©CopyrightJASSS

FedericoCecconi,MarcoCampenni,GiuliaAndrighettoandRosariaConte(2010)

WhatDoAgent-BasedandEquation-BasedModellingTellUsAboutSocialConventions:TheClashBetweenABMandEBMinaCongestionGameFramework

JournalofArtificialSocietiesandSocialSimulation13(1)6<http://jasss.soc.surrey.ac.uk/13/1/6.html>

Received:13-Mar-2009Accepted:24-Dec-2009Published:31-Jan-2010

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Abstract

Inthisworksimulation-basedandanalyticalresultsontheemergencesteadystatesintraffic-likeinteractionsarepresentedanddiscussed.Theobjectiveofthepaperistwofold:i)investigatingtheroleofsocialconventionsincoordinationproblemsituations,andmorespecificallyincongestiongames;ii)comparingsimulation-basedandanalyticalresultstofigureoutwhatthesemethodologiescantellusonthesubjectmatter.OurmainissueisthatAgent-BasedModelling(ABM)andtheEquation-BasedModelling(EBM)arenotalternative,butinsomecircumstancescomplementary,andsuggestsomefeaturesdistinguishingthesetwowaysofmodelingthatgobeyondthepracticalconsiderationsprovidedbyParunakH.V.D.,RobertSavitandRickL.Riolo.Ourmodelisbasedontheinteractionofstrategiesofheterogeneousagentswhohavetocrossajunction.Ineachjunctionthereareonlyfourinputs,eachofwhichispassableonlyinthedirectionoftheintersectionandcanbeoccupiedonlybyanagentoneatatime.TheresultsgeneratedbyABMsimulationsprovidestructureddatafordevelopingtheanalyticalmodelthroughwhichgeneralizingthesimulationresultsandmakepredictions.ABMsimulationsareartifactsthatgenerateempiricaldataonthebasisofthevariables,properties,localrulesandcriticalfactorsthemodelerdecidestoimplementintothemodel;inthiswaysimulationsallowgeneratingcontrolleddata,usefultotestthetheoryandreducethecomplexity,whileEBMallowstoclosethem,makingthuspossibletofalsifythem.

Agent-BasedModelling,Equation-BasedModelling,CongestionGame,ModelofSocialPhenomena

Introduction

Tenyearsago,duringthefirstMulti-AgentSystemsandAgent-BasedSimulationWorkshop(MABS'98),ParunakH.V.D.,RobertSavitandRickL.RiolodiscussedthesimilaritiesanddifferencesbetweentheAgent-BasedModelling(ABM)andtheEquation-BasedModelling(EBM),developingsomecriteriaforselectingoneortheotherapproach(Parunaketal.1998).Theyclaimedthatdespitesharingsomecommonconcerns,ABMandEBMdifferintwoways:thefundamentalrelationshipsamongtheentitiestheymodel,andthelevelatwhichtheyoperate.Theauthorsobservedthatthesetwodistinctionsaretendencies,ratherthanhardandfastrules,andindicatethatthetwoapproachescanbeusefullycombined.

Duringthelasttenyears,alivelydebateonthesubjectmatterhasbeendeveloped(see Epstein2007).Anoverallreviewisbeyondthescopeofthiswork.Inthiscontribution,wewilladdressatheoreticalissueconcerningtheemergenceofsocialconventionsandwewillexplorehowandtowhatextentanintegratedapproachofABMandEBMmethodologiescanhelpushandleit.Theaimofthisworkistoexploretheemergenceofsteadystatesincongestiongames(Rosenthal1973;Milchtaich1996;ChmuraandPitz2007 ),andmorespecifically,toinvestigatetheemergenceofaprecedenceruleintraffic-likeinteractions(seealso SenandAiriau2007).

Acongestiongameisagamewhereeachplayer'spayoffisnon-increasingoverthenumberofotherplayerschoosingthesamestrategy.WeuseaWikipediaexampletoshowthisconcept:

forinstance,adrivercouldtakeU.S.Route101orInterstate280fromSanFranciscotoSanJose.While101isshorter,280isconsideredmorescenic,sodriversmighthavedifferentpreferencesbetweenthetwoindependentofthetrafficflow.Buteachadditionalcaroneitherroutewillslightlyincreasethedrivetimeonthatroute,soadditionaltrafficcreatesnegativenetworkexternalities

Acelebratedcongestiongameisthe MinorityGame,wheretheonlyobjectiveforallplayersistobepartofthesmalleroftwogroups( ChalletandZhang1997,1998;ChmuraandPitz2006 ).Awell-knownexampleoftheminoritygameisthe ElFarolBar problemproposedbyW.BrianArthur( 1994),inwhichapopulationofagentshavetodecidewhethertogotothebareachweek,usingapredictorofthenextattendance.Allagentsliketogotothebarunlessitistoocrowded(i.e.whenmorethat60%oftheagentsgo).Butsincethereisnosinglepredictorthatcanworkforeverybodyatthesametime,thereisnodeductivelyrationalsolution.Howcanagentsachievethisgoaltogotothebar,avoidingcrowd?

Ithasbeensuggestedthatincoordinationproblemsituations(likecongestiongames),aconventionisapareto-optimalrationalchoice(see Gilbert1981;Lewis1969;Sugden2004;Young1993),i.e.achoiceinwhichanychangetomakeanypersonbetteroffisimpossiblewithoutmakingsomeoneelseworseoff,therebyinthesekindofcircumstancesitshouldbetheconventionalsolutiontoemerge.

Inthispaper,wesuggestadifferentsolutiontothesubjectmatter,arguingthatinspecificcoordinationproblemsituations,morespecificallyin4-strategygamestructures,aconventionalequilibriumisnotgrantedtoemerge.Thus,theobjectiveofthepaperistwofold:a)showingthatincoordinationproblemsituations,rationalagentsmayalsoconvergeonnon-conventionalequilibriums;b)comparingandintegratingsimulation-basedandanalyticalresultsinordertodrawsomeconclusionsaboutwhatthesemethodologiescantellusonthesubjectmatter.

Thepaperisdividedintosevensections:sections2and3describeageneraltechniquetowriteananalyticmodelforapopulationgame(likeacongestiongame),i.e.thereplicator-projectordynamic.Insection4wedescribeourmodelandanalgorithmtoimplementthereplicatordynamicontoit.Ourmodelisbasedontheinteractionofstrategiesofheterogeneousagentswhohavetocrossajunctiontryingtoavoidcollisions.Ineachjunctionthereareonlyfourinputs,eachofwhichispassableonlyinthedirectionoftheintersectionandcanbeoccupiedonlybyanagentoneatatime.Theagentscanperceivethepresenceofotheragentsonlyiftheycomefromorthogonaldirections.Therearefourtypesofagents,eachcharacterizedbyadifferentstrategy:twoofthesestrategiesconsiderbindingonitsdecisionthepresenceofotheragentsfromaparticulardirection(e.g.byobservingtheirright)tocrossthejunction.Wecallthesestrategies(andtheseagents)conditioned.Insections5and6wedescribetheresultsofsimulationandanalyticprocedures.Finally,insection7weargumentabouttherelationshipbetweensimulationandanalyticresultsandwepointoutsomepossibleoutcomes.

Thereplicatordynamics

Congestiongamesinvolvelargenumbersofsimpleagents,eachofthemplayingoneoutofafinitenumberofrules.Ingeneral,acongestiongamedealswithasocietyPofpdifferentpopulationofagents;nevertheless,ourmodelwilluseonlyasinglepopulation(casep=1).Wecallthesegamespopulationgames.Duringthegame,eachagentchoosesastrategyfromthesetS={1,...,n}wherenisthetotalnumberofstrategies(behaviors).Nowwewilldescribethereplicatordynamicsforpopulationgames.

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LetΣbethespacecontainingallthevectoroffrequencydistributionsoftheagentsplaying,

(1)

forexamplewecouldhaveapopulationwithtwodifferentstrategies,n=2;thestrategy(behavior)1istokeeptheright,thestrategy2istokeeptheleft.x1isthefrequencyofagentsthatarekeepingtheright,x2isthefrequencyofagentskeeping-the-left.Oncewefix PandS,thesetsofpopulationsandstrategies,wecanidentifyagamewithitsownpayofffunction,P:Σ_Rn,i.e.amapassigningavectorofpayofftoeachfrequencydistribution,oneforeachstrategyineachpopulation.Wedefineπithepayoffoftheagentswithstrategy(behavior) i.

Wecancomputesuchapayoffmapintwodifferentways.

a. (a)Coordinationgame.Thepayofffunctionisaconstantmatrix nbyncontainingthepayoffresultsderivingfromtheinteractionsbetweentwostrategies:wecallthiskindofgameRPSsGame(RPSsstandsforRock-Paper-Scissor,tocelebratethewellknown3-strategiesgameswhererockbreaksscissor,scissorcutspaperandpaperwrapsuprock).

b. (b)Congestiongame.Thepayofffunctionisafunctionofbothfrequencydistributionsandsomekindofcombinatoryamongagents.Insuchagametheagentsinteractinacomplexway(interactionswithmorethantwoagentsareallowed).Forinstance,letusconsideracollectionoftownsconnectedbyanetworkoflinks(say,highways).Agentsneedtocommutefromacitytoanother,choosingapath.Anagent'spayofffromchoosingapathisthenegationofthedelayonthispath.Thedelayonapathisthesumofthedelaysonitslinks,andthedelayonalinkisafunctionofthenumberofagentsusingthatlink.

Areplicatordynamicinapopulationgameisaprocessofchangeovertimeinthefrequencydistributionsofstrategies:strategieswithhigherpayoffsreproducefaster(Cecconietal2007;CecconiandZappacosta2008 ).Inmathematicalterms,thedynamicisatrajectoryonthespaceoffrequencies.Weusetwofigures(figure1andfigure2)fromSandholmetal.(2008)toshowthatthereplicatordynamiccouldbedescribedingeometricterms,likeatrajectorywithsomeconstraints.Thispointisthebaseofprojectordynamicmodeling.

Figure1.Weshowthedynamicofapopulationgame.Theredarrowsindicatethedirectionthatthefrequenciesofthestrategiesfollow.Thepurplearrowsindicatetheprojectionoftheredarrowsontoasubspaceofthestrategies:inthiscase,thesubspaceisaline,

correspondingtothepointswherethesumoffrequenciesis1.TheHawkDoveGameisontheright.WeshowthatintheHawkDoveGame(butnoninasimpleCoordinationGame)thereisastablepoint(TheimagecomescourteouslyofSandholm,Dokumaci&

Lahkar,fromSANDHOLM,W.H.,Dokumaci,E.,Lahkar,R.(2008)"Theprojectiondynamicandthereplicatordynamic",GamesandEconomicBehavior,64-2:666-683).

Wedefineπtheaveragepayoffforthewholepopulation

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Thereplicatordynamicis

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i.e.underreplicatordynamic,thefrequencyofabehaviorincreaseswhenitspayoffsexceedtheaverage.Foreachtimestept,thesystemofdifferentialequations(3)definesavectorfield(seefigure1).WecallvectorfieldafieldthatachievesthereplicatordynamicF.WeimplementFbyaglobalimitationalgorithm.Duringthegame,eachagenttestswhetheranotherindividualinthepopulationhasagreaterpayoffthanitsown.Themainruleinthismodelisthatanagentwithalowerpayoffimitatesthebehaviorofanotheragentwithagreaterpayoff.Inourmodel,wedon'thavemutationintheimitationprocess.ThistheoreticalframeworkdirectlyderivesfromtheformalizationdonebyEkman(2001).

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Figure2.Weshowtheprojectordynamicofapopulationgamewith3strategies(TheimagecomescourteouslyofSandholm,Dokumaci&Lahkar,fromSANDHOLM,W.H.,Dokumaci,E.,Lahkar,R.(2008)"Theprojectiondynamicandthereplicatordynamic",

GamesandEconomicBehavior,64-2:666-683).

Thegeometryofpopulationgames

Projectordynamic.

Thedynamicdescribedby(3)issubjecttoaconstraintsincethenumberofagentsremainsconstant,i.e.thesumofthe ximustbe1.Wecould,thus,describethedynamicofthegamewithadifferentapproach,thesocalledProjectorDynamic(Sandholmetal.2008).Thekeypointofthisapproachisthatthevectorfield Fdescribingthepopulationgameisprojectedontoanewvectorfield

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where

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Informally,weprojectthevectorfrom Fontoaline(ifn=2),ontoabidimensionalsimplex(if n=3,seefigure2)andsoon.Theprojectionofavectorialfieldisastandardprocedure.YoucanfindanalgorithminSandholmetal.(2008).

ProjectordynamicwithtwostrategiesintoRPSsgame.

Wecanexplaintheprojectordynamicframeworkwithtwostrategiesusinga2-strategies RPSsgame,theHawkDovegame.InHawkDovegamewefillthepayoffmatrixPHD(payoffmatrixforaRPSsgames)usingtherules:

1. whenHawkmeetsDoveHawkwina;2. whenHawkmeetsHawkbothlosebwithb<a;3. whenDovemeetDovebothwinb;4. whenDovemeetsHawk,Dovegets0.

wecancomputethecomponentsoftheprojectionofPHDfortheHawkDovegame.Aswesaidbefore,P HDdenotesthepayoffmatrixfortheHawkDovegame

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Ifxisthevectoroffrequencies,wehave

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hencefrom(7)wecancomputethecomponentsoftheprojectionofPHDfortheHawkDovegame

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Therelationobtainedfortheprojectionletusstatethatthegamehasastablestationarypoint(seefigure1,ontheright)

Projectordynamicintocongestiongame.

InthepreviousSubsection,wedescribedtheprojectordynamicin RPSsGame,withconstant2-agentsinteractionspayoffmatrix.Inourmodel(congestiongame),weadapttheprojectordynamicframeworktocomputeapayoffmatrixthatstartsfromapayofffunctionwithtwoparameters,i.e.thedistributionofthestrategiesattimetandthecombinationofstrategiesatthesametime.

Forexample,wesetagamewhere

1. 2hawksvis-à-vis1dovegetadifferentpayofffromtheoppositesituation(2dovesand1hawk)2. agentsplayingthesamestrategiesinteract(asinRPSsgameHawk-HawkandDove-Dove).

WefillthepayoffmatrixPCG(payoffmatrixforcongestiongame)usingthefollowingrules

i. wefillthediagonalusingthesamestrategyinteractions,thatistheinteractionsbetweenthesamestrategies,forexampleHawk-HawkandDove-Dove.ii. wefillothercellsusingthedifferentstrategyinteractions,thatistheinteractionbetweenthecorrespondingstrategyinaconstantpayoffmatrix(inthesimple

caseHawk-DoveweconsidertheinteractionbetweenHawk-Dove).

Themainfeatureofcongestiongameis:whenwefillthepayoffmatrixwecomputetheprobabilityfortheinteractions,forexampletheprobabilitytohave2hawksvs.1doves.

Themodel

Thegeneralframework:ajunction

LetPbeasocietyconsistingofasinglepopulationwith Nindividuals.Weconsideragamewhereeachindividualfollowsoneoutoffourbehaviors, WatchRight,WatchLeft,DoveandHawk.Wedenotethebehaviorwiththeindex I.Thefrequencyofindividualswithinthesocietywithbehavior iisxi.

Wedefinethebehaviorsandcomputethepayoffsbecauseofthepresenceof3differentalgorithms:the conditioned,thecompliantandtheaggressivealgorithm.TheconditionedalgorithmleadstoWatchRightandWatchLeftbehaviors.ThecompliantalgorithmdescribestheDove'sbehavior;theaggressivealgorithmdescribestheHawk'sbehavior.Weuseadiscretetimemodel,followingtheschedulinginalgorithm1,whereindividualsarepositionedrandomlyoveradiscretebi-dimensionallatticeL.Weusearegularlattice,withCcell.WeconsidereachcellofLacrossroadwithfourinputs(seefigure3).

WedescribethecrossroadsusingcounterclockwisedirectionandstartingfromWest,ifweareusingabird's-eyeview.Ifweareusinganindividualview(i.e.thecrossroadasitappearstotheindividual)weusetheright,leftandforwardconventions(firstweindicatewhatIfindonmyright,thenonmyleft,finallyinfrontofme).Hence,inFigure3(a)weshow(1)anindividualrunningEast(withWatchLeftbehavior);(2)thedirectionNorth-Southisempty;(3)anindividualwithHawkbehaviorisrunningtoWest;(4)anindividualwithDovebehaviorisrunningtoNorth.WecallparallelthedirectionsNorth-SouthandSouth-North,andthedirectionsWest-EastandEast-West.Wedefineorthogonaltheothercombinations.

Figure3.TwosnapshotsofL.In(a)wehave3individuals,in(b)only2;weputagentsontheinputofthejunction,thentheytrytocross.Thereisnomovementfromajunctiontoanother.

Themodelhassomeconstraints:

theindividualsthatarealoneinacelldonotgetpayoff;amaximumoffourindividualscanoccupyonesinglecell;onlyoneindividualperdirectionisadmitted(itisnotpossiblethat,forexample,twoHawksarerunningintheNorth-to-Southdirection);onlyorthogonalindividuals(wemeanindividualscomingfromorthogonaldirections)matter.

Nowwesketchtheconditionedalgorithm(seealgorithm2),theaggressivealgorithm(seealgorithm3)andthecompliantalgorithm(seealgorithm4).

Algorithm1.Theschedulingofsimulation.Wedefineas Crossroadsthecellswithmorethan1individual.

for t = 1 to T do

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for a = 1 to N do a := P to set a over L end for for c = 1 to Crossroads do Compute payoffs end forfor a = 1 to N do Imitation dynamic end forend for

Algorithm2.Conditioned.Aistheindividual.Xindicatesthedirectiontomonitor. CRASHindicatesthatAgoesintothecrossroadatthesametimeofanotherorthogonalindividual.STOPindicatesthatAdoesnotgointocrossroad.GOAHEADindicatesthattheindividualcrossesthecrossroad. Hisahawkindividual.DisaDoveindividual.Cisaconditionedindividual.WedefinenotblockedCaconditionedindividualwiththefreemonitoreddirection.

If X is occupied then STOP else if there is (orthogonal H) or (orthogonal not blocked C) then CRASH else GOAHEAD end if end if

Algorithm3.Aggressive.

if there is (orthogonal H) or (orthogonal not blocked C) then CRASH else GOAHEAD end if

Algorithm4.Compliant.

if there is an orthogonal individual then STOP else GOAHEAD end if

Payoffcomputations

Torealizeananalyticmodelforourcongestiongame,wewritedownanexpressionfor πi,thepayofffor i'sstrategy.Morespecifically,wehavetowritedownamatrixPCG,4×4.Theelementsofthismatrixarefunctionsoffrequencyofthestrategiesandtheprobabilityofthecombinationsofthestrategiesoverthetime.The

computationofπileadsdirectlytofillthePCGmatrix,usingruleslikeinsection3.3,i.e.… wefillthediagonalusingthesamestrategyinteractions(forexampleWatchRightVsWatchRight),…wefilltheothercellsusingthedifferentstrategies'interactions (forexampleHawkVsDoveVsWatchRight).

Wecallthebehaviorsi,r=1,l=2,d=3,h=4 .Thegeneralrulesare:ifagentSTOPS,thepayoffissetto 0.IfagentsCRASHtheirpayoffsaresetto -1.IfanagentGOESAHEAD,itspayoffis1.

Weletxi(t)denotetheamountofindividualswithbehaviors iinperiodt.Clearly,wehave x1+x2+x3+x4=N,whereNisthetotalnumberofplayers.Wealsolet Cdenotethetotalnumberofcells.WestudythemodelwithdensityC=N.Theprobabilitythatonecellcontains kindividualsis

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Wecallakthenumberofcellswith kindividuals.Wecancomputethepayoffforeachbehaviorasthesumoftheproductsbetween akandthesumofthepayoffsforeachcombination,withkfixed.Forexamplefork=2 wehave10combinations,h={rr,ll,dd,hh,rl,rd,rh,ld,lh,dh} .Only8combinationsgivesomepayoffsthataredifferentfromzero,rr,ll,dd,rd,rh,ld,lh,dh.hhdoesnotgivepayoff,becausewehaveparallel(=) hhwithpayoffs2andorthogonal(||)hhwithpayoffs-2. rldoesnotgivepayoffbecauseforparallelcombinationsweobtain2,andforthe2orthogonalcombinations(probabilities1/2and1/2)weobtain0and-2.

WecallKhithematrixofpayoffsdifferentfromzeroforbehavior iandcombinationh.Theprobabilityofcombinationhwithh1+h2+h3+h4=k conditionedtofrequenciesisamultinomialdistribution

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Finally,wecandefinethepayoffforstrategy iforstrategyi

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Figure4.Asnapshotofthesimulatorinterface

Figure4isasnapshotofthesimulatorinterfaceimplementedinNetLogo.Inthisinterfaceeachdifferentstrategyisrepresentedbyadifferentcolor:

1. blueagentsareright-watchers;2. grayagentsareleft-watchers;3. redagentsarehawks;4. greenagentsaredoves.

Thecentralitemrepresentsthetoroidalworldinwhichagents(randomly)move.Theslidersontheleftareusedtoset,atthebeginningofthesimulation,thenumberofagentsforeachstrategy.Thetwoitemsconcerningtheplots(ontherightandontheleft,respectively)areusedtomonitortheaverageutilityobtainedbyeachstrategyandthenumberofsurvivingagentsforeachstrategy(seehttp://jasss.soc.surrey.ac.uk/13/1/6/AwkDoveModel.htmlforanappletofthemodel).

Table1showstheparametersofthesimulations.

Table1:Thesetofparametersofsimulations

Parameter Valueright-watchers from50to200left-watchers from50to200doves from50to200hawks from50to200#ofTicks 1000Typeofimitation global

Simulationresults

Werun256simulationsvaryingthesizeofeachsub-populationofagentsbetween50and200,thusthepopulation'srangevariesfrom200to800agents.Eachsimulationincludes1000ticks(seetable1fortheparameters).Infigure5,6,and7weshowsomeindividualrunsofthesimulations.

Inaccordancewiththedescriptionabovemadeofallofthepossiblesteadystates,wefoundthefollowingresults:

1. onesub-populationsurvives:a. Right-watchers:3times(1.18%);b. Left-watchers:5times(1.95%);

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Figure5.Anindividualrunofthesimulations.Inthiscase,thesimulationstartswith100right-watchers,100left-watchers,100dovesand50hawksandonlyleft-watcherssurvive)

2. twosub-populationssurvive:a. hawksanddoves:48times(18.75%);b. right-watchersanddoves:15times(5.86%);c. left-watchersanddoves:19times(7.42%);

Figure6.Anotherindividualrun.Inthiscase,thesimulationstartswith100right-watchers,100left-watchers,100dovesand100hawksandbothdovesandhawkssurvive)

3. threesub-populationssurvive:a. right-watchersandhawksanddoves:85times(33.20%);b. left-watchersandhawksanddoves:81times(31.64%);

Figure7.Inthiscase,thesimulationstartswith100right-watchers,100left-watchers,100dovesand100hawksandonlyright-watchersdie

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Simulationsmayendupwithmultiplesteadystates(seefigure8;thetreeshowsthedistributionofsteadystates.Therearenooutcomesthatdonotinvolveanysub-population:thenumberofagentsremainsconstant):

Onesub-populationsurvives:inthissituationthesub-populationcanonlybeeitherrightwatchingorleftwatchingsuggestingthat,toputitinLewis'sterms,theseareself-sufficientalternativesolutionstoproblemsofcoordination.Beingthetwosolutionsequivalent,thechoicebetweenthemispurelyarbitrary.Thereby,onceonesolutionhasbeenselected,theotheronecannotcoexist.Twosub-populationssurvive.Inthiscase,

asithappenswiththeEvolutionarilystablestrategy(ESS)(see Smith1974;Gilbert1981),ifonesub-populationishawks,theotheronewillnecessarilybedoves;hawksdonotsurvivewitheitherrightorleftwatchingsub-populations;if,onthecontrary,oneofthesub-populationsisdoves,theotheronemaybeanyofthethreeremainingsubpopulations,i.e.rightwatchers,leftwatchersorhawks.Indeed,unconditionedstrategiesarenotsymmetrical:theformercansurviveonlybyexploitingthemostaltruisticstrategy.

Threesub-populationssurvivebyincludingeitherrightwatchersorleftwatcherssub-populations,butneverbothatonce.Foursub-populationscannotsurvive(thisisaconsequenceofthepreviousitem).

Figure8.Thetreeshowsthedistributionofsteadystates.Thewidthofthearrowsindicatethefrequencyofthesteadystate(fatarrowmeanshighfrequency)

Summingallofthepercentages,wecancalculateallofthecasesinwhicheachsub-population(aloneorwithothers)survives.Wecoulddefinetheadaptabilityofasub-populationthepossibilityofasub-populationtosurvivewithanotherintoasteadystate.Wecanproposeahierarchyofadaptability:

1. doves:96.87%;2. hawks:83.59%;3. right-watchers:40.24%;4. left-watchers:41.01%.

Analyticresults

Wecanextractinformationfromtheanalyticmodelinthreedifferentways.(1)wecannumericallysolvethedifferentialequationssystem;(2)wecandrawthevectorialfielddescribedbytheODEsystem;(3)finally,wecancomputethesteadystatesandthegradientsaroundthem,tovaluateifthesteadystatesarestableornot.Figure9,10,11showsomeexamples.

Figure9.Anumericalsolutionoftheanalyticmodel.TheredcurveshowsthefrequencyofHawks,theblueshowsthefrequencyofDoves.Thegrayandtheblackcurvesshowthefrequenciesofconventionalstrategies.

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Figure10.Weshowthedirectionofthevectorfieldforthex1=right-watchersandx2=left-watchers.Inthefigureattheleft,wesetthenumberofhawkstozero.Inthefigureattheright,wesetthenumberofdovestozero.Weseethatwithnohawks,thesteadystatewith

twosub-populationsisfeasible

Figure11.Wehaveamapfromtheinitialstateswithdifferentnumerousnesstothesteadystates.Inthefigureweshowthat,startingwithanaveragenumberofdovesandamediumN,wehavefourpathsonfivetotwo-subpopulationsteadystates.(x1,x3,x4means:

right-watchers,dovesandhawks.Wesetleft-watcherstozero)

Usingthesemethods,weobtainedsomeanalyticresults,fittingwiththesimulationoutcomes:

1. TheConditionedstrategyWatchRightandWatchLeftareincompatible.Therelationobtainedfortheprojectiondynamicletsusstatethatthegameswithonlyonestrategyhaveanunstablestationarypoint(thederivativeofthefrequencieshasoppositesign,seefigure9).Theinitialstateisalmostuniform.Weshowthat,

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afteradelay,aconventionalstrategywins(inthiscasethesteadystateisone-subpopulation):thedynamicshowthattheconventionalstrategiesareincompatible.

2. TheConditionedstrategiesononehandandtheAggressivestrategyontheotherareincompatible.Viceversa,wefindastablesteadystatewithConditionedstrategiescombinedwiththeCompliantstrategy(seefigure10).

Theanalyticmodelsuggestssomeotherhints:

Figure9showsthatwehaveadelaybeforeseparationbetweentheConditionedstrategies.(thedelayisobservedduringthesimulationalso).Thus,wecouldarguethatthedelayisnotastochasticeffectcomingfromthefluctuations,duringtheagent-basedsimulation.Thedelayistheeffectofthenon-linearityoftheinteractions.BytheEBM,wehaveageneralmap(thefigure11isjustanexample)ofthepathsfromtheinitialstates,withdifferentfrequencies,tothefinalsteadystates.Themapshowsacorrespondencewiththesteadystatesdistributionfromthesimulations(0.05onesub-population,0.3twosub-populations,0.65threesub-populations).

Discussion

Wetrytosummarizewhatmightbetheaddedvalueofthiswork(a)fromthestandpointofunderstandingtheroleofconventionsincongestiongames(b)inthecomparisonbetweenABMandEBMapproachinthestudyofsocialphenomena.

Regardingsection(a),inourABMmodelweinvestigatetheroleofconditionedstrategies-whichcanbeseenassocialconventionsinLewis'ssense-inthesolutionofacongestiongame,incomparisonandincombinationwithunconditionedstrategies.

Whatdotheseresultstellusaboutsocialconventions?Inlinewithevolutionarygametheory,thisstudyhelpstopredictwhichstrategiesachieveastablestate.Besides,itallowsnewpropertiesofstrategiestobedetected.Inparticular,weareabletodiscriminatenotonlybetweenunconditionedandconditionedstrategies,butalsobetweenincompatibleequivalentstrategies,i.e.strategiesthathavethesamepayoffsforalloftheplayers,andcomplementarynon-equivalentones,i.e.strategiesthatbenefitagentstoadifferentdegree(seethesectiononSimulationresults).Theformerareself-sufficient,thelatterarenot.Hence,althoughwecannotpredictwhichspecificequilibriumwillbeachieved,wecantellthefinalcompositionofthepopulationgivenanth-strategyequilibrium.

Butisittruethatwecannotpredictwhichspecificequilibriumwillbeachieved?Notcompletely.Infact,usingEBMwecanpredictifasteadystateshouldcontainsomecombinationofstrategies,withwhichprobabilityasteadystatefollowsfromsomeinitialdistribution,andthestabilityofsteadystates(seetheresultsinthesectiononAnalyticresults).

Thepoint(b)ismorecontroversial.Therealquestionis:woulditbepossibletodoABMwithoutEBM?And,ontheotherhand,woulditbepossibletorealizetheabove-mentionedanalyticalmodelwithoutthesimulationdata?Probablynot.Weclaimthat,forwhatconcernsthesocialphenomenonweareinterestedin,i.e.thesolutiontoacoordinationprobleminalarge-scalepopulationofheterogeneousagentsininteraction,simulationdataaredecisive.Theformeralreadyincludefeaturesthatarenecessaryforrealizingthemathematicalmodel.Nevertheless,weclaimthatthesetwoclassesofmodelsarenotalternative,butinsomecircumstancescomplementary.Letustrytodefendthisargument.

ThemaindifferencebetweenABMandEBMisinthecapacitytograspdifferentstochasticaspectsofthephenomena.ABMdescribesstochasticfluctuations.EBMdescribesthestatisticsofthefluctuations,forexamplethemean,andtheshapeofthedistribution.

Theresultsgeneratedbythesimulationsmayfittheanalyticalmodel:theyprovide insilicodatafordevelopingtheanalyticalmodeltogeneralizethesimulationresultsandtomakepredictions(Conte2002;Bonabeau2002).

Wealsoclaimthatneitherthesimulationsnortheanalyticalmodelalonehelpedusexplainwhythesedatahavebeenproduced:inotherwords,noneofthetwomethodologieshelpsusunderstandthedatawehavegenerated.Goingbacktoourmodel,resultsshowthatplayingwithfourqualitativelydifferentpopulations,notonlyconventionsemerge-i.e.equivalentarbitrarysolutionstoproblemsofcoordinationrealizedbytherightwatchingorleftwatchingpopulations-butalsoexploitationsolutions,representedbythecoexistenceof"hawks"and"doves".Inotherwords,droppingthe2-strategygamelogics,hegemonicinGameTheory,itisbynomeansguaranteedthataconventionalequilibriumwillemerge.

Wecantrysomeexplanation,usingtheanalyticpayoffmatrix.Forexample,thecomputationofpayoffshowsthatthereisagradualdeteriorationofconditionedstrategiesproceedingfromnon-crowdedtocrowdedworld.Inotherwords,ifagentsliveinanenvironmentwherethelikelihoodof4-agentsjunctionishigh,theadvantagetomaintainpositivepayoffsusingconventionalstrategiesdisappears.

Now,letusconsidersomefactsemergingfromtheABMframework.Supposeonesub-populationsurvives:inthissituationthesub-populationcanonlybeeither"rightwatching"or"leftwatching",suggestingthat,toputitinLewis'sterms,theseareself-sufficientalternativesolutionstoproblemsofcoordination.Weshowthatthesesteadystatesoccurrarely.Why?Onepossibleanswer(suggestedbytheEBMframework)couldbethatthenumberofpathsfrominitialstatestoonesub-populationsteadystateislow.

Considernowthecaseinwhichtwosub-populationssurvive:inthiscase,asithappenswithEvolutionarilyStableStrategies(ESS),ifonesub-populationis"hawks",theotheronewillnecessarilybe"doves";"hawks"willnotsurvivewith"rightwatching"or"leftwatching"sub-populations.Thesetwolatterresultsseemtosuggestthatunconditionedstrategiesarenotsymmetrical:althoughhawksanddovesarenotself-sufficient,theformercansurviveonlybyexploitingthemostaltruisticstrategy.Onthecontrary,thelatterismoreadaptive,sinceitmaysurviveininteractionwithanyothersubpopulation.

Hence,foracongestiongameliketheonediscussedinthispaper,wecouldtrytodefineahierarchyofadaptability.Wefinddovesonthetop:dovescouldlivewithhawksandconditionedagents;hawkscouldlivewithdovesand,atthebottom,wefindtheconditionedagents.Infact,aconditionedstrategychasesawaytheotherconditionedones.Hawksanddovesarenon-equivalentstrategies:theyhavedifferentpayoffs,whosecombinationcanbestablebecauseofthecomplementarinessofthetwostrategies.

Threesub-populationssurvivebyincludingeither"rightwatchers"or"leftwatchers",butnotbothofthematonce.

Atthispoint,onemaycomeupwitharatherobviousquestion:isitpossibletopredict,ingeneral,thefinalstatesfromthepopulationsize?Thereaderisremindedthatwehaveanalyticallymodeledallofthepossibleinteractionsinourscenario,andwehavethencalculatedthepointofequilibriumasafunctionofN,thedimensionofthepopulation(atleastintheory).Theanalyticmodelallowstherelationshipsbetweenthestrategiesandtheirpayoffstobedescribedthroughclosedformequationsandthesimulationresultstobegeneralized,allowingaccuratepredictions.However,thisisanopenpointfordiscussion.

Finally,acrucialfeatureconcernsABM:simulationresultsareorganizedinsomehierarchicalstructuresincetheyaregeneratedbyouralgorithms.Forexample,inourmodel,theresultsincorporatepeculiarsymmetries:i.e.equivalentstrategiescannotcoexist,whilenon-equivalentonescan.WeareattemptingtoexplicatetheseresultsbyEBM.

TheresultsgeneratedbyABMsimulationsprovidestructureddatafordevelopingtheanalyticalmodelthroughwhichgeneralizingthesimulationresultsandmakepredictions.ABMsimulationsareartifactsthatgenerateempiricaldataonthebasisofthevariables,properties,localrulesandcriticalfactorsthemodelerdecidestoimplementintothemodel;inthiswaysimulationsallowgeneratingcontrolleddata,usefultotestthetheoryandreducethecomplexity,whileEBMallowstoclosethem,makingthuspossibletofalsifythem.

Inbrief,Parunak,SavitandRioloclaimedthatABMandEBMdifferin:i)relationshipsamongtheentitiestheymodel,ii)thelevelatwhichtheyoperate.AlookbackwardtothestepswefollowedtorealizetheanalyticalmodelprovidesususefultipsonthecomparisonbetweenABMandEBM.Thesetwoclassesofmodelsarenotalternative,butinsomecircumstancescomplementary,andsuggestsomefeaturesdistinguishingthesetwowaysofmodelingthatgobeyondthepracticalconsiderationsprovidedbyParunak,SavitandRiolo.

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Acknowledgements

ThisworkwassupportedbytheEMILproject(IST-033841),fundedbytheFutureandEmergingTechnologiesprogramoftheEuropeanCommission,intheframeworkoftheinitiative"SimulatingEmergentPropertiesinComplexSystems".

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