 10 12

a v e r a g e n u m b e r o f w i n n e r s

memory

S = 2S = 4S = 8S = 16S = 32random

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Ref. 6 is an example of such a model. The agents in this

model start with equal amount of assets. They invest the

portion of their own assets, with the constant rate (r), before

each game. The winner (i.e. minority agents) receives a

portion of the sum of invested asset in proportion to the

invested amount. We call this model as an Investing Minority

Game (IMG)

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

G i n i ' s c o e f f i c i e n t

Time Fig.5 Time series of Gini’s coefficient (G) for the standard minority game(RMG, N=201,S=4,M=8). Gini’a coefficient quickly converges to 0.01 which

is much too small compared to the observed value, 0.5G ≅ , indicating that the

wealth is evenly distributed among agents.

0

0.2

0.4

0.6

0.8

1

01000 2000 3000 4000 5000 6000 7000 8000 9000 10000


Time

Fig.6 Time series of Gini’s coefficient (G) for the reward-driven minority game(RMG, N=201,S=4,M=8). G converges to 0.2 which is approximately one half of the observed value [8].

However, the investing version of MG (IMG) proposed in

Ref. 6 cannot save the MG-oriented artificial market. The

coefficient G quickly saturates around the value very close to1 as shown in Fig. 7. Parameters chosen to take data for Fig.

7 are N=10001, S=4, M=8, and r=0.3, 0.1, 0.01,

corresponding to the upper, middle, lower line, respectively.

We use N=10001 to show the result of IMG and TMG, mainly

because N=201 is not enough to see the scale-invariant nature

of the Pareto distribution, though the qualitative features of G

in Fig.5 and Fig.6 are about the same as the case of N=201.

While choosing the smaller investment rate r, say r=0.01, can

slow down the growth of G and lowers the saturated value

slightly compared to the cases like r=0.1 or r=0.3, G

converges to the values much too large compared to the

observed value in all the three cases. Fig. 8 shows the result of

choosing r=0.001, where the convergence of G takes more

time but eventually saturates towards 1. This fact corresponds

to the fact that the society made out of IMG seems too biased

in terms of wealth distribution.

0

0.2

0.4

0.6

0.8

1

0 20000 40000 60000 80000 100000


Time

r = 0.01r = 0.1

r = 0.3

Fig.7 Time series of Gini’s coefficient (G) for the investing minority game (IMG,

N=10001,S=4,M=8). G converges quickly to 1 for large investment rate (r=0.1,0.3), and more slowly for smaller investment rate (r=0.01).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 200000 400000 600000 800000 1e+006


Time Fig.8 Time series of Gini’s coefficient (G) for the investing minority game (IMG,

N=10001,S=4,M=8) for the case of very small investment rate, r=0.001. Geventually converges to 1 after a long time.

To remedy this situation, we consider further modification

of MG that pulls back the biased wealth distribution created

by the investment activity. We apply a fixed rate (Z) of tax to

all the winners. The rate Z is applied for the share to be

distributed to the winners and the sum of the collected tax isre-distributed to all the winners evenly. We call this version

of MG as the investing version of the standard minority game

with tax-redistribution effect, TMG. The time series of Gini’s

coefficients obtained from the simulation experiments are

shown in Fig.9 for r=0.01 and Fig.10 for r = 0.3, in which five

lines corresponding to Z=0.01, 0.05, 0.1, 0.2, 0.3 are plotted

from the top to the bottom. The best fit to the real-world case

is obtained for the case of Z=r.

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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 10000 20000 30000 40000 50000


Time Fig.9 Time series of Gini’s coefficient for the investing minority game with tax

effect (TMG, r=0.01, N=10001,S=4,M=8) for various rates of Tax. The fivelines correspond to Z=0.01, 0.05, 0.1, 0.2, 0.3, from the top to the bottom. Thecoefficient G saturates around 0.45, close to the observed value for Z=r.

0.2

0.4

0.6

0.8

1

0 10000 20000 30000 40000 50000


Time Fig.10 Time series of Gini’s coefficient for the investing minority game with taxeffect (TMG, r=0. 3, N=10001,S=4,M=8) for various rates of Tax. The fivelines correspond to Z=0.01, 0.05, 0.1, 0.2, 0.3, from the top to the bottom. Thecoefficient G takes the value close to the observed value G=0.5 for Z=r.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10000 20000 30000 40000 50000


Time

Fig.11 Time series of Gini’s coefficient f for the investing minority game withtax effect (TMG, Z=0.1, N=10001,S=4,M=8) for various values of investment

rate r, r=0.3, 0.2, 0.1, 0.05, 0.01, from the top to the bottom. The coefficient Gtakes the realistic value of 0.4 for r=Z.

Fig.11 shows the same result from the different perspective,

namely, the time series of G for r=0.1 (Fig.11) is plotted in

five lines corresponding to r=0.3, 0.2, 0.1, 0.05, 0.01, from

the top to the bottom. The coefficient G takes the value close

to the observed value G=0.5 for r=Z

It is worthwhile at this moment to compare the essence of

SMG and TMG (as well as IMG). The original form of MG

(SMG) guarantees the same amount of reward no matter how

large the size of minority is, as shown in Fig. 12. On the other

hand, TMG and IMG give a fixed amount of reward to the

winner’s group and each winner receives the total reward

divided by the number of winners, as shown in Fig.13.

win

winlose

Each winner

receives one unit

of reward

lose

The winner

receives one

unit of reward

SMG

Fig.12 Rewarding scheme of SMG. Each winner receives the same amount of reward. The resulting wealth distribution is very even, corresponding to G<<1,

and large Pareto’s index (=48)

win

winlose

Each winner

receives 1/3 unit

of asset

lose

The winner

receives one

unit of asset

TMG,IMG

Fig.13 Rewarding scheme of TMG, IMG. Winners share the reward.Accordingly each winner receives the total reward divided by the number of winning agents. The resulting wealth distribution is in good agreement with the

real value in terms of G=0.5, and the reasonable Pareto’s index (=1.8)

V. PARETO’S DISTRIBUTION

Wealth distribution of top few percent of wealthy members

is known to have a scaling property, which is indicated by the

power-law behavior in the accumulated distribution [7, 8].

More recent empirical studies based on the real data partly

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have confirmed the above result [9, 10] and narrowed down

the size of Pareto’s index within the range around 1-4. The

results of SMG, RMG and TMG are shown in Fig. 14, 15, 16,

respectively. The scale-invariant property and the size of

Pareto’s index are nicely reproduced in the case of TMG as

shown in Fig. 16.

0.001

0.01

0.1

1

a c c u m u l a t e d d i s t r i b u t i o n

wealth

-48

4000 5200

Fig.14 Accumulated distribution of wealth among agents in the investingminority game with tax effect (SMG, N=201,S=4,M=8)

0.001

0.01

0.1

1

1000 10000

a c c u m u l a t e d d i s t r i b u t i o n

wealth

30000

-3.57

Fig.15 Accumulated distribution of wealth among agents in the investingminority game with tax effect (RMG, N=201,S=4,M=8)

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000 1000 00 1e +006

a

c c u m u l a t e d d i s t r i b u t i o n

wealth

-1.82

Fig.16 Accumulated distribution of wealth among agents in the investing

minority game with tax effect (TMG, r=0.3, Z=0.3, N=10001,S=4,M=8)

VI. CONCLUSION

We have investigated the wealth distribution problem of an

artificial society playing minority games as the major activity

of the society. The original version of minority game (SMG)

has a problem of giving the same reward to the winners

regardless of the size of minority. Gini’s coefficient G is

extremely small compared to the realistic size. To remedy this

situation, we have considered two different versions of MG.

One is to introduce a biased rewarding scheme named

reward-driven minority game (RMG) which gives more

rewards to the winners and less penalties to the losers. Gini’s

coefficient of RMG saturates around G=0.2 which is close to

one half of the real statistical value of G=0.5.

Another way of altering the model is to explicitly introduce

a new variable of agents’ asset in order to allow direct

exchanges of assets among agents. This version of MG,

named investing MG (IMG), creates accelerated

concentration of wealth so that Gini’s coefficient (G)

eventually saturates to a very large number near 1, which is

too large compared to the real value. Changing r,

representing the rate of invested amount out of the total asset,

does not help the situation. We further modified the model to

introduce re-distribution of the earned amount in IMG, i.e.,

tax effect. This new model called taxed MG (TMG)

successfully reproduce the realistic value of Gini’s coefficient

at the cost of adjusting two parameters, investment rate, r and

the tax rate, Z. Moreover, our new model TMG has proved,

via simulation, to reproduce the scaling behavior (a straight

line in log-log plot) in the accumulated wealth distribution

and the corresponding Pareto’s index comes out to be

consistent with the measured values.

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[7] V. Pareto, “Cours d’Economique Politique”, McMillan, London 1897.[8] E.W. Montroll and M.F. Shlesinger, Journal of Statistical Physics, 1984;[9] H. Aoyama, W. Souma, Y. Nagahara, H.P. Okazaki, H. Takayasu, and

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minority game as a model for the artificial financial markets

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