Table Of ContentPromote cooperation by localised small-world communication
Zi-Gang Huang,1 Sheng-Jun Wang,1 Xin-Jian Xu,2 and Ying-Hai Wang1∗
1Institute of Theoretical Physics, Lanzhou University, Lanzhou Gansu 730000, China
2Departamento de F´ısica da Universidade de Aveiro, 3810-193 Aveiro, Portugal
(Dated: February 1, 2008)
The emergence and maintenance of cooperation within sizable groups of unrelated humans offer
many challenges for our understanding. We propose that the humans’ capacity of communication,
suchashowmanyandhowfarawaythefellowsonecanbuildupmutualcommunication,mayaffect
theevolutionofcooperation. Westudythisissuebymeansofthepublicgoodsgame(PGG)witha
8 two-layered network of contacts. Players obtain payoffs from five-person public goods interactions
0 on a square lattice (the interaction layer). Also, they update strategies after communicating with
0 neighbours in learning layer, where two players build up mutual communication with a power law
2 probability depending on their spatial distance. Our simulation results indicate that the evolution
of cooperation is indeed sensitive to how players choose others to communicate with, including the
n
amount as well as the locations. The tendency of localised communication is proved to be a new
a
mechanism to promote cooperation.
J
5
PACSnumbers: 02.50.Le,89.75.Hc,87.23.Ge
1
]
h I. INTRODUCTION tive behavior in the absence of any strategic complex-
p ity [11, 12, 13, 19]. In spatial evolutionaryPGG, the co-
- operatorscansurvivebyformingcompactclusters,which
c In behavioral sciences, evolutionary biology and more
minimizes the exploitation by defectors [15]. Further-
o recently in economics, understanding conditions for the
s more, Szabo´ and Hauert et al. have recently discussed
emergence and maintenance of cooperative behavior
. the effects of compulsory and voluntary interactions of
s
among unrelated and selfish individuals becomes a cen-
c players in evolutionary PGG, with the structured pop-
si tral issue [1, 2]. In the investigationof this problem, the ulations bound to regular lattices [15, 16, 20] as well as
most popular framework is game theory together with
y with the well-mixedpopulation [16, 21, 22, 23]. The fac-
h its extensions involving evolutionary context [3, 4]. The tors such as the voluntary participation [20], and small
p public goods game (PGG), which attracted much atten-
density of population [22], are found to be capable of
[ tion from economists, is a general paradigm to explain
boosting cooperation. More recently, Huang et al. have
cooperativebehaviorthroughgroupinteractions[5]. The
2 proposed an extended public goods interaction model to
PGGmodelischaracterisedbygroupsofcooperatorsdo-
v studytheevolutionofcooperationinheterogeneouspop-
2 ing better than groups of defectors, but defectors always ulation, and proved that scale-free networks of contacts
2 outperforming the cooperators in their group. In typical
can lead to more competitive cooperation [24].
3 examples, the individual contributions are multiplied by
0 a factor r and then divided equally among all players. In real world, people always wish to make decisions
9. With r smaller than the group size, this is an example based on a comprehensive knowledge of the pertinent
0 of a social dilemma: every individual player is better off background, such as the historical performance of each
7 defecting than cooperating, no matter what the other alternative choice [25], which they may obtain by learn-
0 players do. Groups would therefore consist of defectors ing from people they are regarding. However, the lim-
v: only and forego the public good. ited eyereach or capacity of humans actually exists and
i induces the unperfect communication. One can merely
X Considerable efforts have been concentrated on explo-
learn from a small set of people (corresponding to the
rationof the originand persistence of cooperation. Dur-
r so-called “role models” in the context of cultural evolu-
a ing the last decades, five rules, namely, kin selection [6],
tion [16, 26]). Furthermore, these role models rarely can
direct reciprocity [7, 8], indirect reciprocity [9, 10], net-
be extended over the whole population, but localised to
work (or spatial) reciprocity [11, 12, 13, 14, 15, 16], and
thelearner’s“vicinity”,whichabstractlyimpliesthepeo-
groupselection[17],havebeenfoundtobenefittheevolu-
ple having similar characteristics as the learner (such as
tionofcooperationinbiologicalandecologicalsystemsas
religious background, age, education, lifestyle, or social
wellaswithinhumansocieties(forareview,see[18]). In
class) that in favour of building up mutual communica-
realistic systems, most interactions among elements are
tion.
spatially localised, which makes spatial or graph mod-
els more meaningful. Unlike the other four rules, spatial Asanaturalextensionofthoseaforementionedfactors,
games (i.e., network reciprocity) can lead to coopera- an new intriguing task is to understand how the lim-
itation of communication capacity of people influences
the cooperative behavior in real world. In this letter,
we will study a spatially extended PGG on two-layered
∗Forcorrespondence:[email protected] graphs, where one layer especially depicts the communi-
2
strategy. The player’scapacityofcommunicationcanbe
measuredbyhowmanyandhowfarawaytherolemodels
it can selected. In order to regulate this capacity, we in-
troduce the LN layerasa variantofthe two-dimensional
small-worldnetwork[29],inwhichconnectionstofurther
neighbours occur with a tunable power law probability.
This network is constructed by adding shortcuts among
thesitesonaL×Lsquarelattice. Withperiodicbound-
arycondition,thelatticedistancebetweentwosites(x,y)
FIG. 1: The average lattice length of the shortcuts l in the
LN network with size N =L×L=512, 1012, and 2012 (a). and (x′,y′) can be written in a two-dimensional fashion
AndclusteringcoefficientEasafunctionofαfortheLNwith as
q =0, 0.5, 1, 2, and 5 (b). The plots are average values over
10 realizations of theLN network. r(x,y),(x′,y′) =∆x+∆y,
with
cation among players, with the aim to find out how the
evolutionofcooperationdependsontheplayers’commu- ∆x=L/2−|(|x−x′|−L/2)|
nications. ∆y =L/2−|(|y−y′|−L/2)|.
Thisvalueisactuallythelengthoftheshortestpathcon-
II. THE MODEL necting these two sites through only lattice links. Each
siteisadditionallylinkedtoqothersites[30]byshortcuts
Inspatialgamemodels,theplayersoccupyingthesites (excludingitsoriginalnearestneighbours). Followingthe
of a graph can follow one of the two pure strategies, co- idea of Kleinberg [29], those other sites are selected in a
operation (C) or defection (D). There are two types biased manner: the probability that site j(xj,yj) is se-
of contacts among players in the evolutionary process: lected to be linked to site i(xi,yi) by one shortcut de-
players collect payoffs from their neighbours by playing pends on the lattice distance between them in the fol-
games with them [12, 13, 14, 15, 19, 20], and then up- lowing way,
date strategies by learning from neighbours. Thus, the
1
graph occupied by the players can be split into two lay- P(r )= r−α , (2)
ers, an interactionlayer and a learning (communication) (xi,yi),(xj,yj) A (xi,yi),(xj,yj)
layer [26, 27, 28]. The former one defines the interaction
where α is a positive exponent and
neighbourhood (IN), i.e., who plays game with whom.
The latteronespecifies learningneighbourhood(LN) for A= r−α , (3)
evolutionaryupdating or,inotherwords,defines who-is- (x,y),(x′,y′)
the-role-model-of-whom. (x′,y′)6=(xi,yi),(Xxi±1,yi),(xi,yi±1)
In our model, the IN layer where players have public isanormalizationfactor. Obviously,theprobabilitythat
goodsinteractionsisa L×Lsquarelattice with periodic i and j are connected is
boundary conditions. The group size of the interaction
neighbours in the PGG therefore is N = 5. The score r−α
achieved in PGG interactions denotesIthe reproductive Pi↔j =1−(1− (xi,yi),(xj,yj))2·q. (4)
A
success,i.e.,theprobabilitythatotherplayerswilladopt
the player’s strategy. This score is assumed to be de- For this network,when α=0, it reduces to the small-
termined merely by a single, typical PGG involving the worldmodelwithrandomshortcuts[31]. As αincreases,
player and its four nearest interaction neighbours [15]. one site’s shortcuts will be clustered in its vicinity and
Thus the payoff of one player i is, twodistantsitesarelesslikelytobeconnected. Here,the
multiple or self-connected edges are not allowed. Such a
rnc −1 if s(i)=C, network has qN shortcuts and an average degree hki ≈
P(i)= nc+nd (1)
(ncr+ncnd if s(i)=D, s4h+or2tqcu.tFsilg,uarned1tshheowclsusttheerianvgercaogeeffilcaiettnitceEleonfgtthhisofkitnhde
where n , and n (with n +n = N = 5) denote, re- graph. We clearly notice the increase of E with α by
c d c d I
spectively, the number of participants choosing C, and reason of the clustered shortcuts.
D,ands(i)denotesthe strategyofplayeri. Thecooper- Inthispaper,wejustconsiderthiskindnetworkasthe
ative investments arenormalisedto unity and r specifies LN of the game players where merely the information
the multiplication factor on the public goods. flow takes place. In this LN network, the neighbours
After each round of the game interactions, for the ref- linked by one edge act as the role model of each other
erence ofthe strategyupdate, eachplayercommunicates via mutual communication, and the degree of each site
withitsrolemodels(neighboursintheLNlayer),inquir- correspondstotheamountofrolemodelseachplayerhas.
ingtheindividualinformationsuchasthepayoffandthe For the sake of convenience, we call those role models
3
FIG. 2: (a1), (a2), and (a3) show the stationary densities of cooperators ρc as a function of multiplication factor r for q =0,
1,2,and5denotedbydifferentsymbols. (b1),(b2),and(b3)plotthecorrespondingphasediagrams. Thetwoboldsolid lines
shows theaverage rc (lower) and rd (upper). The straight dot line in themiddle is r=5.0. The threedash lines from bottom
totoppresentthecontoursρc =0.25, 0.5,and0.75,respectively. Thereddash-dotlinesin(b1)and(b2)showthecrosspoints
of those ρc curves of q 6= 0 with the curves of q = 0. (c1), (c2), and (c3) show the strategy update rate R in the dynamical
equilibrium stateas afunction of ρc. Theresultsare averaged over10 realizations of theN =512 systems with α=0(left), 2
(middle), and 5 (right).
introduced by shortcuts the additional role models III. SIMULATION RESULTS
(shortly ARMs). Thus, each player has 2q ARMs on
average. Also, the biased effect of nonzero α, giving rise
to the clustering of shortcuts, can be understood as the We study above model by Monte-Carlo (MC) simu-
“localization” of players’ communication. lations started from a random initial distribution of C
andDstrategies. Afterappropriaterelaxationtimes,the
Followingpreviousstudies[15,19],theevolutionofthe
system can converge to a dynamical equilibrium state.
presentsystemisgovernedbyrandomsequentialstrategy
We characterise this state by the stationary density of
adoptions, that is, the randomly chosen player i adopts
cooperators ρ averaged over the last 5000 MC steps of
oneofits(randomlychosen)rolemodelj’sstrategywith c
the 25000totalsamplingsteps. Figures2 (a1),(a2),and
a probability depending on the payoff difference
(a3) show the dependence of stationary density of coop-
erators ρ on the multiplication factor r for the systems
c
1 with α = 0, 2, and 5, respectively. The simulation data
W[s(j)→s(i)]= , (5)
resultfromanaverageovereithertenrealizationsofinde-
1+exp{[P(i)−P(j)+τ]/κ}
pendent initial strategy distributions or ten realizations
of the LN networks. For q = 0, i.e., when the IN and
where τ > 0 denotes the cost of strategy change, and LN are identical, we recover the results of ref. [15]: be-
κ characterises the noise introduced to permit irrational low the threshold value r < r = 4.526(1) cooperators
c
choices. For κ = 0 the neighbouring strategy s(j) is quickly vanish (the absorbing homogenous state with all
adopted deterministically provided the payoff difference defectors), whereas for high r > r defectors go extinct
d
exceedsthecostofstrategychange,i.e.,P(j)−P(i)>τ. (the absorbing homogenous state with all cooperators);
For κ > 0, strategies performing worse are also adopted forintermediater,strategiesCandDcoexistindynami-
withacertainprobability,e.g.,duetoimperfectinforma- calequilibrium. Ther (r )indicatesthevalueofrwhere
c d
tion. It is proved that the dynamics remains unaffected cooperators (defectors) vanish. In the case that q 6= 0,
qualitativelywhenchangingκ,andτ withinrealisticlim- i.e.,wheneachplayercancommunicatewithmoreARMs
its. Following the previous work [15], we simply fix the via shortcuts, the quantitative properties of the station-
valueofκandτ tobe0.1,andconcentrateonthegeneral ary density ρ are different [see the curves with different
c
dynamical properties affected by the structure of LN. q in figs. 2 (a1), (a2) and (a3)].
4
Onecanfindfromfigs.2(a1),and(a2)that,eachofthe
ρ curvewith q 6=0 hasa crosspoint(denoted by r )
c cross
with the curve of q = 0. The density of cooperators ρ
c
for the system with larger q is comparatively smaller at
r <r region,butlargerρ atr >r region. That
cross c cross
is to say, for the systems with α = 0 and 2, the more
available role models will favour cooperators when r is
large, and favour defectors in contrast when r is small.
However,thesystemwithα=5[figs.2(a3)]isobviously
differentfromthatofα=0and2,thatis,theARMscan
favour cooperators in the whole range of r with respect
to the q =0 system, moreover,defectors vanish at much
lower r.
We plot the phase diagrams of the corresponding sys-
tems in figs. 2 (b1), (b2), and (b3), respectively. The
twoboldsolidcurvesrespectivelyshowthe r andr av-
c d
eraged over 10 realizations, which divide the region into
FIG. 3: Snapshots of typical distributions of cooperators
absorbinghomogenousstatesofC(upper)andD(lower), (black)anddefectors(white)onsquarelatticeofN =2012for
aswellasthecoexistenceregimeofthetwostrategies(in-
multiplicationfactorr=4.9(upper)and4.6(lower). (a)and
termediate). That is, for fixed number of ARMs, when (c)correspondtothecasewithidenticalINandLN,while(b)
varyingthefactorr,twophasetransitionsoccurbetween and (d) correspond to thecase of LN with α=2 and q=2.
the coexistence region and homogeneous states of C or
D.Inthecoexistenceregime,thethreedashlinespresent
the contours ρ = 0.25, 0.5, and 0.75, respectively. The
c of the competition between C and D, which is charac-
straight dot line in the middle corresponds to r = 5.0,
terised by the rate of the effective strategy update R
largerthanwhich the socialdilemma raisedby the PGG
(the fraction of players who adopt the opposite strategy
with N = 5 is relaxed in the sense that each unity in-
I in each generation averaged over 1000 MC steps). For
vestment has a positive net return. As q = 0 defectors
the sakeof comparison,the rate R are plotted as a func-
may exist in the system and exploit cooperators until r
tion of ρ in figs. 2 (c1), (c2), and (c3) for the systems
c
isverylarger. However,fromtherapiddeclineofr with
d with α = 0, 2, and 5, respectively. For a given station-
increasing q we know that, with the aid of the ARMs,
ary density ρ , the system with larger q results in larger
c
defectors vanish at much smaller r. More, interestingly,
rate R, which implies that more ARMs induce more in-
even if r < 5.0, defectors can be eliminated as long as
tense competition. This is due to the additional mutual
q > 3.49 for α = 0, q > 2.42 for α = 2, or q > 1.41 for
contacts between C and D, or, in other words the larger
α=5. Also, the regionof homogenous C becomes wider
surface between C and D in the LN network, induced
as the value of α increases.
by the additional shortcuts. Also, R exhibits a bell-like
Thecrosspointsrcross ofthoseρc curvesofq 6=0with form, because the surface between C and D will shrink
the curve of q = 0 in fig. 2 (a1) [and (a2)] are marked with the density difference of the two strategies. Addi-
by the (red) dash-dot line in the phase diagram. The tionally, for a given value of q, the system with larger α
coexistenceregimeofCandDthus isseparatedintotwo are found to result in smaller rate R.
regions by this line, with ARMs favouring cooperation Figure 3 shows the snapshots of the system with iden-
in the upper region, but favouring defection in the lower tical IN and LN (left hand), as well as the system with
region. It can be seen that, the cross points rcross is not q = 2 and α = 2 LN (right hand), respectively. For the
sensitive to the parameter q. For the system with α=2 caseofidenticalINandLN[fig.3(a)(c)],itisknownfrom
the upper region favouring cooperation is comparatively PGGandothercooperationgamesthat,cooperatorsper-
wider than that with α = 0. Moreover, for the system sist through forming compact C clusters and thereby re-
with α=5[see figs.2(a3)and(b3)], thoseρc curvesare ducing exploitationby defectors [12, 15, 19, 24? ]. How-
notintersectant,andtheARMsfavourcooperatorsinthe ever, for the availability of ARMs [fig. 3(b)(d)], the C
wholerangeofr. Thecurvesofrc,rd,andothercontours clusters break into smaller pieces. Furthermore, one in-
of ρc for various α systems are proved to asymptotically teresting phenomenon distinctly exhibited in fig. 3 (d)
approach r = 5.0 with increasing q, and finally collapse is, some rare cooperators may take place at the sites
into one there (r = 5.0) for q → ∞, which implies that totally surrounded by defectors, where it is extremely
each player can learn from the whole population. That difficult for C to survive because of the very low pay-
is to say,as a result ofthe dynamicalprocess the system offs. These sites are found to be connected by shortcuts
withq →∞willendupinthe absorbingDstate forany to the well-paid C role models who are located in the
r smaller than 5.0,where the first-orderphase transition C clusters. Therefore, we can say that, the C strategy
from D to C takes place. is adopted there when players blindly imitate the suc-
We also notice that the ARMs may affect the severity cessful role models without regard to their own actual
5
r > 4.6), however, the nonmonotonic behaviors of ρ
c
occur for the cases of smaller r. This result is proved
to remain unaffected qualitatively for different vales of q
within realistic limits. As it is well known, the cooper-
ators located along the boundary of the C clusters (the
so-called boundary C in the following text) outweigh
the losses against defectors by gains from interactions
within the C cluster [14, 19]. These boundary C gain
low payoffs and thus would be sensitive to the introduc-
tion of the ARMs, i.e., what kind of role models they
meet with via shortcuts really matters. The system at
the coexistence state is composed of cooperators as well
astwokindsofdefectors,thedefectorlocatedrightatthe
boundaryofCclusterwhichwenamedD ,andthedefec-
1
FIG. 4: Stationary densities of cooperators ρc as a function tor surrounded by the same-strategy interaction neigh-
of αwith r rangesfrom 4.5 to5.05. Theresults areaveraged
bours which we named D . One D would accumulate
over10 realizations of the system with N =2012 and q=2. 2 1
comparatively higher payoff by exploiting the boundary
C,andthenactasanattractiverolemodelofitslearning
neighbours. However,oneD gainszeropayoff,andthus
‘habitats’. This may correspond to the phenomenon in 2
will not affect the stability of others’ strategy. They put
societythat,somepeoplefollowlikesheeptothemanner
up fluctuating individuals if they are linked to well-paid
of others ignoring the question whether it fits into their
C by shortcuts, as that discussed above. In our model,
own social surrounding. Interestingly, this unreasonable
the players will choose ARMs nearer and nearer to its
behavior simply introduces mutations to the population
vicinity as α increases from 0. When α is around 2.5,
ofD.Wethereforesuggestthat,itmayprovideonemech-
the lattice lengths of these shortcuts (see fig. 1) approx-
anism to favour cooperation, for the invasions of C may
imately reachthe generalsize of the C clusters, then the
form mutual protection and thus survive in the popula-
probabilityfortheboundaryCtomeetwiththewell-paid
tion of D, although the rare C therein are unstable and
D players becomes larger,which would pose a high risk
hardtopropagatebythemselves. Inaddition,thesesites 1
tothestabilityoftheboundaryC.TheCclusterbecomes
exhibit one path to put up the fluctuating individuals [?
constricted and thus cooperation is depressed when the
], who alternatively spend some time as cooperators and
boundary C imitates D . This effect can be clearly ob-
some time as defectors. 1
servedfromthecomparativelysmallerρ aroundα=2.5
Finally, we focus our attention on how the localised c
at small r (see fig. 4). However, when the value of r
communicationaffectstheevolutionofcooperation. The
is large, this effect is weakened by the C clusters highly
stationary density of cooperators ρ at different values
c crowded [see fig. 3(b)].
of r are plotted as a function of α in fig. 4. We can
see that the values of ρ keep almost invariable in the
c
region 0 ≤ α < 1. This can be naturally understood
IV. CONCLUSION
from the results in fig. 1, as well as the analysis in the
ref. [32, 33, 34] that, in this region the LN network be-
haves as a small world with the topological properties In summary, we have shown how the communication
changing slowly with α. However, in the large α region, among players may affect the evolution of cooperation
whichimplies thatthe LN is distinctly localised,cooper- by means of a two-layered PGG model. The learning
ation is favouredmore comparedto the case with α=0. layer is constructed as a Kleinberg small-world network,
We haveknown that, the cooperatorsprotect eachother where twoplayerscancommunicatewith probabilityde-
and survive by forming clusters. Thus, when the short- pendingontheirspatialEuclideanlatticedistanceinthe
cuts are clustered by larger α, one cooperator is more power-law form controlled by an exponent α. The play-
likely to select role models within its own C cluster via ers’ capacity of communication is characterised by the
shortcuts. That is to say, the mutual protection among number and the distance of the role models from whom
cooperators is enhanced by large α, and thereby cooper- they can learn the individual information. The biased
ation is favoured. This mechanism is obviously reflected effect of nonzero α, which gives rise to the preferential
by the monotonic increase of ρ with α at large α re- selection of role models near the vicinity, corresponds
c
gion. Furthermore, the best communication strategy for to the localizationofplayers’communicationinducedby
players to promote cooperation is α → ∞, which corre- limited capacity. Our simulation results indicate that,
spondstotheextremelylocalisedselectionofARMs,i.e., the communication among players plays a highly impor-
the LN network with the shortcuts merely connects the tant role in the evolution of cooperation: the density of
next-nearest neighbours through lattice links. C is cruciallyinfluencedby the number (q) andthe loca-
It is also notable in fig. 4 that, the ρ monotonically tion of the ARMs; the coexistence region of C and D is
c
increases with α when r is comparatively large (about reduced and r (r ) is expected to tend to 5 (the group
c d
6
size N ) if q increases; in addition, for certain density of the emergence and persistence of cooperation in society.
I
C, more available ARMs result in more intense competi- The other feature is that a population of cheaters would
tionbetweenCandD.Moreover,thecommunicationvia be invaded by cooperators by the presence of long-range
shortcuts (i.e. the ARMs) are found to introduce muta- social connections.
tion to the population of D, which might be helpful to
the emergence of C. It is also notable that, the localised
communication with large α can favour cooperators.
Acknowledgments
Our model gives a crude simulation of real social be-
havior. However, it does catch a few features of poten-
tialinterest. Themostinterestingfeatureisthelocalised We thank Dr. Zhi-Xi Wu for helpful discussion.
communication favouring cooperation. We suggest that, This work was supported by the Natural Science Foun-
the limitation of humans’ capacity to build up commu- dation of China (No. 10775060). X.-J. Xu acknowl-
nication system, which results in the localised selection edges financialsupport fromFCT (Portugal),GrantNo.
of role models, would be a new mechanism supporting SFRH/BPD/30425/2006.
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