Collective behavior and evolutionary games - An introduction
aa r X i v : . [ phy s i c s . s o c - ph ] J un Collective behavior and evolutionary games – An introduction
Matjaˇz Perc ∗ and Paolo Grigolini † Faculty of Natural Sciences and Mathematics, University of Maribor, Koroˇska cesta 160, SI-2000 Maribor, Slovenia Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA
This is an introduction to the special issue titled “Collec-tive behavior and evolutionary games” that is in the makingat Chaos, Solitons & Fractals. The term collective behaviorcovers many different phenomena in nature and society. Frombird flocks and fish swarms to social movements and herdingeffects [1–5], it is the lack of a central planner that makes thespontaneous emergence of sometimes beautifully ordered andseemingly meticulously designed behavior all the more sensa-tional and intriguing. The goal of the special issue is to attractsubmissions that identify unifying principles that describe theessential aspects of collective behavior, and which thus allowfor a better interpretation and foster the understanding of thecomplexity arising in such systems. As the title of the spe-cial issue suggests, the later may come from the realm of evo-lutionary games, but this is certainly not a necessity, neitherfor this special issue, and certainly not in general. Interdisci-plinary work on all aspects of collective behavior, regardlessof background and motivation, and including synchronization[6–8] and human cognition [9], is very welcome.
1. Evolutionary games
Evolutionary games [10–15] are, nevertheless, particularlylikely to display some form of collective behavior, especiallywhen played on structured populations [16, 17], and hencehave been chosen to co-headline the special issue. Some back-ground information and basic considerations follow.Consider that players can choose either to cooperate or todefect. Mutual cooperation yields the reward R to both play-ers, mutual defection leads to punishment P of both players,while the mixed choice gives the cooperator the sucker’s pay-off S and the defector the temptation T . Typically R = 1 and P = 0 are considered fixed, while the remaining twopayoffs can occupy − ≤ S ≤ and ≤ T ≤ . If T > R > P > S we have the prisoner’s dilemma game,while
T > R > S > P yields the snowdrift game [18].Without much loss of generality, this parametrization is of-ten further simplified for the prisoner’s dilemma game, so that T = b is the only free parameter while R = 1 and P = S = 0 are left constant. However, since the condition P > S is nolonger fulfilled, this version is usually referred to as the weakprisoner’s dilemma game. For the snowdrift game one can, ina similar fashion, introduce r ∈ [0 , such that T = 1 + r and S = 1 − r , where r is the cost-to-benefit ratio and constitutesa diagonal in the snowdrift quadrant of the T − S plane. ∗ Electronic address: [email protected] † Electronic address: [email protected]
In the prisoner’s dilemma game defectors dominate coop-erators, so that in well-mixed populations natural selectionalways favors the former. In the snowdrift game [19], onthe other hand, a coexistence of cooperators and defectors ispossible even under well-mixed conditions, and spatial struc-ture may even hinder the evolution of cooperation [20]. Theprisoner’s dilemma is in fact the most stringent cooperativedilemma, where for cooperation to arise a mechanism for theevolution of cooperation is needed [21]. This leads us to theyear 1992, when Nowak and May [22] observed the sponta-neous formation of cooperative clusters on a square lattice,which enabled cooperators to survive in the presence of de-fectors, even in the realm of the prisoner’s dilemma game.The mechanism is now most frequently referred to as networkreciprocity or spatial reciprocity, and it became very popu-lar in the wake of the progress made in network science andrelated interdisciplinary fields of research [23–26]. The pop-ularity was amplified further by the discovery that scale-freenetworks provide a unifying framework for the evolution ofcooperation [27] – a finding that subsequently motivated re-search on many different interaction networks [16], includingsuch that coevolve as the game evolves [28–32].The prisoner’s dilemma and the snowdrift game are exam-ples of pairwise interaction games. At each instance of thegame, two players engage and receive payoffs based on theirstrategies. However, there are also games that are governedby group interactions, the most frequently studied of which isthe public goods game [33]. The basic setup with cooperatorsand defectors as the two competing strategies on a lattice canbe described as follows [34]. Initially, N = L players are ar-ranged into overlapping groups of size G such that everyoneis surrounded by its k = G − neighbors and thus belongsto g = G different groups, where L is the linear system sizeand k the degree (or coordination number) of the lattice. Co-operators contribute a fixed amount a , normally consideredbeing equal to without loss of generality, to the commonpool while defectors contribute nothing. Finally, the sum of allcontributions in each group is multiplied by the synergy factor r > and the resulting public goods are distributed equallyamongst all the group members. Despite obvious similaritieswith the prisoner’s dilemma game (note that a public goodsgame in a group of size G corresponds to G − pairwise pris-oner’s dilemma interactions), the outcomes of the two gametypes may differ significantly, especially in the details of col-lective behavior emerging on structured populations [35].
2. Strategic complexity and more games
Significantly adding to the complexity of solutions areadditional competing strategies that complement the tradi-tional cooperators and defectors, such as loners or volunteers[36, 37], players that reward or punish [38–48], or conditionalcooperators and punishers [49, 50], to name but a few recentlystudied examples. These typically give rise to intricate phasediagrams, where continuous and discontinuous phase transi-tions delineate different stable solutions, ranging from singleand two-strategy stationary states to rock-paper-scissors typecyclic dominance that can emerge in strikingly different ways.Figure 1 features characteristic snapshots of four representa-tive examples.Besides traditionally studied pairwise social dilemmas,such as the prisoner’s dilemma and the snowdrift game, andthe public goods game which is governed by group interac-tions, many other games have recently been studied as well.Examples include the related collective-risk social dilemmas[51–54] and stag-hunt dilemmas [55], as well as the ultima-tum game [56–65]. Depending on the setup, most notablyon whether the interactions among players are well-mixed orstructured [44, 45], but also on whether the strategy spaceis discrete or continuous [57, 64, 65], these games exhibitequally complex behavior, and they invite further researchalong the lines outlined for the more traditionally studied evo-lutionary games described above.
3. Simulations versus reality
Monte Carlo simulations are the predominant mode of anal-ysis of evolutionary games on structured populations. Follow-ing the distribution of competing strategies uniformly at ran-dom, an elementary step entails randomly selecting a playerand one of its neighbors, calculating the payoffs of both play-ers, and finally attempting strategy adoption. The later is ex-ecuted depending on the payoff difference, along with someuncertainty in the decision making to account for imperfectinformation and errors in judging the opponent. The temper-ature K in the Fermi function [66] is a popular choice to ad-just the intensity of selection, and it is also frequently consid-ered as a free parameter in determining the phase diagramsof games governed by pairwise interactions [67] (note that forgames governed by group interactions the impact of K is qual-itatively different and in fact less significant [34]). Repeatingthe elementary step N times gives a chance once on averageto every player to update its strategy, and thus constitutes onefull Monte Carlo step.Although simulations of games on structured populationsare still far ahead of empirical studies and economic experi-ments [68], recent seminal advances based on large-scale hu-man experiments suggest further efforts are needed to recon-cile theory with reality [69–71]. According to the latter, net-work reciprocity does not account for why we so often choosesocially responsible actions over defection, at least not in therealm of the prisoner’s dilemma game. On the other hand,there is also evidence in support of cooperative behavior inhuman social networks [72, 73], as well as in support of thefact that dynamic social networks do promote cooperation inexperiments with humans [74]. These findings, together withthe massive amount of theoretical work that has been pub- lished in the past decades, promise exciting times ahead. Ourhope is that this special issue will successfully capture someof this vibrancy and excitement, and in doing so hopefully rec-ommend the journal to both readers and prospective authors.
4. Future research
In terms of advisable future directions for research, at leastin terms of evolutionary games, interdependent (or multiplex)networks certainly deserve mentioning. Not only are our so-cial interactions limited and thus best described by models en-tailing networks rather than by well-mixed models, it is alsoa fact that these networks are often interdependent. It has re-cently been shown that even seemingly irrelevant changes inone network can have catastrophic and very much unexpectedconsequences in another network [78–82], and since the evo-lution of cooperation in human societies also proceeds on suchinterdependent networks, it is of significant interest to deter-mine to what extent the interdependence influences the out-come of evolutionary games. Existing works that have stud-ied the evolution of cooperation on interdependent networksconcluded that the interdependence can be exploited success-fully to promote cooperation [83–86], for example throughthe means of interdependent network reciprocity [87] or in-formation sharing [88], but also that too much interdepen-dence is not good either. In particular, individual networksmust also be sufficiently independent to remain functional ifthe evolution of cooperation in the other network goes terriblywrong. Also of interest are evolutionary games on bipartitenetworks [89, 90], where group structure is considered sepa-rately from the network structure, and thus enables a deeperunderstanding of the evolution of cooperation in games thatare governed by group interactions. It seems that the evolutionof cooperation on both interdependent and bipartite networkshas reached fruition to a degree that the next step might be toconsider coevolution between cooperation and either interde-pendence or bipartiteness. Lastly we also refer to [91], whereSection 4 features 10 interesting open problems that certainlymerit attention.Finally, we would like the potential authors to explore alsothe challenging issue of a possible connection between soci-ology and neurophysiology. In same cases [36] game theory,which is widely applied in sociology, generates patterns rem-iniscent of those produced by the Ising model thereby sug-gesting a possible connection with criticality [92], which isbecoming an increasingly popular hypothesis in neurophys-iology, especially for brain dynamics [93, 94], where thisassumption generates theoretical results yielding a surpris-ingly good agreement with the experimental observation ofreal brain [95]. The potential authors may also contributesignificant advances to understand the real nature of neuro-physiological criticality, whose connections with the critical-ity of physical systems are not yet satisfactorily established[96], although criticality-induced dynamics are proven to beresponsible for a network evolution fitting the main subjectof this special issue as well as the crucial neurophysiologicalhypothesis of Hebbian learning [97]. The decision making
FIG. 1: Spatial patterns, emerging as a consequence of the spontaneous emergence of cyclic dominance between the competing strategies. Topleft: Dynamically generated cyclic dominance in the spatial prisoner’s dilemma game [75]. Light yellow (blue) are cooperators (defectors)whose learning capacity is minimal, while dark yellow (dark blue) are cooperators (defectors) whose learning capacity is maximal. Top right:Cyclic dominance in the spatial public goods game with pool-punishment [45]. Black, white and blue are defectors ( D ), pure cooperator ( C )and pool-punishers ( O ), respectively. Within the depicted (D+C+O) c phase, there are significantly different interfaces between the coexistingphases, which give rise to the anomalous “survival of the weakest”. Pure cooperators behave as predators of pool-punishers, who in turnkeep defectors in check, who in turn predate on pure cooperators. Bottom left: Cyclic dominance in the spatial ultimatum game with discretestrategies [65]. The dominance is not between three strategies, but rather between two strategies ( E depicted blue and E depicted green) andan alliance of two strategies ( E + A , where A is depicted black). Although similarly complex phases have been reported before in spatialecological models [76] and in the spatial public goods game with pool punishment [45], the observation of qualitatively similar behavior inthe ultimatum game enforces the notion that such exotic solutions may be significantly more common than initially assumed, especially insystems describing human behavior. Bottom right: Cyclical dominance between cooperators (white), defectors (black) and peer-punishers(orange) in the hard peer-punishment limit [77]. If punishment is sufficiently expensive and taxing on the defectors, this reduces the income ofboth defectors and peer-punishers. Along the interface, players can thus increase their payoff by choosing to cooperate, which manifests as theformation of white “monolayers” separating defectors and peer-punishers. We refer to the original works for further details about the studiedevolutionary games. model [97], a dynamical model sharing [98] the same criti-cality properties as those adopted to study the brain dynamics[92], generates an interesting phenomenon that the authors in[99] used to explain the Arab Spring events. This is a socio-logical phenomenon, where a small number of individual pro-duces substantial changes in social consensus [99], in agree-ment with similar results based on the adoption of game the-ory [100]. The theoretical reasons of this surprising agreementis one of the problems that hopefully some contributors to thisspecial issue may solve. We hope that papers on this issuemay help to establish a connection between criticality [101]and swarm intelligence [1] and hopefully between cognition[9] and consciousness [102].To conclude, we note that this special issue is also aboutto feature future research. In order to avoid delays that aresometimes associated with waiting for a special issue to be-come complete before it is published, we have adopted analternative approach. The special issue will be updated con-tinuously from the publication of this introduction onwards,meaning that new papers will be published immediately after acceptance. The issue will hopefully grow in size on a regularbasis, with the last papers being accepted no later than August30th for the special issue to be closed by the end of 2013. Thedown side of this approach is that we cannot feature the tra-ditional brief summaries of each individual work that will bepublished, but we hope that this is more than made up for bythe immediate availability of the latest research. Please staytuned, and consider contributing to “Collective behavior andevolutionary games”. Acknowledgments
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Rev. E 72 (2005) 047107.[68] C. F. Camerer, Behavioral Game Theory: Experiments inStrategic Interaction, Princeton University Press, Princeton,2003.[69] J. Gruji´c, T. R¨ohl, D. Semmann, M. Milinksi, A. Traulsen,Consistent strategy updating in spatial and non-spatial behav-ioral experiments does not promote cooperation in social net-works, PLoS ONE 7 (2012) e47718.[70] C. Gracia-L´azaro, A. Ferrer, G. Ruiz, A. Taranc´on, J. Cuesta,A. S´anchez, Y. Moreno, Heterogeneous networks do not pro-mote cooperation when humans play a prisoner’s dilemma,Proc. Natl. Acad. Sci. USA 109 (2012) 12922–12926.[71] C. Gracia-L´azaro, J. Cuesta, A. S´anchez, Y. Moreno, Humanbehavior in prisoner’s dilemma experiments suppresses net-work reciprocity, Sci. Rep. 2 (2012) 325.[72] J. H. Fowler, N. A. Christakis, Cooperative behavior cascadesin human social networks, Proc. Natl. Acad. Sci. USA 107(2010) 5334–5338.[73] C. L. Apicella, F. W. Marlowe, J. H. Fowler, N. A. Christakis,Social networks and cooperation in hunter-gatherers, Nature481 (2012) 497–501.[74] D. G. Rand, S. Arbesman, N. A. Christakis, Dynamic socialnetworks promote cooperation in experiments with humans,Proc. Natl. Acad. Sci. USA 108 (2011) 19193–19198.[75] A. Szolnoki, Z. Wang, J. Wang, X. Zhu, Dynamically gen-erated cyclic dominance in spatial prisoner’s dilemma games,Phys. Rev. E 82 (2010) 036110.[76] P. Szab´o, T. Cz´ar´an, G. Szab´o, Competing associations in bac-terial warfare with two toxins, J. Theor. Biol. 248 (2007) 736–744.[77] A. Szolnoki, G. Szab´o, L. Czak´o, Competition of individualand institutional punishments in spatial public goods games,Phys. Rev. E 84 (2011) 046106.[78] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin,Catastrophic cascade of failures in interdependent networks,Nature 464 (2010) 1025–1028.-person stag huntdilemmas, Proc. R. Soc. Lond. B 276 (2009) 315–321.[56] W. G¨uth, R. Schmittberger, B. Schwarze, An experimentalanalysis of ultimatum bargaining, J. Econ. Behav. Org. 3 (1982) 367–388.[57] M. A. Nowak, K. M. Page, K. Sigmund, Fairness versus rea-son in the ultimatum game, Science 289 (2000) 1773–1775.[58] K. Sigmund, E. Fehr, M. A. Nowak, The economics of fairplay, Sci. Am. 286 (2002) 82–87.[59] M. N. Kuperman, S. Risau-Gusman, The effect of topologyon the spatial ultimatum game, Eur. Phys. J. B 62 (2008) 233–238.[60] V. M. Equ´ıluz, C. Tessone, Critical behavior in an evolutionaryultimatum game with social structure, Adv. Complex Systems12 (2009) 221–232.[61] R. Sinatra, J. Iranzo, J. G´omez-Garde˜nes, L. M. Flor´ıa, V. La-tora, Y. Moreno, The ultimatum game in complex networks, J.Stat. Mech. (2009) P09012.[62] B. Xianyu, J. Yang, Evolutionary ultimatum game on complexnetworks under incomplete information, Physica A 389 (2010)1115–1123.[63] J. Gao, Z. Li, T. Wu, L. Wang, The coevolutionary ultimatumgame, EPL 93 (2011) 48003.[64] J. Iranzo, J. Rom´an, A. S´anchez, The spatial ultimatum gamerevisited, J. Theor. Biol. 278 (2011) 1–10.[65] A. Szolnoki, M. Perc, G. Szab´o, Defense mechanisms of em-pathetic players in the spatial ultimatum game, Phys. Rev.Lett. 109 (2012) 078701.[66] G. Szab´o, C. T˝oke, Evolutionary prisoner’s dilemma game ona square lattice, Phys. Rev. E 58 (1998) 69–73.[67] G. Szab´o, J. Vukov, A. Szolnoki, Phase diagrams for an evolu-tionary prisoner’s dilemma game on two-dimensional lattices,Phys. Rev. E 72 (2005) 047107.[68] C. F. Camerer, Behavioral Game Theory: Experiments inStrategic Interaction, Princeton University Press, Princeton,2003.[69] J. Gruji´c, T. R¨ohl, D. Semmann, M. Milinksi, A. Traulsen,Consistent strategy updating in spatial and non-spatial behav-ioral experiments does not promote cooperation in social net-works, PLoS ONE 7 (2012) e47718.[70] C. Gracia-L´azaro, A. Ferrer, G. Ruiz, A. Taranc´on, J. Cuesta,A. S´anchez, Y. Moreno, Heterogeneous networks do not pro-mote cooperation when humans play a prisoner’s dilemma,Proc. Natl. Acad. Sci. USA 109 (2012) 12922–12926.[71] C. Gracia-L´azaro, J. Cuesta, A. S´anchez, Y. Moreno, Humanbehavior in prisoner’s dilemma experiments suppresses net-work reciprocity, Sci. Rep. 2 (2012) 325.[72] J. H. Fowler, N. A. Christakis, Cooperative behavior cascadesin human social networks, Proc. Natl. Acad. Sci. USA 107(2010) 5334–5338.[73] C. L. Apicella, F. W. Marlowe, J. H. Fowler, N. A. Christakis,Social networks and cooperation in hunter-gatherers, Nature481 (2012) 497–501.[74] D. G. Rand, S. Arbesman, N. A. Christakis, Dynamic socialnetworks promote cooperation in experiments with humans,Proc. Natl. Acad. Sci. USA 108 (2011) 19193–19198.[75] A. Szolnoki, Z. Wang, J. Wang, X. Zhu, Dynamically gen-erated cyclic dominance in spatial prisoner’s dilemma games,Phys. Rev. E 82 (2010) 036110.[76] P. Szab´o, T. Cz´ar´an, G. Szab´o, Competing associations in bac-terial warfare with two toxins, J. Theor. Biol. 248 (2007) 736–744.[77] A. Szolnoki, G. Szab´o, L. Czak´o, Competition of individualand institutional punishments in spatial public goods games,Phys. Rev. E 84 (2011) 046106.[78] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin,Catastrophic cascade of failures in interdependent networks,Nature 464 (2010) 1025–1028.