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Cooperation in spatial prisoner’s dilemma with two types of players for increasing number

of neighbors

Gy?rgy Szabóand Attila Szolnoki

Research Institute for Technical Physics and Materials Science,P.O.Box49,H-1525Budapest,Hungary

?Received17July2008;published14January2009?

We study a spatial two-strategy?cooperation and defection?prisoner’s dilemma game with two types?A and

B?of players located on the sites of a square lattice.The evolution of strategy distribution is governed by

iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff

difference and also on the type of the neighbor.The strategy adoption probability is reduced by a prefactor

?w?1?from the players of type B.We consider the competition between two opposite effects when increasing

the number of neighbors?k=4,8,and24?.Within a range of the portion of in?uential players?type A?the

inhomogeneous activity in strategy transfer yields a relevant increase?dependent on w?in the density of

cooperators.The noise dependence of this phenomenon is also discussed by evaluating phase diagrams.

DOI:10.1103/PhysRevE.79.016106PACS number?s?:89.65.?s,89.75.Fb,https://www.wendangku.net/doc/0611007006.html,,05.50.?q

The investigation of the spatial evolutionary prisoner’s dilemma?PD?games expands progressively since Nowak and May?1?reported the maintenance of cooperative behav-ior among sel?sh players.In these models the PD game?2?represents a pair interaction between two players who can

either cooperate?C?or defect?D?and their income depends

on both choices in a way forcing both rational?sel?sh?play-

ers to choose D while they would share equally the maxi-

mum total payoff for mutual cooperation.

In the?rst model the players are located on a square lat-

tice,they can follow one of the two pure strategies?D and C?

and their income comes from PD games with the neighbors.

During a synchronized strategy update the players adopt the

strategy from the neighbor receiving the highest score.After

this pioneering work many modi?ed versions of the original

model have been suggested and studied?for recent surveys

see Refs.?3,4??.Let us mention only a couple of examples:

in some models a larger set of strategies was used?5–7?,in

the evolutionary rules noises?8–10?were introduced that can

help the cooperative behavior?11–13?,and the spatial struc-

ture was also extended by locating the players on different

graphs giving a better description about the connections in

human societies?14–17?.In the last years the concept of

interaction and learning graphs have been distinguished ?18–20?and the research of the coevolution of strategy dis-tribution and these graphs has also become a promising topic ?21–23?.It is found,furthermore,that different types of per-sonality?24,25?and inhomogeneous activity in the strategy

adoption can also support cooperation?15,26–28?particu-

larly if some distinguished players have higher in?uence to

spread their strategies?29,30?.

In the latter case the relevant increase in the frequency of

cooperators is related to a phenomenon described previously

by Santos et al.?31,32?who studied evolutionary PD games

on scale-free networks with an evolutionary rule exploiting

the high income for players?called hubs?who have a large

number of neighbors.As a result,on the scale-free networks

the strategy of hubs becomes an example to be followed by

their neighborhood.Thus,the hubs as in?uential players face

the consequence of the imitation of their own strategy that

increases?decreases?the income of cooperative?defective?hubs.After a short transient process this phenomenon favors

the spreading of cooperation because the in?uential players

can also adopt strategy from each other for suitable connec-

tivity structures.Evidently,in the absence of links between

in?uential players the mentioned mechanism cannot help co-

operators to beat defectors?31–33?.Recent studies?30,34?

have indicated that the presence of linked in?uential players

on scale-free graphs can ef?ciently promote cooperation ?even for normalized payoffs?if the capability of strategy spreading differs from player to player.These results raise

many interesting questions about the impact of the size of a

neighborhood on the frequency of cooperators for inhomo-

geneous activity in the strategy transfer.

In the present work we study the competition between

two opposite effects emerging if the average number of

neighbors is increased.On one hand,the above described

mechanism?supporting the spreading of cooperation for in-

homogeneous strategy transfer capability?is enhanced when

choosing larger and larger k.On the other hand,the increase

of the number of neighbors k is bene?cial for defectors on

regular networks?32,35–40?.Here it is worth mentioning

that the mean-?eld approximation?predicting the extinction

of cooperators in the evolutionary PD games?3,4??gives a

simple explanation of this phenomenon.The scope of the

present paper is to explore the impact of these two opposite

effects by comparing results obtained for three different sizes

of neighborhoods.More precisely,the studied types of neigh-

borhood are the von Neumann neighborhood including only the nearest neighbors?k=4?,the Moore neighborhood with nearest and next-nearest neighbors?k=8?,and the case of k =24where players within a5?5box of sites are neighbors of the central player?self-interaction is excluded?.Monte Carlo?MC?simulations are used to study systematically the effects of payoff,number of neighbors,and inhomogeneous capability of strategy transfer?for a?xed noise level?on the average number of cooperators.

For these evolutionary PD game models two types of players?n x=A or B?are located on the sites x of a square lattice with a concentration of?and?1???and their random initial distribution remains unchanged?quenched?during the simulations.The income of player x comes from one-shot PD games with her neighbors,that is,

PHYSICAL REVIEW E79,016106?2009?

P x=?y??

s x·A·s y,?1?

where the sum runs over all neighboring sites??x?of player x,the payoff matrix is de?ned as suggested by Nowak and

May?1?,

A=?0b01?,1?b?2,?2?

and the defective and cooperative strategies are denoted by

unit vectors as

s x=D=?10?or C=?01?.?3?

The evolution of strategy distribution is governed by random sequential strategy update representing strategy adoption from a randomly chosen site y to one of its neighbors x with a probability

W?s x←s y?=w y

1+exp??P x?P y?/kK?

?4?

dependent on the difference of normalized payoffs?e.g.,?P x?P y?/k?for later convenience of comparisons.For this strategy adoption probability the meaning of the parameter K is analogous to the temperature as introduced in the kinetic Ising model and characterizes the magnitude of payoff noise affecting the decision of player x?9,10?.The multiplicative factor w y denotes the strategy transfer capability of player y,

w y=?1,if n y=A

w,if n y=B

,0?w?1.?5?

In this notation players of type A represent those individuals who can easily convince their neighbors to adopt the strategy they are just following.This personal feature can be related to age,reputation,etc.

For all the three cases studied here the simulations are performed on an L?L square of sites with periodic boundary conditions.The evolution of the spatial distribution of the C and D strategies starts from an uncorrelated initial state where cooperators and defectors are present with the same probability.When repeating the above described elementary steps the system develops into a?nal stationary state char-acterized by the average density of cooperators???.After a suitable relaxation time t r?is determined by averaging the density of cooperators over a time t a.Typical?maximum?values of parameters used in our simulations are the follow-ing:L=400?1600?and t r?t a=104?106?MCS?during one MC step?MCS?each player has a chance once on average to adopt one of the neighboring strategies?.Pronounced long relaxations were observed at the large noise limit.

Before discussing the behaviors of the above systems we brie?y recall some general features of the homogeneous sys-tem??=0or1??41?.The average?total?payoff increases monotonously with?independent of the initial strategy dis-tribution.Furthermore,in each homogeneous system the value of?decreases monotonously from1to0if b is in-creased within a region of b?b c1?k??K??b?b c2?k??K??,where the strategies C and D coexist.For all the three types of neigh-borhoods b c1?k??K??b c2?k??K??tends to1from below?above?if K→?.In other words,in the strong noise limit?K→??the systems reproduce the behavior of the mean-?eld model,that is,?drops suddenly from1to0at b=1.In the opposite case ?K→0?the limit values of b c1?k?and b c2?k?depend on k.When decreasing K the upper critical value of b tends monoto-nously to a value b c1?k??0?larger than1if k=8or24.On the contrary,for k=4,the function b c2?4??K?has a local maximum at K=K opt?0.08?b c2?4??K=K opt??1.08?and approaches1if K→0.

In the light of the above results we?rst study the density of cooperators???when varying the portion of players of type A for?xed values of payoff?b?,strategy transfer capa-bility?w?,and noise?K?.The latter was chosen to present optimal cooperation for the k=4system?i.e.,KХK opt?.The MC data are compared in Fig.1for the three types of neigh-borhood.For the sake of comparison,we selected such a high value of b,which prevents cooperation in the homoge-neous model of k=24.Figure1shows that the highest den-sity of cooperators can be observed?at?=0or1?for k=8 where the overlapping triangles in the connectivity structure support the spreading?maintenance?of cooperation as dis-cussed in Ref.?12?.The further increase of k,however, yields a decrease in both?and b c2?k??32?tending to the be-havior of mean-?eld model.This is the reason why coopera-tors become extinct in the?nal stationary state for the homo-geneous system at k=24.

Figure1demonstrates clearly the existence of an opti-mum composition?de?ned by the maximum in??of the players A and B.The presence of distinguished players re-sults in a relatively higher impact on cooperation level for larger k.In agreement with the expectations,the more neigh-bors the players have,the smaller portion of in?uential play-ers?type A?are capable to achieve the highest increase in?. The resultant asymmetry can be observed in the function ????for k=24.For the largest neighborhood our simulations have clearly indicated that cooperators can remain alive only within a region of?with boundaries dependent on w and K.It is expected that this region,?1?w,K?????2?w,K?, shrinks if we increase k further.

As the largest effect is found for the largest neighborhood, henceforth our attention will be focused on the system of k =24.Figure2illustrates the increase of the density of coop-0

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0.4

0.6

0.8

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FIG.1.Density of cooperators as a function of the portion of A players if b=1.05,K=0.1,and w=0.1for three different neighbor-hoods:k=4?open squares?,8?closed squares?,and24?closed triangles?.

GY?RGY SZABóAND ATTILA SZOLNOKI PHYSICAL REVIEW E79,016106?2009?

erators when varying the composition of players A and B for several values of w at a ?xed payoff and noise level.When the difference is small,typically when 1/w ?2,the coopera-tors cannot remain alive at the given payoffs and noise inde-pendently of the actual composition of A and B players.If the ratio 1/w is increased then the cooperators can survive within the above mentioned region of ?.This interval be-comes wider and wider while the maximum value of ?in-creases monotonously until reaching its saturation value ??=1?.Consequently,we can observe four subsequent transi-tions in Fig.2if ?increased for suf?ciently high values of the ratio 1/w .Apparently the density of cooperators tends to a limit pro?le if 1/w →?.We have to emphasize that the rigorous analysis of the asymptotic behavior becomes dif?-cult because the transient time increases with the ratio 1/w particularly at small values of ?.

We have also studied the effect of the variation of w on the cooperation level at different payoffs ?b ?.To avoid addi-tional effects the noise level is ?xed at a composition ??=0.2?close to its optimum value.The results,summarized in Fig.3,illustrates that the curves ??b ?shift to larger b values if the ratio 1/w is increased.?For comparison,the left curve shows the results obtained in the homogeneous system.?The plotted results refer to a shift proportional to ln ?1/w ?.Due to the above mentioned increasing run time,if we choose larger values of 1/w ,the more rigorous ?numerical ?con?rmation of this trend goes beyond the scope of the present work.Instead

of it we have concentrated on the effect of noise for the two extreme neighborhoods ?k =4and 24?at a ?xed portion of players A and B .For this purpose we have performed sys-tematic MC simulations to determine the critical values b c 1and b c 2for a ?xed ratio of strategy transfer capability ?1/w =50?.

Figure 4can be interpreted as a phase diagram where the connected data represent phase boundaries.Between the up-per and lower critical points strategies C and D coexist.Above ?below ?this region only defectors ?cooperators ?re-main alive in the ?nal stationary states.For both cases the system behavior is not affected by the spatial inhomogene-ities in the low noise limit,in agreement with the previous results ?29?.In other words,the relevant improvement in the maintenance of cooperation appears in the noisy systems even for the limit K →?.In contrary to the prediction of mean-?eld theory the present data indicate clearly that coop-erators and defectors can coexist within a region of b if K goes to in?nity,that is,b c 1and b c 2tend to two distinct limit values for both types of neighborhoods.This latter feature has already been con?rmed qualitatively by the pair approxi-mation for k =4?29?.Notice,furthermore,that the larger neighborhood yields a larger increase in the value of b c 1and b c 2when applying optimum composition of players A and B for both systems.

In summary,within the framework of evolutionary PD games,the present investigation of the effect of the inhomo-geneous strategy transfer capability on the cooperative be-havior has indicated a relevant increase in the density of cooperators if the fraction of in?uential players was close to the optimum value dependent on the number of neighbors ?range of interaction ?if two types of strategy transfer capa-bility ?represented by the players A and B ?are distinguished.It is found that the larger neighborhood with a smaller frac-tion of in?uential players ?type A ?can be more bene?cial for the whole system due to the imitation mechanism rewarding ?punishing ?cooperation ?defection ?for the in?uential play-ers.The improvement of cooperation increases with the ratio of strategy transfer capability ?1/w ?between players of type A and B .Furthermore,the maintenance of cooperation is

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0.810

0.2

0.4

0.6

0.8

νFIG.2.Density of cooperators vs ?for ?ve different values of the reduced strategy transfer capability ?w =0.01,0.02,0.05,0.1,and 0.2from top to bottom ?at b =1.05,k =24,and K =0.1.

00.20.40.60.81

FIG.3.Density of cooperators as a function of b for different values of 1/w ?w =1,

0.2,0.05,0.02,and 0.005from left to right ?at ?xed noise level ?K =0.1?,composition ??=0.2?,and neighborhood ?k =24?.

0.91.01.11.21.300.51 1.52

b c

FIG.4.The upper and lower critical values of b for k =24and ?=0.2?open squares connected with dashed lines ?.Results for k =4and ?=0.5are denoted by closed squares ?connected with solid lines ?at w =0.02.The dotted line illustrates the prediction of mean-?eld approximation in the homogeneous system ?b c 1?MF ?=b c 2?MF ?

=1for arbitrary K ?.

COOPERATION IN SPATIAL PRISONER’S DILEMMA …

PHYSICAL REVIEW E 79,016106?2009?

supported remarkably by the mentioned effect in the high noise limit where the region of coexistence is broadened and shifted to higher values of b.For a small fraction of players A one can think that the competition between the in?uential players surrounded by their followers can be characterized by an effective?rescaled?payoff matrix favoring cooperation ?as it appears on evolving networks?21??while their compe-titions are disturbed by those players of type B who do not belong to the neighborhood of any in?uential player.These latter B players can mediate an interaction between the in?u-ential players and/or preserve the defective behavior.Further research is requested to clarify the relevance of these oppo-site effects.

This work was supported by the Hungarian National Re-search Fund?Grant No.K-73449?.

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GY?RGY SZABóAND ATTILA SZOLNOKI PHYSICAL REVIEW E79,016106?2009?