Social diversity and promotion of cooperation in the spatial prisoner's dilemma game

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by  Matjaz Perc

Socialdiversity andpromotion ofcooperationinthe spatialprisoner’s dilemma game
Matjazˇ Perc1 and Attila Szolnoki2
1Department of Physics, Faculty of Natural Sciences and Mathematics, University of
Maribor, Korosˇka cesta 160, SI-2000 Maribor, Slovenia
2ResearchInstituteforTechnicalPhysicsandMaterialsScience, P.O.Box49, H-1525Budapest, Hungary
Thediversityinwealthandsocialstatusispresentnotonlyamonghumans,butthroughouttheanimalworld.
Weaccountforthisobservationbygeneratingrandomvariablesthatdeterminethesocialdiversityofplayers
engagingintheprisoner’sdilemmagame.Herethetermsocialdiversityisusedtoaddressextrinsicfactorsthat
determinethemappingofgamepayoffstoindividualfitness.Thesefactorsmayincreaseordecreasethefitness
8 ofaplayerdependingonitslocationonthespatialgrid. Weconsiderdifferentdistributionsofextrinsicfactors
0 thatdeterminethesocialdiversityofplayers,andfindthatthepower-lawdistributionenablesthebestpromotion
0 ofcooperation. Thefacilitationofthecooperativestrategyreliesmostlyontheinhomogeneoussocialstateof
2 players,resultingintheformationofcooperativeclusterswhichareruledbysociallyhigh-rankingplayersthat
n areabletoprevail against thedefectors evenwhen thereisalargetemptationtodefect. Toconfirmthis, we
a alsostudytheimpactofspatiallycorrelatedsocialdiversityandfindthatcooperationdeterioratesasthespatial
J correlation length increases. Our results suggest that the distribution of wealth and social status might have
7 playedacrucialrolebytheevolutionofcooperationamongstegoisticindividuals.
1
PACSnumbers:02.50.Le,05.40.Ca,87.23.Ge

]
h
p I. INTRODUCTION oftheinteractionnetworkmightbothhaveabeneficialeffect
- ontheevolutionofcooperationintheprisoner’sdilemmaand
c
o thesnowdriftgame. Foracomprehensivereviewofthisfield
The prisoner’s dilemma game, consisting of cooperation
s ofresearchsee[14].
. anddefectionasthetwocompetingstrategies,isconsidereda
s Besidesstudiesaddressingnetworkcomplexityasasome-
c paradigmforstudyingtheemergenceofcooperationbetween
what direct extension of [4], several approaches have also
i selfish individuals [1]. Since the game promises a defect-
s been proposed that warrant the promotion of cooperation
y ing individual the highest income if facing a cooperator the
withinnearestneighborinteractions.Examplesincludestrate-
h prevalence of cooperation within this theoretical framework
p presents a formidable challenge. Indeed, the classical well- gic complexity [15], direct and indirect reciprocity [16],
[ mixedprisoner’sdilemmagamecompletelyfailstosustainco- asymmetry of learning and teaching activities [17], random

2 operation[2],whichisoftenatoddswithrealitywheremutual diffusionofagentsonthegrid[18], aswellasfine-tuningof
noise and uncertainties by strategy adoption [19]. The im-
v cooperation may also be the final outcome of the game [3].
pactofasymmetricinfluence,introducedviaaspecialplayer
6 A seminal theoretical mechanism for cooperationwithin the
4 prisoner’sdilemmagamewasintroducedbyNowakandMay with a finite density of directed random links to others, on
7 thedynamicsoftheprisoner’sdilemmagameonsmall-world
[4],whoshowedthatthespatialstructureandnearestneighbor
1 networks has also been studied [20]. It is worth noting that
interactionsenablecooperatorstoformclustersonthespatial
8. gridandsoprotectthemselvesagainstexploitationbydefec- stochasticity in general, either being introduced directly or
0 tors. Adecadelater,thistheoreticalpredictionhasbeencon- emerging spontaneously due to finite population sizes [21],
7 has been found crucial by several aspects of various evolu-
firmedbybiologicalexperiments[5]. Nonetheless,thesome-
0 tionaryprocesses.
what fragile ability of the spatial structure to support coop-
:
v eration[6], alongwith the difficultiesassociated with payoff Inthispaper,wewishtoextendthescopeofstochasticef-
Xi rankings in experimental and field work [7], has made it a fectsontheevolutionofcooperationinthespatialprisoner’s
commonstartingpointforfurtherrefinementsofcooperation dilemma game by introducingthe social diversity of players
r
a facilitatingmechanisms.Inparticular,thespecifictopologyof astheirextrinsicallydeterminedproperty. Thisisrealizedby
networksdefiningtheinteractionsamongplayershasrecently introducingscaling factors that determine the mapping from
received substantial attention [8], and specifically scale-free game payoffs to the fitness of each individual. The scaling
graphs[9]havebeenrecognizedasextremelypotentpromot- factorsrepresentextrinsicdifferencesamongstplayers,andas
ersofcooperativebehaviorintheprisoner’sdilemmaaswell such determine their social status and the overall social di-
as the snowdrift game [10]. Although the promotion of co- versity on the spatial grid. Importantly, the scaling factors
operation by the scale-free topology has been found robust aredrawnrandomlyfromdifferentdistributionsandaredeter-
onseveralfactors[11],themechanismhasrecentlybeencon- minedonlyonceatthebeginningofthegame.Positivescaling
testedviatheintroductionofnormalizedpayoffsorso-called factors increase the magnitudeof payoffsa particular player
participationcosts[12]. Moreover,theinterplaybetweenthe isable to exchange,while negativefactorshavethe opposite
evolutionofcooperationaswellasthatoftheinteractionnet- effect. It is important to note that the average of all scaling
work has also been studied [13], and it has been discovered factorsis exactlyzero, so thatthere is nonetcontributionof
thatintentionalrewiringinaccordancewiththepreferenceand socialdiversitytothetotalpayoffofthepopulation.Presently,
fitness of each individualas well as simple randomrewiring we consider scaling factors drawn from the uniform, expo-
2
nential,andscale-freedistribution,andthusdistinguishthree 1
differentcasesofsocialdiversity. Weinvestigatehowthein-
troductionofsocialdiversity,andinparticularitsdistribution
and amplitude, affect the evolution of cooperation amongst 0.5
playersonthespatialgrid. Wereportbelowthatthecoopera-
tionisenhancedmarkedlyastheamplitudeofsocialdiversity
increases,andmoreover,thatthebiggestenhancementisob- ξ 0
tainedifthesocialdiversityfollowsapower-lawdistribution.
(p) (e) (u)
Weattributetheenhancementofcooperationtotheemergence
ofcooperativeclusters,whicharecontrolledbyhigh-ranking -0.5
playersthatareabletoprevailagainstthedefectorsevenifthe
temptationtodefectislarge.Indeed,theroleoftheseclusters
-1
issimilartotheroleofpurecooperatorneighborhoodsaround
0 0.2 0.4 0.6 0.8 1
hubs on scale-free networks [22], or to the part of imitating
χ
followerssurroundingmaster playersin the enhanced teach-
ingactivitymodel[17],aswillbeclarifiedlater. Tohighlight
FIG.1: Distributionsofscalingfactors(ξ)originatingfromtheuni-
the importance of the strongly diverse social rank, we also
formlydistributedrandom variable (χ). Thethreefunctions corre-
study the impact of spatially correlated social diversity, and
spondtouniform(u),exponential(e),andpower-law(p)distribution
find that the facilitation of the cooperative strategy deterio- whena = 1. Foreasiercomparisononlythe[−1,1]intervalofξis
ratesasthecorrelationlengthincreases. Thus,afinitespatial plotted.
correlationofsocialdiversityhinderstheformationofstrong
cooperativeclusters,inturnlendingsupporttothevalidityof
thereportedcooperation-facilitatingmechanisminthespatial applicable since the introductionof social diversity via scal-
prisoner’sdilemmagame. ing factors only acts as a mapping of the original payoffsto
The remainderof thispaper is structuredas follows. Sec- individualfitness,aswillbedescribednext.
tion II is devoted to the description of the spatial prisoner’s Tointroducesocialdiversityweuserescaledpayoffsofthe
dilemmagameandthepropertiesofscalingfactorsdetermin- form X′ = X(1 + ξ), where X is either T, R, P or S,
ing socialdiversityof playersonthe spatialgrid. In Section ξ = min(ξi,ξj), and ξi is a scaling factor drawn randomly
IIIwepresenttheresultsofnumericalsimulations,andinSec- froma givendistributionfor eachparticipatingplayeri only
tionIVwesummarizetheresultsandoutlinesomebiological oncebeforethestartofthesimulation.Notethattheminimum
implicationsofourfindings. of the two randomscaling factors involvedat everyinstance
ofthegameisusedforpayoffsofbothinvolvedplayersiand
j toensurethattheprisoner’sdilemmapayoffrankingispre-
II. MODELDEFINITION served. Itisworthmentioningthatthemaximumofbothval-
ueswouldhavehad the same effect, butbelowpresentedre-
We consider an evolutionary two-strategy prisoner’s sultsarevirtuallyindependentofthistechnicality,andmore-
dilemma game with players located on vertices of a two- over, it seems reasonable to assume that the higher-ranking
dimensionalsquarelatticeofsizeL×Lwithperiodicbound- player (the one with the larger value of ξ) will adjust to the
ary conditions. Each individual is allowed to interact only weaker one since the latter simply cannot match the stakes
with its four nearest neighbors, and self-interactions are ex- ofthegameotherwise.Withinthisstudyweconsidertheuni-
cluded. A player located on the lattice site i can change its form(alsoknownasrectangular),exponential,andpower-law
strategy s after each full iteration cycle of the game. The distributedsocialdiversitydefinedbythefollowingfunctions:
i
performanceofplayeriiscomparedwiththatofarandomly
ξ =a (−2χ+1) (2)
chosenneighborjandtheprobabilitythatitsstrategychanges
tos isgivenby[23] ξ =a (−logχ−1) (3)
j
n
ξ =a (χ−1/n− ) where2≤n∈N. (4)
1 n−1
W(s ←s )= , (1)
i j
1+exp[(P −P )/K]
i j Here χ are uniformly distributed random numbers from the
1
where K = 0.1 characterizes the uncertainty related to the unitinterval,andR ξ(χ)dχ = 0 inallcases, sothattheav-
0
strategyadoptionprocess,servingtoavoidtrappedconditions erage of ξ over all the players is zero. It is easy to see that
andenablingsmoothtransitionstowardsstationarystates.The thelargestdifferencebetweentheexponentialandpower-law
payoffsP andP ofbothplayersacquiredduringeachitera- function can be obtained if n = 2 in the latter, as shown in
i j
tioncyclearecalculatedinaccordancewiththestandardpris- Fig.1. Henceforthweusen = 2whenthepower-lawdistri-
oner’sdilemmascheme[4],accordingtowhichthetemptation butionisapplied. Theparameteradeterminestheamplitude
todefectT = b > 1, rewardformutualcooperationR = 1, ofundulationofthescalingfactors,andhencethedispersion
punishmentformutualdefectionP =0,andthesucker’spay- of social diversity, and can occupy any value from the unit
off S = 0. Importantly, throughout this work the original interval. In particular, a = 0 returns the original payoffs,
prisoner’s dilemma payoff ranking (T > R > P > S) is whereasa = 1 isthe maximallyallowedvaluethatstillpre-
3
1
20
(p) 0
10
-1
0
20 1
ξi 10 (e) ξi 0

-1
0
20
1
(u)
10 0

0 -1
0 50 100 150 200 250 300 0 50 100 150 200 250 300
L L
FIG. 2: Spatial distributions of social diversity obtained for a = FIG. 3: Spatial distributions of social diversity constituted by ran-
1whenuniform(u), exponential (e), andpower-law(p)distributed domlydrawnscalingfactorsfromauniformdistributionbya = 1
valuesofscalingfactorswereused.Linesshowanexemplarycross- andspatialcorrelationlengthλ = 1(bottom),λ = 3(middle),and
sectionofthetwo-dimensionalgrid. λ = 7 (top). Lines show an exemplary cross-section of the two-
dimensionalgrid.
serves (1 + ξ ) ≥ 0 for all i, thus preventing possible vio-
i
lations of the prisoner’s dilemma payoff ranking. Note that To avoid finite-size effects, the simulations were carried out
(1+ξ ) < 0couldinduceR > T,whichwoulddirectlyvi- for a population of 300 × 300 players, and the equilibrium
i
olate the rules of the prisoner’s dilemma game. The spatial frequencies of cooperators were obtained by averaging over
distributionsofthethreeconsideredcasesobtainedbya = 1 104 iterations after a transient of 105 iteration cycles of the
are demonstrated in Fig. 2 where cross-sections with linear game. Thefiguresshowingvaluesofcooperatordensitieson
system size L = 300 are plotted. Clearly, the uniform dis- thespatialgrid(FC)resultedfromanaverageover30simula-
tribution provides the gentlest dispersion of social diversity, tionswithdifferentrealizationsofsocialdiversityasspecified
whereasthelargestsegregationofplayersiswarrantedbythe by the appropriateparameters. These simulation parameters
power-law distributed scaling factors where some values are yieldatleast±3%accuratevaluesofFC inallfigures.
very high at the expense of extended regions of very small
ξ resulting in these players having much smaller payoffs as
compared to the non-scaled case. The exponential distribu- III. NUMERICALRESULTS
tion of scaling factors yields a dispersion of social diversity
thatisbetweentheuniformandthepower-lawcase,ascanbe Next, we study how differentdistributionsof social diver-
inferredfromFig.2,aswellasindirectlyalsofromFig.1. sityaffecttheevolutionofcooperationvianumericalsimula-
In order to explain the main features of the reported re- tionsoftheabove-describedspatialprisoner’sdilemmagame.
sults, we also use spatially correlatedsocial diversityso that Figure4featurescolor-codedF forallthreetypesofsocial
C
thescalingfactorsξi satisfythecorrelationfunctionhξiξji∝ diversityintherelevantb−aparameterspace. Evidently,re-
exp(| i−j | /λ), where λ is the spatial correlation length. gardlessofthedistribution,increasingvaluesofaclearlypro-
In this case we constrain our study to the case where ξ are mote cooperation as the threshold of cooperation extinction
i
drawnfromauniformdistributionwithintheinterval[−1,1]. increasesfromb = 1.01in theabsenceofsocialdiversityto
An efficient algorithm for the generation of spatially corre- b = 1.20 (uniformly distributed social diversity), b = 1.28
latedrandomnumberswithaprescribedcorrelationlengthis (exponentially distributed social diversity), and b = 1.34
givenin[24].Theeffectofdifferentvaluesofλonthescaling (power-law distributed social diversity). Moreover, there al-
factorsdeterminingsocialdiversityisdemonstratedinFig.3. waysexistsabroadrangeintheparameterspacewithinwhich
Clearly,therandomundulationsbecomemoreandmorecor- cooperatorsrulecompletely;anon-existentfeatureifa=0.
relatedacrossneighborsonthespatialgridasλincreases. We argue that the above-reported facilitation of coopera-
Before the start of each game simulation, both strategies tionisduetotheinducedinhomogeneoussocialstateofplay-
populatethespatialgriduniformlyandthescalingfactorsare ers that fosters cooperative clusters around players with the
drawnrandomlyfromagivendistributiontodeterminetheso- largestvaluesof ξ . Indeed, assoon as cooperatorsovertake
i
cialdiversityofparticipatingplayers. Aftertheseinitialcon- theseprimespotsofthe gridtheystartto spreaddueto their
ditionsareset,thespatialprisoner’sdilemmagameisiterated cluster-formingnature.Notethatthelatterfeatureisnotasso-
forwardintimeusingasynchronousupdatescheme,thuslet- ciated with defectorswho thereforefail to take the same ad-
ting all individuals interact pairwise with their four nearest vantageoutofsocialdiversity,andarethusdefeated. Asimi-
neighbors. After every iteration cycle of the game, all play- larbehaviorunderliesalsothecooperation-facilitatingmech-
ers simultaneously update their strategy according to Eq. 1. anismreportedforthescale-freenetworks,wheretheplayers
4

1.4 1.4 1.4
1.3 1.3 1.3
b b b
1.2 1.2 1.2
1.1 1.1 1.1
1 1 1
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
a a a
FIG.4:Color-codedFC fortheuniform(left),exponential(middle),andpower-law(right)distributedsocialdiversityontheb−aparameter
space.Thecolorscaleislinear,whitedepicting0.0andblack1.0valuesofFC.
withthelargestconnectivitydominatethegame. Sincecoop- 1
erators are much better equippedfor permanentlysustaining uncorrelated
λ = 1
theoccupationofhubs,thescale-freenetworksprovideauni- λ = 2
0.8 λ = 4
fying framework for the evolution of cooperation [10]. An λ = 7
even better similarity can be established with the model in- a = 0
corporatingtheinhomogeneousteachingactivity[17]. Inthe 0.6
latter, some players are blocked and therefore unable to do- FC
nate their strategies, which resultsin homogeneouscoopera- 0.4
tivedomainsaroundplayerswithfullteachingcapabilities.In
the presentlystudiedmodel, the introductionof socialdiver-
0.2
sitypartlyresultsinasimilarsuppressionofselectedplayers,

especiallysoforthepower-lawdistributedcasewherethema-
0
jority of players are low-ranking, as demonstrated in Fig. 1.
1 1.05 1.1 1.15 1.2 1.25
These players can form an obedient domain around a high-
b
rankingplayerandsoprevailagainstdefectors. Ontheother
hand, a high-rankingdefector will be weakened by the low-
FIG.5: FC asafunctionofbfordifferentvaluesofλ. Thea = 0
rankingplayerswhofollowitsdestructivestrategy.
curveisdepictedforreference.
The above-described feedback works only if the high-
IV. SUMMARY
rankingdefectorscanbelinkedsolelybytheirfollowerneigh-
borhoods;namelytheremustnotbeastrongconnectionwith
other high-rankingplayers. In the opposite case, the hetero-
geneous strategy distribution supports the survival of defec- In this paper, we show that social diversity is an efficient
tors. Tosupportourexplanation,westudytheimpactofspa- promoter of cooperation in the spatial prisoner’s dilemma
tially correlatedsocialdiversityon the evolutionof coopera- game. Thefacilitativeeffectincreaseswith thedispersionof
tion. Accordingtoourargument,thetransitionfromuncorre- social diversity and deteriorates with the increase of its spa-
lated to spatially correlated diversity should be marked with tial correlation. Accordingly, spatially uncorrelated power-
the deteriorationofcooperation. We thusintroducespatially law distributed social diversity providesthe biggest boost to
correlatedsocialdiversityasdescribedinSectionII,andstudy the cooperativestrategy. We arguethat the facilitative effect
theeffectofdifferentλontheevolutionofcooperation. Fig- is conceptuallysimilar to the one reportedpreviouslyfor in-
ure 5 featuresthe results. In agreementwith our conjecture, homogeneousteaching activities, and also has the same root
thefacilitationofcooperationdeterioratesfastasλincreases. asthemechanismapplicablebyscale-freenetworks,namely:
Indeed, a near linear decrease of the critical b marking the akin to hubs or players with enhanced teaching activity, in
extinction of cooperators can be established, and moreover, thismodel,high-rankingplayerscanformrobustcooperative
bylargeλthecooperatorscanoutperformdefectorsonlyfor clusterswithlow-rankingobedientneighbors. Thepresented
slightlyhighervaluesofbthanintheabsenceofsocialdiver- results suggest that the distribution of wealth plays a crucial
sity(a = 0). Thisresultsupportsthevalidityofthereported rolebytheevolutionofcooperationamongstegoisticindivid-
cooperation-facilitatingmechanism, and hopefullypaves the uals,anditseemsreasonabletoinvestigatefurtherwhethera
way for additionalstudies incorporatingsocial diversity into co-evolutionof both might yield new insights and foster the
thetheoreticalframeworkofevolutionarygametheory. understandingoftheformationofcomplexsocieties.
5
Acknowledgments search Fund (T-47003) (A. S.). Discussions with Gyo¨rgy
Szabo´ are gratefully acknowledged. We also thank Girish
This work was supported by the Slovenian Research Nathanforusefulcomments.
Agency (Z1-9629) (M. P.) and the Hungarian National Re-
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