Imitation dynamics in a game of traffic

0.15 MB·English

by  Gabriel H. Paissan

EPJmanuscriptNo.
(willbeinsertedbytheeditor)
Imitation dynamics in a game of traffic
G.H.Paissan1 andG.Abramson2a
1 ConsejoNacionaldeInvestigacionesCientíficasyTécnicas,CentroAtómicoBarilocheandCentroRegionalUniversitarioBariloche-UNCo,
8400Bariloche,RíoNegro,Argentina
2 ConsejoNacionaldeInvestigacionesCientíficasyTécnicas,CentroAtómicoBarilocheandInstitutoBalseiro,8400Bariloche,RíoNegro,
Argentina
3
1
Received:date/Revisedversion:date
0
2
Abstract. Westudyamodeloftrafficwheredriversadoptdifferentbehavioralstrategies.Thesecanbecooperativeor
n
defectiveaccordingtoadriverabidingornotbyatrafficrule.Driverscanchangetheirstrategybyimitatingthemajority,
a
J witharulethat depends on thestrategieswithwhichtheyhaveinteracted. Theseinteractionsoccurat intersections,
wherevehiclespayatemporalcostaccordingtotheirstrategy.Weanalyzetheconditionsunderwhichdifferentstrategy
7
compositionsrepresentanadvantageinthesystemvelocity.Wefoundthatthecooperators’meanspeedishigherthan
1
thedefectors’evenwhenthevehicledensityislarge.However,defectorscanobtainbenefitsintheirmeanspeedwhen
] theyareaminorityinanessentiallycooperative population. Thepresenceofacoreofeducated drivers,whopersist
O firmly in a cooperative behavior, optimizes the speed in the system, especially for intermediate values of vehicular
A densityandhighertemporalcosts.
.
n PACS. 89.65.-a Tansportation;urbantraffic–02.50.Le Gametheory–02.70.-c Computationaltechniques;simula-
i tions
l
n
[
1 Introduction
fluid—characterized by a spatial density and a velocity field.
1
On a different vein, “microscopic” models that treat the mo-
v
0 Vehiculartrafficdynamicshasreceivedconsiderableattention tion of each vehicle separately have been developedas a par-
3 since,atleast,themiddleofthetwentiethcentury.Inthedecades allel line of research. Microscopicmodelsemphasizethe role
1 thatfolloweddifferentpointsofviewhavebeenusedtotackle played by the (non linear) interactions between vehicles. The
4 differentfeaturesofthemanyproblemsassociatedwithtraffic. interestofphysicistsintrafficproblemsreceivedanenormous
1. The interest of these hardly needs justification: the many as- boostwiththeworkofNagelandSchreckenberg[5].Thereex-
0 pectsoftraffichaveenormousimpactinourcivilization,with istverydetailedmicroscopicmodelsthatservethepurpose of
3 applications in many fields, ranging from engineering to the analyzing mostly local situations. Such level of detail is still
1 social sciences. The volume of traffic flow has quickly over- prohibitive in the study of large scale features of traffic flow.
: passedthecapacityofthecitiesandhighways,onceandagain, Nagel-Schreckenbergmodelsandothers,ontheotherhand,ex-
v
i countryaftercountry,makingtheunderstandingofitsdynamic ploitthe moremanageablerepresentationoftraffic as cellular
X an imperative in many societies. The problems of infrastruc- automata.
r ture and urbandevelopmentable to accommodatean increas- In this work we present a bidimensional model of traffic
a
ingflowrangeresideatoneendofthespectrum.Attheother where drivers operate with different behavioral strategies. In
endthereareaspectsofeducation,socialplanningandlawen-
thisframeworkwe introducea dynamicofimitationbasedon
forcement,aimingathelpingtheflowinthebestinterestofthe the types of strategies faced by driversat each intersection of
society. Many other ancillary problems lay in between: con- a city. Our aim is to get an insight about how the interaction
gestion,pollutionmanagement(includingnoiseandvibration), betweendriverswith distinctivebehaviorscan influencea 2D
optimization of energetic resources, economics, reduction of
[6] traffic flow. Specifically, we study a system where part of
accidents and casualties, parking, public transportation alter- theagentsrespectsa traffic rule,while othersignoreit. It is a
natives,remotemonitoring,robotization,etc. verycommonsituationinmanycountrieswithpoortrafficedu-
The complex spectrum of mathematical traffic models is cation,fromwhichanimitationbehavior,“doastheothersdo,”
well documented in the reviews by Helbing and Chowdhury usually emerges.In such a context,we explorethe effectthat
[1,2].TheinfluentialworksofLighthillandWhitham[3],and the introductionof a core of “well educated” drivers, abiding
then of Richards [4], are based on a macroscopic approach bythelaw,affectsthecollectivebehaviorandtheefficiencyof
in which traffic is considered a continuous medium—like a thesystemintermsof,forexample,speed.
Sendoffprintrequeststo:GabrielH.Paissan Inthe vastliteratureontraffictherearefewstudieswhich
a address:[email protected] use the tools of game theory. Yet, the behaviorof driverscan
2 G.H.Paissan,G.Abramson:Imitationdynamicsinagameoftraffic
easilyandadequatelybetreatedasastrategyinthegamethe- tions.Thisruleiswidelyusedincountrieswithright-handtraf-
oretical sense. The framework we follow in the present work fic,andappliedatallintersectionswhereitisnotoverriddenby
is reminiscentof otherstudies in the dynamicsof pedestrians prioritysignsortrafficsignals,neitherofwhicharepresentin
[7,8,9].Inparticular,Baeketal.[10]haveanalyzedacellular our model. Drivers are either cooperators (who abide by the
automatonwhereagentsmovealongapassagewayinbothdi- rule,andyieldtodriverscomingfromtheirright)or defectors
rections.Whenencounteringotheragentstheytakeasteptothe (whoignoretherule).
rightwithprobabilitypandtotheleftwithprobability1−p.In When two vehicles approach an intersection at the same
thisway,theyplayacoordinationgameinwhichtwostrategies timethereisaninteractionthatdeterminestheorderofcross-
areconsidered:trafficruleabidersandtrafficruleignorers. ing and the time involved. There may be four differentsitua-
Other game theoretical approaches have been applied on tions according to the drivers’ strategies, as follows. (i) Both
networksoftransportationapproachedfromaneconomicpoint drivers are cooperators. One of them gives way to the other,
of view [11,12]. Certainly, there are manyinstances were the whohasrightofwayaccordingtopriorityoftheright.(ii)The
mathematicsofcooperatinganddefectingstrategies,ofimita- driver coming from the right is a defector, and the other is a
tion and education,can be applied in traffic systems. The ed- cooperator.Thissituationisanalogousto(i),sincethecooper-
ucation of traffic rulesis perhapsthe epitomewhere the need atoryieldstothedefector,andthedefectorjustignorestherule.
ofreachingaconsensusofcooperationinthesystemproduces (iii) The cooperatorapproachesfrom the right, and the other
a globalbenefit. A similar problemin a more abstract setting driverisadefector.Thisisthereverseof(ii).Thedefectordoes
hasbeenstudiedin[13],andweaimtoapplyingsomeofthose notyield,while the cooperatorassumeshis rightto cross and
ideashere. attemptstodoit.Inthiscasewesupposethatsometimeislost
inthefakepass.(iv)Bothdriversaredefectors.Inthissituation
wesupposeacollisionthatentailsalongerlossoftimethanin
2 Model case(iii).Thetimethatadriverneedstocrosstheintersection
canbeusedasameasureofthecostinvolvedinthegamesitua-
With the purpose of obtaining an insight of the mechanisms tion,analogoustothepayoffoftheusualformulationofformal
operatingbehindtheemergentcollectivepatterns,we analyze games.The loss of time that occursin situations(iii) and (iv)
asimplemodelthatcontainsenoughingredientsfromthereal canbeasubstantialcontributiontothecumulativecost,andcan
systems. The model city is a square with straight streets ar- beinterpretedaspassfakes,finesorevencrashes.Thecrosses
ranged in a regular lattice. For the sake of simplicity in the actuallyhappenonlyiftheslotjustaheadoftheintersectionis
description, imagine that the streets run either in the North- free.Figure1showsacartoonsummarizingthedescriptionof
Southdirection(thelongitudinalstreets)ortheEast-West(the thesystem.
transversalstreets).Allstreetsareone-way,single-laneroads. The costs involved in rules (i) to (iv) are summarized in
The direction of traffic alternates in both sets of streets, as is the following table, that show the time in units of simulation
usualinmanyrealworldsituations. steps.Aleft(right)driverisadriverapproachingtheintersec-
The vehicles are modelled as a cellular automaton. A car tionfromtheleft(right)oftheotherdriver.Adriverthatyields
can occupythe space betweentwo intersections,and advance losesatimestep,incurringintoacostof2forcrossing.Thepa-
oneblockateachtime stepof theautomatondynamics.Dou- rametersaandbcharacterizethedelayincurredintoincaseof
ble occupancy of the blocks is not allowed: the cars can ad- moreseriousinteractions.Observethat,intheinteraction(iii),
vanceonlyiftheblockaheadisempty.Interactionofcarsoccur weassumethatbothdriverslosethesametime,a.Thisisasim-
onlyattheintersections,inamannerthatwillbedescribedbe- plificationofasituationthatcouldbe,ofcourse,morecomplex
low.Wekeepaconstantnumberofvehiclesbyallowingthose inreallife.Forexample,thecooperatormightstopcompletely,
driversthatexitthelatticefromabordertoreenterthesystem. theoffendingdefectorcross,andthenthecooperatorcontinue
To preventartifacts that a periodic boundaryconditionwould his way. This could be implemented in the present formula-
produceintherathersmallsystemsweusetomimicthestreets tion with an additionalparameter c>a for the cooperator.A
ofarealworldcity,thesecarsarerandomlyreturnedtoempty sensiblechoice,inthepresentcellularautomatonformulation,
slotsattheborderofanystreet,regardlessthestreetfromwhich wouldbec=a+1.Theresultsdonotchangesignificantlyin
theycameout. such a case, reflecting a robustness of the modelwith respect
The dynamics proceeds in discrete steps. At each step all todetailsthatmightbedifficulttoquantify.Ontheotherhand,
theintersectionsofthecityareupdatedinarandomorder.The onedoesnotsensiblyexpectthatc>a+1.Evenifonewould
effectofthisisanasynchronousupdateofthepositionsofthe allowit,athoroughreformulationoftheruleswouldbeneces-
vehicles,whichpreventssomeartifactsthatasequentialupdate sary to accommodatefor the possibility of a vehicle standing
produces,especiallyathighdensity.Asimulationrunconsists still even when the interacting one has followed his way. In
of the repetition of this update step nL2 times (with usually thepresentworkweprefertokeepthemodelrulessimple,as
n=100),whichprovidesareasonablenumberofinteractions formulated.
sbteattweeaefntedrraivsehrosr.tWtreanosbiseenrtv,ewdhthicahttahlleoflwoewdruesatcohetaskaesmtaetiaosnuarrey- hLhefhtdhrivherhhhhRihghhtdhrivhehr C D
ments and time averages during the course of the simulation. H H
H 1 H 1
Theresultsshownbelowcorrespond,furthermore,toensemble C 2 HHH 2 HHH
averagesof100suchsimulationruns. H H
H a H b
There is only one traffic rule in the system: drivers shall D a HHH b HHH
give way to vehicles approaching from the right at intersec-
G.H.Paissan,G.Abramson:Imitationdynamicsinagameoftraffic 3
with25%cooperators.Theupperthree,ontheotherhand,cor-
respond to a very cooperative system with 75% cooperators.
6
These two situations provide a good characterization of both
cooperativeandnon-cooperativeensembles.
2 5
1
0.7 25% cooperators
3 < >
> 0.6 < C>
4 D
0.5 < D>
<
>, 75% cooperators
8 C 0.4 < >
< < C>
7 10 >, 0.3 < D>
< 0.2
9
0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig.1.Sketchofadistributionofvehiclesinamodelcity.Cooperators Fig.2.Meanvelocityofthesystem,ofcooperatorsandofdefectors
and defectors are represented by white and grey ovals respectively. (hvi,hvCiandhvDirespectively)asafunctionofthedensityofvehi-
Thearrowsshowthedirectionofmotionofthecars.Dashedarrows cles,d .Weshowtwodifferentcooperationscenarios,asindicated in
indicatedriversthatgivewaytotheother;solidarrowscorrespondto thelegend:25%and75%ofcooperatorsinthepopulation.Thecosts
driversthat cross (ortrytocross) theintersection. Thefour interac- involvedintheinteractionsarea=3,b=100.Dataareaveragedover
tions shown (besides the “free” drivers 9 and 10) correspond to the 100realizations.
fourtypesofencounterofstrategiesthatcanoccuratintersections.
AsFig.2shows,thevelocitydecreasesmonotonicallywith
3 Results vehicledensity.This is so forthe system averagevelocity hvi
aswellasforthecooperatorsandthedefectors.Observe,also,
thatthe averagevelocityofcooperatorsis greaterthanthatof
Letusbeginwiththeanalysisofsystemswithquenchedstrate-
defectorsatalldensities.Thisreflectsthefactthatcooperators
gies.Eachdriverreceives,atthebeginning,a strategythatre-
are less prone to the higher costs involved in the violation of
mains unchangedduring the course of the simulation. Strate-
the right hand rule (b≫a>2). Moreover, observe also the
giesare randomlydrawn,uniformlywith a probability p for
C
effectofsystemcompositionontheaveragevelocity.Whenthe
cooperationanditscomplement1−p fordefection.Theden-
C
systemismorecooperative,defectorsalsogetabenefit.Indeed,
sityofcooperatorsisthen p ,uniformlydistributedinthecity.
C
The simulations runs we show below consist of 100L2 time the velocity of defectors is greater in a system with 75% of
steps, in a square city of L2/2 intersections, and with a time cooperatorsthan that of cooperatorsin a system with 25% of
cooperators—atleastforintermediatedensities.Wewillreturn
step that updates, randomly and asynchronously,as many in-
tersections. After a transient of L2 time steps we measure the tothesematterslateron.
We have also computed the distribution of time-averaged
instantaneousaveragevelocityinthesystem,definedasthera-
individualvelocities,andcheckedthatthemeanvaluesshown
tio of movingvehiclesto theirtotalnumber,andcomputethe
inFig.2areagoodcharacterizationofthem.Figure3showsa
timeaverageofit,denotedhvibelow.Averagevelocityofco-
typicalexample.Thedistributionsarebellshapedaroundtheir
operatorsandofdefectors(hv iandhv i)aredefinedsimilarly.
C D
meanvalues,withnarrowdispersion.Onlywhenthemeanval-
Besides,wealsomeasuretheaveragevelocityofeachvehicle
uesofcooperatorsanddefectorsareveryclose (asit happens
astheratioofthenumberofstepsitmakesduringthesimula-
whenthedensityishigh)thedistributionsshowsomeoverlap
tion,tothetotalduration.Fromthisensemblewecomputethe
asweseeintheinsertoftheFigure3.
distributionsP(v )andP(v ),characterizingthemovementof
C D
thetwoclassesofdrivers.
3.2 Imitation dynamics
3.1 Density dependence Theresultspresentedintheprevioussectionprovideafirststep
inouranalysisofthedynamicsofcooperationanddefectionin
Figure 2 shows the average velocities as a function of vehi- thesimpletrafficmodel.Wewillnowintroducethekeyingre-
cle density d in a 20×20 square lattice. Two sets of curves dient of strategy imitation. Let us suppose that there are two
areshown,correspondingtodifferentstrategycompositionsof idiosyncrasiesinthedrivers.Someofthemadopttheirdriving
the system. The lowest three curves correspond to a system strategynotbecauseofconvictionorbeliefintheconvenience
4 G.H.Paissan,G.Abramson:Imitationdynamicsinagameoftraffic
thoseinabigcitydonot.Inthisspiritwehavemadet depend
0.20 0.20 onthesizeofthecity,asL2 simulationsteps.
P(v )D 0.16 =0.2 P(v ), P(v )CD000...011505 =0.8 edreseatfaceTbhclhetiosesrhsnae.nusImnaedeqbreuiapiclleaaibnlnrdcrieeeusnmubtlleytitsnwosefhtheotnehwetchtohienmainttpiutoahmlseibctdioeoyrnnndoaioftmficoiosnctossrpa,otetefhrgaiemiteossiryt.sasTttaienohmdne
), C 0.12 0.00 0.020 0.022 0.024 0.026 arevaelriazgaetionnusm,bweirthofthceooppaeraramtoertserasnwdedeufseecdto,rasreat5t6h%eeannddof44th%e
(v <vS> respectively. Moreover, these same values of equilibrium are
P 0.08 observed even in extreme cases when a single cooperator or
defector is introduced at the beginning of the simulations in
populationsinwhichallotherdrivershavetheoppositestrategy
0.04
(besidesfortuitousextinctionsofoneofthepopulationswhen
itisverysmall,duetostochasticfluctuations).
0.00 Thisapproachtoequilibriumcanbeunderstoodinamean
0.08 0.10 0.12 0.14 0.16 fieldapproximationbyanalyzingamasterequationfortheprob-
<v > abilitydensities.Considerthefollowingequationfortheprob-
S
Fig. 3. Distribution of the velocity of cooperators (full curve) and abilitydensityofcooperators:
defectors(dotted curve) fordensities d =0.2inthemainpanel and
d =0.8intheinsert.Thecostsarea=3,b=100.Dataareaveraged dr C =W(C|D)r −W(D|C)r , (5)
D C
over100realizations. dt
where W(S|S′) is the transition probability for a driver with
ofrespectingthetrafficrule,butbyimitationofotherdrivers. strategy S′ to switch to S, and r C and r D = 1−r C are the
Itisacommonhabitinmanysocietieswithlittleenforcement probabilitydensitiesofcooperatorsanddefectorsrespectively
of the rules and a low level of educationof them: “Do as the attimet.
otherdo.”Besidesthese“imitators,”acoreofdriversmaywell Giventhelackofdiscriminationofcooperatorsdrivingon
bebonafidecooperators,representingasmallgroupofdrivers the left at intersections, in a well mixed approximation their
that have been well educated in the traffic rules, and are con- probability to switch to defection is W(D|C) ≈1/2. On the
vinced of their application. These “core cooperators” do not otherside,defectorsbuildupaprobabilityofswitchingto co-
imitatethestrategyofotherdrivers,andalwayscooperate. operation based on their encounters with cooperators, so we
Weimplementthisimitationdynamicsinthefollowingway. have W(C|D) = r C. With these, Eq. (5) becomes a logistic
Imitating driverscan change their strategiesbased on the fre- equationoftheform:
quencyofencounterswithdriversthatdisplayoneortheother
dr
strategy. Since it is unreasonable that drivers change strategy C =r (1/2−r ). (6)
C C
too often, the imitation takes place every t simulation steps, dt
based on the frequency of interactions during the previous t ThestableequilibriumsolutionofEq.(6)isr ∗=1/2,closeto
C
steps.Thefractionofencounterswitheitherstrategydefinethe theobservedstationaryvalue.Incidentaly,observethata par-
individualprobabilitiesofimitation: tialdiscriminationofthecooperators,ofafractione <1ofthe
defectorsintheirencounters,canbetakenintoaccountwitha
f
P = C , (1) transitionprobabilityoftheformW(D|C)=er +(1−e )/2.
C D
f +f
C D It is easy to show that the ensuing logistic equation also has
PD=1−pC, (2) r ∗=1/2asonlyattractor.Thetimescaleoftheevolution,how-
C
ever,becomesdependentone as2(1−e )−1.
where f is the number of interactions with strategy s during
the prevsioust steps. Observe that the left driverparticipating Besides the imitators, we set a fraction fCC of drivers to
formacoreofcooperators.Theyrespectthetrafficrulesstrictly,
in encountersof type (i) and (ii)—a cooperator—cannotinfer
and we intend to study their effectin the dynamics.Afterthe
thestrategyofthe rightdriver.Theyalwaysgiveway,regard-
systemhasreachedastationarystatetheyarerandomlychosen,
less of whether the right driver is a cooperator or a defector.
setascooperators,andwillnotchangetheirstrategyforthere-
In these situations, the cooperator adds 0.5 both to f as f .
C D mainingcourseoftherun.Theyrepresentasetofdriverswho
In any othertype of encountereach driveris able to infer the
havebeeneducatedtoabidebythelaw,disregardingthebehav-
strategyusedbytheotherdriver,andwilladd1to f orto f
C D ioroftherestofthesystem.Weareinterestedinthereactionof
asappropriate.Based onthedefinedprobabilities,drivers can
thesystemasawhole,asaresultoftheimitationofstrategies,
changetheirstrategybyimitatingthemajority,iftheyplaythe
anditsdependenceonthesizeofthiscoreofcooperation.
otherstrategy:
Asaglobalmeasureoftheinfluenceofthecorewedefine
C→D withprobabilityP , (3) acoefficient:
D v −v
D→C withprobabilityP . (4) D v= a b, (7)
C v
b
Itisreasonabletoassumethatdriversinasmalltowntend where v and v are respectively the average system veloci-
b a
to interact with the same drivers over and over again, while ties calculated before and after the establishment of the core.
G.H.Paissan,G.Abramson:Imitationdynamicsinagameoftraffic 5
Figure 4 shows the effectof the core size on D v fordifferent
vehicle densities. It is clear that the existence of the core of 2.4
cooperatorsrepresentsa benefit for the system as a whole, in b=100
termsofthe velocityof thetraffic flow.Moreover,the growth 2.0 b=50
ofD vismonotonous,withoutanyindicationofatransitionfor b=10
b=4
some critical core size. This behavioris verygeneral,and we 1.6
havecheckeditforawiderangeofmodelparameters. v
Themeaningofthisresultmaybeimportantforthedesign 1.2
of education plans. It implies that not only it is necessary to
educate and convince as many drivers as possible of the im-
0.8
portanceoftherespectoftherules,butalsothattheimitation
ofbehaviordoesnotproduceacollectivecriticalturnoverofa
0.4
defectivesystem.
Observe,also,thattheactualvalueofthegrowthofperfor-
0.0
mancemaybesometimesdubious,since thegrowthofveloc-
0.0 0.2 0.4 0.6 0.8 1.0
itycanbemarginalinsomesituations,particularlyforlowand
high densities of traffic. This is an indication that other mea-
Fig. 5. Coefficient D v as a function of vehicle density, for different
suresarenecessarytoachievespecificgoalsoftrafficfluidity,
costsof the interactionbetween defectors. Theother parametersare
besidestheeducationofacoreofdrivers.
a=3and fCC=0.5.Errorbarsrepresentthemeansquaredeviation
ofthedatain100realizations.

4
haviors are possible: drivers either respect or ignore the rule

thatgivespriorityto thevehicledrivingontheright.Thetwo
0.2
classicalstrategiesoftwo-strategiesformalgames—cooperation
3 0.35
and defection—areimmediatelyassociated with those. In our
0.5
framework,pairsofdriverscaninteractatcrossingswhereeach
v 0.65
vehiclepayatemporalcostaccordingtoitsstrategy.
2 0.80
We studied first the average velocity achieved in systems
0.95
withdifferentandfixedproportionsofcooperatorsanddefec-
tors.We foundthatthevelocitydecreasesmonotonicallywith
1
density, driving away the possibility of a sharp transition at
anytrafficdensity.Evenforhighvehicledensities,theaverage
speedofcooperatorsisgreaterthanthatofdefectors,indepen-
0 dently of what strategy is a majority in the population. This
0.0 0.2 0.4 0.6 0.8 1.0
seems to imply that cooperatorsalways benefit when abiding
f
CC by the law. However, in a population essentially cooperative,
Fig.4.CoefficientD v,measuringtheincreaseofthesystemvelocity
defectors can exploit this situation for their own benefit, ob-
aftertheintroductionofacoreofcooperators,versustherelativesize
taininghigherspeedsthanthoseachievedwhenthepopulation
ofthecoreinthepopulation.Thecurvesshowthebehaviorfordiffer-
isessentiallydefective.
entvaluesofthevehicledensityd ,asshowninthelegend.Theinitial
Additionally,we introduceda dynamic of imitation. Each
fractionofcooperatorsinthesystemis25%inallcases.Thecostpa-
driver—at a slower time scale than the update of the state of
rametersarea=3,b=100.Dataareaveragedover100realizations.
thesystem—canchangetheirstrategywithaprobabilitycalcu-
latedfromtheirperceptionofthestrategyofthosetheymeetat
AsafinalcommentonFig.4,observethattheincreaseof thecrossroads.Thisimitationrepresentstheattitudeofdrivers
thevelocityisnotmonotonousinthedensity:thegreatestval- who have not received a proper education in traffic rules; in-
uesofD vcorrespondtointermediatevaluesofd .Toshowthis stead, they try to adjust their behavior to that of the rest of
dependenceweplot,inFig.5,D vasafunctionofd forasin- thesystem.We haveexploredtheeffectofestablishingacore
gle value of the core size, f =0.5. The plot shows that the of law-abiding drivers in this environment, in the form of a
CC
increaseoftheaveragevelocitiesofthesystemaregreaternear sub-populationofcooperatorsthatdonotchangetheirbehav-
andinexcessof0.5occupancyofthelattice.Moreover,greater iorduringthedynamicalevolution.
penaltieson defectordriversentail a substantial improvement The existence of a core of cooperators represent a bene-
oftheseaveragespeedsinthesystem. fit for the whole system, in the sense that higher speeds are
achieved.This benefitis gradualand continuousin its depen-
denceonthesizeofthecoreset.Thatis,thecombinationofthe
4 Conclusions core togetherwith the emulation attitude providedby the dy-
namicofimitation,doesnotproduceacriticalturnoverofade-
Westudiedacellularautomatamodelofvehiculartrafficbased fectingsystem.Thisisapointtotakeintoconsiderationinthe
ongametheory.Oursimplemodelassumesthatonlytwo be- formulation of driver’s education programs (even for drivers
6 G.H.Paissan,G.Abramson:Imitationdynamicsinagameoftraffic
thatalreadyhaveapermitorlicense)orroadsafetycampaigns.
Remarkably, this benefit reaches a maximum at intermediate
densitiesoftraffic,anditisgreaterwhenthepenaltiesimpose
highercosts.Theresultsareverygeneralinthesimple model
reported in the present work, which tries to capture a set of
minimal important details of the actual system. They will be
furtherexploredwith the additionof more complexrulesand
interactionsinfutureinvestigations.Moreover,someofthedy-
namicalfeaturesobservedinthemodelcouldbeconsideredas
suggestionsforfieldobservations.
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