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of Stock Markets 

Shu-Heng Chen

AI-ECON Research Group

Departmentof Economics

National Chengchi University

Taipei, Taiwan11623

TEL: 886-2-27386874

FAX: 886-2-29361913

E-mail: chchen@nccu.edu.tw

Chia-Hsuan Yeh

AI-ECON Research Group

Departmentof Economics

NationalChengchi University

Taipei, Taiwan11623

TEL: 886-2-29387308

FAX: 886-2-29390344

E-mail: g3258501@grad.cc.nccu.edu.tw

Abstract

Inthis paper, wepropose anew architectureto studyarti cialstock markets. This architecture

restsonamechanism called\school" whichis aprocedure to mapthephenotypeto thegenotypeor,

inplainEnglish, to uncoverthe secretofsuccess. We proposeanagent-based model of \school", and

considerschoolasanevolvingpopulationdrivenbysingle-populationGP(SGP).Thearchitecturealso

takesintoconsiderationtraders' searchbehavior. Bysimulated annealing,traders'searchdensity can

be connectedto psychological factors, suchas peer pressure oreconomic factors suchasthe standard

of living. Thismarketarchitecturewas thenimplementedinastandardarti cial stock market. Our

econometricstudyoftheresultantarti cialtimeseriesevidencesthatthereturnseriesisindependently

and identicallydistributed(iid), andhencesupportsthe eÆcientmarkethypothesis(EMH).Whatis

interesting thoughisthat thisiid serieswasgenerated bytraders, whodonotbelieveintheEMHat

all. Infact, ourstudyindicates thatmany ofourtraders wereable to nd usefulsignalsquite often

frombusinessschool,eventhoughthesesignalswereshort-lived.

KeyWords: Agent-BasedComputationalEconomics,SocialLearning,GeneticProgramming,

BusinessSchool, Arti cialStock Markets, SimulatedAnnealing, Peer Pressure.

1 Background and Motivation

Overthepastfewyears,geneticalgorithms(GAs)aswellasgeneticprogramminghavegraduallybecome

amajor tool in agent-based computational economics (ABCE). According to Holland and Miller (1991),

there aretwostyles ofGAs orGPin ABCE, namely, single-population GAs/GP (SGA/SGP) and

multi-populationGAs(GP)(MGA/MGP). SGA/SGPrepresentseachagentasachromosome oratree,andthe

wholepopulationofchromosomesandtreesaretreatedasasocietyofmarketparticipantsorgameplayers.

TheevolutionofthissocietycanthenbeimplementedbyrunningcanonicalGAs/GP.Arifovic(1995,1996),

Miller (1996), Vila (1997), Arifovic, Bullardand Du y (1997), Bullard and Du y (1998a,1998b, 1999),

Staudinger (1998)are examples of SGA, while Andrews and Prager (1994), Chen and Yeh (1996, 1997,

1998),andChen,YehandDu y(1996)areexamplesofSGP.MGA/MGP,incontrast,representseachagent

asasociety of minds (Minsky, 1987). Therefore,GAs orGPis actually runinside each agent. Since, in



ThepapertobepresentedattheSpecialSessionon\EvolutionaryComputationinEconomicsandFinance",TheFifth

International Conference of the Society for Computational Economics (CEF'99), BostonCollege campus, Chestnut Hill,

Massachusetts,U.S.A.June24-26,1999. ResearchsupportfromNSCgrantsNo.88-2415-H-004-008andNo. 87-I-2911-15is

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should not be mistaken as the applications of parallel and distributed GAs/GP, where communications

among\islands"doexist. Examplesof MGAcanbefoundin Palmeret al. (1994),Tayler(1995),Arthur

etal. (1997),Price(1997),Heymann, PearzzoandSchuschny(1998).

WhilethesetwostylesofGAs/GPmaynotbemuchdi erentinengineeringapplications,theydoanswer

di erentlyforthefundamental issue: \wholearns what fromwhom? " (Herreiner, 1998). First,agentsin

theSGA/GParchitectureusuallylearnfromotheragents'experiences,whereasagentsin theMGA/MGP

architectureonlylearnfromtheirownexperience. Second,agents'interactionsintheSGA/SGP

architec-turearedirectandthroughimitation,whileagents'interactionsintheMGA/MGParchitectureareindirect

andaremainlythroughmeditation. Itisduetothisdi erencethatSGA/SGPisalsocalledsociallearning

and MGA/MGP individual learning (Vriend, 1998). At thecurrentstate, the SGA/SGP architecture is

much morepopularthantheMGA/MGParchitecturein ABCE.

In addition to its easy implementation, the reasonfor the dominanceof SGA/SGP in ABCEis that

economistswouldliketosee theirgeneticoperators(reproduction, crossover,andmutation)implemented

withinaframeworkofsociallearningsothatthepopulationdynamicsdeliveredbythesegeneticoperators

canbe directly interpretedasmarket dynamics. In particular, someinterestingprocesses, such as

imita-tion,\followingtheherd",rumorsdissemination,canbemoree ectivelyencapsulatedintotheSGA/SGP

architecture.

However, it has been recently questioned by many economists whether SGA/SGP can represent a

sensiblelearningprocessatall. OneofthemaincriticismsisgivenbyHarrald(1998),whopointedoutthe

traditionaldistinctionbetweenthephenotypeandgenotypeinbiologyanddoubtedwhethertheadaptation

can be directly operated on the genotype via the phenotype in social processes. Back to Herreiner's

issue,ifweassumethatagentsonlyimitate others'actions (phenotype) withoutadopting theirstrategies

(genotype), then SGA/SGP may beimmune from Harrald'scriticism. However,imitating other agents'

actions area very minor partof agents' interactions. In many situations, such as nancial markets and

prisoners'dilemmagames,itwouldbehopelesstoevolveanyinterestingagentsiftheyareassumedtobe

abletolearnonlyto\buyandhold"or\cooperate anddefect". Moreimportantly,whatconcernsusishow

theylearn the strategies behindthese actions. But,unless wealsoassumethatstrategiesareobservable,it

would bediÆcult toexpectthat theyareimitable. Unfortunately, inreality,strategiesare ingeneralnot

observable. For instance, it is verydiÆcult to know theforecasting models used bytraders in nancial

markets. Tosomeextent,theyaresecrets. Whatisobservableis,instead,onlyasequenceoftradingactions.

Therefore,Harrald'scriticismisine ect challengingallseriousapplicationsofSGA/SGPin ABCE.

AlthoughHarrald'scriticismiswell-acknowledged,wehaveseennosolutionproposedtotacklethisissue

yet. Atthisstage,the onlyalternativeo eredis MGA/MGP. Infact,itis interestingto notethat many

applicationswhichheavilyrelyonevolutionoperatedonthegenotype(strategies)tendtouseMGA/MGP.

Modeling nancial agents is acasein point. What is ironicis that this typeof application is in essence

dealingwithhumaninteractionand thusrequires an explicitmodeling ofimitation, speculationand herd

behavior. Asaresult,MGA/MGPisnotreallyasatisfactoryresponsetoHarrald'scriticism.

Inthispaper,weplanto proposeanewarchitectureand henceasolutiontoHarrald'scriticism. This

architecture rests ona missing mechanism, which wethink is akeyto Harrald'scriticism. Themissing

mechanismis whatwecall \school". Why\school"? ToanswerHarrald'scriticism,onemustresolvethe

issue\howcanunobservablestrategiesbeactuallyimitable"? Thepointishow. Therefore,bythequestion,

whatismissinginSGA/SGPisafunction toshowhow,andthatfunctioniswhat wecall\school". Here,

\school"istreatedasaprocedure,aproceduretomapthephenotypeto thegenotype,orinplainEnglish,

touncoverthesecretof success. This notionof\school" goeswellwithwhat schoolusually meansin our

mind. However,itcoversmore. Itcan bemassmedia, nationallibrary,information suppliers,andso on.

WarrenBu ettmaynotbegenerousenoughtosharehissecretsofacquiringwealth,buttherearehundreds

ofbooksandconsultantsthatwouldbemorethanhappytodothisforus. Allthesekindsofactivitiesare

called\schooling". Therefore,ifwesupplementSGA/GPwithafunction\school",thenHarrald'scriticism

can,in principle,besolved.

Nevertheless, to add \school" to an evolving population is not that obvious. Based on our earlier

description,\school"isexpectedtobeacollectionofmostupdatedstudiesabouttheevolvingpopulation

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anevolvingpopulationdrivenbysingle-populationGP(SGP).Inotherwords,\school"mainlyconsistsof

faculty members(agents) whoare competing with each other to survive (gettenureor researchgrants),

andhencethesurvival-of-the- ttestprincipleisemployedtodrivetheevolutionoffacultythewayitdrives

theevolutionofmarketparticipants.

Tosurvivewell,afacultymembermustdoherbesttoanswerwhatisthekeytosuccessinthe evolving

market. Ofcourse, asthe market evolves,theansweralso needstoberevisedand updated. Thevalidity

oftheanswerisdeterminedbyhowwellmarketparticipantsaccepttheanswer.

Once \school"isconstructedwiththeagent-basedmarket,theSGPused toevolvethemarketis now

alsorun in thecontext ofschool. The advantageofthis setup isthat, while theSGP used to evolvethe

market su ers fromHarrald'scriticism, theSGP usedto evolve\school" doesnot. The reasonissimple.

Tobeasuccessfulmember,onemustpublish asmuchassheknowsandcannotkeepanything secret. In

this case, observabilityand imitability (replicatability) is notan assumption but arule. In other words,

there isno distinctionbetweenthe genotypeand phenotypein \school". Hence, Harrald'scriticismdoes

notapplyand SGPcanbe\safely"usedtoevolve\school".

Now,whathappenstotheoriginalSGPusedtoevolvethemarket? Thisbringsupthesecondadvantage

ofourapproach. Sincethefunctionofschoolistokeeptrackofstrategies(genotypes)ofmarketparticipants

andtocontinuouslygeneratenewandpromisingones,anyagentwhohaspressuretoimitateotheragents'

strategiesorto lookforeven betterstrategiescannowjust consult\school"and seewhether shehasany

goodlucktohavearewardingsearch. So,theoriginaloperationofSGPinthemarketcannowbereplaced

by SGP in \school" and a search procedure driven by the survival pressure of agents. Agents can still

haveinteraction onthephenotype in themarket, but theirinteraction onthegenotype isnowindirectly

operatedin\school".

An interestingaspect ofthis approach isto explicitlymodelthe interactionbetween \school" andthe

marketbyintroducingaco-evolutionmodel. Tosurvive,schoolmustadapt tomarket dynamics. Onthe

otherhand,marketdynamicsgeneratestudentsfor\school"who,inturn,bringtheknowledgelearnedfrom

\school" back to themarket,and that knowledge may havefurther impact on market dynamics. While

agent-basedmodelingisabottom-upapproach,onemayuseasystemoftwononlineardi erenceequations,

governingthedynamicsof\school"andthemarket,asatop-down\summary".

The di erence between our proposed architecture and SGA/SGP and MGA/MGP is also illustrated

in Figures1to 3. Figure 1depictsthemarket architecturerepresentedbySGA/SGP. Thetopof Figure

1 is the market as a single object, and the bottom is a population of directly interacting heterogeneous

agents. Thedirectinteraction ischaracterizedbythesymbol\$"amongthem. Bythisarchitecture,the

information(knowledge)aboutthemarketisopenlydistributedamongallagents. Nothingiskeptsecret. In

betweenisasymbol\="(equivalentto),whichmeansthatmarketdynamicsisequivalenttotheevolution

ofthispopulationofdirectlyinteractingagents.

Figure2givesthemarketarchitecturerepresentedbymulti-populationGP.Themarketremainsatthe

top,but there are twoessentialdi erences asopposed to the previous gure. First, thesinglesymbol=

is replaced by a series of ()s. Under these ()s is a population of indirectly interacting agents. By

\indirectly",wemean thatthese agentsareinteractingonlythroughabulletinboard. Imaginethat each

agentsitsin heroÆce andwatchestheworldfrom theweb. Theyhavenodirectcontactwith oneother,

physically, and in some sense, mentallyas well. The information (knowledge) about the market is now

privatelydistributedamongagents. Eachagenthasherownworldandkeepsherownsecrets.

Thepointhereisthatotheragents'mindsarenotdirectlyobservable,andhencenotimitable. Within

each agent'smind, there isasocietyofminds. Theevolutionofthis societyisdrivenbyGP.Within this

architecture,agentsbasicallylearnfromherownexperience,andnotfromotheragents'experiences. Thus,

itisatypicalmodel ofindividual learning.

Figure 3representsthearchitectureof ourproposed modi cation. Again,the market isplacedat the

top. Atthebottomto theright,itissomethingbetweenFigure1andFigure2. Inthephenotype, agents'

interactionisdirectandidenticaltowhatFigure1shows,whereasinthegenotype,itlookssomethinglike

Figure2,wherethere isnodirect interaction. Theoriginalconnectionbetweenmarketsandagentsisnow

replacedbytheconnectionbetweenagentsandschoolshownatthebottomtotheleft. Insideschool,there

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Market

Figure 1 : The Market Architecture

Represented by Single-Population

GAs/GP (SGA/SGP)

Market

Figure 2 : The Market Architecture Represented by

Multi-Population GAs/GP (MGA/MGP)

Market

School

Figure 3 : The Market Architecture Represented by

Single-Population GAs/GP with "School"

Thekeyelementsof ourproposed architecture entitled\MS-GP" (standingfor GPimplementedwith

\School"intheMarket)aretheprocedures\school"andthesearch. Weshallconcretizetheseprocedures

withan application to the arti cial stock market. The arti cialstock market is anewbut growing eld.

Somewondersandmissionsofthisresearchareahavebeenwell-documentedbyLeBaron(1998). Inhis

ar-ticle,LeBarondistinguishestherecentmodelsofcomplexheterogeneityfromthoseofsimpleheterogeneity.

Theuseofheterogeneousagentsis certainlynotnewto nance,and thereisalonghistoryto

building heterogeneous agent rational expectations models. What is attempted in this set of

computationalframeworksistoattacktheproblemofverycomplexheterogeneitywhichleaves

the boundary of what canbe handled analytically. Traders are made upfrom a verydiverse

setoftypesandbehaviors. Tomakethesituationmorecomplexthepopulationofagenttypes,

or the individual behaviors themselves, are allowed to change overtime in response to past

performance. (p.1)

Oneofthemissionsoftheseagent-basedcomputationalmodelsistoreplicatetimeseriesfeaturesofreal

markets. Whileitwillcontinuetobepursued,thefocusofthisnewprojectwillbemuchmorefundamental.

Ascalibrationtechniquesadvance,wemayexpectthatsoonerorlateragent-based nancialmodelswillbe

sopowerfulthatreplicatingtimeseriesfeaturesofrealmarketswillnotbethatdaunting. Infact,LeBaron

himselfhasmadethefollowingobservation:

Validation remains acritical issue if arti cial nancial markets are going to prove successful

in helping explain thedynamics of real markets. This remainsa veryweakareafor theclass

of models describedhere. Furthercalibration techniques and tighter test will be necessary....

However, there are somekeyissues which a ect these markets in particular. First, theyare

loadedwithparameterswhichmightbeutilizedto tanyfeaturethatisdesiredinactualdata....

(Ibid,pp. 19-20,Italicsadded)

Judgingfromtheresultsofrecentprogressesintheliteratureofarti cialstockmarket,thatmomentwill

comeinacoupleofyears. Whenthatmomentdoescome,onemaystarttoquestionhowthesecalibration

techniques can be justi ed, which leads to the foundation of this research: can we regard GAs/GP as

asuitable model of learning behavior within society?. Theanswercanhardly bepositiveorconvincingif

Harrald'scriticismhasnotbeenwelltaken. Wethereforeconsiderthisphaseofresearchmorefundamental.

Bysaying that,this research tendsto modify GPin amanner such that it hasa closerconnection with

humanlearningand adaptation.

MS-GP brings back search behavior, a subject which was once intensively studied in economics but

hasbeenlargelyignored in theconventional GAs/GPeconomicliterature. As weshallsee later,through

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\school"enablesustoinvestigatetheroleof\school"orthevalueof\education"intheevolutionofavery

speci c socialprocess. Thestatisticsgeneratedfromsimulations,suchasthetimeseriesof thenumberof

\students" registered,thenumber of\students"whoreceivesfutile orfruitful lessons at\school" canall

helpusunderstandhow\school",orinformationindustryin general,coevolveswithsociety.

In Section 2, weshall present the analyticalmodel on which our arti cial market is constructed. In

Section3,aconcreteapplicationoftheinstitutionalGPtothearti cialstockmarketisdetailed. Section4

providestheexperimentaldesign. Experimentresultsandeconometricanalysesofthese designsaregiven

inSection 5followedbyconcludingremarksin Section6.

2 The Analytical Model

Thebasic framework of thearti cial stock market consideredin this paper is thestandardasset pricing

model (GrosssmanandStiglitz,1980). Themarketdynamics canbe describedasaninteraction ofmany

heterogeneousagents,eachof them, basedonherforecastofthefuture,havingthegoalto maximizeher

expected utility. Technically, there are two major components of this market, namely, traders and their

interactions.

2.0.1 Model ofTraders

Thetrader partincludestraders' objectives andtheiradaptation. Weshallstartfromtraders' motivesby

introducingtheirutilityfunctions. Forsimplicity,weassumethatalltraderssharethesameutilityfunction.

Morespeci cally,thisfunctionisassumedtobeaconstantabsoluteriskaversion(CARA)utilityfunction,

U(W i;t )= exp( W i;t ) (1) whereW i;t

isthewealthoftraderiat timeperiodt,andisthedegreeofrelativeriskaversion. Traders

canaccumulate theirwealth bymaking investments. There are twoassetsavailablefor tradersto invest.

Oneistherisklessinterest-bearingassetcalledmoney, andtheotheristheriskyassetknownasthestock.

Inotherwords,at eachpointintime,eachtraderhastwowaystokeepherwealth,i.e.,

W i;t =M i;t +P t h i;t (2) where M i;t and h i;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

),atrader'stotalwealthW

i;t+1 isthus W i;t+1 =(1+r)M i;t +h i;t (P t+1 +D t+1 ) (3) whereP t

isthepriceofthestockattimeperiodtandD

t

isper-sharecashdividendspaidbythecompanies

issuingthe stocks. D

t

canfollowastochastic processnotknownto traders. Given thiswealth dynamics,

thegoalofeachtraderistomyopically maximizetheone-periodexpectedutilityfunction,

E i;t (U(W i;t+1 ))=E( exp( W t+1 )jI i;t ) (4) subjectto W i;t+1 =(1+r)M i;t +h i;t (P t+1 +D t+1 ); (5) whereE i;t

(:)istraderi'sconditional expectationsofW

t+1

givenherinformationupto t(the information

setI

i;t

),andristherisklessinterestrate.

It is well known that under CAR A utility and Gaussian distribution for forecasts, trader i's desire

demand,h 

i;t+1

forholdingsharesofriskyassetislinearin theexpectedexcessreturn:

h  i;t = E i;t (P t+1 +D t+1 ) (1+r)P t  2 i;t ; (6) where 2 i;t

istheconditionalvarianceof(P

t+1 +D t+1 )givenI i;t .

Oneof theessentialelementsof agent-basedarti cial stockmarkets istheformation ofE

i;t

(:), which

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Givenh 

i;t

,themarketmechanismisdescribedasfollows. Letb

i;t

bethenumberof sharestraderiwould

liketo submit abid to buy at period t,and let o

i;t

be the number traderi would liketo o er to sell at

periodt. It isclearthat

b i;t =  h  i;t h i;t 1 ; h  i;t h i;t 1 ; 0; otherwise: (7) and o i;t =  h i;t 1 h  i;t ; h  i;t <h i;t 1 ; 0; otherwise: (8) Furthermore,let B t = N X i=1 b i;t ; (9) and O t = N X i=1 o i;t (10)

be the totals of the bids and o ers for the stock at time t. Following Palmer et al (1994), we use the

followingsimplerationingscheme:

h i;t = 8 < : h i;t 1 +b i;t o i;t ; if B t =O t ; h i;t 1 + O t Bt b i;t o i;t ; if B t >O t ; h i;t 1 +b i;t B t Ot o i;t ; if B t <O t : (11)

Allthese casescanbesubsumedinto

h i;t =h i;t 1 + V t B t b i;t V t O t o i;t (12) whereV t min(B t ;O T

)isthevolumeoftradeinthestock. BasedonPalmeretal'srationingscheme,we

canhaveaverysimplepriceadjustmentscheme, basedsolelyontheexcessdemandB

t O t : P t+1 =P t (1+ (B t O t )) (13)

where isafunction ofthedi erencebetweenB

t andO

t

. canbeinterpretedasspeedofadjustmentof

prices. Oneofthe functions weconsideris:

(B t O t )=  tanh( 1 (B t O t )) ifB t O t ; tanh( 2 (B t O t )) ifB t <O t (14)

wheretanhisthehyperbolic tangentfunction:

tanh(x) e x e x e x +e x (15) Since P t

cannot be negative, weallow the speed of adjustment to beasymmetric to excessdemand and

excesssupply.

The priceadjustment processintroduced aboveimplicitlyassumesthat thetotalnumberof shares of

thestockcirculatedin themarketis xed,i.e.,

H t = X i h i;t =H: (16)

Inaddition,weassumethatdividendsandinterestsareallpaidbycash,so

M t+1 = X i M i;t+1 =M t r+H t D t : (17)

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Inthissection,weshalladdresstheformationoftraders'expectations,E i;t (P t+1 +D t+1 )and 2 i;t . Motivated

bythemartingale hypothesis in nance,weshallassumethefollowingfunction formforE

i;t (:). E i;t (P t+1 +D t+1 )=(P t +D t )(1+ 1 tanh( 2 f i;t )) (18)

The virtueof this function form is that,if f

i;t

=0,then the traderactually validates themartingale

hypothesis. Therefore, from the cardinality of set fi j f

i;t

= 0g, denoted by k

t

, wecan know how well

the eÆcientmarkethypothesisisacceptedamong traders. Thepopulationoffunctions f

i;t

(i=1;:::;N)is

determined by the geneticprogrammingprocedure Business Schooland Search in Business School

giveninthefollowingtwosubsections.

As to the subjective risk equation, originally we followed Arthur et al. (1997) to use the following

updatingscheme.  2 i;t =(1  3 ) 2 i;t 1 + 3 [(P t +D t E i;t 1 (P t +D t )) 2 ]: (19)

Withoutfurther restrictions,thisupdate makesthesubjectivemeasureriskrangebetween0andin nity.

Since h  i;t = E i;t (P t+1 +D t+1 ) (1+r)P t  2 i;t ; (20) itisclearthat h  i;t =  0; if 2 i;t =1; 1; if 2 i;t =0: (21)

As a consequence, all traders will tend to \leave" the market (h 

i;t

= 0;8i) due to incredible large

subjective risks. For example, in onepilotsimulationof ourearlierstudy, the subjectiverisksof traders

aredistributed around 240(Gen. 50),750(Gen. 100),1200 (Gen. 150) and3900 (Gen. 500). However,

thephenomenon characterizedby

h i;t !0; and  2 i;t !1 (22)

is not self-ful lling, because the increasing sequence of  2

i;t

impliesa decreasing sequence of P

t . If P

t is

continuously decreasing, there is not much volatility and uncertainty, and traders have no reason to be

subjecttoincreasingsubjectiverisks. Therefore,Equation(17)isnotdirectlyapplicabletoourmodel,and

wehavemodi editintothefollowingform,

 2 i;t =(1  3 ) 2 t 1jn1 + 3 [(P t +D t E i;t 1 (P t +D t )) 2 ]: (23) where  2 tjn1 = P n1 1 j=0 [P t j P tjn1 ] 2 n 1 1 (24) and P n1 t = P n 1 1 j=0 P t j n 1 (25) Inotherwords, 2 t 1jn1

issimplythehistorical volatilitybasedonthepastn

1

observations.

2.3 Business School and Single-Population GP

Themajorcomponentofarti cialstockmarketsistheadaptivetraders,whocanberegardedasanevolving

population. SinceArifovic(1994),geneticalgorithmhasbeenemployedtodrivetheevolvingpopulationof

agentsin economics. ChenandYeh(1996)generalizedthis approachbyusing geneticprogramming. The

styleof GPused in Chenand Yeh(1996) isknown assingle-population GPin agent-based computational

economics,whichisdi erentfrommulti-populationGP.Insingle-populationGP,eachtreecanberegarded

asa forecasting model used by an agent; hence, the adaptation of agents(in terms of their forecasting

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Ourmodi edversionischaracterizedtheanadditionofabusiness school tothearti cialstockmarket.

Thebusinessschoolinourmodelfunctionsasusualbusinessschoolsintherealworld. Itmainlyconsists

offaculty,andtheirdi erentkindsofmodels(schoolsofthoughts).LetFbethenumberoffacultymembers

(forecastingmodels). Thesemodelsarepropagatedviaacompetitionprocessdrivenbythefacultythrough

publications. In this academic world, ascholar canill a ord to keepsomething serious to herself if she

wantstobewell-acknowledged. If weconsider business schoolacollectionof forecastingmodels,then we

maywellusesingle-populationGPtomodel itsadaptation.

Nonetheless, scholarsandtraders maycareaboutdi erentthings. Therefore,in thisproject,di erent

tnessfunctions arechosento takecare ofsuchadistinction. Forscholars,the tness function ischosen

purelyfrom ascienti cviewpoint, say, forecastingaccuracy. Forexample,onemaychoosemean absolute

percentage error (MAPE) as the tness function (Table 1). Single-population GP is then conducted in

astandard way. Each faculty member (forecasting model) is represented by atree. The faculty will be

evaluatedwithaprespeci edschedule,sayonceforeverym

1

tradingdays. Thereviewprocedureproceeds

asfollows.

At theevaluation date, say t,each forecastingmodel (facultymember) will bereviewed bya visitor.

Thevisitorisanothermodelwhichisgeneratedrandomlyfromthecollectionoftheexistingmodelsinthe

businessschoolatt 1,denotedbyGP

i;t 1

,byoneofthefollowingthreegeneticoperators,reproduction,

crossover and mutation, each with probability p

r , p

c , and p

m

(Table 1). In the case of reproduction or

mutation, we rst randomlyselecttwoGP trees,say, gp

j;t 1

and gp

k ;t 1

. TheMAPE ofthese twotrees

overthelastm

2

days'forecastsarecalculated. Atournamentselection isthenapplied tothese twotrees.

TheonewithlowerMAPE, saygp

j;t 1

, is selected. Wethenapply Schwefel's1+1 strategy overthehost

gp

i;t 1

andthe visitorgp

j;t 1

(in thecaseof reproduction)orgp 0

j;t 1

(in thecase ofmutation) basedon

thecriterionMAPE,andgp

i;t

istheoutcomeof this1+1competition(Schwefel,1995).

Inthecaseofcrossover,we rstrandomlyselecttwopairsoftrees,say(gp

j1;t 1 ;gp j2;t 1 )and(gp k1;t 1 ;gp k2;t 1 ).

Thetournamentselectionisappliedseparatelytoeachpair,andthewinnersarechosentobeparents. The

children, say (gp

1 ;gp

2

), are born. One of them is randomly selected to compete with gp

i;t 1

, and the

winnerisgp

i;t

. ThefollowingisapseudoprogramoftheprocedureBusinessSchool(AlsoseeFlowchart

1). Table1isanexampleofthespeci cationofthecontrolparameterstoevolvethebusinessschool.

Procedure[Business School]

0. begin 1. CalculateMAPE(gp i;t ) 2. A=Random(R,C,M)with(p r ;p c ;p m ) 3. IfA=C,gotostep(11). 4. (gp 1 ;gp 2 )=(Random(GP t 1 ),Random(GP t 1 )) 5. CalculateMAPE(gp 1

)andMAPE(gp

2 ).

6. gp

new

=TournamentSelection(MAPE(gp

1 );MAPE(gp 2 )) 7. IfA=R,gotostep(17). 8. gp new Mutation(gp new ) 9. CalculateMAPE(gp new ) 10. Go tostep(17)

11. Randomly selecttwopairsoftreesfromGP

t 1

12. CalculateMAPEofthese twopairsofGPtrees

13. gp

1

=TournamentSelection(PAIR1)

14. gp

2

=TournamentSelection(PAIR2)

15. (gp 1 ;gp 2 ) Crossover(gp 1 ;gp 2 ) 16. gp new =Random(gp 1 ;gp 2 ) 17. gp i;t

=TournamentSelection(MAPE(gp

i;t 1

);MAPE(gp

new ))

(9)

Flowchart 1 : Evolution of the Business School

Yes

No

P

R

P

M

P

C

No

Yes

No Yes

T

M

: Tournament selection according to MAPE

S(T

M

,i) : Selection procedure according to tournament selection with the criterion

MAPE

based on i (genetic operator)

i : R, M, C represent reproduction, mutation and crossover respectively

m

2

: Evaluation cycle

F : Number of faculty member

Gen := Gen + 1

Gen mod m

2

= 1 ?

i = 1

Reproduction

S(T

M

,R)

Mutation

S(T

M

, M)

Crossover

S(T

M

,C)

Validating by MAPE

i := F ?

i := i + 1

Keep the original model(s)

2.4 Traders and Business School

Given the adaptive process of the business school, the adaptive process of traders can be described as

a sequence of twodecisions. First, should she go back to the business school to take classes? Second,

shouldshefollowthelessons learnedatschool? Intherealworld,the rstdecisionsomehowcanbemore

psychologicalandhassomethingtodowithpeerpressure. Onewaytomodelthein uenceofpeerpressure

istosupposethateachtraderwillexaminehowwellshehasperformedoverthelastn

2

tradingdays,when

comparedwithothertraders. Supposethattradersarerankedbythenet changeofwealthoverthelastn

2

tradingdays. LetW n

2

i;t

bethisnetchange ofwealthoftraderiat timeperiodt,i.e.,

W n2 i;t W i;t W i;t n2 ; (26) and,letR i;t

beherrank. Then,theprobabilitythattraderiwillgotobusinessschoolattheendofperiod

tisassumedtobedeterminedby p i;t = R i;t N : (27)

Thechoiceofthefunctionp

i;t

isquiteintuitive. Itsimplymeansthat

p i;t <p j;t ;if R i;t <R j;t : (28)

Inwords,the traderswhocome outtopshallsu er lesspeerpressure,and hencehavelessmotivation to

gobacktoschoolthanthosewhoarerankedatthebottom.

Inadditiontopeerpressure,atradermayalsodecidetogobacktoschooloutofasenseofself-realization.

Letthegrowthrateofwealthoverthelastn

2 daysbe Æ n2 i;t = W i;t W i;t n 2 jW i;t n2 j ; (29)

(10)

i;t

day,thenitisassumedthat

q i;t = 1 1+exp Æ n 2 i;t : (30)

Thechoiceofthisdensityfunctionisalsostraightforward. Noticethat

lim Æ n 2 i;t !1 q i;t =0; (31) and lim Æ n 2 i;t ! 1 q i;t =1: (32)

Therefore,thetraderswhohavemadegreatprogresswillnaturallybemorecon dentandhencehavelittle

needforschooling,whereasthosewhosu erdevastatingregressionwillhaveastrongdesireforschooling.

Insum,fortraderi,thedecisiontogotoschoolcanbeconsideredasaresultofatwo-stageindependent

Bernoulli experiments. The success probability ofthe rstexperiment is p

i;t

. If theoutcome ofthe rst

experiment is success, the trader will go to school. If, however, the outcome of the rst experiment is

failure,thetraderwillcontinuetocarryoutthesecondexperimentwiththesuccessprobabilityq

i;t . Ifthe

outcomeofthesecondexperimentissuccess,thenthetraderwillalso gotoschool. Otherwise,thetrader

willquitschool. Ifweletr

i;t

betheprobabilitythattraderidecidestogotoschool,then

r i;t = p i;t +(1 p i;t )q i;t = R i;t N + N R i;t N 1 1+exp Æ k i;t (33)

Onceatraderdecidestogotoschool,shehastomakeadecisiononwhatkindsofclassestotake. Since

weassume that business school, at period t,consists of 500faculty members (forecastingmodels),let us

denotethembygp

j;t

(j=1;2;:::;F.) Theclass-takingbehavioroftradersisassumedtofollowthefollowing

sequentialsearchprocess. Thetraderwillrandomlyselectoneforecastingmodel gp

j;t

(j=1;:::;F)witha

uniformdistribution. Shewillthenvalidatethismodelbyusingitto tthestockpriceanddividendsover

thelastn

3

tradingdays,andcomparetheresult(MAPE)withheroriginalmodel. Ifitoutperformstheold

model,shewilldiscardtheoldmodel,andputthenewoneintopractice. Otherwise,shewillstartanother

randomselection,anddoitagainandagainuntileithershehasasuccessfulsearchorshecontinuouslyfail

I 

times. Thefollowingis apseudoprogramof theprocedure Visiting the Business School(Alsosee

Flowchart2).

Procedure[VistingBusiness School]

0. begin 1. CalculateMAPE(f i;t ) 2. I 1 3. Randomlyselectagp j;t (U[1;500]) 4. CalculateMAPE(gp j;t ) 5. IfMAPE(gp j;t )<MAPE(f i;t ),goto Step(10) 6. I I +1 7. IfI <I  ,goto step(3) 8. f i;t+1 =f i;t 9. Goto Step(11) 10. f i;t+1 =gp j;t 11. end

Equation (33) and the procedure Visting Business School give the distinguishing feature of our

adaptive traders. As wementioned earlier, there is no direct interaction among traders in terms of the

(11)

Flowchart 2 : Traders' Search Process in the Business School

Yes

Yes

No Yes

No No

Yes No

No Yes

i = Individual ?

N : Number of traders

Gen := Gen + 1

Computing wealth (W

i,t

) for each

individual and ranking them (R

i,t

)

Probability of visiting

business school for trader i,

due to peer pressure,

p

i,t

= R

i,t

/ Individual

Probability of visiting business

school for trader i, due to a

sense of self-realization,

q

i,t

= 1 / (1 + e

ìi,t

) , where

ì

i,t

= (W

i,t

- W

i,t-1

) / |W

i,t-1

|

Keep the original model

i = N ?

Randomly select a new model

from business school with a

uniform distribution

Validating by MAPE

I = 1

I := I + 1

i := i + 1

I := I* ?

i := 1

applicablehere. Inotherwords,ourtradersarenotGP(GA)-based. Instead,theiradaptationbehavioris

modeled by anexplicit search process. Thesearch process startswith a decision to search or not. This

decisionisstochastic,i,e.,thetraderatanypointintimecannotbesurewhethersheshouldstartsearching,

andtheuncertaintyofthisdecisionisfurthermodeledviaatechniqueknownassimulatedannealing(SA).

Insum,itisasocietycomposing ofSA-basedtradersandSGP-basedfaculty,whocoevlovewithdi erent

tnessfunctions(objectivefunctions).

3 Experimental Designs

Oneoftheformidabletasksforagent-basedcomputationaleconomicsisthedesign oftraders. AsLeBaron

(1998)pointedout: \Thecomputationalrealmhastheadvantagesanddisadvantagesofawideopenspace

inwhichtodesigntraders,andnewresearchersshouldbeawareofthedauntingdesignquestionsthatthey

will face. Most of these questions still remain relatively unexploited at this time. (p.18)" Nevertheless,

oneshould noticethat thisissueisnotcon nedtoABCE,andiswidelysharedbyallresearchinbounded

rationality. Forexample,Sargent(1993)stated\Thisareaiswildernessbecausetheresearchfacessomany

choicesafterhedecidestoforgothedisciplineprovidedbyequilibriumtheorizing. (p.2)"

LeBaron's and Sargent's descriptionof this wildernesscan be further exempli ed by Table 1. Facing

(12)

The Stock Market

Sharesofthestock(H) 100

InitialMoneysupply(M

1 ) 100 Interestrate(r) 0.1[0.0001] StochasticProcess(D t ) U(5:01;14:99)[U(0:0051;0:0149)]

Priceadjustmentfunction tanh

Priceadjustment( 1 ) 10 5 Priceadjustment( 2 ) 0.210 5 BusinessSchool

Numberoffacultymembers(F) 500

Numberoftreescreatedbythefullmethod 50

Numberoftreescreatedbythegrowmethod 50

Functionset f+; ;Sin;Cos;Exp;R log;Abs;Sqrtg

Terminalset fP t ;P t 1 ;;P t 10 ; P t 1 +D t 1 ;;P t 10 +D t 10 g

Selectionscheme Tournamentselection

Tournamentsize 2

Probabilityofcreatingatreebyreproduction 0.10

Probabilityofcreatingatreebycrossover 0.70

Probabilityofcreatingatreebymutation 0.20

Probabilityofmutation 0.0033

Probabilityofleafselectionunder crossover 0.5

Mutationscheme TreeMutation

Replacementscheme (1+1)Strategy

Maximumdepthoftree 17

Numberofgenerations 20,000

MaximumnumberinthedomainofExp 1700

Criterionof tness(Facultymembers) MAPE

Evaluationcycle(m

1

) 20(10,40)

SampleSize(MAPE)(m

2

) 10

Traders

NumberofTraders(N) 500

DegreeofRRA() 0.5

Criterionof tness(Traders) Incrementsin wealth (Income)

Samplesizeof 2 tjn1 (n 1 ) 10 Evaluationcycle(n 2 ) 1 Samplesize(n 3 ) 10 Searchintensity(I  ) 5(1, 10)  1 0.5  2 10 5  3 0.0133

ThenumberoftreescreatedbythefullmethodorgrowmethodisthenumberoftreesinitializedinGeneration0

(13)

AggregateVariables

Stockprice P

t

Tradingvolumes V

t

Totalsofthebids B

t

Totalsoftheo ers O

t

#of martingalebelievers N

1;t

#of tradersregisteredtoBusinessSchool N

2;t

#of traderswithsuccessfulsearchinBusinessSchool N

3;t

The Wealth Share ofDi erent Classes

1stlowest20percentile S 0:2;t 2ndlowest20percentile S 0:4;t 3rdlowest 20percentile S 0:6;t 4thlowest20percentile S 0:8;t 5thlowest20percentile S 1;t IndividualTrader Forecasts f i;t Subjectiverisks  i;t Bidtobuy b i;t O ertosell o i;t Wealth W i;t Income W 1 i;t

Rankofpro t-earningperformance R

i;t Complexity(depthoff i;t ) k i;t Complexity(#ofnodesoff i;t )  i;t

resultsmaynotberobusttoalldesigns. Hence,in additiontorun manyruns inasingledesign,itisalso

crucialtotestmanydi erentdesigns,i.e.,totestmanytableslikeTable1,whilewithdi erentparameters.

This paper, however,has a very limitedscope, i.e., to illustrate therich dynamics our MS-GP arti cial

stockmarket can possibly o er, and thequestions it cane ectively deal with asingle pilot experiment.

Therefore,whilewedo ne-tunesomeofourparameterslistedinTable1,wedonotintentionallycalibrate

ourparametersforthepurposeof replicatingthestylizedfactsof nancialtimeseries. 1

Thesimulationresultsofourarti cialstockmarketaremainlyaseriesoftimeseriesvariablesoftraders

(microstructure)andthemarket. TheyaresummarizedinTable2.

4 Simulation Results

Basedontheexperimentaldesigngivenabove(Table1),asinglerunwith14,000generationswasconducted.

Noticethatthenumberofgenerationsisalsothetimescaleofsimulation,i.e.,GEN =t. Inotherwords,

wearesimultaneouslyevolvingthepopulationoftraderswhilederivingthepriceP

t

. Inthefollowing,we

shallpresentourresultsinanordertoansweraseriesof questionsraisedin Pagan(1996). 2

1. Arepricesandreturnsnormallydistributed?

2. Doespricesfollowarandomwalk?

1

AccordingtoLeBaron(1998),oneofthemissionsofthe agent-based modelingof nancial marketsistoreplicatetime

seriesfeaturesofrealmarkets.Lux(1995,1998),LuxandMarchesi(1998),Chen,LuxandMarchesi(1999)haveshowedhow

thesestylizedfactscanbereplicatedinaspeci cstyleofagent-basedmodels.

2

(14)

Periods P  Skewness Kurtosis Jarqu-Bera p-value 1-2000 84.07 4.82 0.34 3.07 40.62 0.00 2001-4000 76.43 5.84 0.65 2.60 153.49 0.00 4001-6000 67.28 1.84 0.94 5.07 654.75 0.00 6001-8000 65.17 3.27 0.67 3.85 212.46 0.00 8001-10000 64.46 2.49 1.16 5.28 887.91 0.00 10001-12000 68.44 5.09 2.24 11.46 7660.11 0.00 12001-14000 74.57 5.48 1.00 3.71 381.93 0.00

Table4: BasicStatisticsoftheArti cialStockReturnSeries

Periods P  Skewness Kurtosis Jarqu-Bera p-value

1-2000 -0.000074 0.015 3.53 23.64 39676.46 0.00 2001-4000 -0.000057 0.010 3.26 18.83 24461.55 0.00 4001-6000 -0.000018 0.007 3.72 25.94 48486.08 0.00 6001-8000 -0.000024 0.007 3.70 25.79 47869.55 0.00 8001-10000 0.000032 0.007 3.69 26.97 52452.04 0.00 10001-12000 0.000169 0.010 6.91 86.56 597871.50 0.00 12001-14000 -0.000154 0.009 4.18 32.80 79867.54 0.00

3. Arereturnsindependentlyandidenticallydistributed?

Inadditiontothe\up"part,agent-basedcomputationalmodelsprovideuswithrichopportunitytostudy

themicrostructure,i.e.,thebehavioralaspectoftraders. Inourarti cialstockmarket,atrader'sbehavior

canbewellkepttrackofbyalistofvariablesgivenin Table2. This listofvariablesenableustoaddress

alot ofinterestingissuesinbehavioral nance.

1. Whatdoesthetradersactuallybelieve? Doesshebelievein theeÆcientmarkethypothesis?

2. Whatexactlyistheforecastingmodel(orthetradingstrategy)employedbythetrader?

3. Howsophisticatedisthetrader? Willshegetmoreandmoresophisticatedastimegoeson?

Inthefollowing,weshallillustratehowthese issuescanbeapproachedbyouragent-basedarti cialstock

market.

First, are price and returnsnormally distributed? Thetime series plotof the stockprice isdrawnin

Figure 4. Overthis long horizon, P

t

uctuates between 55 and 105. The basic statistics of this series,

fP

t g

14000

t=1

,issummarizedin Table3. Giventhepriceseries,thereturnseriesis derivedasusual,

r t =ln(P t ) ln(P t 1 ): (34)

Figure5is atimeseriesof stockreturn,and Table4givesthebasicstatisticsof thisreturnseries. From

thesetwotables, neitherthestockpriceseries fP

t

gnorreturnseries fr

t

gisnormal. Thenullhypothesis

thatthese seriesare normalarerejectedby theJarqu-Berastatisticsinall periods. Thefat-tail property

isespeciallystrikinginthereturnseries. Thisresultisconsistentwithoneofstylizedfacts documentedin

(15)

Periods DFofP t (p,q) 1-2000 -0.285 (0,0) 2001-4000 -0.288 (0,0) 4001-6000 -0.150 (0,0) 6001-8000 -0.180 (0,0) 8001-10000 0.173 (0,0) 10001-12000 0.680 (0,0) 12001-14000 -0.753 (0,0)

TheMacKinnoncriticalvaluesforrejectionofhypothesisofaunitrootat99%(95%)signi cancelevelis-2.5668(-1.9395).

Table6: BDS Test

Periods DIM=2 DIM=3 DIM=4 DIM=5 Reject

1-2000 -0.36 -0.20 -0.14 -0.18 No 2001-4000 -0.16 0.13 0.40 0.57 No 4001-6000 1.34 1.35 1.22 1.24 No 6001-8000 0.89 0.99 1.18 1.35 No 8001-10000 1.93 2.38 2.64 2.69 Yes 10001-12000 0.85 0.92 0.96 0.87 No 12001-14000 0.29 0.21 0.37 0.66 No

Theteststatisticisasymptoticallynormalwithmean0andstandarddeviation1. Thesigni cancelevelofthetestissetat

0.95.

Second,doespricesfollow arandom walk? Or,moretechnically,doesthepriceserieshaveaunit root?

Thestandardtoolto testforthepresenceofaunitrootis thecelebratedDickey-Fuller(DF)test(Dickey

andFuller,1981). TheDFtestconsistsofrunningaregressionofthe rstdi erenceofthelogpricesseries

againsttheserieslaggedonce.

ln(P t )=ln(P t ) ln(P t 1 )= 1 ln(P t 1 ) (35)

The nullhypothesis is that

1

is zero, i.e., ln(P

t

) containsa unit root. If

1

is signi cantly di erent

fromzerothenthenullhypothesisisrejected. AscanbeseenfromthesecondcolumnofTable5,fromthe

totalnumberof7periodsnoneleadstoarejectionofthepresenceofaunitroot. All ofthis doessuggest

that P

t

does follow a random walk. This resultalso agree with oneof the stylized facts documentedin

Pagan(1996).

Third, are returnsindependentlyand identicallydistributed? Here,wefollowedtheprocedure ofChen,

Lux and Marchesi (1999). This procedure is composed of two steps, namely, the PSC ltering and the

BDS testing. We rstapplied the Rissanen'spredictivestochastic complexity (PSC) to lter thelinear

process. The third column ofTable5givesus theAR MA(p;q) processextracted from thereturn series.

Interestinglyenough,allthesesevenperiodsarelinearlyindependent(p=0;q=0). Thisresultcorresponds

totheclassicalversionofthe eÆcientmarkethypothesis. 3

Oncethelinearsignalsare ltered,anysignalsleftintheresidualseriesmustbenonlinear. Therefore,

oneof the most frequentlyused statistic, the BDS test, is applied to the residuals from the PSC lter.

Since noneof the seven returnseries have linearsignals, theBDS test isdirectly applied to the original

returnseries. TherearetwoparametersrequiredtoconducttheBDStest. Oneisthedistanceparameter(

3

ChenandKuo(1999)simulatedacobwebmodelwithGP-basedproducersandspeculators. In38outoftheir40cases,

theorderpandqidenti edbytheirPSC lterissimply(0;0). OnewonderthatwhetherornotGP-basedagentscannormally

interactinsuchaneÆcientwaythatlinearpredictabilityoftheiraggregatebehaviorisalmostimpossible. Thisiscertainly

(16)

to the rst choice, and hence,weonlyreport the resultwith =1. As to theembedding dimension, we

triedDIM=2;3;4;;5,andtheresultisgivenin Table6. SincetheBDStestisasymptoticallynormal,it

isquiteeasyto haveaneyeballcheckontheresults.

WhatisalittlesurprisingisthatthenullhypothesisofIID(identicallyandindependentlydistributed)

is rejected in 6out of 7periods. The only period whose returnseries hasnonlinear signalsis Period 5.

Putting theresult of PSC ltering and BDS testing together, our returnseries is eÆcient to the degree

that,85%ofthetime,itcanberegardedasaiidseries. But,iftheseriesisindeedindependent(nosignals

at all), what is the incentive for traders to search? Clearly, here, wehavecome to the issues raised by

GrossmanandStiglitz20yearsago(GrossmanandStiglitz,1981).

Oneoftheadvantagesagent-basedcomputationaleconomics(thebottom-upapproach)isthatitallows

ustoobservewhattraders areactuallythinkinganddoing. Aretheymartingale believers? Thatis,dothey

believethat E t (P t+1 +D t+1 )=P t +D t ? (36)

If theydonot believein the martingalehypothesis, do theysearchintensively? In other words, dothey

go to schooland can still learn somethinguseful in such an iid-seriesenvironment? To answerthe rst

question,thetimeseries ofN

1;t

in Table2isdrawn in Figure6. The gureisdrawnonlyupto the rst

1000 tradingdays, because after that the groupof believers goes extinct. Hence, while econometricians

mayclaimthatthereturnseriesisiid,traderssimplydonotbuyit.

Thisnaturallybringsupthesecondquestion: if theydo notbelieve inthe martingale hypothesis, what

dothey actually do? Figure7isthetime seriesplotofthenumberoftraderswithsuccessfulsearch,N

3;t .

Dueto thedensityoftheplotandthewiderangeof uctuation,this gureis somewhatcomplicatedand

diÆculttoread. We,therefore,reporttheaverageofN

3;t

overdi erentperiodsoftradingdaysinTable7.

FromTable7,itcan beseenthat thenumberoftraderswith successfulsearch,ontheaverage, uctuates

about200. Ataroughestimate,40%ofthetradersbene tfrom businessschoolpertradingday. Clearly,

searchinbusiness schoolisnotfutile.

It isinterestingto knowwhat kindof usefullessons traderslearnfrombusiness school. Is ittheBDS

test,eÆcientmarkethypothesis orthemartingalemodel? Theanswerisnoneoftheabove. Basedonour

designgivenin Section 3, what business schoolo ersis acollectionof forecastingmodels fgp

i;t

g, which

canwellcapturetherecentmovementofthestockpriceanddividends. Therefore,whileinthelong-runthe

returnseriesisiid,tradersunder survivalpressuresdonotcaremuchaboutthislong-runproperty. What

motivatesthem to search and helps them to surviveis in e ect brief signals. A similar observation was

madebyPeters(1991):

The evidence calls into question the EÆcient Market Hypothesis, which underlies the linear

mathematicsusedinmostcapitalmarkettheory. Italsolendsvaliditytoanumberofinvestment

strategiesthatshouldnotworkifmarketsareeÆcient,.... This ndingisofparticularimportance

for practitioners, because experience hasshown that these strategies do work when properly

applied,eventhoughtheorytellsustheyshouldnotworkinarandom-walkenvironment. (Italics

added.)

Anotherwaytosee whattradersmaylearnfrombusiness schoolis toexaminetheforecastingmodels

they employ. However, this is a very large database, and is diÆcult to deal with directly. But, since

allforecastingmodelsare in theformat ofLISP trees, wecanat least askhow complex these forecasting

models are. Todoso,wegivetwode nitions ofthecomplexityof aGP-tree. The rstde nition isbased

onthenumber ofnodesappearingin thetree,whilethesecondisbasedonthedepthofthetree. Oneach

tradingday,wehaveapro leoftheevolvedGP-treesfor500traders,ff

i;t

g. Thecomplexityofeachtreeis

computed. Letk

i;t

bethe numberof nodesofthemodelf

i;t and i;t bethedepthoff i;t . Wethenaverage asfollows. k t = P 500 i k i;t 500 ; and  t = P 500 i  i;t 500 : (37)

Figures 8 and 9 are the time series plots of k

t and 

t

. One interestinghypothesis one maymake is

(17)

Figure 4 : Time Series Plot of the Stock Price

45

55

65

75

85

95

105

1

1001

2001

3001

4001

5001

6001

7001

8001

9001

10001

11001

12001

13001

Trading Day

Stock Price

Figure 5 : Time Series Plot of Stock Returns

-0.05

0

0.05

0.1

0.15

0.2

1

1001

2001

3001

4001

5001

6001

7001

8001

9001

10001

11001

12001

13001

Trading Day

Stock Returns

Figure 6 : The Number of Traders with Martingale Strategies on Each Trading Day

0

10

20

30

40

50

60

1

1001

2001

Trading Day

Number of Traders

(18)

Strategies Periods N 3 k  1-2000 209.13 17.85 8.14 2001-4000 189.03 28.14 9.66 4001-6000 218.53 54.34 13.29 6001-8000 215.91 59.51 14.13 8001-10000 220.78 76.60 14.74 10001-12000 206.80 69.22 13.97 12001-14000 185.40 50.58 12.94

N3 istheaverageofN3;ttakenovereachperiod.kandaretheaverageofkt andttakenovereachperiod.

Table8: AverageoftheNumberofTraderswithSuccessfulSearchonthehdayafterBusinessSchoolHas

UpdatedtheInformation

h N 3;h h N 3;h h N 3;h h N 3;h 1 308.52 6 208.88 11 189.87 16 183.49 2 270.24 7 200.80 12 188.04 17 184.49 3 246.39 8 196.56 13 187.81 18 186.54 4 230.82 9 193.27 14 187.94 19 193.39 5 218.86 10 191.47 15 184.61 20 185.39

words, traders will evolveto be more and moresophisticated astime goes on. However, this is not the

casehere. Both guresevidencethat,whiletraderscanevolvetowardahigherdegreeofsophistication,at

somepointin time,theycan besimpleaswell(Alsosee Table7). Despitetherejectionof themonotone

hypothesis, weseenoevidence thattraders'behaviorwillconvergeto thesimplemartingalemodel.

Figures 7,8and9together leaveus animpression that tradersin ourarti cialstockmarket arevery

adaptive. Aboutthisphenomenon,Arthur(1992)conducted asurvivaltestonit.

We ndnoevidencethat marketbehavioreversettlesdown;thepopulationofpredictors

con-tinuallyco-evolves.Onewaytotestthisistotakeagentsoutofthesystemandinjectthemin

againlater on. If market behavior isstationary theyshould be able todo as well inthe future

as they are doing today. But we nd that when we \freeze" a successful agent's predictors

earlyonand injecttheagentintothesystemmuchlater, theformerlysuccessfulagentisnow

adinosaur. Hispredictionsareunadaptedandperformpoorly. Thesystemhas changed. (p.24,

IalicsAdded)

Authur's interestingexperimentcanbeconsideredasameasureof thespeedofchangein asystem. If

asystemchanges inaveryfastmanner,then knowledgeaboutthesystemhastobeupdatedin asimilar

pace; otherwise, theknowledgeacquired shall soon become obsolete. To seehowfast ourarti cial stock

market changes, we made an experimentsimilar to Arthur's survivaltest. Since in our arti cial market

businessschoolisupdateevery20periods(m

1

=20,Table1),wecanmeasurethehowfasttheknowledge

becomeobsoletebycalculatingthenumberoftraderswithsuccessfulsearchonthehthdayafterbusiness

(19)

Figure 7 : The Number of Traders with Successful Search on Each Trading Day

20

70

120

170

220

270

320

370

420

470

1

1001

2001

3001

4001

5001

6001

7001

8001

9001

10001

11001

12001

13001

Trading Day

Number of Traders

Figure 8 : Traders' Complexity : The Average of the Number of Nodes of GP-Trees

0

10

20

30

40

50

60

70

80

90

100

110

120

1

1001

2001

3001

4001

5001

6001

7001

8001

9001

10001

11001

12001

13001

TradingDay

Average Number of Nodes

Figure 9 : Traders' Complexity : The Average of Depth of GP-Trees

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1

1001

2001

3001

4001

5001

6001

7001

8001

9001

10001

11001

12001

13001

Trading Day

Average Depth

(20)

helpfulforthesearchingtraders. Therefore,thenumberoftraderswithsuccessfulsearchshouldbestrikingly

highonthatday,andthefartheritisfromtheupdating,thelessthechanceofhavingasuccessfulsearch.

Moreprecisely,denote N

3;t byN

3;h

i

,wheret=(i)20+h,andlet

N 3;h = P 14000=20 i=1 N 3;hi 14000=20 ; (38)

thenArthur's survivaltestcan bereformulatedasfollows. N

3;h

isamonotonic decreasing function of h.

Tosee whether this propertywill apply to oursystem, Table8reportsthestatistics N

3;h

. This series of

numbersstartswithapeakat308,andquicklygoesdownbelow300andthendropsfurther below200as

hincreases. This result simplysays that when moreandmore people knowsthesecret, there canbeno

longeranysecret.

Thelastresultalsoshowstheco-evolvingcomplexdynamicsbetweenbusinessschoolandthemarket. To

survive,schoolmustadapttomarketdynamics. Ontheotherhand,marketdynamicsgeneratestudentsfor

\school"who,inturn,bringtheknowledgelearnedfrom\school"backtothemarket,andthat knowledge

mayhavefurther impactonmarketdynamics. Thepatterns discoveredbybusiness schoolareeventually

annihilated by the traders who learn and make a living on these patterns. However, on the process of

annihilatingthesepatterns,newpatternsarefurthergeneratedforschooltodiscover,andthisprocessgoes

onandon. Onemaycallthis processaself-destruction-and-organizationprocess.

5 Concluding Remarks

Thesinglepilot experimenthas demonstratedthe rich dynamics that ourMS-GP arti cial stockmarket

cangenerate. We alsoshowthe relevance of this rich dynamics to nancial econometrics andbehavioral

nance. Forthelatter, weaddressPeters' criticismontheeÆcientmarkethypothesisaswellasArthur's

survivaltestwithourdynamicsofmicrostructure. Itisinterestingtonotethat, whileeconometricianson

thetop may claim that ourarti cial market is eÆcient, our traderson the bottom donot act asif they

believe in the eÆcientmarket hypothesis. This result seemsto be consistent withour experience of the

realworld,andisoneoftheinterestingfeaturesonematexpect fromthebottom-upapproach.

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數據

Figure 1 : The Market Architecture Represented by Single-Population GAs/GP (SGA/SGP)
Figure 5 is a time series of stock return, and T able 4 gives the basic statistics of this return series
Table 6: BDS T est
Figure 4 : Time Series Plot of the Stock Price 455565758595105 1 1001 2001 3001 4001 5001 6001 7001 8001 9001 10001 11001 12001 13001 Trading DayStock Price
+2

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