Table Of ContentEvolutionary prisoner’s dilemmagamewithdynamicpreferential selection
Zhi-Xi Wu1, Xin-Jian Xu2, Zi-Gang Huang1, Sheng-JunWang1, and Ying-Hai Wang1
1Instituteof Theoretical Physics, Lanzhou University, Lanzhou Gansu 730000, China
2DepartmentofElectronicEngineering, CityUniversityofHongKong, Kowloon,HongKongSAR,China
(Dated:February2,2008)
Westudyamodifiedprisoner’sdilemmagametakingplaceontwo-dimensional disorderedsquarelattices.
The players are pure strategists and can either cooperate or defect with their immediate neighbors. In the
generations each player update itsstrategy by following one of the neighboring strategies with a probability
7
dependent on thepayoff difference. Theneighbor selectionobeys adynamic preferential rule, i.e., themore
0
frequentlyaneighbor’sstrategywasadoptedbythefocalplayerinthepreviousrounds,thelargerprobabilityit
0
willbechosentorefertointhesubsequentrounds.Itisfoundthatcooperationissubstantiallypromoteddueto
2
thissimpleselectionmechanism.Correspondinganalysisisprovidedbytheinvestigationsofthedistributionof
n players’impactweights,persistence,andaswellascorrelationfunction.
a
J
PACSnumbers:02.50.Le,05.50.+q,87.23.Cc,89.65.-s
8
] I. INTRODUCTION of one of the neighboring strategies is allowed with a prob-
h
abilitydependenton the payoffdifference. Thisrevisedver-
p
siontookintoaccounttheirrationalchoicesoftheplayers. It
- Game theory and its evolutionary context are efficient
c was observed that a stable absorbing state of all C emerged
frameworkstostudycomplexbehaviorsofbiological,ecolog-
o
whenthetemptationtodefectisbelowacertaincriticalvalue
s ical, social and economic systems [1, 2, 3, 4]. Of particular
. renown is the evolutionary prisoner’s dilemma game (PDG) (noise-dependent). Vainstein and Arenzon studied the PDG
s
on a diluted lattice [14] and found that cooperation is eas-
c whichhasattractedmuchattentionintheoreticalandexperi-
si mentalstudies[3,4,5,6,7]. IntheoriginalPDG,playerscan ier to maintain due to blocking the spreading of defection
y maketwochoices:eithertocooperatewiththeirco-playersor caused by the empty sites on the lattice. The investigations
h to defect. Theyare offeredsome payoffsdependenton their of the PDG on random graphs have shown that cooperation
p is strongly enhanced for lattices with fluctuating connectiv-
choices,whichcanbeexpressedbya 2×2payoffmatrixin
[ ity in comparisonwith fixed connectivitylattice [15]. In the
agreementwiththefourpossibilities. Theplayersgetrewards
past few years, the PDG has been studied on different so-
3 R(orpunishmentP)ifbothchoosetocooperate(ordefect).
v Ifoneplayercooperateswhiletheotherdefects,thentheco- cial network models. It was found that cooperation can be
4 operator(C)getsthelowestpayoffS(sucker’spayoff),while maintainedonthesenetworksinawiderangeofnetworkpa-
7 rameters [16, 17, 18, 19, 20, 21, 22]. In addition, several
the defector (D) gains the highest payoff T (the temptation
1 mechanismswere also introducedto sustain high concentra-
todefect). Thustheelementsofthepayoffmatrixsatisfythe
4
tion of cooperators,such as voluntaryparticipation[21], dy-
0 conditions: T > R > P > S and2R > T +S,sothatlead
namicnetworkmodel[23]anddynamicpayoffmatrices[24],
5 toaso-calleddilemmasituationwheremutualcooperationis
andso on. Very recently,Santoset al. studied the PDG and
0 beneficial in a long perspective but egoism can produce big
/ short-termprofits. snowdrift games on scale-free networks. Their results indi-
s
catedthattheheterogeneityofnetworksfavorstheemergence
c OneofthemostinterestingitemsonthePDGistostudyun-
i ofcooperation[25].
s derwhatconditionsmutualcooperationwillemergeandsus-
y tain stably or how to facilitate the cooperation of the whole Though many intriguing fruits have been obtained in un-
h population[2, 3, 4]. In the evolutionary PDG, the state that derstanding the emergence of cooperation in spatial PDG
p all playersare defectorshas been provedto be an evolution- [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22],
:
v ary stable state [4], which has inspired numerous investiga- it would also be interesting to explore other mechanisms to
i tionsofsuitableextensionsthatenablecooperationtopersist. enhance the cooperative behavior. In the present work, we
X
Nowak and May [8] introduced a spatial evolutionary PDG make further studies of the evolutionary PDG using Szabo´-
ar modelinwhichindividualslocatedonalatticeplaywiththeir To˝keversion[13]ontwo-dimensionaldisorderedsquarelat-
neighbors and with themselves. The dynamics is governed tices. Considering individuals heterogeneously affected by
byadeterministicrule:ineachsubsequentround,individuals theirneighborsinsociety,weintroduceimpactweighttoeach
adoptthestrategythathasgainedthehighestpayoffamongits one of the interacting players. Based on the player’s impact
neighbors(includingalso themselves)in the previousround. weights, a dynamic preferential selection (DPS) mechanism
Ithasbeenshownthatthespatialeffectspromotesubstantially isincorporatedintothespatialPDGtomodeldynamicbehav-
the survival of cooperators[8, 9, 10, 11, 12]. The enhance- iors of human communities. It will be shown that the DPS
mentof cooperationon lattice-like spatial structure is robust promotessubstantiallythecooperativebehavioroftheplayers
evenifsomespecifieddistanceinteractionsareallowed[11]. which may provide a new perspective in understanding the
Szabo´ and To˝ke extendedthe deterministic dynamicsto a persistenceofcooperationinrealisticworld.
stochastic evolutionary one [13]: rather than following the In the nextsection, evolutionaryrulesof the gameare ex-
most successful neighbor’s strategy straightly, the adoption plained. In Sec. III, simulation results of our model imple-
2
mented on disordered lattices are presented and, in Sec. IV, whereκcharacterizesthenoiseintroducedtopermitirrational
theeffectsoftheDPSontheevolutionofcooperationisstud- choices. Itshouldbenotedthatthetransformationisonlyaf-
iedindetail. Conclusionsaregiveninthelastsection. fectedbytheirpayoffdifference.Wesetthevalueofκto0.1,
similartothatinRefs. [13,21,22,26]. Generatingarandom
numberr fromauniformdistributionintheinterval(0,1),if
II. THEMODEL r < W (or r > W), the player j’s strategy is accepted (or
discarded)by the playeri andA (t) is revised accordingto
ij
We consider the evolutionary PDG with players located thefollowingrule
on the two-dimensional disordered square lattice which is
A (t+1)=A (t)(1±α), (3)
characterizedby a total φ portion of randomlyrewired links ij ij
with fixed number of neighbors of each site (in the limit
wheretheminuscorrespondstothecaseofr > W,denoting
φ → 1.0, the “disordered lattice”is in fact a regular random
nostrategyupdatingfortheithplayer.Initially,allA (0)are
ij
graph[21,26]). Thisapproachexcludesthoseeffectscoming
assignedto1.0. Theparameterα inEq. (3)can bedepicted
fromthefluctuationofthenumberofneighborsasithappens
asamultiplicationfactorwhichcharacterizesqualitativelythe
on diluted lattices [14], random graphs [15], and scale-free
relativechangeoftheimpactweightineachround. Largerα
networks [25]. Similar consideration has also been made in
correspondstostrongerstrengthening(orlosing)ofimpact.In
Refs. [22, 26]. The players are pure strategists and can fol-
subsequentinvestigations,thevalueofαisconstrainedinthe
lowonlytwosimplestrategies:C (cooperate)andD(defect).
range(0,1)to ensureno negativeelementsin A (t), which
ij
Each player plays a PDG with itself and with its neighbors,
meansthatallopinions(CandDinthepresentcase)haveop-
andcollectspayoffsdependentonthepayoff-matrixparame-
portunitiestobefollowed. Ifstrategiesoftheplayersspread
ters. Thetotalpayoffsofa certainplayeristhe sumoverall
successfully,theycanbecalledas“winners”regardlessofthe
interactions. Followingcommonpractices[8,13,16,18],the
payoffs(forexample,duetothenoise,theworseperformance
elementsof the payoffmatrix can be rescaled, i.e., choosing
strategyalsohasacertainprobabilitytotakeoverabetterper-
R = 1,P = S = 0,andT = b(1.0 < b < 2.0)torepresent
formance one). Thus, this rule (Eq. (3)) could be termed
theadvantageofdefectorsovercooperators.Wehavechecked
as: “win-strengthen, lose-weaken”. The more frequently a
thatthepropertiesofthesimulationresultsdonotchangefor
player’sstrategyfollowedbyits neighbors,the largerproba-
S =−ǫ(0.0<ǫ≪1.0).
bilityitwillbepickeduptorefertointhesubsequentgenera-
Insociety,somespecialpersonsmayinfluenceothersmuch
tions,andviceversa.
strongerthantheaverageindividual[18]. Inotherwords,dif-
ferent neighbors would have different impacts on one’s be-
havior. Ingeneral,onecanexpectthattheinfluencebetween
III. SIMULATIONRESULTS
twoindividualswouldbeasymmetricandtime-dependent.To
model this situation, we define a quantity A (t), which de-
ij Inwhatfollowsthreegroupsofsystemswillbeconsidered:
scribestheimpactweightofthejthplayerontheithplayerat
(i) regularsquare lattices (φ = 0.0 with α = 0.0, 0.01, 0.1,
time t andpossesses the asymmetricproperty,i.e., A (t) 6=
ij and0.3, respectively);(ii)weakdisorderedlattices(φ = 0.1
A (t). Inthisway,wehopetocatchsomegeneraleffectsof
ji with the same values of α as before); (iii) strong disordered
thedynamicasymmetricinfluenceamonginteractingplayers
lattices (φ = 1.0 with the same values of α as the former).
onthedynamicalbehaviorofthegame.
Ontheonehand(asabyproduct),wewanttounderstandhow
Aftereachroundofthegame,playersareallowedtoinspect
theunderlyingdisorderedstructuresaffecttheevolutionofthe
theirneighbors’payoffsandchangetheirstrategiesinthenext
PDG;onthe otherhand(asthemainaim),we trytoexplore
round according to the comparisons. The randomly chosen
the potentiality of the DPS (in combination with the disor-
playerirevisesitsstrategybyselectingoneofitsneighborsj
deredstructures)ontheevolutionofcooperation.
withaprobabilityγintermsofapreferentialselectionrule:
Allthesimulationsareperformedinsystemswith300×300
Aij(t) players. Startingfroma randominitialstate with equalfrac-
γ = , (1)
ij P A (t) tionofC andD, we iteratethe modelwith synchronousup-
k∈Ωi ik
date. Theevolutioneventuallyleadsto dynamicequilibrium
where Ω is the communitycomposingof the nearestneigh-
i stateswithsmallfluctuationsofthecooperatordensityaround
bors of i. Eq. (1) means that the stronger the impact of
anaveragevalue.Theparameterα,whichdeterminestheevo-
a neighbor is, the larger probability it is selected to com-
lutionofimpactweights,isnotcritical,andconvergencehas
parewith. Ifandonlyif theirstrategiesare different,the ith
beenverifiedforallvaluesofαbetween0and1.Akeyquan-
player’sstateaswellastheneighbor’simpactweightwillbe
tityisthedensityofcooperators,ρ ,whichisdefinedasthe
C
updated,otherwisenothinghappens(nostrategytransforma-
averagefractionofplayersadoptingthestrategyCintheequi-
tionandimpactweightupdate).Followingideasdevelopedby libriumstate.Thetotalsamplingtimeswere105Monte-Carlo
Szabo´ etal. [13, 21, 22, 26], giventhetotalpayoffs(E and
i (MC) steps [27], and the equilibrium density of cooperators
Ej fortheithandjthplayers,respectively)fromtheprevious wasobtainedbyaveragingoverthelast104 steps. Eachdata
round,theplayeriadoptsj’sstrategywiththeprobability
point(inFigs. 1and2)resultsfromanaverageover10real-
1 izations for the same type of disorderedlattices specified by
W = , (2)
1+exp[−(E −E )/κ] theparameterφ.
j i
3
Thedependenceofρ onbintheequilibriumstatefordif-
C
ferent values of φ and α, are shown in Fig. 1. The main
featuresin the steady state are similar to the results reported 1.0
= 0.0
in Ref. [13], i.e., there exist two different absorbing states 0.8 = 0.0 0.01
(ρC = 1andρC = 0)separatedbytheactivephase(coexis- 0.6 0.1
tenceofCandD).Wehavefoundnumericallythatρ =1in 0.3
C 0.4
allthecasesweareinterestedinwhenb<5/4,whichcanbe
regardedasahomogeneouscooperationstate. Sinceourmain 0.2 (a)
1.0
aim goes beyond this trivial steady-state, we will only con-
= 0.1
centrate on the region of b > 5/4, where new features may 0.8
C
emerge. 0.6
First, we consider the model without DPS (α = 0.0). In 0.4
thecaseofφ = 0.0,whichcorrespondstotheregularsquare 0.2 (b)
lattice, we recover the results of the stochastic model [13], 1.0
i.e. ρC decreases monotonically with the increase of b and 0.8 = 1.0
vanishesatathresholdb . Whenlongrangelinksemergeon
c 0.6
thelattice (φ = 0.1),thelevelofcooperationispromoted(a
0.4
largerthresholdb inFig. 1(b)thanthatinFig. 1(a)),whichis
c
0.2 (c)
differentfrompreviousreportsthatlocalinteractionspromote
0.0
the sustainment of cooperation(see Refs. [8, 9, 10, 12] and
1.4 1.6 1.8 2.0
the references therein). Particularly, in the case of φ = 1.0
b
(correspondingtoaregularrandomgraph),wherethespatial
correlation is very weak, cooperationis enhanced further (if
FIG.1:(coloronline).Averagedensityofcooperators,ρc,asafunc-
it is just measured by the value of b ). Even in the case of
c tionofthetemptationtodefectbintheequilibriumstate. Filledand
b>2.0,minorfractionalcooperatorscanbefoundinaseaof opensymbolscorrespondtothecasesofrandomselectionandpref-
defectors. erentialselection,respectively.
Theseresultscan beunderstoodinthe followingway. On
the one hand, in our model, the state transformation of the
playersconformstoaFermifunction(seeEq. (2))suggested however,thetendencyisreversedandthefractionofcooper-
bySzabo´ etal. [13,21,22,26]. Usingthisupdaterule,they ators is somewhatlower than the correspondingvalue in the
have also found thatlong range interactionsenhancethe co- case of α = 0.0. There exists a cross pointfor b separating
operationwhenthe PDG (withnoself-interactionsincluded) thetworegionswhichdependsonα: forα = 0.01,b ≈ 1.5.
was performedon regularrandomgraphs[21]. On the other Fortheothertwovaluesofα(0.1and0.3),theDPSenhances
hand, self-interactionsare includedin the presentwork. Re- greatlythecooperationwithrespecttootherwise.Inthecases
cently,HauertandDoebelihavestudiedanotherfamousevo- ofdisorderedstructuralpopulations(φ=0.1and1.0),forall
lutionarygame, snowdriftgame, on differenttypesof lattice thevaluesofαconsideredhere,thecooperativebehaviorcan
[28]. They have found that the spatial structure eliminates besubstantiallypromoted(seeFigs. 1(b)and1(c))andmain-
cooperationforintermediateandhighcost-to-benefitratioof tainedevenintheextremedefectioncircumstance(b > 2.0),
cooperationbecausebenefitsofcostlycooperativeactsaccrue wherethegameisnotaproperPrisoner’sDilemma.Itisstrik-
not only to others but also to the cooperator itself [28, 29]. inglyinterestingthat,fordisorderedstructuralpopulations,all
As to our PDG model, each player C plays with itself be- thecurvesnearlycollapseintoasinglecurve(showninFigs.
sidesitsnearestneighbors,whichindicatesthatitwillgainat 1(b)and1(c))whentheDPSisintroducedintothegame.The
leastRpayoffevenintheworstcase(surroundingbyfourde- visibledifferenceexistsonlyintheregionwherecooperators
fectors). Inadifferentinterpretation,besidestheirneighbors, aregoingtoextinction.Thelargerαgivesrisetolargerthresh-
thecooperators’investmentwillbenefitthemselvestoo. Note old b . The data collapsing is also found in well structured
c
thatthehighcost-to-benefitratioofcooperationinsnowdrift populationsforlargeα(seeFig. 1(a)).
gamecorrespondstolargevaluesofbinthePDG.Consider- From these observed features, we argue that, whether the
ing these two factors, it is not surprising that the disordered underlying “lattice”is well structured or disordered, as soon
structurecanpromotecooperationinthepresentmodel. as the multiplication factor α is finite and sufficiently large,
We now focus our attention on the influence of the DPS the enhancementof cooperationwill be to some extentreal-
on the game evolution. The results obtained for α = ized. Despite ofthefactthatthe disorderedstructureprefers
0.01,0.1,0.3aresummarizedinFig. 1. Althoughqualitative tofavortheemergenceofcooperation,themuchstrongeren-
behaviors are similar to those of the random selection case, hancementofcooperativebehaviorforα > 0.0thanthatfor
therearesomeremarkablequantitativedifferences.Thesimu- α = 0.0 implies that the DPS playsa crucialrole in the dy-
lationresultsdependstronglyonboththeparametersφandα. namics.
Forwellstructuredpopulations(φ = 0,regularsquarelattice In Fig. 2, we show the plots of ρ as a functionof α for
C
case)withα=0.01,theDPSpromotescooperationforsmall threespecialvaluesofb,toemphasizethechangesinbehav-
bincomparisonwith therandomselectioncase. Forlargeb, iorsasthelatticevaries. Thecoloreddashedlinescorrespond
4
1.0 -1 -1
10 b = 1.48 10 b = 1.75
0.8 b = 1.48 (a) -2 = 0.0 -2 = 0.0
10 10
0.6 P(A) 10-3 1 0-3
0.4 -4 -4
10 10
0.2 10-5 (a) 10-5 (b)
1.0
-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15
C 00..68 b = 1.75 (b) 1100--21 (c) b == 01..19375 1100--21 (d) b == 12..00
0.4 P(A) 10-3 1 0-3
0.2 10-4 10-4 = 0.01
0.1
01..80 = 0.0 b = 1.9375 (c) 10-5-15 -10 -5 0 5 10 15 10-5-15 -10 -5 0 0.3 5 10 15
0.1
0.6 1.0 log10Aij(t) log10Aij(t)
0.4
0.2 FIG.3: (coloronline). Histograms(over105 timesteps)ofimpact
0.0 weightsofplayersintheequilibriumstate. Onlythedatawhichfall
0.01 0.1 intotheregion[10−16,1016]areshown.Thepower-lawdecayingbe-
haviordeterminedbytheintrinsicnatureofEq.(3)isvisibleinboth
theverylargelimitandtheverysmalllimitoftheimpactweights.
FIG. 2: (color online). Average density of cooperators, ρc, as a
function of the multiplication factor, α, for three special values of
b(coloredsymbols). Thecoloredlinescorrespondtorandomselec- forthecaseof0.1 < φ < 1.0. Thisproblemshoulddeserve
tioncases(α = 0.0): dashed(black)forφ = 0.0,dotted(red)for tomakefurtherinvestigationsinthefuturework.
φ=0.1,anddash-dotted(blue)forφ=1.0,respectively. To end up this section, let us review the novelfeatures of
the results obtained by computer simulations. When intro-
ducingtheDPStotheevolutionaryPDG,wehavefoundthat
to the results obtainedfor the randomselection case and the thecooperativebehaviorinthepopulationsissubstantiallyen-
symbols for the DPS case. Different colors denote different hanced with respect to the random selection case. The en-
disordered degree of the lattice: black for φ = 0.0, red for hancementis robustagainstwhethertheunderlyinglattice is
φ = 0.1 and blue for φ = 1.0, respectively. Note that only wellstructuredordisordered.Thoughtheevolutionaryresults
the curvesfor φ = 0.0 have a clear sensitivity to the values dependon both φ and α, as long as the underlyinglattice is
of α (black squares in Figs. 2(a) and 2(b)). The temptation finitelydisorderedandthemultiplicationfactorαisalsosuf-
todefect,b, hasalsoaninfluenceontheresults. Inthecases ficiently large (e.g., α > 0.003 in the present studied cases
ofsmallα withφ = 0.0, forlow temptation(b = 1.48),the [31]),theresultingcooperationlevelseemsinsensitivetoboth
cooperationlevelisslightlyhigherfortheDPScasethanthat φandαforfixedvaluesofb.
for the random selection case; whereas for high temptation
(b=1.75),thephenomenonisreversed.TheSimilardiscrep-
ancycanalsobefoundforlargevaluesofαwithφ=0.0:the IV. ANALYSISANDDISCUSSION
spatial structure promotes slightly the cooperation level for
b=1.48(seeFig.2(a)),andyetinhibitsthatforb=1.75(see Before discussing the further details as to why the above
Fig. 2(b)). Thetransitionpointseemstobearoundα≈0.06. evolutionaryresultsmayoccur,webrieflyreviewthoseresults
Sinceregularlatticesarenotrealisticrepresentationsofmost obtainedfor the evolutionaryPDG performedon the regular
actualpopulationstructures[30], especially human(because lattice. Generally, we are more interested in the region near
ofthemobilityanddispersalabilityoftheindividuals),wedo the extinction threshold of cooperators, from which we can
notcommentfurtheronthesetrivialresultsandfocusourat- figureoutclearlycooperators’evolutionandsurvivinginthe
tentiononthoseissueswheredisorderedstructuresarepresent sea of defectors. For this issue, Hauert et al. [21, 28] have
(φ=0.1and1.0). foundthatcooperatorscansurvivebyforminglarge,compact
Now taking a look at Figs. 2(a), (b) and (c), we clearly clusterswhichminimizetheexploitationbydefectors. Along
observe that for any finite values of α we are interested in, the boundary, cooperators can outweigh their losses against
givenfixedvaluesofbandφ, thecooperationlevelisalmost defectorsbygainsfrominteractionswithinthecluster. Simi-
independentof α, and the relative deviation of the results is larphenomenawereobservedearlierbySzabo´ andTo˝ke[13].
lessthan5%.Despiteofthispoint,wewouldliketopointout The compact clusters composing of cooperators do random
thatthe enhancementof cooperationappearsalso insensitive walksandspreadoutinthebackgroundofdefectors.Though,
tothespecialvaluesofφ,i.e.,moredisorderedstructureshave occasionally a cluster splits into two or two clusters collide,
little essential influence on the evolutionary results, at least, merge, or annihilate and vanish [21], the compactstructures
5
00..1102 b === 010...0705 001...680 P(t,tw) 00..0180 b === 010...07015 001...680 00..0180 b === 201...000 001...680
P(t, t)w 000...000468 000...024 QC(t, tw) 00..0046 000...024 00..0046 000...024
(t, t), Cw 00..00021 (a) 10 100 PC1(0t0, 0tw) 00..00021 (b) 10 100 1000 00..00021 (c) 10 100 1000
Q
P(t, t), Cw 000...456 t - tw 00..45 t - tw 0.3 t - tw
0.3 b = 1.75 1.00 0.3 b = 1.75 1.0 0.2 b = 2.0 1.00
= 0.0 0.95 = 0.1 0.9 = 1.0 0.95
0.2 = 0.1 0.90 0.2 = 0.01 0.8 0.1 = 0.3 0.90
0.1 (d) 0.85 0.1 (e) 0.7 (f) 0.85
0.0 0.0 0.0
1 10 100 1000 1 10 100 1000 1 10 100 1000
t - tw t - tw t - tw
FIG.4:(coloronline).PersistencefunctionPc(t,tw)andcorrelationfunctionQc(t,tw)fordifferentcombinationsofb,φandαasafunction
of timeinterval t−tw where tw = 95000. Inset: timeevolution of thepersistence function P(t,tw) for both strategiesC and D. That
thepersistencePc(t,tw)goestozerointhelongtimelimitindicatestherandomwalkandannihilationofcooperators,whereasthenon-zero
plateauofitimpliesthestablemaintenanceofthecommunitiesofthecooperators. TheplotsofQc(t,tw)alsogivethesamehints, which
sustainsthelevelofthedatumtimetw forthestablemaintenancecase,anddisplaysthemean-fieldbehaviorfortherandomwalkcase. See
thetextfordetails.
ofthesurvivingclusterscanremaininprincipleunchanged. alargeenoughfractionofinfluentialco-players(beingcoop-
erators)toformstablecoreinpopulations.
Theproblemcanbeviewedmoreexplicitly:whentheplay-
ers’ strategy update is implementedby randomlyselecting a InFigure3, we plotprobabilitydistributionsofA (t)for
ij
neighborto comparewith accordingto Eq. (3), the compact fourdifferentcombinationsofbandφandwithcorresponding
structuresfavorthoseinternalcooperatorstoformingastable threevaluesofα: b = 1.48, φ = 0.0 (Fig. 3(a)); b = 1.75,
core, since the componentsinteractonly with players taking φ = 0.0 (Fig. 3(b)); b = 1.9375, φ = 0.1 (Fig. 3(c));
overstrategyC.Theexistenceofthiscoreenablesthosecoop- and b = 2.0, φ = 1.0 (Fig. 3(d)), respectively. As indi-
eratorsonsurfacetohaveenoughchannelstocollectpayoffs cated,thereareverylargebroaddistributionsofA (t)except
ij
sothattheycanresisttheinvadingofdefectors,whichinre- for two combinations, (b = 1.48, φ = 0.0, α = 0.01) and
turnreinforcethestabilityofthecore. Themutualprotection (b = 1.75,φ = 0.0,α = 0.01). Thesetwocombinationsdo
(orreciprocity)ofthecooperatorsenablesthecompactstruc- notfavortheemergenceofcooperation(seeFig. 1(a)forthe
turestosustainstably,andhencecontributestothepersistence symbolsofredcircles)incontrasttoothercombinations,and
ofcooperation. also do not provide sufficient conditions for players to form
stable clusters which can be seen in the following part. It is
Motivatedbytheaboveinterpretations,someconjectureon
worthy to point out that the combinations of well structured
the origin of those results of ourmodelwith the presenceof
populations with sufficiently large α values (Figs. 3(a) and
DPSdepictedinSec. IIImaybeappropriatehere. We think
3(b))ordisorderedpopulationswithfiniteαvalues(Figs.3(c)
thattheDPScouldinducetheemergenceofpairsofinfluential
and3(d))resultinginbroadhistogramsofA (t)indeedfacil-
co-players,andwhensomeofthemarecooperators,theywill ij
itate the cooperative behavior compared with corresponding
“always”refer to their influential neighbors’ strategies in the
randomselection cases(α = 0.0), whichhave beenverified
subsequentgenerations,thencommunitiesconsistingoftheir
byprevioussimulations(seeFigs. 1and2).
neighborsandthemselvesmayformandsurvivestablyinthe
backgroundof defectors, which would contributeto the per- Toprovethatsomefractionalplayersmayformstableclus-
sistence of cooperation. Those influential co-players taking ters in the presence of the DPS in a more detailed way, we
overstrategyC canberegardedasthecoreofthecommuni- measured the persistence function P (t,t ), the fraction of
C w
ties(orclusters). Totestthisspeculation,wechecktheprob- cooperators that do not change strategy between an initial
abilitydistributionofA (t)intheequilibriumstate,whichis waiting time t , and the time t ≥ t [14]. In addition, the
ij w w
expectedtohaveabroaddistributionsothattherecouldarise correlationfunctionQ (t,t ), which characterizesthe frac-
C w
6
FIG.5: Rightpanel: illustrationofcooperatorsanddefectorsattimet = 9.5×104. Middlepanel: snapshotshowingplayersthatremain
unchangedbetweenMCtimet=9.5×104and105:thepinnedcooperators,defectors,andtheactiveplayersaredenotedbydarkgrayboxes,
lightgrayboxes,andwhitevacancy,respectively. Rightpanel: onlythepinnedcooperatorsareshown. Parameters: φ= 0.0,b= 1.75,α=
0.1.
tionofcooperatorsattimetthathavearisenattimet inspite small α (Fig. 4(b)), the random walks and annihilation of
w
offinitelymanytimesflippingofthestrategyduringthetwo cooperators causes the long time correlation to be indepen-
timeinterval,isalsoexplored.Inadistinctview,theresultsof dent of the initial state, which can be calculated roughly by
these two quantitiesas a functionofthetime intervalt−t amean-fieldmethod. Sincethespreadofcooperatorscanbe
w
(measuredinMCsteps,thetimet canbeselectedarbitrarily regardedas walking randomlyin such cases, the probability
w
aslongasthe systemshadattainedequilibrium)are summa- of revisiting those sites which had been visited before will
rizedinFig. 4onalog-linearscalefort =95000. be proportional to the average density of cooperators. As-
w
suming that the fraction of cooperators is equal to ρ(t ) at
For the cases of well structured populations with suffi- w
time t (the system has already attained equilibrium before
ciently largeα (Fig. 4(d))ordisorderedpopulationswith fi- w
this time), the mean-field approximation gives the relation,
niteα(Figs. 4(e)and4(f)),afteraninitialdecrease,theper-
Q (t,t ) ≈ ρ × ρ(t ), in the long time limit, t → ∞.
sistenceattains,forlargetimeintervals,aplateauwhosevalue C w C w
AsshowninFigs. 4(a),4(b)and4(c),theanalysisisinwell
dependsontheparametersb,φandγ.Interestingly,foranon-
agreementwithnumericalsimulations.
Hamiltonianmodelwestudiedhere,thedecayingbehaviorof
Foranintuitiveunderstandingofthesurvivingofcoopera-
thepersistenceintheequilibriumisexponentialatlargetime
tors,inFig. 5wepresentsnapshotsofthesystemintheequi-
intervals,justlikethathasbeenobservedcommonlyinHamil-
libriumstateundertheconditionofφ=0,b=1.75,α=0.1,
tonian models and in site diluted lattice model [32]. If the
showingbothactiveandpinnedplayers. Thesesnapshotsare
persistencedoesnotgoto zero, thereisa fractionof sitesof
a50×50portionofthefull300×300lattice.Cooperatorsand
cooperatorsthatfliponlyfinitelymanytimes(blocking),and
defectorsareshownindarkgrayandlightgrayboxes,respec-
domain wall movementsare constrained (pinning)[14]. For
tively.Thewhitevacancydenotesactiveplayers.Thistypical
thepresentsystemswestudied,itindicatesthatcommunities
configurationisalmoststableandthesmallamountofactive
ofcooperatorsexiststablyinthebackgroundofdefectors. In
sitesisconfinedtoafewregions.Itisshownthatmostcooper-
thecasesofrandomselection(nomatterwhetherthepopula-
atorclustersremainunchangedconfirmingpreviousanalysis.
tions are well structured or disordered, Figs. 4(a) and 4(c))
orwellstructuredpopulationswith verysmallα(Fig. 4(b)), At last, it should be pointed out that it is only qualita-
however, the persistence goes to zero in the long time inter- tivelydepictedthatthelargebroaddistributionsoftheimpact
val limit, which means that all the cooperators are renewed weightsfavor the formationof stable clusters. The quantita-
completely after finite waiting time. This is reminiscent of tive analysis on this issue, however, is unimplemented. The
therandomwalkandannihilation[13]. Thepersistencefunc- workalongthislineisinprogress.
tionP(t,t )forbothstrategiesC andDshownininsetsalso
w
displaysthesimilarcharacteristics.
ThetimedependenceofthecorrelationfunctionQ (t,t ) V. CONCLUSIONS
C w
also reflects the same evolvingbehaviorof the systems. For
well structured populations with sufficiently large α (Fig. Inthiswork,wehaveexploredthegeneralquestionofco-
4(d))or disorderedpopulationswith finite α (Figs. 4(e) and operationformationandsustainmentfromthe perspectiveof
4(f)),Q (t,t )fluctuatesweaklyaroundtheaveragefraction the coevolutionbetween the dynamicsof players’ states and
C w
of cooperators, ρ , indicating the stable maintenance of the theirinter-influentialrelationships. Byconsideringasymmet-
C
communities of cooperators. For the random selection case ricandheterogenousinfluentialeffectsinmanynaturalpopu-
(Figs. 4(a)and4(c))orwellstructuredpopulationswithvery lations,wedefinedimpactweightsforanypairsofneighbor-
7
ingindividuals,whichdescribestheinfluenceofoneplayeron tributetotheemergenceandpersistenceofcooperation. One
anotherandevolvestimely.Basedonthisquantity,adynamic interestingresultisthattheenhancementofcooperation,once
preferentialselection(DPS)mechanismwasintroducedtothe realized,isinsensitivetosomespecialvaluesofφandα. That
dynamics of an evolutionary PDG. To simulate the mobility is to say, the enhancementof cooperationin the presence of
anddispersalabilityofindividuals,thedisorderedstructureof the DPS is general and robust, especially for finitely disor-
populationswasalsotakenintoaccount. deredstructuralpopulations.Anotherintriguingresultisthat,
Althoughdisorderedstructurehasbeenprovedtofavorthe giventheappropriatemultiplicationfactorα,cooperatorscan
emergence of cooperation, it was found that the DPS plays alwayspersistinthewholerangeof1.0 < b < 2.0fordisor-
a crucial role in determining the evolutionary results of the deredstructuralpopulations,providinganescapehatchoutof
game. In fact, for well structured populations with strong statesofmutualdefectionpredictedbybothclassicalandevo-
multiplicationeffects(largeα)ordisorderedstructuralpopu- lutionarygametheory. Thus,theDPSiscapableofchanging
lationswiththepresenceoftheDPS,thecooperativebehavior the doomedfate of cooperatorsand ensuringthem to persist
can be substantially promoted. These findingsare presented even under harsh conditions when b → 2.0. Although it is
forsome specific sets of parametervaluesof b, φ andα, but simple, the presentmodeloffersan efficientway to simulate
they are qualitatively the same for a broad range of values. realsocialbehaviors,andmightshedlightsinunderstanding
ByanalyzingtheprobabilitydistributionofA (t)inthelong theevolutionofcooperationincomplexsocialsystemswhere
ij
timelimit,thepersistencefunctionP (t,t )andthecorrela- asymmetry,heterogeneity,andfeedbackareubiquitous.
C w
tionfunctionQ (t,t )forthecooperatorsintheequilibrium
C w
state, we found that the DPS gives rise to very large broad This work was supported by the Fundamental Research
distributions of the impact weights, which favor the influen- FundforPhysicsandMathematicsofLanzhouUniversityun-
tialcooperatorstoformstablecommunities,andtherebycon- derGrantNo. Lzu05008.
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