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Benefits of tolerance in public goods games

Attila Szolnoki1,∗ and Xiaojie Chen2,†

1Institute of Technical Physics and Materials Science, Centre for Energy Research,Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary

2School of Mathematical Sciences, University of ElectronicScience and Technology of China, Chengdu 611731, China

Leaving the joint enterprise when defection is unveiled is always a viable option to avoid being exploited.Although loner strategy helps the population not to be trapped into the tragedy of the commons state, it couldoffer only a modest income for non-participants. In this paper we demonstrate that showing some tolerancetoward defectors could not only save cooperation in harsh environments, but in fact results in a surprisingly highaverage payoff for group members in public goods games. Phase diagrams and the underlying spatial patternsreveal the high complexity of evolving states where cyclic dominant strategies or two-strategy alliances cancharacterize the final state of evolution. We identify microscopic mechanisms which are responsible for thesuperiority of global solutions containing tolerant players. This phenomenon is robust and can be observed bothin well-mixed and in structured populations highlighting the importance of tolerance in our everyday life.

PACS numbers:

I. INTRODUCTION

It is difficult to overestimate the importance of coopera-tion among players who are motivated to search for maximalindividual income during their interactions with competitors[1]. Although mutual cooperation would provide the opti-mal income for the whole community, a higher payoff can bereached individually by exploiting others. This conflict, sum-marized in several social dilemmas [2], can be identified asthe key problem in a broad range of research fields [3–8].

Staying with a specific example, it is always disappoint-ing to realize when some of our partners defect in a workinggroup, which significantly lowers the income of cooperatormembers. A natural reaction could be to punish the traitor, butthe institution of punishment raises further questions, whichsometimes just transfers the basic problem to another level[9–14]. An alternative response from betrayed cooperatorscould be to stop further cooperation and not to participate inthe joint venture anymore. Accordingly, cooperators may be-come “loners” because the latter strategy can offer a modest,but at least guaranteed, payoff to them. Previous works re-vealed that the option of voluntary participation in commonventures could be an effective way to avoid being exploitedbecause it introduces a cyclic dominance between competingstrategies of defectors, cooperators, and loners [15–17].Asa consequence, the cooperator state can survive even in harshconditions when a low synergy factor would result in a full de-fector state in a two-strategy system where participation in apublic goods game is compulsory. There is, however, a disap-pointing feature of the new, three-strategy solution. Namely,the average payoff is unable to exceed the income of a loner’sstrategy, hence participating in a public goods game does notnecessarily provide an attractive option for competing players[15, 18].

This failure suggests that perhaps it is not the best option for

∗Electronic address: szolnoki@mfa.kfki.hu†Electronic address: xiaojiechen@uestc.edu.cn

cooperators to leave the group when defectors emerge becauseby switching to a loner state they lose all benefits of mutualcooperation immediately. In this way the original dilemmacan be transformed into a new form where cooperator playersshould decide how many defectors they tolerate in their groupbefore leaving the group for a modest, but guaranteed payoff.

To explore this new dilemma we introduce a four-strategymodel of a public goods game in which besides the uncondi-tional defector (D), cooperator (C), and loner (L) strategiesthere is a so-called tolerant or mixed (M ) strategy, that be-haves as a cooperator as long as the number of defectors re-mains below a threshold value in the group but it switches toloner a state otherwise. By following this approach we cancheck the viability of this mixed strategy and clarify if thereis an optimal level of tolerance which provides the highest in-come for the whole population.

Beyond these fundamental questions there is an additionalaspect which makes the proposed model even more interest-ing. On one hand, the coexistence ofC, D, andL strategies isbased on the previously mentioned cyclic dominance betweencompeting strategies, which is a well identified general mech-anism to maintain diversity [19–25]. On the other hand, byconsideringM players we introduce a strategy which is lessharmful to defectors because they may coexist. Intuitively,one may expect that such intervention is beneficial to defec-tion, but, as we demonstrate in this paper, the opposite effectcan be observed.

The organization of this paper is as follows. We present thedefinition of the model in the next section. Results obtainedby means of the replicator equation in well-mixed populationsare summarized in Section III, which is followed by the pre-sentation of Monte Carlo results obtained in structured pop-ulations. Finally we conclude with an argument for broadervalidity of our observations and a discussion of their implica-tions in Section IV.

2

II. PUBLIC GOODS GAME WITH TOLERANT PLAYERS

We consider a public goods game where the game is playedin groups of sizeG. Following the standard model [15], eachplayer is set as an unconditional cooperator (C), an uncondi-tional defector (D), or a loner (L). Whereas each cooperatorcontributes an amountc to the common pool, defectors con-tribute nothing but exploit others’ efforts. Loners do not par-ticipate in the joint enterprise, instead, they prefer a moderate,but guaranteed,σ income. Beyond these well-known strate-gies we consider an additional, so-called mixed (M ) strategy.The latter players are principally cooperators who contributeto the common pool but permanently monitor the status ofother players in the group at an additional cost ofγ. This extraknowledge allows them to realize if the level of defection ex-ceeds a certain level in the group. As a reaction, they becomeloners and stop contributing to the common pool. Designatingthen the number of unconditional cooperators, defectors, and“mixed” players among the otherG−1 players in the group asnC , nD andnM , the payoff of the four competing strategiesare the following:

ΠD =r(nC + δnM )

nC + nD + 1 + δnM

(1)

ΠC =r(nC + 1 + δnM )

nC + nD + 1 + δnM

− 1 (2)

ΠL = σ (3)

ΠM = δ

[

r(nC + 1 + nM )

nC + nD + 1 + nM

− 1

]

+ (1− δ)σ − γ . (4)

Here, according to the broadly accepted notation,r depicts thesynergy factor, characterizing the benefit of mutual coopera-tion, whereasσ is the loner’s payoff. It should be emphasizedthat ther > 1 synergy factor is applied only if there are morethan one contributor to the common pool, otherwiser = 1is used. In this way we can avoid an artificial support of alonely cooperator against loners and prevent single individ-uals playing a public goods game with themselves [26]. Fur-thermore, without loss of generality, cooperators’ contributionto the common pool is considered to bec = 1, as Eqs. (2) and(4) indicate. Lastly, in close agreement with previous works[15, 18], the payoff of loners is chosen asσ = 1, but we stressthat using other values would not change our main findings.

As we already noted, it is a fundamental point that anMplayer uses a more sophisticated strategy by checking the sta-tus of other players in the group. Accordingly, such a playerbehaves as a loner and refuses participating in the publicgoods game if the number of defectors reaches a criticalHthreshold in the group. Otherwise, when the total number ofdefectors is below theH threshold,M cooperates and con-tributes to the common pool similarly to unconditional coop-erators. The possible “switch” of a player’s status, or say-ing differently the adoption to change a neighborhood, can behandled technically via aδ factor, which isδ = 0 if nD ≥ Hor δ = 1 if nD < H .

Formally, strategyL can be considered as a cost-free, veryspecial mixed player who applies zero threshold, hence he al-ways avoids participating in a joint venture independent ofthe

strategies of other group members. In general, the value ofHcharacterizes the level of tolerance ofM players. Namely, thehigherH is applied, the more defectors are accepted in thegroup without refusing cooperation fromM players. As anextreme case, formallyH = G denotes the situation whenMplayers remain in an unconditional cooperator state. Hencewe may say that the concept of “tolerance” builds a bridge be-tween loner and unconditionally cooperator behaviors. Closeto the latter end,H = G − 1 represents the case when anMplayer seems to be almost “endlessly tolerant” and becomesa loner only if all the others in the group are defectors, hencecooperation becomes unambiguously pointless to him.

Evidently, the extra knowledge ofM players needs addi-tional efforts from their side, which can be implemented viaan additional costγ. This cost should always be considered,no matter whetherM plays aC or L strategy, as indicated inEq. (4). The presence of this permanent cost also means thatM players have no obvious advantage either overC or overLstrategies.

In the following we consider both well-mixed and struc-tured populations.

III. RESULTS

A. Well-mixed populations

In a well-mixed system the fraction ofC,D,L, andMplayers can be denoted byx, y, z, andw respectively. Evi-dently, they are not independent but are normalized and al-ways fulfill the equationx + y + z + w = 1. Consequently,the strategy evolution can be studied by using replicator dy-namics [27]:

x = x(PC − P )

y = y(PD − P ) (5)

z = z(PL − P ) ,

where dots denote the derivatives with respect to timet. Herethe average payoffP for the whole population is given by

P = xPC + yPD + zPL + wPM , (6)

wherePi (i = C,D,L,M ) designates the average payoff foreach strategy:

Pi =∑

nC ,nD,nL,nM

(G− 1)!

nC !nD!nL!nM !xnCynDznLwnMΠi (7)

where0 ≤ ni ≤ G−1 and∑

ni = G−1 are always fulfilled.For the sake of comparison with the case of a struc-

tured population we suppose that players form groups ofsizeG = 5 randomly and consider the impact of differentH = 1, . . . , G− 1 threshold values of tolerance.

Our principal goal is to compare the results of a well-mixedand spatially structured population, therefore we will launchthe evolution from a random initial state where all compet-ing strategies are present with equal weight. The replicator

3

L M

CD

M

M

(a)

LM M

D C

M

(b)

FIG. 1: (Color online) Replicator dynamics on the boundary facesof the simplexS4 using theG = 5 group size. Filled circles repre-sent stable fixed points whereas open circles represent unstable fixedpoints. Parameter values areH = 2, r = 3.5, γ = 0.6 for panel (a),whereasH = 4, r = 3.8, γ = 0.04 for panel (b). Flow diagramssuggest that both(D,C,L) and (D,C,M) strategies can form arock-paper-scissors cycle, but the stable two-strategy (D,M ) phasealso emerges in dependence on the initial fraction of strategies.

dynamics, however, may depend sensitively on the initial fre-quencies of strategies. This behavior is illustrated in Fig. 1where we have plotted two representative flow diagrams inthe unit simplexS4 at two branches of parameter values.

The top panel illustrates the case when theH = 2 thresh-old value is applied at a significantly highγ cost of inspec-tion when the synergy factor is moderate. Here, as is alreadyknown from a previous work [15], thew = 0 face containsa fix point which is surrounded by periodic orbits. On thez = 0 face, however, there is a stable limit circle which isthe composition of(D,C,M) strategies. Furthermore, a sta-ble two-strategy fix point can also be detected on thex = 0face. In the bottom panel, which was taken at theH = 4threshold level, we can observe that the(D,M) solution re-mains stable whereas the rock-scissors-paper-type(D,C,M)solution disappears. Naturally, the portrayal of replicator dy-namics can also depend on the appliedr andγ parameters,but the presented plots are representative in a broad intervalof parameters.

In the following we focus on the evolution from a random

initial state where all strategies are present with equal weight,but we scan the wholer − γ parameter plane. Interestingly,whenH = 1 then strategyM cannot survive at any finitevalues ofγ. Here,(D,C,L) strategies form the well-knownrock-scissors-paper type solution in the2 < r < 5 region[15]. At low γ values, however,M players may crowd outloners first from a random state, which is followed by the ex-tinction of defectors. Finally, whenM remains alone withunconditionalC players,M is defeated by the latter strategy.This time evolution is similar to the “The Moor has done hisduty, the Moor may go” effect previously observed in a relatedmodel where punishing strategies were studied [28]. Never-theless, we should emphasize that the only stable solution isthe mentioned three-strategy(D,C,L) state atH = 1.

By increasing the tolerance level, however, we can observenew types of solutions. Namely, strategyM can replaceLplayers and forms another solution whereD, C, andM play-ers dominate each other cyclically. As the top panel of Fig. 2illustrates, this(D,C,M) state can be dominant even at a sig-nificantly highγ cost if synergy factorr is high enough. Thelatter condition, when mutual cooperation pays more, is es-sential, otherwise the benefit of mutual cooperation could notcompensate the additional cost of strategyM .

The previously mentioned two-strategy solution can evolvestarting from a complete, four-strategy initial state. Here DandM players form a two-strategy alliance against other com-peting strategies [29, 30]. Note that unconditional cooperatorswould beat strategyM in the absence of defectors, but thepresence of latter players manifests the advantage of a mixedstrategy. This solution, as we emphasize in the subsequentsection, is of prime importance to understand why toleranceemerges during an evolutionary process. Lastly, we brieflynote that there is a specific combination ofD,L, andM play-ers which could prevail in the whole system, albeit at a verylimited parameter region.

Qualitatively similar behavior can be observed for otherH > 1 threshold values, too, but solutions containing mixedplayers become less vital as we increaseH . The bottom panelof Fig. 2, obtained atH = 4, illustrates that the benefit ofmixed strategy is less likely at such a high tolerance level andthe area where strategyM survives on theγ − r parameterplane shrinks significantly.

It is an important consequence that the introduction of toler-ance does not only result in the individual success of strategyM , but also has a favorable impact on the general well-beingof the whole population. This effect can be illustrated nicelyif we compare the average payoff values obtained at differentthreshold levels. Figure 3 highlights that adopting a nonzero,but moderate tolerance towards defection could elevate signif-icantly the global income of players. As we already noted, theusage of the minimalH = 1 tolerance level does not allowM players to survive, hence the system becomes equivalentto the well-known three-strategy model [15]. Here the pres-ence ofL players can help to avoid the tragedy of the commonstate [31], but the average payoff cannot exceed the loners’in-come that isσ = 1 in the present case. At a higher tolerancethreshold, the average income in the whole population can in-crease significantly due to the presence of tolerant players.

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FIG. 2: (Color online) Fullr− γ phase diagrams for the well-mixedsystem in the case ofG = 5 group size. In panel (a)H = 2 whereasin panel (b)H = 4 threshold values are applied. Solid lines representcontinuous, while dashed lines indicate discontinuous transitions be-tween stable solutions. At moderateγ valuesM players replace lon-ers by forming a cyclic dominant coexistence withC andD players.M players are more viable at an intermediate threshold value.Inter-estingly, strategyM can form a two-strategy alliance withD whichcan dominate the evolution at specific parameter values.

This enhancement is particularly conspicuous forH = 2. Forcomparison, the dashed grey line shows the average incomein the idealistic state when all players are in an unconditionalcooperator state. It suggests that the usage of tolerant playerscould be especially efficient at lowr values when cooperatorswould face a harsh environment otherwise.

B. Structured populations

Considering a structured population where players havefixed neighborhood offers not just a more appropriate ap-proach to some real-life situations but it often provides noveland sometimes unexpected behaviors which are absent in awell-mixed system [32–34]. To explore and clarify the pos-sible differences between emerging solutions we considera structured population where players are distributed on asquare graph and form groups with their nearest neighbors(G = 5). It also means that a player is involved not only in

0

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aver

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payo

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synergy factor r

H = 2

H = 3

H = 4

three-strategy

FIG. 3: (Color online) Average payoffs in stable stationarystates independence of synergy factorr as obtained for different thresholdvalues of tolerance atγ = 0.04. Players are in a well-mixed popula-tion where they form groups of sizeG = 5 randomly. For compari-son, we also show the maximum reachable average payoff that can beobtained in the state, marked by a dashed grey line, where allplay-ers are in unconditional cooperator states. Note that in thecase ofH = 1 the stable solution is the traditional three-strategy state whenD,C, andL players form a cyclic dominant solution [15]. Here theaverage payoff cannot exceed the loner’sσ = 1 income. By using abit higher tolerance level, however, a significant improvement can bereached, which is comparable to the value obtained in an idealistic(all C) state.

the game where it is a focal player but also in the games of hisneighbors. Therefore a player may participate inG = 5 pub-lic goods games, and the total payoff should be accumulatedaccordingly.

During the strategy update protocol, we apply strategy im-itation based on pairwise comparison of competing strategies[34]. Namely, a playerx will adopt the strategy of a neighbor-ing playery with a probability

Γ(Πx −Πy) =1

1 + exp((Πx −Πy)/K), (8)

whereK is the noise parameter. Without loss of generalitywe will use a representativeK = 0.5 value, which ensuresthat strategies of better-performing players are adopted almostalways by their neighbors, although adopting the strategy of aplayer that performs worse is not impossible.

In an elementary step, we choose a player and his neigh-bor randomly. If their strategies are different then the strategyimitation is executed with the probability defined by Eq. 8.In a complete Monte Carlo step (MCS) every player has onechance on average to update his strategy. To get reliable phasediagrams, which are valid in the large system size limit, thesystem size was chosen from400× 400 to 6400× 6400, andthe relaxation time was between 20000 and 100000 MCS. Tofurther improve accuracy, the results of the stationary statewere averaged over 10 independent realizations for each setof parameter values.

We should stress that the evolution of strategies in a struc-tured population is highly independent of the initial stateif all

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DML 0.0

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n γ

synergy factor r

L DCL DC

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DM

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DML

H = 4

FIG. 4: (Color online) Fullr − γ phase diagrams for the spatial public goods game where players are distributed on a square lattice formingG = 5 size of groups. DifferentH threshold values are indicated. Solid lines represent continuous, whereas dashed lines indicate discontinuoustransitions between stable solutions. The comparison of diagrams shows that a moderate tolerance, an intermediate threshold value ofH , allowsM players to prevail even at a significantly high cost value.

competing strategies are present. The only critical conditionis the sufficiently large system size which prevents finite-sizeeffects and allows a stable solution to emerge somewhere ina space from a random initial distribution. Later this solutioncan invade the whole space and remains stable.

In our model there are three key parameters, namely ther, γ, andH threshold level. To demonstrate their impactson the stable solutions we have presented the resulting phasediagrams for the four possible threshold values in Fig. 4.

In general, as expected, strategyM always dies out if theinspection cost is too large and we get back the well-knownthree-strategy(D,C,L) model. In this case unconditional co-operators can survive abover ≥ 2.19 due to cyclic dominanceand form a three-strategy (D,C,L) phase. If the synergy fac-tor is high enough and strategyC is capable to coexist withD due to network reciprocity then the previously mentionedcyclic dominance is broken, which will result in the extinctionof strategyL. Consequently, a two-strategy (D,C) phase re-mains where the fraction of defectors decreases gradually aswe increaser.

Significantly different behavior can be obtained if the extraγ cost ofM players is reasonably moderate. As a general ob-servation, which is partly against mean-field results, strategy

M becomes viable, but the composition of the stable solutiondepends sensitively on the threshold value of tolerance. Thelowest nonzeroH = 1 value represents a special case becausehereM can only survive withD in the presence ofC players.Due to this low threshold anM player changes from theCto theL state immediately when it recognizes the presence ofa defector in the group, hence the previous mentioned cyclicdominance is established, but instead of the(D,C,L) cyclethe strategiesM → D → C → M will form the stable three-strategy solution. In other words,L is simply replaced byMwho takes the role of the former strategy. The advantage ofa three-strategy solution over the other state depends on theaverage rotation speed between cyclic members: If the inva-sion rate is faster, then it can stabilize a solution [30]. Byincreasingr we may observe a reentrant transition between(D,C,L) → (D,C,M) → (D,C,L) phases, which is againa general behavior when the average invasion rates within acycle can be adjusted by varying a control parameter [35].

If we increaseH and allowM players to “tolerate” thepresence of defectors further then a new kind of solutionemerges, which was already observed in the well-mixed sys-tem. In this caseM can coexist withD without the presenceof a third party. As we will show later, this(D,M) solu-

6

tion can be specially efficient to reach a state when a highaverage payoff can be reached for the whole population. Be-sides the mentioned two-strategy solution, there are parametervalues where all four competing strategies coexist, and thereare some specific cases whenM crowds out unconditionalCbut stay together withL in the presence ofD. HereD andM players are still capable of forming a two-strategy solu-tion, butL players can invade defectors. As a result, smallL patches emerge temporarily, but they are vulnerable againstthe invasion ofM players, who are capable of utilizing net-work reciprocity, which closes the cycle.

Figure 4 highlights that the new kind of solution can alsoemerge in a structured population. In particular, we canobserve a stable coexistence of four strategies (marked byDCML in phase diagrams) which is absent in a well-mixedsystem. Such a kind of coexistence of competing strategies isa general feature of structured populations which is a straight-forward consequence of the limited interactions of players[32].

The comparison of phase diagrams obtained for differenttolerance thresholds highlights that there is an optimal inter-mediate tolerance level which provides the best condition forM players. In this case strategyM can survive even at a sig-nificantly high inspection cost. Note that anM player shouldalways bear this cost but has to invest also in the common poolwhen it cooperates. AtH = 2, for instance,M should paynearly double cost of the unconditionalC strategy, still, it cancrowd out bothC andL strategies. On the other hand, such ahigh “peak” is missing both atH = 1 and atH = 4, whichcan be considered as extreme (too high or too low) thresholdvalues.

Based on the comparison of phase diagrams we can con-clude that neither too small nor too high tolerance will helpM players survive and they become extinct at relatively smallγ values. This observation agrees with our previous experi-ences obtained for a well-mixed population. It is worth stress-ing, however, that a tolerant strategy prevails more easilyin astructured population andM players can survive even at ex-tremely high additional costγ.

In the following we provide an intuitive explanation whytolerance can offer a viable way to handle defection. It shouldbe emphasized that the three-strategy(D,C,L) phase is al-ways a solution in the low-r region [15]. To understand thesuperiority of the(D,M) phase we will start the evolutionfrom a special, prepared initial state where both the cyclicallydominated phase and the stable coexistence ofD andM play-ers could evolve calmly in a restricted area first. Panel (a)of Fig. 5 illustrates the final result of these isolated evolu-tions. After, we let the borders open, and the battle of so-lutions starts. The elementary steps of this competition areidentified in panel (b), which is zoomed out for clarity. In thissnapshot we can distinguish three different cases of how thethree-strategy solution meets with the external two-strategy(D,M) phase. If aC domain, marked by dark blue, is at thefrontier then unconditional cooperators start spreading in thesea ofM . [These invasions are marked by “I” in panel (b).]The success ofC, however, is temporary, because defectors,marked by red, will follow them and gradually invade the in-

D C M L

(a) (b) (c)

(d)

I

IIII

I

III

IV

IV

FIG. 5: (Color online) The competition of two possible solutions atr = 2.7, γ = 0.15, usingH = 2 on a200 × 200 square lattice. De-fectors are denoted by red (middle grey), unconditional cooperatorsare denoted by dark blue (dark grey), tolerant players are denoted bylight blue (light grey), whereas loners are denoted by a green color(dotted lighted grey), as indicated by the legend on the top.At thespecific parameter values both the cyclic dominant(D,C,L) andtwo-strategy(D,M) phases could be possible solutions. Panel (a)illustrates a prepared initial state where a cyclic dominant solutionis embraced by the other stable solution. In panel (b) we openedthe borders and allowed solutions to compete for space. Eventuallythe (D,M) solution crowds out the other phase, as is illustrated inpanel (c), and finally the two-strategy phase prevails (not shown).Panel (d) shows the enlarged part of panel (b) to illustrate the micro-scopic mechanisms that are responsible for the successful invasionof the (D,M) solution. Further details are given in the main text.Snapshots were taken at 0, 70, 210MCSs.

vaders. [This stage is marked by “II” in panel (b).] After,whenD players remain alone withM players then the latter(marked by light blue) will regulate defectors and lower theirconcentration to a minimal level. The second option of howcompeting solutions meet is when aD spot from the(D,C,L)phase meets with the external(D,M) phase. [This is markedby “III” in panel (b).] In this case the previously described“regulation” process starts immediately, which will decreasethe area of the(D,C,L) phase. Finally, when anL domain(marked by green) is at the interface then it will shrink im-

7

mediately becauseM is able to utilize the positive impact ofnetwork reciprocity. (This process is marked by “IV” in thepanel.) Altogether, the three elementary processes will reducethe area of the middle zone. As the total area of the three-strategy phase shrinks, it becomes more vulnerable againstanexternal invasion because the local oscillations of(D,C,L)strategies are significant in small patches. (Note that in themiddle zoneL’s would only survive if defectors feed themdue to the cyclic dominance.) Consequently, when the widthof the middle zone becomes comparable to the typical sizeof patches then the three-strategy phase can be easily trappedinto a homogeneous state. This stage is illustrated in panel(c)in Fig. 5. After, independent of which strategy is present atthefrontier, the phase becomes an easy prey for the two-strategy(D,M) phase. Finally this solution will invade the whole sys-tem (not shown). We stress again that the system will termi-nate into the same state if we start the evolution from a randomstate independent of the initial fractions of strategies.

In agreement with the well-mixed case, the application ofa moderate tolerance level does not only help strategyM tosurvive, but it also has a useful impact on the average payoffofthe whole population. This observation is specially importantbecause a previous work highlighted that the introduction ofloner strategy is unable to solve the original problem of thepublic goods game and the average payoff cannot exceed theincome of the loner’s strategy [15]. Therefore, to participatein the joint venture is not an attractive option for loners.

0

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aver

age

payo

ff

synergy factor r

H = 1H = 2H = 3H = 4

three-strategy

FIG. 6: (Color online) Average payoff in dependence of a synergyfactor using different thresholds of tolerance whenG = 5. The costof inspection isγ = 0.04 for all cases. For comparison, the result ofthe traditional three-strategy model is also plotted whereonly pureD,C andL players are present. The highest collective payoff ismarked by the dashed grey line which can only be reached in theidealistic case if all players cooperate unconditionally in the group.

The concept of tolerance, however, can resolve thisdilemma. Figure 6 illustrates the average payoff in depen-dence of the synergy factor for different threshold values oftolerance at a reasonable cost of inspection. (Note that qual-itatively similar behavior can be obtained for higher cost val-ues.) For comparison, the average payoff is also plotted inthe traditional model where only the pure (D,C,L) strategiesare present. In the latter case the growth of the general pay-

off is being hindered by the presence of strategyL no matterhow we apply higherr. The average income of players canonly increase significantly when loners die out. In the lat-ter case, whenr is high enough, the network reciprocity canlower the fraction of defectors efficiently which will be fol-lowed by the general rise in payoff. To evaluate properly thepayoff values due to tolerance we have also plotted the high-est collective payoff value (marked by the dashed grey line)which can be obtained only if all players cooperate uncondi-tionally in the group. Figure 6 shows that the system can bevery close to this idealistic state even ifM players have tobear an extra cost. In agreement with our previous observa-tion in well-mixed systems, this effect is the strongest at low rvalues, where cooperation would be unlikely otherwise. Thisfeature suggests that the application of tolerance becomesex-tremely useful when the other cooperator supporting mecha-nism, based on network reciprocity, becomes fragile.

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three-strategy

FIG. 7: (Color online) The fraction of defector players in dependenceof a synergy factor for those threshold values where the applicationof tolerance is capable of suppressing the vitality of defectors. Forboth casesγ = 0.04 was applied as for Fig. 6. For comparison, thefraction of defectors is also plotted when only unconditional L andC strategies fight against defection. Note that defectors could onlygrow notably when strategyM dies out.

Rather counterintuitively, the concept of tolerance of de-fection is capable of minimizing the occurrence of defectors.This effect is illustrated in Fig. 7 where we have plotted thefraction of defectors for the cases when the appropriately cho-sen tolerance level can result in a notably high average pay-off. For comparison, we have also plotted the level of de-fection in the reference three-strategy(C,D,L) model. Asthe plot shows, we can reach only limited impact on reduc-ing defection by applying unconditional loners. The latterstrategy gives a too “drastic” response to defection and can-not utilize the positive effect of network reciprocity. Tolerantplayers, however, use both sides of the coin: Punish defectorsby switching to the loner state if it is inevitable, but remain acooperator until the last chance. Consequently, they can reacha competitive payoff, which could also be attractive for otherplayers that will reduce the defection level implicitly.

The clear advantage of an intermediate tolerance level canbe illustrated nicely if we apply larger groups where even

8

more differentH threshold values are available. Largergroups can be easily formed if we extend the interaction rangefrom the von Neumann to the Moore neighborhood whereplayers are arranged intoG = 9 group size with their nearestand next-nearest neighbors. For comparison, in Figure 8 wehave also plotted the related payoff values for a well-mixedsystem where the same group size was applied. These plotsdemonstrate clearly that having tolerant players in the pop-ulation is beneficial to the whole society. Furthermore, in aspatial system, where bonds are limited and are maintainedfor a long time, even a higher tolerance threshold could be thebest compromise which provides the highest average payofffor players at some parameter values. Naturally, this effectcan be even more pronounced for larger group sizes which aretypical for human systems [36–38].

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8

aver

age

payo

ff

tolerance threshold H

1

2

3

4

0 2 4 6 8

FIG. 8: (Color online) Average payoff in dependence of the thresh-old value of tolerance for different synergy factors when a Moore-neighborhood is applied. The inset shows the payoff values for awell-mixed case where the sameG = 9 group size is considered.For both topologies we used the fixedγ = 0.1 inspection cost. Theapplied synergy factors arer = 2.5, 3, 3.5, 4, 4.5 and5 from bottomto top. Note thatH = 0 corresponds to the three-strategy(D,C,L)model whereM players are unable to survive. Because of discretevalues ofH , the dashed lines are just guides to the eye. (The errorbars are smaller than the symbol sizes.)

IV. SUMMARY AND CONCLUSION

It is our everyday experience that tolerance embraces uswhereas the absence of it has serious consequences on thewhole community. Our simple model can provide an intu-itive explanation for its evolutionary origin: Albeit it mightbe costly, but it pays to monitor our neighborhood and reacton how the cooperation level changes around us. Even if werecognize some defection in our group we should show toler-ance towards it because by quitting out from the joint venturewe would loose the possible benefit of mutual cooperation.But, of course, we should not be tolerant endlessly becausesuch an attitude takes the system back to the original versionof the dilemma where uncontrolled defectors can exploit un-conditional cooperators easily. Instead, a delicately adjusted

threshold of tolerance can utilize the advantage of both theunconditional cooperator and the loner strategies. Namely, amoderate tolerance threshold helps to utilize the synergy im-pact of mutual cooperation but it can also keep defection ata bearable level, which altogether can provide a reasonablewelfare for the whole community.

It is worth stressing that unconditional cooperator strategydoes not necessarily represent a “naive” approach from play-ers. There can be those who are generally generous towardsothers but do not want to invest extra effort to inspect others’acts continuously. Being tolerant, however, involves not onlyjust a forgiving approach towards others but also assumes apermanent monitoring of the neighborhood.

It has been studied intensively how players can avoid beingexploited in social dilemmas. One option could be to breakadverse ties or leaving an unsatisfactory neighborhood andbuild new connections on social networks [39–46]. Theseworks focused on the evolving interaction graph and con-cluded that emerging local homogeneities have a decisive im-portance on the evolution of cooperation. Indeed, focusingon the similarity of partners or tag-based support is a well-known mechanism, which could provide a clear advantage forcooperation [47–50]. But some tolerance, according to thepresent study, might be beneficial, which has crucial impor-tance especially when the average group size in a communityis considerably large.

Our paper underlines that the positive impact of toleranceis robust and can be observed both in well-mixed and in struc-tured populations. The effect, however, is more pronouncedin a spatial system because network reciprocity augments thebasic mechanism. The supporting influence of spatiality couldexplain the widespread emergence of tolerant behavior [51].

One may claim that strategyM is conceptually similar toa tit-for-tat strategy [12, 52–54]. Indeed, there is some sim-ilarity becauseM players can behave differently in differentsituations, but the concept of tolerance offers a more sophis-ticated reaction that is more beneficial to the whole commu-nity. Our last figure illustrates that the best response to varyingconditions could be different and sometimes more, sometimesless tolerance provides higher average income, hence a sim-ple reactive strategy would be too rude to respond adequately,especially in the case when multi-point or group interactionis considered [55]. A logically similar approach could be thepossibility of conditional cooperation or conditional participa-tion in joint efforts [56–59], but the present paper revealsthesignificant role of additional cost, which was partly ignored inprevious works.

Being tolerant to a certain point can also be considered asa threshold game where there is a nonlinear relation betweenthe benefit of the whole group and the proportion of coop-erators [60–63]. Our model, however, is conceptually closerto conditional strategies where the decision is made on a per-sonal level which provides an optimal choice not only for anindividual but also for the whole community.

Lastly, we note that several pioneering works demonstratedthe utility of punishment but also highlighted its side effects[64, 65]. Namely, it could be effective to control defection,but simultaneously, the usage of punishment may lower the

9

income of both punisher players and those who are punished.On the other hand, reward has a cooperation supporting im-pact but it also requires an additional source (of reward) toapply it [66–71]. The presently discussed mechanism, how-ever, offers a simple, but still effective, way on how we cantame defection without losing well-being.

Acknowledgments

This research was supported by the Hungarian National Re-search Fund (Grant K-101490) and by the Fundamental Re-search Funds of the Central Universities of China.

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