Minority Games: Interacting Agents in Financial Markets

361 Pages·2005·16.54 MB·English

by  Damien Challet

MinorityGames
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Minority Games
DAMIEN CHALLET, MATTEO MARSILI,
YI-CHENG ZHANG
1
3
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Foreword
Once in a while a model problem appears that is simple to describe but offers a
wealth of lessons. If the model problem is a classic—like the Ising model or the
Prisoner’s Dilemma—it both opens up our insight and gives us analytical pathways
intoanintriguingworld.Challet,MarsiliandZhang’sMinorityGameproblemissuch
amodel.Itisaclassic.Thegameitselfiseasytovisualize.Anumberofagentswishto
takesomeparticularaction,buttheydonotbenefitifthemajorityofotherstakeittoo.
Thecircumstanceswherethisoccursmayseemtobeveryspecific,butinfacttheyarise
everywhere in the economy when agents must commit in advance to something that
showsdiminishingreturns.(Diminishingreturns,thatis,tothenumberscommitting).
Financialtradesmustbecommittedtoinadvance,andthebenefitsofmanystrategies
may be arbitraged away if others use them too. Hence the minority situation arises
alsoinfinancialmarkets.
Damien Challet andYi-Cheng Zhang conceived of the Minority Game in 1997.
They, and their co-worker Matteo Marsili who joined them later, were intrigued not
only by the problem itself, but by the possibility of applying statistical mechanics
techniques to economics and game theory more generally. Over the previous two
decades, physics had become adept at analyzing systems with many particles inter-
acting heterogeneously, and markets clearly consisted of such particles—investors.
But these economic “particles” reacted by attempting to forecast the outcome that
would be caused by the forecasts of other particles, and to react to that in advance.
Theywerestrategic.Thiswasarecognizableworldforphysicistsbutanintriguingly
different one, and the minority game provided a pathway into it. The problem was
quicklytakenupbyotherphysicists.Itsformulationwassimplifiedandprogressively
redefined to be closer to what physics was used to dealing with.Versions appeared,
results appeared, and advances in understanding appeared. This book chronicles the
progress of thought and the considerable insights gained. And it collects the papers
thatbecamethemainmilestones.
vi Foreword
Onedistinctivefeatureofthisworkisitsemphasisonphasesinparameterspace—
different qualitative properties of the outcome that obtain under different parameter
sets.Delineatingphasesissecondnaturetophysicistsbutnottouseconomists,andwe
canlearnmuchfromthispractice.Intheminoritygame,agentsaligntheirstrategiesto
theconditionofthemarket:theywanttochoosethestrategythatonaverageminimizes
their chance of being in the majority. As a result, strategies co-organize themselves
soastominimizecollectivedissatisfaction.Butthiscollectiveresulthastwophases.
With few players, the recent history of the game contains some useful information
thatstrategiescanexploit.Butoncethenumberofplayerspassesacriticalvalue,all
useful information has been used up. The properties of the outcome differ in these
tworegimes.Information,infact,iscentraltothesituation.Anditsrolecanbeexpli-
citlyobserved:playersactasiftominimizeaHamiltonianthatisitselfameasureof
availableinformation.Sotheprocessbyhowinformationgetseatenupbyplayersis
madeexplicit,andthisisanotherusefulpathwayintoexploringfinancialmarkets.
Becausesomeinformationisleftunexploitediffewplayersareinthemarket,andall
informationisusedupasplayersincreasepastacriticalnumber,Challet,Marsiliand
Zhangpointoutthatplayersmayenterthemarketuntilthiscriticalnumberisreached.
The market will therefore display self-ordered criticality. This is an important con-
jecture,andnotjustatheoreticalone.Ittellsusthatspeculativeinvestors—technical
traders,atleast—willseekoutthinmarketswherepossibleandavoiddeepones.Mar-
kets may therefore hover on the edge of efficiency—a significant insight that can be
madeexplicitwiththetechniquesusedhereandonethatisworthfurtherinvestigation.
TheMinorityGamegrewoutofmyElFarolbarproblem, andtofillinsomepre-
history I should say a few words about that. Legend is indeed correct: in 1988 on
ThursdaynightsGalwaymusicianGerryCartyplayedIrishmusicatElFarol,andthe
barwasnotpleasantifcrowded.EachweekImulledwhetheritwasworthshowingup,
andImulledthatothersalsomulled.Itoccurredtomethatsituationwassomethinglike
theforecastingequivalentofprisoner’sdilemma.Fewshowingupwasbest;butitwas
ineveryone’sinteresttobeinthatfew.Butmymotivationwasnottofindaforecasting
prisoner’sdilemma.Iwasinterestedatthetimeinrationalexpectations(orforecasts)
and their meaning for economics. In solving forecasting problems, economics had
founditusefultoimaginethateveryonehadthesameforecastingmachineandused
it. You could then ask what forecasting machine would lead to agents’actions that
would produce aggregate outcomes that on average would validate the machine’s
forecasts. This would be the rational expectations solution—the forecasting method
that“rational”agentswouldchoose.Asananalyticalstrategythisworked,atleastin
the cases studied. But it bothered me. If someone were not smart enough to figure
theproperforecasttheywouldskewtheoutcome.Thenforecastsshoulddeviate;then
I should deviate. Then others should too, and the situation would unravel. Rational
Foreword vii
expectationsmayhavebeengoodtheory,butitwasnotgoodscience.IrealizedthatEl
Farolmadethisplain.Ifwepostulateda“correct”bar-attendanceforecastingmachine
thateveryonecoordinatedon,theiractionswouldcollectivelyinvalidateit.Therefore
therecouldbenosuchcorrectmachine.Butifnodeductiveforecastingsolutionwas
possible, what should agents in the economy do? The problem was behaviorally ill-
defined. And so were most situations involving the future in economics, I realized.
Thisfascinatedme.
In1992IstrippedtheElFarolsituationtoitsessentialsandprogrammedit.Iwrote
theproblemupandpresenteditattheJanuary1994AmericanEconomicAssociation
meetings in a session on complexity in the economy chaired by Paul Krugman. The
paper was received politely enough, but my discussant was irritated. He pointed out
theproblemhadasolutioninmixedNashstrategies:thebar-goerscouldtosscoinsto
reachasatisfactoryoutcome.Ihadthoughtofthat,buttomethatmissedthepoint.To
me, El Farol was not a problem of how to arrive at a coordinated solution (although
the Minority Game very much is). I saw it as a conundrum for economics: How
do you proceed analytically when there is no deductive, rational solution? Defining
the problem as a game lost this—all games have at least one Nash mixed-strategy
equilibrium—soIresistedanygame-theoreticformulation.Mypaperdulyappeared,
but economists didn’t quite know what to make of it. My colleague at Santa Fe, Per
Bak, did know however. He saw the manuscript and began to fax it to his physics
friends. The physics community took it up, and in the hands of Challet, Marsili and
Zhang,itinspiredsomethingdifferentthanIexpected—theMinorityGame.ElFarol
emphasized (for me) the difficulties of formulating economic behavior in ill-defined
problems.TheMinorityGameemphasizessomethingdifferent: theefficiencyofthe
solution. This is as should be. The investigation reveals explicitly how strategies
co-adapt and how efficiency is related to information. This opens an important door
tounderstandingfinancialmarkets.
AsIwritethis,therearenowover200papersontheMinorityGame,andagrowing
communityofeconophysicistswhohavebecomedeeplyimmersedinthedynamicsof
markets—especially financial markets. Economists wonder at times whether all this
workinphysicsisnotjustalengthyexerciseinphysicistslearningwhateconomists
know. It is certainly not—and indeed this book shows it is not. Modern physics can
offermuchtoeconomics.Notjustdifferenttoolsanddifferentmethodsofanalysis,but
differentconceptssuchasphasetransitions,criticalvalues,andpowerlaws.Andnot
justtheanalysisofpatternatstasis,buttheanalysisofpatternsinformation.Economics
needsthis.Infacteconomicsischangingcurrentlyfromanemphasisonequilibrium
andhomogeneitytoanemphasisontheformationofpatternandheterogeneity.And
so,economicsinduecoursewouldbeforcedtousethekindoftoolsthatarebrought
tobearinthisbook.Luckilyphysicsistheretosupplythem.
viii Foreword
Thepapershereandthetextareanimportantpartofamovement—lookingatthe
economy under heterogeneity. Challet, Marsili and Zhang and the others described
herehavedonemuchtopowerthismovement, andIcongratulatethemforthis. Itis
indeedabenefittophysicistsandeconomiststohavethisworkcollectedinoneplace.
W.BrianArthur
SantaFe,NewMexico.
Preface
In the twenty-first century, science seems to be taking on a more interdisciplinary
approach.Wallsbetweenthedisciplinesarebecomingporous.Anaturalresultofthis
processisthatmanyresearcherswithdifferentbackgroundsfindthemselvesworking
on the same subject, each having his own goals and dictionary, each bringing their
ownmethods,concepts,andnotations.
Financial markets have been a subject on which the interest of many researchers
from different disciplines have converged. In the past decade many economists and
physicistsalikehavebeenbusilystudyingmarket-relatedphenomena.Thisunpreced-
ented surge in interest in understanding the mechanism of financial markets can be
attributednotonlytotheirevermoreprominentimpactonsociety,butalsotothenow
mucheasieraccesstoverylargequantityofdata.
Inthemeantime,physicsisundergoingsomeprofoundchangesaswell:asignificant
numberofphysicistspreviouslyworkingontraditionalhardsciencetopicsareshifting
their attention to interdisciplinary areas bordering on biology, economics, and other
social sciences. The main force behind this outflow is not so much that physicists
have lost interest in physics, but the realization that there are incredibly interesting
complex phenomena taking place in other disciplines which seem now within reach
ofthepowerfultheoreticaltoolswhichhavebeensuccessfulinphysics.
Thereisnothingnewaboutphysicistsbeinginterestedineconomicsorotherdiscip-
linessuchasbiologyorcomputerscience.Ineconomics,thisisatrendwhichismore
than a century old, many first-class economists and several Nobel laureates having
a degree in physics in their earlier carrier. What is new is that nowadays physicists
studyinginterdisciplinarysubjectsremainphysicistsanddonotbecomeeconomists,
biologists, or computer scientists. They are interested in the wonders which such
complexsystemsasafinancialmarketoraproteinunveil,andinunderstandingthese
systemsfromadifferentperspective.
Physicists feel uneasy about the formal mathematical approach used by theor-
etical economists. Their view of a financial market may be close to a black box