Social dilemmas in off-lattice populations
SSocial dilemmas in off-lattice populations
B.F. de Oliveira a , A. Szolnoki b a Departamento de F´ısica, Universidade Estadual de Maring´a, 87020-900 Maring´a, PR, Brazil b Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, H-1525 Budapest, Hungary
Abstract
Exploring the possible consequences of spatial reciprocity on the evolution of cooperation is an intensivelystudied research avenue. Related works assumed a certain interaction graph of competing players and studiedhow particular topologies may influence the dynamical behavior. In this paper we apply a numerically moredemanding off-lattice population approach which could be potentially relevant especially in microbiologicalenvironments. As expected, results are conceptually similar to those which were obtained for lattice-typeinteraction graphs, but some spectacular differences can also be revealed. On one hand, in off-latticepopulations spatial reciprocity may work more efficiently than for a lattice-based system. On the otherhand, competing strategies may separate from each other in the continuous space concept, which gives achance for cooperators to survive even at relatively high temptation values. Furthermore, the lack of strictneighborhood results in soft borders between competing patches which jeopardizes the long term stabilityof homogeneous domains. We survey the major social dilemma games based on pair interactions of playersand reveal all analogies and differences compared to on-lattice simulations.
1. INTRODUCTION
In a multi-agent system the assumption when ev-ery member interacts with all others randomly canbe handled analytically, hence it could always bea starting point to study the evolution of coopera-tion among self-interest players [1, 2, 3, 4, 5]. Theabsence of stable connections, however, is a highlysimplified working hypothesis because in almost ev-ery real-life examples individuals have fixed, or atleast temporarily stable neighbors [6, 7]. This ob-servation can be modeled by assuming an interac-tion graph where players have limited and stablepartners, which fact determines their potential fit-ness and the dynamical process in the applied topol-ogy [8, 9, 10]. As expected, this modification maychange the system behavior significantly which wasconfirmed by thousands of research papers in thelast two decades [11, 12, 13, 14, 15, 16, 17, 18, 19].Notably, the graph approach does not always re-flect faithfully the interactions of individuals. Forinstance, in a microbial environment an off-latticeapproach seems to be more appropriate modelingtechnique, where individuals still interact with alimited number of partners, but their actual dis-tance, which may change continuously, determines the interaction strength [20, 21, 22]. In the lastyears several experimental works have been pub-lished where an off-lattice model seems to be a morerealistic assumption [23, 24, 25].We must stress, however, that off-lattice simula-tions are more demanding technically and requiressignificantly larger numerical efforts comparing toon-lattice or in more general graph-based simula-tions. Therefore it is not surprising that previousworks were restricted to the latter case exclusively[26, 27, 28, 29, 30, 31, 32, 33]. In our present workwe focus on off-lattice simulations and explore theirspecific characters. Importantly, we use the origi-nal social dilemma games to identify the similaritiesand potential differences between on- and off-latticeenvironments and only change the dynamical ruleswhich can be applied for off-lattice simulations di-rectly. We will show that the behavior of systemsin off-lattice environment is conceptually similar tothose observed for models of lattice-type interac-tion graphs. There are, however, some differenceswhich warn us to treat the conclusions of lattice-based models with a special care when we want toadopt them directly to microbiological or relatedsystems.In the following we specify the model and the mi-
Preprint submitted to Chaos Solitons and Fractals February 9, 2021 a r X i v : . [ q - b i o . P E ] F e b roscopic rules in more detail. After we present itsconsequences for prisoner’s dilemma game, which isextended to the related social dilemmas includingsnow-drift and stag-hunt games. Finally we con-clude with the summary of the results and a dis-cussion of their implications in the last section.
2. MODELING DILEMMAS IN OFF-LATTICE ENVIRONMENT
As we emphasized, we consider the traditionalsocial dilemma games where players interact withtheir partners and collect payoff elements from ev-ery specific pair interaction [34]. A player’s statecan be described as a cooperator or defector. Whentwo cooperator players meet then both of them ob-tain a payoff value 1, while the meeting of defectorsyields zero payoff value for each participant. Theinteraction of a cooperator and a defector playersprovides a value T (temptation) for the latter and S value (sucker’s payoff) for the former strategy. Inthis way, the actual values of T − S pair determinethe character of the social dilemma. By keeping thetraditional parametrization we use T > S = 0 fixed to describe the (weak)prisoner’s dilemma game. Furthermore, a parame-ter 0 ≤ r ≤ T = 1 + r and S = 1 − r servesas a control parameter to span the snow-drift gameregion of the T − S parameter plane. Last, the stag-hunt game region is covered by the same 0 ≤ r ≤ S = − r and T = r [35].In an off-lattice environment N players are dis-tributed randomly on a square-shaped box of linearsize L = 1. As usual for spatial populations, peri-odic boundary conditions are applied. For every k player the horizontal x k and vertical y k coordinatesare continuous variables. A player k and a player m interact if they are within the interaction range,namely their distance is less than l i . Note that tocalculate the proper distance we consider the men-tioned periodic boundary conditions.To introduce an evolutionary dynamics we as-sume that a player’s strategy may change accordingto the broadly applied pairwise comparison imita-tion dynamics [36]. More precisely, a player k willchange its strategy to the opposite strategy repre-sented by a neighboring player m with a probability w that depends on the difference of the Π k and Π m payoff values collected by the mentioned players: w = 11 + exp[(Π k − Π m ) /K ] . In this Fermi-function-type formula parameter K represents a noise level, where the zero limit makesthe change deterministic for positive difference andforbidden in the reversed case, while large K limitprovides a random strategy change independentlyof the actual payoff values of players.It is important to stress that in an off-lattice sim-ulation we cannot follow precisely the usual wayof strategy adoption applied in graph-based sim-ulations. In particular, we cannot simply replacethe strategy of the target player with the new oneand leave its position unchanged because this wouldmake the evolutionary outcome highly sensitive onthe initial spatial distribution of players. Instead,we remove the player who wants to change its strat-egy and add a newborn player with the new strat-egy somewhere randomly within an l b distance ofthe model player. This modification is similar tothe so-called death-birth dynamics, but still keepsthe essence of pairwise comparison and makes pos-sible the match with graph-based simulations [37].In the default case l b is chosen to be equal to l i ,but we also discuss smaller and larger ranges whenadding a newborn player to replace the old one.As we will show the former option has no, whilethe latter change does have relevant consequenceon the results. In general, it is important to em-phasize that the evolutionary outcome could be sig-nificantly different even if we use the same param-eter values of the model. For instance the systemmay be trapped into a homogeneous full defectorstate, but we may also observe a coexistence phaseby using another seed value of our random numbergenerator. Therefore to obtain a reliable observa-tion about the system behavior it is vital to averagethe results of individual runs. At every parametervalues we distributed the competing strategies andthe positions of N players randomly and monitoredhow the fractions of strategies change in time. Afterwe repeated every simulation 1000 times to obtainthe requested accuracy of averaged value.
3. RESULTS
First we summarize our observations obtained forprisoner’s dilemma game and after we briefly out-line the results of other dilemma games. But be-fore jumping deep into the prisoner’s dilemma casewe first present the results obtained for differentlengths of the interaction range. This helps us toget impressions and evaluate properly the charac-teristic length scale of the l i parameter. Figure 12 . . . .
81 0 .
98 1 1 .
02 1 .
04 1 . N =5000 K =0 . ρ C T l i = l b =0 . l i = l b =0 . l i = l b =0 . Figure 1: Cooperation level in dependence of temptation T for prisoner’s dilemma. The applied characteristic lengthvalues are indicated for each curves. If l i = l b ≥ . shows the cooperation level in dependence tempta-tion value for different characteristic length scales.Here we used the default model where the inter-action ( l i ) and reproduction ( l b ) ranges are equal,while the consequences of unequal scales will be dis-cussed later. Our first observation is the criticaltemptation level where cooperators dies out is verysimilar to those values we observed for simulationson lattice-type interaction graphs [38]. There is,however, a conceptual difference from the systemssimulated on lattices. As our present plot shows,cooperators become fully dominant as we approach T = 1 temptation value, which is missing in alattice-type environment. For example on squarelattice cooperators control only the two thirds of thewhole population even at T = 1 because the strictdegree number rule of nodes allows defectors to co-exist with cooperators [39]. But in our off-latticeenvironment there is no such artificial constraintand the advantage of spatial reciprocity enjoyed bycooperator strategy manifests entirely. Our plotalso warns us that by choosing too large character-istic length scale the system behavior becomes prac-tically identical to the one observed for well-mixedpopulation. In particular, if l i = l b ≥ . T = 1 .
03 for N = 7500, while . . . .
81 0 .
98 1 1 .
02 1 .
04 1 .
06 1 .
08 1 . . l i = l b =0 . K =0 . ρ C T N = 2500 N = 5000 N = 7500 Figure 2: Cooperation level as a function of temptation valuefor different density of players. The total number of inhab-itants are marked in the legend. Rare population providesbetter chance for cooperation strategy. the same temptation value provides a cooperatordominance for the N = 2500 case. We should em-phasize that the improvement of cooperation levelis conceptually different from the one observed onlattice simulations where there was an optimal in-termediate concentration of players which ensuredthe highest cooperation level [40, 41, 42]. While thelastly mentioned phenomenon was strongly relatedto the percolation threshold of the specific latticestructure, it has a different explanation in off-latticeenvironment.But before discussing its origin, let us comparethe time evolutions of cooperation level for an off-lattice and a square-lattice environment. Impor-tantly, all other parameters, including the tempta-tion value, the noise level, the number of playersare identical to both cases. Figure 3 demonstratesclearly that there is a strong fluctuation in off-lattice environment compared to lattice-based sim-ulations and the system travels between the full co- . . . .
810 500 1000 1500 2000 ρ C t off-latticeon-lattice Figure 3: Time evolution of cooperation level obtainedfor off-lattice and on square lattice simulations where allother parameters, including temptation value, noise, and to-tal number of inhabitants are equal. In particular, T =1 . , K = 0 . , N = 5000. It is salient that the fluctuationsfor off-lattice population are significantly larger than in caseof on-lattice population. a) (b) Figure 4: Spatial distribution of players in a coexistencephase. In both panels defectors are marked by red whilecooperators are denoted by blue color. In a rare population,shown in panel (a), strategies can be separated easily result-ing in a frozen final state. This phenomenon makes possiblefor cooperators to survive even at a relatively high tempta-tion value, as shown in Fig. 2. When the average densityof population is high, illustrated in panel (b), then compet-ing strategies maintain a dynamical equilibrium. The latterstate is common in on-lattice populations. Parameters are N = 2500 , b = 1 .
03 for panel (a), and N = 7500 , b = 1 .
01 forpanel (b). In both cases l i = l b = 0 .
05 were applied. operator and full defector states permanently. Thedescribed phenomenon can be monitored in the at-tached video where the color of population changesalmost periodically [43]. This animation also helpsus to understand the origin of this heavy oscilla-tion. Importantly, the borders separating homoge-neous domains are not as sharp as for graph-basedsimulations. This soft intermediate zone allows de-fectors to crack the phalanx of cooperators whichwould be robust and steady in a lattice population.As a result, homogeneous blue patches of cooper-ators diminish eventually giving room for red de-fector players. A homogeneous red patch, however,becomes vulnerable, too, because in the absence ofcooperators defectors cannot exploit their neighborsand they are unable to collect competitive payoff.The above mentioned separating zone becomesempty for small N values which has a crucial conse-quence especially for higher temptation value. Herea homogeneous island may become isolated fromthe rest of the population, hence resulting in a co-existence of competing strategies. In this frozenstate cooperators can survive even if the relativelyhigh temptation value would dictate a full defectorstate on a lattice.The two types of coexistence are illustrated inFig. 4. In panel (a) we have plotted the above men-tioned frozen state which can be observed for small N and high T values. Technically the competingstrategy “coexist”, albeit they have no proper inter- . . . .
81 0 .
98 1 1 .
02 1 .
04 1 .
06 1 .
08 1 . . l i = l b =0 . K =0 . P c o e x . T N = 2500 N = 5000 N = 7500 Figure 5: The chance of coexistence between competingstrategies in dependence of the temptation value for differentnumbers of inhabitants. actions. Panel (b) illustrates a snapshot for a morecrowded population at a smaller temptation valuewhere the coexistence of strategies is a dynamicalprocess similar to those we described regarding theanimation.The above described differences between the pat-tern formation in rare and crowded population canbe made more quantitative if we plot the proba-bility of coexisting state for populations containingdifferent numbers of inhabitants. This is shown inFig. 5, where we can see that crowded populationmakes the coexistence harder. Either cooperators,or defectors prevail depending on the temptationvalue. Less busy population, however, can easily re-sult in isolated patches, hence providing an escaperoute for cooperators at harsh temptation values.Next we briefly summarize the possible impactof noise parameter in off-lattice environment. Thisquestion could be specially interesting, because ear-lier observations revealed that some character of theinteraction topology could be a decisive factor howthe critical threshold value depends on the noiseparameter [44]. More precisely, if there are over-lapping triangles in the interaction graphs then thecritical threshold value where cooperators die out isa decaying function of noise parameter. In the lackof it the mentioned function is non-monotonous andhas an optimum at an intermediate noise strength.This optimal level can be the result of a selectionprocess [45].In on off-lattice simulation, in the absence of acharacteristic interaction topology, we observed adifferent situation which represents a new type ofbehavior. As Fig. 6 demonstrates, in off-latticeenvironment a higher noise level ensures a betterchance for cooperators to survive. This effect isagain related to the above described isolation pro-cess. At higher noise level the prompt invasion4 . . . .
81 1 1 .
02 1 .
04 1 .
06 1 . l i = l b =0 . N =5000 ρ C T K = 0 . K = 1 . K = 2 . Figure 6: Noise dependence of cooperation level for pris-oner’s dilemma game at fixed N and l values. By increasingthe uncertainty of imitation the cooperation becomes morelikely even at higher temptation values. of defectors at larger temptation is not straight-forward, which makes a chance for cooperators tosurvive the first attack. Later they have a higherchance to be isolated, hence to maintain a modestcooperation level.In the following we utilize the liberty of our modeland allow the defined length scales to be differ-ent. Our key observations are summarized in Fig. 7.First, we should note that the comparison is a bitmisleading because it suggests that l b < l i destroyssignificantly the cooperation level. But the properreason of this behavior is the pretty high value of l i = 0 .
25, which brings the system towards thewell-mixed condition. Normally, when l i is below0.1 then the application if smaller l b value does notchange the cooperation level significantly.However, this is not valid when l b exceeds thecurrent l i value. A typical curve is shown in Fig. 7,which suggests that the increase of l b has two-foldconsequences. When the temptation is moderateand the cooperator strategy would be dominant . . . .
81 0 .
98 1 1 .
02 1 .
04 1 .
06 1 .
08 1 . . N =5000 K =0 . ρ C T l i =0 . , l b =0 . l i =0 . , l b =0 . l i =0 . , l b =0 . Figure 7: Cooperation level in dependence of temptationvalue for smaller and larger range of birth process as indi-cated in the legend. The chance of remote birth of newcomerweakens the cooperation for moderate T , but increase it forhigher T values. This phenomenon is robust for other l b values, too. . . . .
81 0 .
98 1 1 .
02 1 .
04 1 .
06 1 .
08 1 . . N =5000 K =0 . P c o e x . T l i =0 . , l b =0 . l i =0 . , l b =0 . l i =0 . , l b =0 . Figure 8: The probability of coexisting state in dependenceof temptation value for normal and enlarged birth range. Formoderate T values enhanced birth distance reduces, whilefor higher temptation values it boosts the chance of a two-strategy final state. then an enhanced l b helps defectors to jump into thebulk of a cooperator island, which becomes morevulnerable in this way. Consequently, the cooper-ation level decays comparing to the l i = l b case.In contrast, for higher temptation the impact hasthe opposite sign. Here the system would evolvetoward a full defector state in the default case. Buta higher l b might help cooperators to “escape” fromdefectors. When such a remote patch becomes ho-mogeneous then it can survive. In other words, in-creasing l b has a similar impact on the system evo-lution to the behavior we observed for smaller N inthe default l i = l b case.The above described argument can be supportednicely if we measure the probability of coexistencestate in dependence of temptation. The comparisoncan be seen in Fig. 8 where we can observe similarbehavior we presented in Fig. 5. Accordingly, themessage is clear: for moderate temptation the en-hanced reproduction range prevents forming homo-geneous (cooperative) state, while for larger temp-tation values it helps to maintain coexistence state,hence to increase the average cooperation level.Summing up our observations, the off-lattice en-vironment provides mostly similar evolutionary tra-jectories to those previously reported for lattice-based simulations, but the consequence of spatialreciprocity could be stronger and the coexistenceof competing strategies is more dynamical in theformer case.These conclusions remain intact when we leavethe prisoner’s dilemma game and consider snow-drift and stag-hunt games. An illustration can beseen in Fig. 9 where we plotted the cooperationlevels in dependence of the control parameter r which makes possible to cross the related quadrant5 . . . .
810 0 . . . . l i = l b =0 . N =5000 K =0 . h ρ C i r snowdriftstag-hunt Figure 9: Cooperation level for snowdrift game and for stag-hunt game where the applied control parameter crosses therelated quadrants of T − S diagonally. Similarly to thelattice-based environments the transition from full C to full D state is gradual in the first case and sharp in the secondcase. of T − S parameter plane diagonally. The greensquare symbols show a gradual decay of cooperationlevel for snow-drift game which agrees with previ-ous observations in lattice populations [46, 47, 48].In sharp contrast to this in the stag hunt game theorange circle symbols sign a sharp transition fromthe full cooperator to the full defector state as wechange the control parameter. But this feature isagain in good agreement with the reported behaviorof on-lattice populations [14, 49, 50, 51].
4. DISCUSSION
The application of graphs to describe interactionsof multi-agent systems becomes an extremely vi-brant and successful theory in the last two decades.Not really surprisingly, evolutionary game theoryhas also enjoyed the benefit of this approach andutilized its concepts and simulation techniques tomodel more realistic dilemma situations [11, 16].Let us stress very clearly that to understand collec-tive behaviors based on graph-based modeling of-fers not just a simpler technique, but it is provedto be appropriate concept in several real-life sys-tems. But there are cases where off-lattice sim-ulations seem to be more appropriate, hence wecan not avoid the numerical difficulties of the lattermodels. We just quote here some microbiologicalsystems, but other situations, like collective move-ment or floating may also require off-lattice model-ing [52, 53, 54, 55].Our present work illustrates nicely that off-latticesystems where interaction are described by socialdilemmas behave conceptually similar to those weobserved on lattice-based populations. Therefore the latter, which are numerically more feasible,could be a reliable tool to explore the collectivebehavior of spatial populations. There are someminor differences, however, which warn us that notall predictions of lattice-based models are robustenough to apply in general. For example, in a sys-tem where the crowding is not really limited the so-called spatial reciprocity may work more strongly.We note that the role of aggregation was also re-ported in another work where the comparison ofon- and off-lattice populations were also studied ina different system [56]. But staying at our socialdilemma systems, the isolation of subgroups couldalso be a phenomenon which has relevant conse-quence on the system behavior.One may claim that we only studied dilemmaswhich are based on pair-interactions of players anda system with multi-point interactions may behavedifferently in off-lattice environment. Indeed, agraph-based population ruled by public goods gamemay behave slightly differently from populationsdriven by prisoner’s dilemma [57]. But on the otherhand, even multi-point interactions were identifiedas a key factor to diminish the differences of graphtopologies [58, 59, 60]. We therefore believe thatconclusions obtained off-lattice simulations remainvalid for games based on multi-point interactions,but future works can confirm it in more detail.
ACKNOWLEDGMENTS
B.F.O. thanks Funda¸c˜ao Arauc´aria, and INCT-FCx (CNPq/FAPESP) for financial and computa-tional support.
References [1] J. Hofbauer, K. Sigmund, The Theory of Evolution andDynamical Systems, Cambridge University Press, Cam-bridge, UK, 1988.[2] M. A. Nowak, Evolutionary Dynamics, Harvard Uni-versity Press, Cambridge, MA, 2006.[3] S. Wang, L. Liu, X. Chen, Tax-based pure punishmentand reward in the public goods game, Phys. Lett. A 386(2021) 126965.[4] Y. Shao, X. Wang, F. Fu, Evolutionary dynamics ofgroup cooperation with asymmetrical environmentalfeedback, EPL 126 (2019) 40005.[5] Z. Xu, R. Li, L. Zhang, The role of memory in humanstrategy updating in optional public goods game, Chaos29 (2019) 043128.[6] M. E. J. Newman, The structure and function of com-plex networks, SIAM Review 45 (2003) 167–256.
7] V. M. Equ´ıluz, M. G. Zimmermann, C. J. Cela-Conde,M. S. Miguel, Cooperation and the emergence of roledifferentiation in the dynamics of social networks, Am.J. Sociology 110 (2005) 977–1008.[8] M. A. Nowak, R. M. May, Evolutionary Games andSpatial Chaos, Nature 359 (1992) 826–829.[9] E. Ahmed, A. S. Elgazzar, On coordination and contin-uous hawk-dove games on small-world networks, Eur.Phys. J. B 18 (2000) 159–162.[10] F. C. Santos, J. M. Pacheco, Scale-free networks providea unifying framework for the emergence of cooperation,Phys. Rev. Lett. 95 (2005) 098104.[11] G. Szab´o, G. F´ath, Evolutionary games on graphs,Phys. Rep. 446 (2007) 97–216.[12] M. A. Amaral, M. A. Javarone, Heterogeneity in evolu-tionary games: an analysis of the risk perception, Proc.R. Soc. A 476 (2020) 20200116.[13] X. Liu, C. Huang, Q. Dai, J. Yan, The effects of the con-formity threshold on cooperation in spatial prisoner’sdilemma games, EPL 128 (2019) 18001.[14] C. P. Roca, J. A. Cuesta, A. S´anchez, Evolutionarygame theory: Temporal and spatial effects beyond repli-cator dynamics, Phys. Life Rev. 6 (2009) 208–249.[15] G. Yang, C. Zhu, W. Zhang, Adaptive and probabilisticstrategy evolution in dynamical networks, Physica A518 (2019) 99–110.[16] M. Perc, J. J. Jordan, D. G. Rand, Z. Wang, S. Boc-caletti, A. Szolnoki, Statistical physics of human coop-eration, Phys. Rep. 687 (2017) 1–51.[17] R.-R. Liu, C.-X. Jia, Z. Rong, Effects of enhancementlevel on evolutionary public goods game with payoff as-pirations, Appl. Math. Comput. 350 (2019) 242–248.[18] Y. Jiao, T. Chen, Q. Chen, The impact of expressingwillingness to cooperate on cooperation in public goodsgame, Chaos, Solit. Fract. 140 (2020) 110258.[19] M. A. Amaral, M. A. Javarone, Strategy equilibrium indilemma games with off-diagonal payoff perturbations,Phys. Rev. E 101 (2020) 062309.[20] M. L. A. Jansen, J. A. Diderich, M. Mashego, A. Has-sane, J. H. de Winde, P. Daran-Lapujade, J. T. Pronk,Prolonged selection in aerobic, glucose-limited chemo-stat cultures of
Saccharomyces cerevisiae causes a par-tial loss of glycolytic capacity, Microbiology 151 (2005)1657–1669.[21] J.-U. Kreft, S. Bonhoeffer, The evolution of groups ofcooperating bacteria and the growth rate versus yieldtrade-off, Microbiology 151 (2005) 637–641.[22] R. Garde, J. Ewald, ´A. T. Kov´acs, S. Schuster, Mod-elling population dynamics in a unicellular social organ-ism community using a minimal model and evolutionarygame theory, Open Biol. 10 (2020) 200206.[23] K. Drescher, C. D. Nadell, H. A. Stone, N. S. Wingreen,B. L. Bassler, Solutions to the Public Goods Dilemmain Bacterial Biofilms, Current Biology 24 (2014) 50–55.[24] K. E. Boyle, H. Monaco, D. van Ditmarsch, M. De-foret, J. B. Xavier, Integration of Metabolic and Quo-rum Sensing Signals Governing the Decision to Coop-erate in a Bacterial Social Trait, PLoS Comput. Biol.11 (2015) e1004279.[25] E. W. Tekwa, D. Nguyen, M. Loreau, A. Gonzalez, De-fector clustering is linked to cooperation in a pathogenicbacterium, Proc. R. Soc. B 284 (2017) 20172001.[26] J. Quan, C. Tang, X. Wang, Reputation-based dis-count effect in imitation on the evolution of cooperationin spatial public goods games, Physica A 563 (2021) 125488.[27] T. Nagatani, G. Ichinose, Diffusively-Coupled Rock-Paper-Scissors Game with Mutation in Scale-Free Hi-erarchical Networks, Complexity 2020 (2020) 6976328.[28] L. Zhang, C. Huang, H. Li, Q. Dai, J. Yang, Cooper-ation guided by imitation, aspiration and conformity-driven dynamics in evolutionary games, Physica A 561(2021) 125260.[29] G. He, L. Zhang, C. Huang, H. Li, Q. Dai, J. Yang,The effects of heterogeneous confidence on cooperationin spatial prisoner’s dilemma game, EPL 132 (2020)48004.[30] K. Li, Y. Mao, Z. Wei, R. Cong, Pool-rewarding in N-person snowdrift game, Chaos, Solit. and Fract. 143(2021) 110591.[31] S. Gao, J. Du, J. Liang, Evolution of cooperation underpunishment, Phys. Rev. E 101 (2020) 062419.[32] J. Quan, Y. Qin, Y. Zhou, X. Wang, J.-B. Yang, How toevaluate one’s behavior toward ’bad’ individuals? Ex-ploring good social norms in promoting cooperation inspatial public goods games, J. Stat. Mech. 2020 (2020)093405.[33] R. Yang, T. Chen, Q. Chen, Promoting cooperation byreputation-based payoff transfer mechanism in publicgoods game, Eur. Phys. J. B 93 (2020) 94.[34] K. Sigmund, The Calculus of Selfishness, PrincetonUniversity Press, Princeton, NJ, 2010.[35] M. Perc, A. Szolnoki, Coevolutionary games – a minireview, BioSystems 99 (2010) 109–125.[36] G. Szab´o, C. T˝oke, Evolutionary prisoner’s dilemmagame on a square lattice, Phys. Rev. E 58 (1998) 69–73.[37] H. Ohtsuki, M. A. Nowak, The replicator equation ongraphs, J. Theor. Biol. 243 (2006) 86–97.[38] G. Szab´o, J. Vukov, A. Szolnoki, Phase diagramsfor an evolutionary prisoner’s dilemma game on two-dimensional lattices, Phys. Rev. E 72 (2005) 047107.[39] M. Doebeli, C. Hauert, Models of cooperation based onPrisoner’s Dilemma and Snowdrift game, Ecol. Lett. 8(2005) 748–766.[40] Z. Wang, A. Szolnoki, M. Perc, If players are sparsesocial dilemmas are too: Importance of percolation forevolution of cooperation, Sci. Rep. 2 (2012) 369.[41] J.-Y. Guan, Z.-X. Wu, Y.-H. Wang, Evolutionary snow-drift game with disordered environments in mobile so-cieties, Chinese Physics 16 (2007) 3566–3570.[42] Z. Wang, A. Szolnoki, M. Perc, Percolation thresholddetermines the optimal population density for publiccooperation, Phys. Rev. E 85 (2012) 037101.[43] https://doi.org/10.6084/m9.figshare.13550189.v1 .[44] J. Vukov, G. Szab´o, A. Szolnoki, Cooperation in thenoisy case: Prisoner’s dilemma game on two types ofregular random graphs, Phys. Rev. E 73 (2006) 067103.[45] A. Szolnoki, J. Vukov, G. Szab´o, Selection of noise levelin strategy adoption for spatial social dilemmas, Phys.Rev. E 80 (2009) 056112.[46] C. Hauert, M. Doebeli, Spatial structure often inhibitsthe evolution of cooperation in the snowdrift game, Na-ture 428 (2004) 643–646.[47] P.-P. Li, J. Ke, Z. Lin, P. Hui, Cooperative behav-ior in evolutionary snowdrift games with the uncondi-tional imitation rule on regular lattices, Phys. Rev. E85 (2012) 021111.[48] F. Shu, X. Liu, K. Fang, H. Chen, Memory-based snow-drift game on a square lattice, Physica A 496 (2018)5–26.[49] M. Starnini, A. S´anchez, J. Poncela, Y. Moreno, Co-ordination and growth: the Stag Hunt game on evolu-tionary networks, J. Stat. Mech. 2011 (2011) P05008.[50] G. Szab´o, A. Szolnoki, Selfishness, fraternity, and other-regarding preference in spatial evolutionary games, J.Theor. Biol. 299 (2012) 81–87.[51] L. Wang, C. Xia, L. Wang, Y. Zhan, An evolving Stag-Hunt game with elimination and reproduction on regu-lar lattices, Chaos, Solitons & Fractals 56 (2013) 69–76.[52] T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen,O. Shochet, Novel Type of Phase Transition in a Sys-tem of Self-Driven Particles, Phys. Rev. Lett. 75 (1995)1226–1229.[53] P. P. Avelino, D. Bazeia, L. Losano, J. Menezes,B. F. de Oliveira, Spatial patterns and biodiversity inoff-lattice simulations of a cyclic three-species Lotka-Volterra model, EPL 121 (2018) 48003.[54] P. P. Avelino, B. F. de Oliveira, J. V. O. Silva, Rock-paper-scissors models with a preferred mobility direc-tion, EPL 132 (2020) 48003.[55] D. Bazeia, M. V. de Moraes, B. F. de Oliveira, Modelfor clustering of living species, EPL 129 (2020) 28002.[56] A. J. Daly, W. Quaghebeur, T. M. A. Depraetere, J. M.Baetens, B. D. Baets, Lattice-based versus lattice-freeindividual-based models: impact on coexistence in com-petitive communities, Natural Computing 18 (2019)855–864.[57] M. Perc, J. G´omez-Garde˜nes, A. Szolnoki, L. M. Flor´ıaand Y. Moreno, Evolutionary dynamics of group inter-actions on structured populations: a review, J. R. Soc.Interface 10 (2013) 20120997.[58] A. Szolnoki, M. Perc, G. Szab´o, Topology-independentimpact of noise on cooperation in spatial public goodsgames, Phys. Rev. E 80 (2009) 056109.[59] A. Szolnoki, J. Vukov, M. Perc, From pairwise to groupinteractions in games of cyclic dominance, Phys. Rev.E 89 (2014) 062125.[60] A. Szolnoki, M. Perc, Vortices determine the dynamicsof biodiversity in cyclical interactions with protectionspillovers, New J. Phys. 17 (2015) 113033.