How to attain maximum profit in minority game?

Physica A 312 (2002) 277–284www.elsevier.com/locate/physa

How to attain maximum pro"t in minority game?H.F. Chau ∗, F.K. Chow

Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong

Received 22 January 2002

Abstract

What is the physical origin of player cooperation in minority game? And how to obtainmaximum global wealth in minority game? We answer the above questions by studying a variantof minority game from which players choose among Nc alternatives according to strategiespicked from a restricted set of strategy space. Our numerical experiment concludes that playercooperation is the result of a suitable size of sampling in the available strategy space. Hence,the overall performance of the game can be improved by suitably adjusting the strategy spacesize. c© 2002 Elsevier Science B.V. All rights reserved.

PACS: 05.65.+b; 02.50.Le; 05.45.−a; 87.23.Ge

Keywords: Minority game; Self-organized systems; Reduced strategy space

Econophysics—the study of economic and economic inspired problems by physicalmeans—is the result of interAow between theoretical economists and physicists. Usingstatistical mechanical and nonlinear physical methods, econophysicists study globalbehaviors of simple-minded models of economic systems making up of adaptive agentswith inductive reasoning. In particular, minority game (MG) [1,2] is an important andperhaps the most extensively studied econophysics model of global collective behaviorin a free market economy. This game was proposed by Challet and Zhang underthe inspiration of the El Farol bar problem introduced by the theoretical economistArthur [3].MG is a toy model of N inductive reasoning players who have to choose one out

of two alternatives independently according to their best working strategies in eachturn. Those who end up in the minority side (that is, the choice with the least num-ber of players) win. Although its rules are remarkably simple, MG shows a surpris-ingly rich self-organized collective behavior. For example, there is a second phase

∗ Corresponding author.

0378-4371/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(02)00860 -9

278 H.F. Chau, F.K. Chow / Physica A 312 (2002) 277–284

transition between a symmetric and an asymmetric phase [4–6]. Since the dynamics ofMG minimizes a global function related to market predictability, we may regard MGas a disordered spin glass system [7,8]. Recently, Hart et al. introduced the so-calledcrowd–anticrowd theory to explain the dynamics of MG [9,10]. Their theory statedthat Auctuations arised in the MG is controlled by the interplay between crowds oflike-minded agents and their perfectly anti-correlated partners. The crowd–anticrowdtheory not only can explain global behavior of MG, it also provides a simple work-ing hypothesis to understand the mechanism of a number of models extended fromthe MG.Numerical simulation as well as the crowd–anticrowd theory tell us that the global

behavior of MG depends on two factors. The "rst one is the product of the number ofplayers N at play and the number of strategies S each player has. The second factor isthe complexity of each strategy measured by 2M+1, where M is the number of the mostrecent historical outcomes that a strategy depends on. Global cooperation, as indicatedby the fact that average number of players winning the game each time is larger thanthe case when all players make their choice randomly, is observed whenever 2M+1 ≈NS [4–6]. In fact, cooperative phenomenon is also seen in our recent generalization ofthe MG in which each player can choose one out of Nc alternatives. More precisely,NMc ≈ NS is a necessary condition for global cooperation between players in ourgeneralization [11].Perhaps, the two most important questions to address are why and when the players

cooperate in MG. In fact, these are the questions that the crowd–anticrowd theory wastrying to answer. On the way of "nding out the answers, Cavagna believed that theonly non-trivial relevant parameter to the dynamics of MG is M [12]. But later on,Challet and Marsili revealed that historical outcomes also determine the dynamics ofMG in general. They also found that information contained in the historical outcomesis irrelevant in the symmetric phase [13].Is it true that global behavior of MG is determined once N; S and M are "xed?

More speci"cally, we ask if it is possible to lock the system in a global cooperativephase for any "xed values of N; S and M . In this way, players, on average, gainmost out of the game. In what follows, we report a simple and elegant way to alterthe complexity of each strategy in MG with "xed N; S and M . By doing so, it ispossible to keep (almost) optimal cooperation amongst the players in almost the entireparameter space.We begin our analysis by "rst constructing a model of MG with Nc alternatives

whose strategy space size equals N 2c for a "xed prime power Nc. We label, for sim-

plicity, the Nc alternatives as the Nc distinct elements in the "nite "eld GF(Nc); and wedenote this variation of MG by MG(Nc; N 2

c ). In MG(Nc; N 2c ), each of the N players is

assigned once and for all S randomly chosen strategies. Each player then chooses oneout of the Nc alternatives independently according to his=her best working strategy ineach turn. The choice chosen by the least non-zero number of players is the minoritychoice of that turn. (In case of a tie, the minority choice is chosen randomly amongstthe choices with least non-zero number of players.) The minority choice of each turnis announced. The wealth of those players who end up in the minority side is addedone point while the wealth of all other players is subtracted by one.

H.F. Chau, F.K. Chow / Physica A 312 (2002) 277–284 279

To evaluate the performance of each strategy, a player uses the virtual score whichis the hypothetical pro"t for using that strategy in playing the game. The strategywith the highest virtual score is considered as the best performing one. (In case ofa tie, one chooses randomly amongst those strategies with highest virtual score.) Theonly public information available to the players is the output of the last M steps. Astrategy s can be represented by a vector s ≡ (s1; s2; s3; : : : ; sL) where L ≡ NMc and siare the choices of the strategy s corresponding to diLerent combination of the outputof the last M steps. In MG(Nc; N 2

c ), strategies are picked from the strategy spaceS = {�ava + �uvu: �a; �u ∈GF(Nc)} of size N 2

c where GF(Nc) denotes the "nite "eldof Nc elements and all arithmetical operations are performed in the "eld GF(Nc). Thetwo spanning strategy vectors va ≡ (va1; va2; : : : ; vaL) and vu ≡ (vu1; vu2; : : : ; vuL) of thelinear space S satisfy the following two technical conditions:

vai �=0 for all i (1)

and by regarding i as a uniform random variable between 1 and L,

Pr(vui = k|vai = j) = 1=Nc for all j; k ∈GF(Nc) (2)

whenever Pr(vai = j) �=0. (We remark that these two technical conditions are satis"edby various choices of va and vu such as vai = 1 and vui = f(imodNc) where f is abijection from ZNc to GF(Nc).)The span of the strategy vector va over GF(Nc) forms a mutually anti-correlated

strategy ensemble Sa since Eq. (1) implies that any two distinct strategies drawn fromSa always choose diLerent alternatives for any given historical outcomes. Hence, theHamming distance between any distinct strategies u 1 �= u 2 in Sa equals

d(u 1; u 2) = L : (3)

In contrast, the span of the strategy vector vu over GF(Nc) forms a mutually uncor-related strategy ensemble Su since Eq. (2) and the fact that �GF(Nc)=GF(Nc) for all�∈GF(Nc) \ {0} imply that any two distinct strategies drawn from Su always choosetheir alternatives independently for any given historical outcomes. In other words, theprobability that any two distinct strategies drawn from Su choose the same alternativeis equal to 1=Nc. Consequently,

d(u 3; u 4) = L(1− 1=Nc) (4)

for any u 3 �= u 4 ∈Su; andd(u 1; u 3) = L(1− 1=Nc) (5)

for any u 1 ∈Sa and u 3 ∈Su \ {(0; 0; : : : ; 0)}.More generally, using Eqs. (3)–(5) as well as the fact that d(a; b)=d(a+ c; b+ c),

we have

d(�a1va + �u1vu; �a2va + �u2vu)

=d([�a1 − �a2] va; [�u2 − �u1]vu)


=

L(1− 1=Nc) if �u1 �= �u2;L if �u1 = �u2 and �a1 �= �a2;0 if �u1 = �u2 and �a1 = �a2:

(6)

That is to say, the strategy space S is composed of Nc distinct mutually anti-correlatedstrategy ensemble (namely, those with same �u); whereas the strategies of each ofthese ensemble are uncorrelated with each other. (We remark that in the language ofcoding theory, S is a linear code of N 2

c elements over GF(Nc) with minimum distanceL(1− 1=Nc).)We expect that the collective behavior of MG(Nc; N 2

c ) should follow the predictionsof the crowd–anticrowd theory as the structure of S matches the assumptions of thetheory. In order to evaluate the performance of players in MG(Nc; N 2

c ), we study themean variance of attendance over all alternatives (or simply the mean variance)

�2 =1Nc

Nc∑i=0

[〈(Ai(t))2〉 − 〈Ai(t)〉2] ; (7)

where the attendance of an alternative Ai(t) is just the number of players chosen thatalternative. (We remark that the variance of the attendance of a single alternative wasstudied for the MG [1].) In fact, the variance of the attendance of an alternative repre-sents the loss of all players in the game. The variance �2, to "rst order approximation,is a function of the control parameter �, which is the ratio of the strategy space size|S| to the number of strategies at play NS, alone [5].To compare the MG(Nc; N 2

c ) with the crowd–anticrowd theory, we "rst have toextend the calculation of the variance by the crowd–anticrowd theory to the case ofNc alternatives. According to the crowd–anticrowd theory, the variance of the atten-dance originates from the independent random walk of each mutually anti-correlatedstrategy ensemble. In each of these strategy ensemble, the action of a strategy iscounter-balanced by its anti-correlated strategies. Therefore, the step size of the randomwalk of a mutually anti-correlated strategy ensemble is equal to the diLerence betweenthe number of players using a single strategy from the mean number of players usingthe strategies in this ensemble [9,10]. This random walk idea can be readily extendedto the case of multiple alternatives. In fact, for the mutually anti-correlated strategyensemble S� = {�vu + �va: �∈GF(Nc)},

step size for A�(�;�)(t) by S�

=

∣∣∣∣∣N�;� −∑

�∈GF(Nc) N�;�Nc

∣∣∣∣∣=

1Nc

∣∣∣∣∣∣∑��=�

(N�;� − N�;�)∣∣∣∣∣∣ ; (8)

where N�;� is the number of players making decision according to the strategy �vu+�vaand �(�; �) is the alternative that are chosen by the strategy �vu+ �va. Thus, the mean


Fig. 1. The mean variance �2 versus the control parameter � ≡ |S|=NS=N 2c =NS in MG(Nc; N 2

c ) with diLerentnumber of strategies S where Nc=37 and M =2. The solid lines are the predictions of the crowd–anticrowdtheory whereas the dashed line indicates the coin-tossed value.

variance predicted by the crowd–anticrowd theory is given by

�2 =

⟨1Nc

∑S�

∑�∈GF(Nc)

1N 2c

∑��=�

(N�;� − N�;�)2⟩; (9)

where∑

S� denotes the sum of the variance over all mutually anti-correlated strategiesensemble, and 〈〉 denotes the average over time. We note that when averaged overboth time and initial choice of strategies, variance of attendance for diLerent alternativesmust equal as there is no preference for any alternative in the game.Fig. 1 shows the mean variance of attendance as a function of the control parameter

� in the MG(Nc; N 2c ) for a typical Nc. For the MG(Nc; N 2

c ), the mean variance ofattendance, �2, exhibits similar behavior as a function of the control parameter � tothat in the MG no matter how many strategies S players have. In particular, wheneverN 2c =NS ≈ 1, the mean variance �2 is smaller than the so-called coin-tossed value.

(Coin-tossed value is the mean variance resulting from players making random choices.)Thus, global cooperation amongst the players is observed in this parameter range.Moreover, Fig. 1 shows that the mean variance predicted by the crowd–anticrowdtheory agrees with our numerical "nding.Further results along this line, including the mean variance of attendance as a function

of the control parameter in MG(Nc; N 2c ) with diLerent strategy space S, will be reported

elsewhere. These results all agree with the crowd–anticrowd theory [14]. Therefore, weconclude that we have successfully build up the MG(Nc; N 2

c ) model whenever Nc is aprime power.Indeed, the MG(Nc; N 2

c ) model can be readily extended to MG(Nc; N kc ) with Nc isequal to a prime power for 36 k6M + 1. We found that the mean variance alsoagrees with the MG and the crowd–anticrowd theory in the MG(Nc; N kc ) [14]. Thus,we can always alter the complexity of each strategy in MG with "xed N; S and M


Fig. 2. The mean variance �2 (square) versus the control parameter � ≡ |SK |=NS=�Nc=NS in MG(Nc; �Nc)with diLerent � where S = 2 and M = 2. The variance of the attendance of a choice (cross) is also shownin the "gure. The solid line indicates the mean variance predicted by the crowd–anticrowd theory whereasthe dashed line indicates the coin-tossed value.

while the cooperative behavior still persist. As a result, we can always keep (almost)optimal cooperation amongst the players in almost the entire parameter space.However, is it possible to construct a MG with Nc alternatives whose strategy space

size is smaller than N 2c that exhibits global cooperation? We give the answer by con-

structing the MG(Nc; �Nc) model where � is an integer ¡Nc.The basic setting of MG(Nc; �Nc) is the same as that of MG(Nc; N 2

c ) except thatthe strategies are drawn from a diLerent strategy space. More precisely, strategies ofMG(Nc; �Nc) are picked from the set SK={�ava+�uvu: �a ∈GF(Nc); �u ∈K ⊂ GF(Nc)}where K contains � elements. Moreover, va and vu satisfy the two technical conditionsin Eqs. (1) and (2). Clearly, the strategy space size of SK equals �Nc.As shown in Fig. 2, the mean variance of attendance �2 in MG(Nc; �Nc) shows

similar behavior as a function of the control parameter � to that in the MG only forsmall �. When � increases, the mean variance �2 in MG(Nc; �Nc) becomes smallerthan that in MG. Nevertheless, the numerical mean variance in MG(Nc; �Nc) doesnot agree with the prediction of the crowd–anticrowd theory except for small �. Theinconsistency is more pronounced when � increases.To account for this discrepancy, we notice that as � → 1+ while keeping all other

parameters "xed, fewer and fewer (or even none) of the strategies in the strategy space


of MG(Nc; �Nc) makes the same choice for the same combination of the output of thelast M steps. Therefore, some of the choices can never be chosen for MG(Nc; �Nc) withsmall � when the number of strategies picked by the players are much smaller than thestrategy space size �Nc. In this circumstances, the attendances of most alternatives areeither one or zero. Consequently, the mean variance �2 in MG(Nc; �Nc) with small � ismuch less than N . In fact, the variance of a choice may even vanished for large �. Suchphenomenon will be more pronounced in MG(Nc; �Nc) with small �. Thus, the meanvariance in the MG(Nc; �Nc) for ��Nc exhibits a radically diLerent behavior from theMG. From the above observation, we know that there is no eLective crowd–anticrowdinteraction whenever ��Nc. And in this case, the dynamics in the MG(Nc; �Nc) is nolonger dominated by the interactions of the anti-correlated strategies. Consequently, thecrowd–anticrowd theory does not correctly predict the mean variance in MG(Nc; �Nc).Nonetheless, we still "nd that in MG(Nc; �Nc); �2 attains a minimum (and hence theaverage number of winning players is maximized) whenever the control parameter� ≡ |SK |=NS = �Nc=NS ≈ 1 for every prime power Nc and 1¡�¡Nc.

Now, we are ready to answer the two questions posted in the abstract. First, inorder to obtain the best overall global wealth, players should switch to the MG(Nc; �Nc)game provided that Nc ¡NS6NM+1

c . More speci"cally, for "xed Nc; N; S and M ,players simply have to agree on an integer � ≈ NS=Nc and the corresponding strat-egy space in order to ensure the best performance of the MG. Second, whenever�¿Nc, the mean variance of attendance �2 agrees well with our extension of thecrowd–anticrowd theory. Thus, we conclude that in MG(Nc; �Nc) with Nc6 �6NMc ,the origin of global cooperation is the self-organization of player’s tendency to chooseanti-correlated strategies in making their decision. The “cancellation” of the actions inthese mutually anti-correlated strategy ensemble leads to a small �2.Finally, we remark that results on the order parameter of MG(Nc; �Nc) will be

reported elsewhere [14]. Readers should note that in case Nc is not a prime power, thepresence of zero divisors in the ring ZNc invalidates the conclusion in Eq. (6). So, itis instructive to "nd a reasonable extension of MG(Nc; N 2

c ) in this case.

Useful discussions with K.H. Ho, P.M. Hui and Kuen Lee is gratefully acknowl-edged. This work is support by the RGC grant of the Hong Kong SAR governmentunder the contract number HKU 7098=00P. H.F.C. is also supported in part by theUniversity of Hong Kong Outstanding Young Researcher Award.

References

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[10] M. Hart, P. JeLeries, N.F. Johnson, P.M. Hui, Eur. Phys. J. B 20 (2001) 547.[11] F.K. Chow, H.F. Chau, cond-mat=0109166.[12] A. Cavagna, Phys. Rev. E 59 (1999) R3783.[13] D. Challet, M. Marsili, Phys. Rev. E 62 (2000) R1862.[14] H.F. Chau, F.K. Chow, in preparation.

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How to attain maximum profit in minority game?