Blocking defector invasion by focusing on the most successful partner
aa r X i v : . [ phy s i c s . s o c - ph ] J un Blocking defector invasion by focusing on the most successful partner
Attila Szolnoki a , Xiaojie Chen b a Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O.Box 49, H-1525 Budapest, Hungary b School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
Abstract
According to the standard protocol of spatial public goods game, a cooperator player invests not only intohis own game but also into the games organized by neighboring partners. In this work, we relax thisassumption by allowing cooperators to decide which neighboring group to prefer instead of supporting themuniformly. In particular, we assume that they select their most successful neighbor and focus externalinvestments exclusively into the related group. We show that this very simple alteration of the dynamicalrule results in a surprisingly positive evolutionary outcome – cooperators prevail even in harsh environmentrepresented by small values of the synergy factor in the game. The microscopic mechanism behind thereported success of the cooperator strategy can be explained by a blocking mechanism which affects thepropagations of competing strategies in a biased way. Our results, which remain intact by using differentinteraction topologies, reveal that it could be beneficial to concentrate individual efforts to reach a higherglobal wellbeing.
Keywords: cooperation, public goods game, strategy invasion
1. Introduction
Explaining the emergence of cooperation among selfish agents who are interested in their best personalinterest is a long-standing problem that has attracted intensive scientific activity in the last two decades[1, 2, 3, 4, 5, 6, 7, 8]. Several mechanisms have already been identified, which highlight the importance ofdifferent forms of reciprocity [9]. As expected, monitoring players by rewarding positive act or punishingbad behavior supports cooperation efficiently [10, 11, 12, 13, 14]. Notably, the punishment can be executedin different ways like by decreasing payoff or via exclusion from mutual benefit [15, 16, 17]. In these cases thedilemma is transformed to another stage because not all cooperators want to bear the cost of an additionalinstitution. Consequently, they become the so-called second-order free-riders and we practically face theoriginal dilemma [18]. Interestingly, this problem can be resolved automatically in structured populationswhere players have limited range of interactions [19, 20]. In spatial systems, those players who bear thecosts of both cooperation and additional institution can separate from simple cooperators hence the formergroup can fight against defection more efficiently [21].However, the most interesting intellectual challenge is to identify those rules and mechanisms which arestrategy-neutral. In the latter cases, when we apply these rules, there is no obvious preliminary reasonto support cooperative acts. An instructive example for a cooperator supporting environment is a highlyheterogeneous interaction graph [22]. By following this research path, the breaking of symmetry, or theintroduction of social diversity among competitors are now believed to be conducive to cooperation [23, 24,25]. For the mentioned cases, a generally valid explanation is that stronger players, who have higher socialinfluence, can spread their strategies in their close neighborhood, resulting in a local coordination in theinvolved patches [26, 27, 28, 29]. Consequently, this type of separation of competing strategies reveals theevolutionary advantage of cooperation.Inhomogeneity can also be introduced in alternative ways. For example, in a public goods game, playersare asked to contribute to a common pool. Their contributions are enlarged by a synergy factor and after it
Preprint submitted to Applied Mathematics and Computation June 16, 2020 s redistributed among group members. In a structured population, a player is involved in different groupsand cooperators are believed to contribute to all related games. However, there are real-life examples whenplayers distribute their contributions in an unequal way.Although the option of heterogeneous investment into different games has been already suggested bysome previous works, these research papers can be grouped along the following directions. In the firstgroup, models focused on heterogeneous interaction graphs. Consequently, they assumed that a player’sinvestment to an external group depends on the degree of the focal player [30, 31]. In this way, the suggestedmicroscopic rule strongly utilizes the heterogeneous topology of interaction graph and does not consider thelimited source of cooperator players. The other groups of models assume sophisticated and demanding skillsof players. In particular, they suggest that a player’s contribution to a specific group depends on their incomereceived from the given group in the previous simulation step [32, 33]. Not really surprisingly, by applyingsuch kind of microscopic rule, we practically enforce the reciprocity mechanism between cooperator players.It is because a cooperator group, where the redistribution is high, can expect additional support from anexternal cooperator. The rest of related models of heterogeneous investment broke the “strategy-neutral”principle directly and considered actual strategy choice of group members. For example, in Ref. [34] thecontribution of a cooperator player into a group depends on the fraction of cooperators within that group.In an analogous work, it was suggested that the investment of a cooperator in a group organized by anotherpartner depends on the reputation of the latter player [35]. As expected, these rules resulted in a highercooperation level because they directly support this strategy.While our present model also considers heterogeneous investment in a public goods game, it does notbelong to any of the previously mentioned paths. In particular, in our case, a player needs no additionalinformation about the topology of interaction graph because his decision does not utilize the proper degreedistribution of neighbors. Furthermore, players are not requested to record their previous incomes fromspecific groups for a decision about their investment. Instead they only need to know the payoff of theirneighbors. The latter, however, should be available for every model, where strategy update is based onthe payoff difference of interacting players. Last, and most importantly, our proposed microscopic rule isstrategy-neutral, because when a player decides about his investment onto a specific group organized by oneof his neighbors, then the actual strategy of his partner has no importance. Surprisingly, this very simplemodel, which requires nothing more than the traditional setup, provides a highly cooperative evolutionaryincome even at small values of synergy parameter which mimics harsh environment in general.In fact, we will show that without increasing the total investment of players it could be beneficial forthe whole population to concentrate individual efforts instead of keeping the uniform, and seemingly moredemocratic investment policy. Before presenting our observations in more detail, we first proceed with theaccurate description of the proposed public goods game with unequal investment rule.
2. Focusing on the most successful partner
For simplicity, we define the applied public goods game on a square lattice with periodic boundaryconditions, but the extension to other topologies is straightforward. In the mentioned case, L × L playersare arranged into overlapping groups of size G = 5 in a way that everyone is connected to its G − g = 1 , . . . G different groups where the first is organizedby the focal player, while the rest G − x is designated either as a defector ( s x = D ), or as a cooperator ( s x = C )randomly. According to the standard protocol, a cooperator player invests a c = 1 amount to each gamewhile defectors contribute nothing. The sum of all contributions in each group is multiplied by the synergyfactor r and the resulting public goods are distributed equally amongst all group members independently oftheir strategies. Notably, the total payoff of every player is the sum of the incomes collected from relatedgames. In the following we assume that a cooperator player considers an alternating investment policywith probability α . We refer to this as selective cooperator ( SC ) state. Otherwise, with probability 1 − α ,he follows the standard investment protocol and invests into every external games equally. In the formercase the cooperator invests into his own game the usual c = 1 amount, but his remaining ( G − · c contribution is distributed unequally. More precisely, the mentioned SC player looks for the specific partner2 n n n n m k k k Figure 1: The focal F player collects income from not only his own game, marked by a yellow set, but also from the gameorganized by his neighbor n . The group of the latter game is marked by a dashed green ellipse. Note, however, if cooperator m is focusing on the best neighbor, then he invests contribution to n ’s game only if the payoff of n player is higher thanthe payoff of k , . . . , k players. If this is the case then m player invests all his external ( G − · c = 4 contribution here.The latter act is marked by an arrow. Similarly, the focal F cooperator always contributes his own game, but his externalinvestment depends on its own state. In normal case, that happens with probability 1 − α , a regular cooperator F contributesto n ’s game by c = 1. With probability α , F focuses on the best neighbor and invests into n ’s game only if the payoff of n exceeds the payoff of n , . . . , n players. Otherwise, F contributes nothing to n ’s game no matter he is in a cooperator state. in his neighborhood with the highest payoff. After selective cooperator invests all his ( G − · c externalcontribution exclusively to the game organized by the most successful neighbor. Given that there are morethan one neighbors with the highest payoff value in the neighborhood then the mentioned cooperator selectsone of them randomly.In Fig. 1 we have summarized the investment policy of our present model. As mentioned, the focal F player collects income from not only his own game, which is marked by a yellow set, but also from thegames organized by his neighbors. One of its external groups is marked by a dashed green ellipse in theplot. Evidently, the payoff collected from the game organized by the neighboring n player depends on thecontribution of m player. If this player is in a SC state, then m contributes to n ’s game only if thepayoff of n player exceeds the payoff values of k , k , and k neighboring players. In the mentioned case m contributes to n ’s game by ( G − · c = 4 amount, as illustrated by an arrow. Otherwise it contributesnothing, no matter m is a cooperator. Notably, a cooperator player always contributes a c = 1 amountinto his own game independently whether he is in an unconditional or selective cooperator state.We should stress that contrary to previous works, our modified investment rule does not utilize theheterogeneous topology of interaction graph, hence it can be applied for homogeneous topology as well. Wealso note that our observations are not restricted to lattice topology, but remain valid on random graphs too.We note that since the investment decision of a selective cooperator requires the knowledge of actual payoffvalues of neighbors, therefore in the zero step we provide a random payoff value for all players from the[ G · ( r − / ±
1] interval. Naturally, in the following steps the players’ payoff values are updated accordingto the proper states of their neighbors. We highlight that the actual initial payoff values have no relevantconsequence on the outcome, they only serve the proper launch of the simulation steps. When we modifiedthese values, we observed identical final state at a specific values of r and α .The rest of the dynamical rule follows the standard procedure. More precisely, during an elementaryMonte Carlo step we choose a player x and one of his nearest neighbors y at random. If the strategies of theseplayers are different, then the related Π x and Π y payoff values are calculated by summing all the incomesacquired in each individual group. Then player y adopts the strategy from player x with a probabilitygiven by the Fermi function w = { y − Π x ) /K ] } − , where K = 0 . L × L = 100 ×
100 to 400 × N = 10 players. In all cases, the stationary valueof cooperation level f C was determined after a typical 10 relaxation steps and f C values were averaged 50independent runs. In the light of results presented first in the next section, we have also studied a slightlymodified model which details will be described in the next section.
3. Results
We first present some representative cooperation levels obtained at different α values in dependence ofthe synergy factor. As Fig. 2 highlights, the introduction of unequal investment policy stimulates a positiveevolutionary outcome. More precisely, by increasing α , means when we increase the chance that a cooperatorbehaves as a selective cooperator, then cooperators survive even at small values of synergy factor r . Forexample, at r ≈ . α = 0 .
5, when there is equal chance that a cooperator distributes his contributionsuniformly or focuses them into a single group, then the system evolves toward a full cooperator state. Notethat at this value of synergy factor the system would always terminate into a full defector state at α = 0,when cooperators disseminate their contributions among their neighbors uniformly. c oop e r a ti on l e v e l , f C synergy factor, r α Figure 2: Cooperation level in dependence on synergy factor for different values of α parameter as indicated in the figurelegend. This plot suggests that cooperation is largely supported if cooperator players prefer to support exclusively the bestneighbor independently of the latter strategy. The results were obtained on square lattice where the linear system size are L = 300 and the error bars are comparable to the symbol size. The complete behavior on the α − r parameter plane is summarized in Fig. 3. This plot confirms that ahigher cooperation level can be achieved when cooperator players prefer to focus their external investmenttoward a specific neighbor, instead of supporting the whole environment uniformly. The best results areobtained when cooperators give up traditional unconditional cooperator state and instead they select theirinvestment target exclusively. We stress that we do not expect any additional effort or cognitive skillfrom players, which are not already available in the traditional model. In particular, players do not needanything to know about the interaction topology, which was a fundamental condition in previous works.Furthermore, they do not have to record their past income originated from earlier games played with theirneighbors. Evidently, to collect the mentioned information would require an extra effort that should beconsidered via an extra cost or lowered payoff value. But in our present model, when cooperators make a4 α rD D+C C Figure 3: Phase diagram, depicting the stable solutions ( C -full cooperator state, D -full defector state and D + C -mixed state)on the α − r parameter plane. In agreement with the previous plot, cooperators can fight more efficiently against defection athigh α values, where they focus their external contributions on a single neighbor, who does the best in their neighborhood. decision about their investment, they only need to know the payoff of their neighbors. This information,however, is available in the traditional model because imitation probability is also based on the payoffdifference of competing partners.We stress that our observations are not limited to square lattice but remain valid for other topologies.Evidently, if the interaction graph is highly heterogeneous, then hubs, who are able to collect high payoff, willsuccessfully attract the investments of neighboring cooperators. Consequently, in this case we can observethe same mechanism as was previously observed for scale-free graphs [38]. Namely, cooperator hubs willcollect high payoff and become strong. Initially, defector hubs can utilize their neighbors, but later, whenthe neighboring players adopts the most successful strategy, then defector hubs become vulnerable. In thisway it is not really surprising that heterogeneous investment supports cooperation on highly heterogeneousgraphs [30, 31].From this point of view, it is more interesting to check homogeneous graphs where there is no relevantdifference between the degree distribution of players, but the topology is not necessarily translation invariantas for square lattice. Motivated by these arguments, in Fig. 4 we present results obtained by using randomregular graph. For proper comparison we used the same k = 4 degree distribution as for square grid[36]. This plot suggests very similar behavior we previously reported in Fig. 2. Consequently, the positiveimpact of selected investment on cooperation level is a more general phenomenon that is not restrictedto translational invariance interaction topologies. Hence we can conclude that the reported effect can beobserved even on homogeneous graphs where there is no significant difference in players’ degree. But moreimportantly, we do not need a highly heterogeneous degree distribution which was an essential condition toobtain a cooperation supporting mechanism previously.Until this point we assumed that cooperators were uniform and they all have a certain chance to switchfrom traditional cooperator into the selective cooperator state, which is controlled by parameter α . The phasediagram, however, illustrates that the best solution can be reached at α = 1, which means that cooperatorsalways behave as selective cooperator in the latter case. An intriguing question can be posed, here. Namely, isit possible to reach a higher global well-being via a selection mechanism? Put differently, how does the systemevolve if we introduce pure cooperators and selective cooperators as permanent strategies simultaneously? Inthe latter case we have a three-strategy model where unconditional defection ( D ), unconditional cooperation( C ) and selective cooperation ( SC ) strategies compete. Their relation is far from trivial because a C and a SC player have identical cost, the only difference is how they distribute their investment. Furthermore, an SC supports just only one of his neighbors, therefore the emergence of networks reciprocity, which is a basicmechanism in structured populations, can hardly evolve among SC players. To clarify this question we havestudied this modified model and found that unconditional cooperators always die out, hence the evolutionaryoutcome depends only on the relation of D and SC strategies. In other words, the system always terminates5 c oop e r a ti on l e v e l , f C synergy factor, r α Figure 4: Cooperation level in dependence on synergy factor for different values of α parameter as indicated in the figurelegend. Here random interaction graph was applied where we used the same k = 4 degree as for square lattice. The cooperatorsupporting consequence of the selected investment protocol remained intact no matter random topology was used. The systemcontained N = 10 players. into the state that was observed for the α = 1 case in the previously studied homogeneous model. Thereforewe can conclude that SC strategy, which is globally beneficial, could be selected during an evolutionaryprocess.To understand the mechanism more deeply, which is responsible for the advantage of SC strategy, inFig. 5 we present a pattern formation process when the system was launched from a prepared initial state.Here the three competing strategies are designated by different colors. Namely, defectors are marked by red,unconditional cooperators by deep blue while selective cooperators are denoted by light blue. Intentionally,we here choose a low r = 3 value of synergy factor that would result in a full defection in the traditional model.For easier comparison we marked by dashed yellow lines the original positions of border lines. In panel (b)and in panel (c) it is easy to see that both defectors and SC players invade unconditional cooperators. Butthe invasion of D strategy is more effective and they invade the majority of space originally occupied by C strategy. When C players die out, the proper final state of the evolution depends only on the relation of D and SC strategies. At the mentioned specific r = 3 value of synergy factor, as already shown in Fig. 3, SC invade defectors and prevail the whole system. Note that this final state is not shown in the plot. (a) (b) (c) (d) Figure 5: Characteristic snapshots of pattern formation starting from a prepared initial state. Here we have three purestrategies, namely defectors (red), unconditional cooperators (dark blue), and selective cooperators (light blue). The initialpositions of frontiers are marked by dashed yellow lines in all panels. Panel (b) illustrates clearly that both defectors andselective cooperators invade unconditional cooperators. But defectors, shown in panel (c), do it more efficiently and conquerthe majority of space originally occupied by unconditional cooperators. Selective cooperators, however, dominate defectors andfinally prevail the whole system (not shown). The snapshots of 180 ×
180 system were taken at 0 (a), 100 (b), 200 (c) and 700(d) full
MCS steps, and the synergy parameter value was r = 3 . For a full explanation, it is instructive to apply an alternative coloring technique that reveals the repre-sentative microscopic process. For this purpose, we use not just the previously introduced colors of strategies6 a) (b) (c) (d)
Figure 6: Pattern formation starting from the same initial setup as for Fig. 5. Here we used a special coloring to mark thoseplayers who have the highest payoff in a group and enjoy the support of a neighboring selective cooperator player. Morespecifically, we mark those defectors, unconditional cooperators, and selective cooperators by black, gray, and white colorrespectively. For clarity we present a smaller 60 ×
60 system where individual players are visible. The mentioned supportedplayers are not present in the borderline between competing domains, but they are generally behind it in the next lines. Theirtypical positions during the evolution are highlighted by yellow ellipses in panels (b-d). but we also add three extra colors. Namely, we mark by black, gray, and white colors those
D, C , and SC players respectively who are supported by a neighboring SC player with an extra large ( G − · c investment.This coloring technique is applied in Fig. 6 where we present a smaller system to make individual playersvisible. Similarly to the previous plot here we used the same r = 3 parameter value, therefore the systemwill terminate again into the full SC state (not shown in the figure). In panel (a) we only see white “highlysupported” players who are distributed randomly in the bulk of SC domain. This is due to the originallyrandomly distributed payoff values in the zero step. Later, when the strategy propagation is launched, frontsstart moving between homogeneous domains. Interestingly, however, it is very rare that black or gray pixelsemerge, which simply means that it almost never happens that an SC player supports a neighboring C or D player. Furthermore, which is also very important, white pixels cannot be detected in the front line. Moreprecisely, at the front separating D and SC domains, it is typical that red and light blue players are facingeach other. Similar situation can be detected at the front between C and SC domains, where black andwhite pixels are hardly detected. The representative feature of the emerging patterns are highlighted byyellow ellipses in the panels (b–c) of Fig. 6.Based on these observations we can easily reveal the key mechanism that is responsible for the successof SC strategy. For simplicity, we describe the elementary step of domain wall propagation between D and SC domains, but conceptually similar explanation can be given for the competition of C and SC strategies.In Fig. 7 we show the competing domains, where neighboring D and SC compare their payoff duringthe strategy invasion. In the traditional model D would exploit a neighboring cooperator partner. First,because an unconditional cooperator would invest directly to the game organized by D . Secondly, D wouldlargely benefit from the game organized by the mentioned cooperator because a substantial contribution iscollected from cooperator members of the mentioned group. But now both elements are missing. First, SC does not invest directly into the game organized by D because SC is more attractive target by having ahigher payoff. Secondly, the income from the game organized by SC is minimal. It is because only SC invests into his own game marked by a solid yellow circle, all other neighboring selective cooperator supporta more successful neighbor. In sum, D has only a modest income in the neighborhood of an SC player.As we argued, SC is weakened at the front, but he has an escape route not to be totally vulnerable. Itis because SC benefits from the very successful game organized by the neighboring SC player. This gameis marked by a dashed yellow circle in the plot. Here not just SC , but also other neighboring SC playersinvest a huge amount into the game, hence all enjoy a large split after multiplication. As a result, SC stillhas a reasonable payoff that is competitive to the payoff of D . Summing up, the introduction of a selectiveinvestment policy results in weakened players in both sided of the front, but defectors suffer more, hencetheir invasions are largely blocked.It is easy to see that similar argument can be given for the competition of C and SC strategies, which ex-plains why SC invades unconditional cooperation no matter they bear the same total cost and the emergence7 SC SC Figure 7: Mechanism that blocks the invasion of defector strategy. Here we used the same color coding as for previous Fig. 6.Accordingly, defector and selective cooperator domains compete where white color marks those players who are supportedexclusively by neighboring SC partners. D cooperator at the border cannot utilize the selective cooperator SC playerbecause SC donates all his external contribution to the game organized by SC player. Furthermore, D has just a minimalbenefit from the game organized by SC player because here the only contribution comes from SC . This group is marked by asolid yellow curve. On the other hand, SC is able to collect a reasonable income from the game organized by the SC player.The latter group is marked by a dashed yellow circle. As a result, SC ’s cumulative payoff can be larger than the payoff of D player even for relatively small r values. of network reciprocity is less obvious for the former strategy.
4. Conclusion
It is clear to see that when someone cooperates then the primary aim is to elevate the general well-beingof the group or community. But one may ask which way serves this goal more efficiently? Is it better tosupport anyone or is there a smarter way to use our efforts? For instance, one may argue that it is uselessto support a group which does not functioning well, where the leader is unsuccessful. On the other hand,our investment may reach the highest impact if we focus on a venture that is organized by a successfulplayer. In this work we have elaborated this idea where we allowed cooperator players to concentrate theirexternal investments to a single group led by the most successful neighbor. Importantly, a cooperator doesnot consider the strategy of the target neighbor, but checks only its payoff. In this way the decision abouta cooperator’s investment requires no extra information comparing to the standard public goods game.Surprisingly, the suggested strategy-neutral investment policy elevates the cooperation level dramaticallyeven at small values of synergy factor, where the traditional model would suggest a full defection state.We note that different forms of heterogeneous investment was already studied by several earlier works[30, 31, 39]. However, they assumed more complicated rules which demand more intellectual effort, henceadditional care from cooperator players. Our present model, however, is the simplest because it assumesnothing additional information than is already available for players during the strategy imitation process.It is worth stressing that the present positive effect also works in homogeneous graphs where some earliermodels, which built on the strong heterogeneity of the interaction graph, would fail to support cooperation.The microscopic mechanism, which explains the success of the suggested protocol, is based not onthe usual reciprocity-based arguments. The latter can be found in various forms in models of structuredpopulations, where limited number of interactions of players offers not just the chance to separate fromthose who exploit others but also enlarge the positive consequence of direct reciprocity [40, 41, 42, 43]. Inour present case, as we argued, the introduced investment policy weaken all fighters who are in the frontline between the competing domains - independently of their actual strategies. But this weakening effectis biased and defectors suffer more from it. As a consequence, they are unable to exploit the vicinity ofcooperators hence they loose their evolutionary advantage. They become less attractive and their invasionis completely blocked. The mentioned weakened cooperators in the front line, however, still have a chanceto enjoy the vicinity of successful cooperators behind them, hence they benefit modestly from the success oftheir neighbors. In sum, weakened cooperators still do better than weakened defectors, hence the directionof strategy propagation can be reversed. 8t is worth noting that the reported cooperator supporting mechanism fits nicely to those observationswhere the introduced strategy-neutral rule has biased impact on the strategy invasion of competing strategies[44, 45, 46, 47, 48, 49]. Hence these mechanisms provide an alternative way to understand to original enigmaand explain why cooperation may prevail among selfish agents.This research was supported by the Hungarian National Research Fund (Grant K-120785) and by theNational Natural Science Foundation of China (Grants No. 61976048 and No. 61503062).
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