Costly hide and seek pays: Unexpected consequences of deceit in a social dilemma
aa r X i v : . [ phy s i c s . s o c - ph ] O c t Costly hide and seek pays: Unexpectedconsequences of deceit in a social dilemma
Attila Szolnoki and Matjaˇz Perc , Institute of Technical Physics and Materials Science, Research Centre for NaturalSciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary Department of Physics, Faculty of Natural Sciences and Mathematics, University ofMaribor, Koroˇska cesta 160, SI-2000 Maribor, Slovenia Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah,Saudi ArabiaE-mail: [email protected], [email protected]
Abstract.
Deliberate deceptiveness intended to gain an advantage is commonplacein human and animal societies. In a social dilemma, an individual may only pretend tobe a cooperator to elicit cooperation from others, while in reality he is a defector. Withthis as motivation, we study a simple variant of the evolutionary prisoner’s dilemmagame entailing deceitful defectors and conditional cooperators that lifts the veil on theimpact of such two-faced behavior. Defectors are able to hide their true intentionsat a personal cost, while conditional cooperators are probabilistically successful atidentifying defectors and act accordingly. By focusing on the evolutionary outcomesin structured populations, we observe a number of unexpected and counterintuitivephenomena. We show that deceitful behavior may fare better if it is costly, and thata higher success rate of identifying defectors does not necessarily favor cooperativebehavior. These results are rooted in the spontaneous emergence of cycling dominanceand spatial patterns that give rise to fascinating phase transitions, which in turn revealthe hidden complexity behind the evolution of deception.PACS numbers: 87.23.Ge, 89.75.Fb, 89.65.-s ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma
1. Introduction
Natural selection favors the fittest under adversity and testing conditions. Accordingto Darwin’s
The Origin of Species , organisms therefore change gradually over time togive rise to the astonishing diversity of life that is on display today [1]. Sometimes,the most effective change is pretending to be someone or something one is not. Inthe animal world, mimicry is common to provide evolutionary advantages through anincreased ability to escape predation or by elevating chances of predatory success [2].The mimics and the species they are trying to fool are in an arms race, each tryingto optimize their chances of survival while having to accommodate additional costs.Beautiful examples of mimicry include the Pandora sphinx moth that looks like a deadleaf to avoid detection, the Flower Mantis that mimic flowers to lure prey, and the manyinsects that have adopted the yellow and black stripes common to bees and wasps to foolothers they are precisely that. Cuckoos are particularly cunning and famous for theirbreeding behavior. A female cuckoo lays its egg in the nest of a completely differentspecies of bird, simply because it wants to avoid spending energy on raising offspring. Animportant mechanism for getting away with such behavior lies in the ability of cuckoosto cleverly deceive their host [3]. First, the egg the cuckoo lays is very similar to the hostspecies eggs, and second, when the cuckoo chick hatches, it mimics the calls made by awhole brood of the host species chicks. In human societies, the ways of deception areof course even more cunning and elaborate. Our advanced intellectual abilities conveyto us an impressive array of different strategies and actions by means of which we mayfool others into a different reality. Obviously, some forms of trickery involve little to noadditional costs, while others impose a significant burden on the practitioners.Here we study the impact of such deceitful behavior on the evolution of cooperationin a social dilemma. Like deceptiveness intended to gain an advantage, situations thatconstitute social dilemmas are common in human and animal societies. In general, asocial dilemma implies that the collective wellbeing is at odds with individual success[4]. Individuals are therefore tempted to defect and maximize their own profit, butat the same time neglecting negative consequences such behavior has for the societyas a whole. A frequently quoted outcome of such selfishness is the “tragedy of thecommons” [5]. Indeed, the evolution of cooperation remains an evolutionary riddle[6, 7], and it is one of the most important challenges to Darwin’s theory of evolution andnatural selection. If during the course of evolution only the fittest survive, why shouldone sacrifice individual fitness for the benefit of unrelated others? While there is nosingle answer to this question, several mechanisms are known that promote cooperativebehavior [8].Evolutionary game theory [9, 10, 11] is long established as the theory of choicefor studying the evolution of cooperation among selfish individuals, and likely the mostfrequently studied social dilemma is the prisoner’s dilemma game [12, 13, 14, 15, 16,17, 18, 19, 20, 21, 22, 23, 24, 25]. Defection is the Nash equilibrium of the game,as it is the optimal strategy regardless of the strategy of the opponent. Beyond the ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma C ) thatcooperate only with other cooperators but defect otherwise, and we introduce deceitfuldefectors ( X ) that only pretend to be cooperative. In this way, we focus only onthe “darker” side of deception, although it is worth emphasizing that prosocial lieswith positive motivation have also been studied [38]. Consequently, we allow defectorsto go beyond pure defection ( D ), thus potentially providing a competitive answer toconditional cooperators. In addition to the temptation to defect r , however, thesemodifications introduce two additional parameters. Namely the probability p that aconditional cooperator will correctly identify a pure defector and avoid being exploited,and the cost γ that deceitful defectors need to bear in order to successfully beliecooperation. For further details we refer to the Model section. The questions we seekto answer within this theoretical framework are: What conditions allow the evolutionof deception? How large can affordable γ values be? And what is the role of theeffectiveness of conditional cooperators in identifying defectors? As we will show, theanswers to these questions are far from trivial. While low detection probabilities helpdefectors and high hiding costs obviously work against the effectiveness of deception,much more unexpectedly, we will also show how deceitful behavior may fare better ifit is costly, and how a higher success rate of identifying defectors does not necessarilyfavor cooperative behavior. These results are due to the spontaneous emergence of cyclicdominance and self-organized pattern formation, both of which give rise to continuousand discontinuous phase transitions that highlight the complexity behind the evolutionof deception.
2. Model
We consider a simple three-strategy social dilemma game where players can be deceitfuldefectors ( X ), conditional cooperators ( C ), or pure defectors ( D ). The payoffs amongstrategies are defined by the matrices ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma A D C XD C SX − γ ( T − γ ) − γ and B D C XD T C S SX − γ ( T − γ ) − γ .We use the payoff matrix A with probability p and the payoff matrix B with probability1 − p . In matrix A the conditional cooperator correctly identifies pure defectors andacts as a defector itself, while in matrix B the conditional cooperator fails to identifypure defectors and thus decides to cooperate. In the latter case, strategies C and D aresimply unconditional cooperation and defection. Importantly, as a specific case of a moregeneral model [37], conditional cooperators always cooperate with deceitful defectors, asthe latter invest γ specifically to that effect. If we would allow conditional cooperatorsto reveal also the deceptiveness of deceitful defectors the cost γ would simply alwaysconstitute an evolutionary disadvantage.Without loosing generality, we use the temptation to defect T = 1 + r andthe sucker’s payoff S = − r , thus building upon the traditional prisoner’s dilemmaformulation of an evolutionary social dilemma game. Here the parameter r > r = 0 . r = 0 .
7, being representative for a moderate and a strong prisoner’sdilemma, respectively.
We perform Monte Carlo simulations of the evolutionary social dilemma on a squarelattice of size L with periodic boundary conditions. The square lattice is the simplestof networks that allows us to go beyond well-mixed populations, and as such it enablesus to take into account the fact that the interactions among competing species are oftenstructured rather than random. By using the square lattice, we also continue a long-standing tradition that has begun with the work of Nowak and May [39], who were thefirst to show that the most striking differences in the outcome of an evolutionary gameemerge when the assumption of a well-mixed population is abandoned for the usage ofa structured population [14, 40, 41, 42, 43, 44, 45].Initially, each player on site x is designated either as a deceitful defector ( s x = X ), aconditional cooperator ( s x = C ) or a pure defector ( s x = D ) with equal probability. TheMonte Carlo simulations comprise the following elementary steps. First, a randomlyselected player x acquires its payoff Π x by playing the game with its four nearestneighbors. Next, one randomly chosen neighbor, denoted by y , also acquires its payoffΠ y in the same way. Lastly, player y adopts the strategy of player x with the probability w ( s x → s y ) = 11 + exp[(Π y − Π x ) / ( K )] , (1)where K determines the level of uncertainty in the Fermi function [14]. The latter canbe attributed to errors in judgment due to mistakes and external influences that affectthe evaluation of the opponent. We use K = 0 . ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma K and the strategy adoption rule, such as choosing the best or the betterperforming neighbor for the imitation, and they remain qualitatively valid also on otherlattices and random networks where the degree distribution remains unchanged but thelinks are uncorrelated.Each full Monte Carlo step (MCS) gives a chance to every player to change itsstrategy once on average. Depending on the proximity to phase transition points andthe typical size of emerging spatial patterns, we have varied the linear size of the latticefrom L = 400 to L = 6000 and the relaxation time from 10 to 10 MCS to obtainsolutions that are valid in the large system size limit, and to ensure that the statisticalerror is comparable with the size of the symbols in the figures. Importantly, even at sucha large system size ( L = 6000), for certain parameter values close to discontinuous phasetransition points, the random initial state may not necessarily yield a relaxation towardsthe most stable solution of the game. To verify the stability of different subsystemsolutions, we have therefore applied also prepared initial states (see for example Fig. 10in [46]), and we have followed the same procedure as applied previously in [47, 48].
3. Results
Before presenting the main results obtained in structured populations, we brieflydescribe the evolutionary outcomes in well-mixed populations, where players interactwith the whole population and choose competitors randomly [37].In the p = 1 limit, where pure defectors are always uncovered, strategy C is superiorto strategy D . This relation, however, is reversed if p < r r . For p values in-between, abistable competition between C and D is possible, whereby the final outcome depends onthe initial concentration of the competing strategies. In other words, C and D cannotcoexist regardless of the value of p . The coexistence of C and X is also impossible,because deceitful defectors dominate cooperators if γ < r , while otherwise strategy C issuperior to strategy X . Lastly, we note that D always beats X because the latter haveto bear the additional cost γ .As a consequence, strategy C prevails in the whole population if the values of p and γ are sufficiently high. Similarly, the full D phase is attainable in the small p limit.A mixed equilibrium, where all three competing strategies coexist, is also possible if(1 − p ) r (1 + r ) < γ < p (1 + r ) and γ < r are fulfilled simultaneously. Althoughthese results already provide useful insight into the impact and evolutionary stability ofdeception, we next focus on studying evolutionary outcomes in structured populations. ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma p r ob a b ilit y , p cost, γ D+C+X C+X D+CCD p r ob a b ilit y , p cost, γ D+C+X C+XD+C C
Figure 1.
Full γ − p phase diagram, as obtained for r = 0 .
3. Solid lines denotecontinuous phase transitions. The vertical resolution hides the intricate structure ofthe phase diagram for intermediate values of γ and p , which we therefore show enlargedin the right panel. Stable solutions include the three-strategy C + D + X phase, two-strategy C + D and C + X phases, as well as the absorbing D and C phase. In structured populations, we first focus on the moderate limit of the prisoner’s dilemmagame that is obtained for r = 0 .
3. The left panel of Fig. 1 shows the full γ − p phase diagram, which describes all possible stable solutions. Evidently, the richnessof solutions is greater than in well-mixed populations. In general, small detectionprobabilities, when conditional cooperators frequently fail to correctly identify puredefectors, are beneficial for the evolution of defection, thus yielding an absorbing D phase as the only stable solution in this region of the phase diagram, regardless of thecost of deception. As the effectiveness of conditional cooperators increases, the pure D phase transforms into a two-strategy D + C phase. This solution, which is absent inwell-mixed populations, is due to the aggregation of cooperators into compact clusters,by means of which a stable coexistence of the two strategies becomes possible withina narrow band of p (see also the right panel of Fig. 1). This is a purely spatial effectthat is rooted in network reciprocity [39]. If both γ and p are large, the D + C phaseterminates into an absorbing C phase, while for sufficiently low values of γ deceitfuldefectors become viable, either through the emergence of a two-strategy C + X phaseor the emergence of a three-strategy C + D + X phase. Within the latter the competingstrategies may dominate each other cyclically, although the stable coexistence of all threestrategies in the D + C + X phase does not always involve cyclic dominance. Severalaspects of these results are counterintuitive and unexpected. Foremost, one would expectthat decreasing values of p will impair the evolution of C , and that increasing values of γ will be detrimental for X . But this is not necessarily the case. In fact, as the value of ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma fr ac ti on s , ρ cost, γ DCX
Figure 2.
Cross-section of the phase diagram depicted in Fig. 1, as obtained for p = 1.Depicted are stationary fractions of the three competing strategies in dependence onthe cost of deceit γ . As the value of γ increases, the three-strategy C + D + X phasefirst gives way to the two-strategy C + X phase, and subsequently to the absorbing C phase. In this cross-section all phase transitions are continuous. We emphasize thatthe rise of the fraction of X as γ increases (before the extinction of D ) is an unexpectedand counterintuitive evolutionary outcome that can only be explained by means of thespontaneous emergence of cyclic dominance amongst all three competing strategies, asillustrated in Fig. 3. p decreases, the first to die out are the deceitful defectors, giving way to a mixed C + D phase. Moreover, as γ increases, the first to vanish from the three-strategy phase arethe pure defectors, thus yielding the C + X phase. If γ > .
401 and p > . C can survive. Interestingly, the two two-strategy phases are always separated bythe three-strategy D + C + X phase, as illustrated clearly in the enlarged part of thephase diagram depicted in the right panel of Fig. 1.Representative cross-sections of the phase diagram provide a more quantitativeinsight into the different phase transitions depicted in Fig. 1. In Fig. 2, we first showhow the fractions of the three strategies vary in dependence on the cost γ at p = 1, whereconditional cooperators are 100% effective in identifying pure defectors. When the costis small, all three strategies coexist in a stable D + C + X phase. As γ increases, deceitfuldefectors initially suffer, but the actual victims turn out to be the pure defectors — themain rivals of the deceitful defectors. Based on the presented results, we may concludethat, up to a certain point, deceitful behavior fares better if it is costly. Put differently,the larger the value of γ , the higher the fraction of strategy X in the stationary state.Only after D die out does the trend reverse, and larger values of γ actually have theexpected impact of lowering the evolutionary success of deceitful defectors, to the pointwhen the latter finally die out to give rise to the absorbing C phase.This evolutionary paradox, namely that deceitful behavior fares better if it is ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma Figure 3.
Consecutive snapshots of the square lattice, illustrating the spontaneousemergence of cyclic dominance from a random initial state between deceitful defectors(green), conditional cooperators (blue), and the pure defectors (red). The snapshotsare taken at 60, 100, 120 and 160 MCS from top left to bottom right, respectively.Invasions proceed according to the C → D → X → C closed loop of dominance.Parameter values are: r = 0 . p = 1, γ = 0 .
02, and L = 100. costly, can only be explained through the self-organized spatial patterns that emergespontaneously and drive cyclic dominance among the three competing strategies. Asshown in Fig. 3, traveling waves indeed emerge, where C beats D , D beats X , and X beats C to close the loop of dominance. The C → D → X → C loop of dominance isclearly inferable from the presented snapshots, as the initial C wave (blue) spreads intothe sea of D (red). The pure defectors, on the other hand, invade the territory of X (green), which in turn spread into the territory of C .Returning to the cross-sections of the phase diagram depicted in Fig. 1, we showin Figs. 4 and 5 how the fractions of the three strategies vary in dependence on theprobability p at γ = 0 . γ = 0 .
31, respectively. If the probability to reveal D is small,then conditional cooperators are unable to survive. Consequently, deceitful defectors donot exist either, as their “targets” ( C ) are not available, and the direct competitionwith D obviously leaves them at a disadvantage due to non-zero γ . As p increases, ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma fr ac ti on s , ρ probability, p DCX
Figure 4.
Cross-section of the phase diagram depicted in Fig. 1, as obtained for γ = 0 .
2. Depicted are stationary fractions of the three competing strategies independence on the probability p . As the value of p increases, the absorbing D phasefirst gives way to the two-strategy C + D phase, and subsequently to the three-strategy C + D + X phase. In this cross-section all phase transitions are continuous. Weemphasize that the rise of the fraction of D as p increases (just after the emergence of X ) is again an unexpected and counterintuitive evolutionary outcome that can onlybe explained by means of the spontaneous emergence of cyclic dominance amongst allthree competing strategies (see main text for details). the absorbing D phase gives way to the two-strategy C + D phase, which is possibledue to network reciprocity and is thus a purely spatial effect. Interestingly, weakening D further by elevating p will initially generate more D players in the stationary state.As the value of p increases further D do begin to decline on the expense of C , buton the other hand, X emerges and serves as an additional target for D . One mightexpect that increasing p further will support C because they will not let D playersexploit them. While the fraction of D indeed decreases, this in turn paves the way fordeceitful defectors, who can finally capitalize on their investment γ . Together, these“plus” and “minus” effects will nullify each other and leave the fraction of conditionalcooperators practically unaffected, despite of their elevated efficiency in detecting puredefectors. Thus, again unexpectedly, a higher success rate of identifying defectors doesnot necessarily favor cooperative behavior.Notably, a qualitatively different final phase is reached if we apply a higher valueof γ , as shown in Fig. 5. Here the fraction of X cannot raise as high, which in turnprovides less targets for pure defectors, who therefore die out more easily. In the absenceof D , however, the conditional cooperators and deceitful defectors can coexist, which isagain made possible by the clustering of cooperators and is thus a purely spatial effect.Naturally, if we increase the cost further, then strategy X cannot survive either, andthe population evolves from an absorbing D to the absorbing C phase via a coexisting ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma fr ac ti on s , ρ probability, p DCX
Figure 5.
Cross-section of the phase diagram depicted in Fig. 1, as obtained for γ = 0 .
31. Depicted are stationary fractions of the three competing strategies independence on the probability p . As the value of p increases, the absorbing D phasefirst gives way to the two-strategy C + D phase, then to the three-strategy C + D + X phase, and finally to the two-strategy C + X phase. Evidently, sufficiently increasingthe value of p may eradicate pure defectors and thus pave the way for deceitful defectorsto capitalize on their investment γ . Due to network reciprocity, however, conditionalcooperators never die out but rather form the C + X phase. C + D phase (cross-section not shown).In the strong limit of the prisoner’s dilemma game that is obtained for r = 0 .
7, weobserve solutions that are qualitatively different from those obtained for r = 0 .
3. Due tothe large temptation to defect, pure defectors and conditional cooperators are unable tocoexist in the absence of deceitful defectors. Instead, below p c = 0 . C + D phase, but gives rise also to the emergence of an absorbing C phase at an intermediate value of p , even if the cost of deception is moderate. Thephase diagram presented in Fig. 6 summarizes these fascinating evolutionary outcomes,which indicate that belying cooperation may actually beget cooperation.If we compare the two phase diagrams in Fig. 1 and Fig. 6, then we also find thatthere exist certain solutions that remain valid independently of the strength of the socialdilemma. In addition to the dominance of D at low values of p and the dominance of C at sufficiently high values of γ and p , our previous observation regarding the optimalvalue of γ when C are efficient also remains valid. Namely, from the point of view of X it is actually better to bear a larger cost of deceit than a small one, because theformer effectively prevents pure defectors to exploit these additional efforts aimed atdeceiving cooperators. In the absence of D , or when they are rare, the evolutionaryadvantage of X can still manifest even at relatively large γ values, especially if r is also ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma p r ob a b ilit y , p cost, γ D+C+X C+XC CD
Figure 6.
Full γ − p phase diagram, as obtained for r = 0 .
7. Solid lines denotecontinuous phase transitions, while dashed lines denote discontinuous phase transitions.As for the r = 0 . C + D + X phase and the two-strategy C + X phase, as well as theabsorbing D and C phase. Evidently, there exist solutions that are independent ofthe strength of the social dilemma, but there also exist significant differences, like thenature of the phase transition points and the “replacement” of the two-strategy C + D phase with the absorbing C phase. large. Moreover, for r = 0 . p can result in an absorbing C phase thatbecomes unstable at higher p values.Also worthy of attention are the phase transitions between the three-strategy C + D + X phase and the absorbing C phase. When the cost of deception is large,then as p decreases the frequency of X decreases gradually, as shown in the top panelof Fig. 7. When X finally vanish, the competition between the remaining D and C terminates in an absorbing C phase. As p decreases further, an abrupt transition toan absorbing D phase occurs. However, if the cost γ is small, a qualitatively differentbehavior can be observed, as shown in the top panel of Fig. 8. In this case, the averagefrequency of X within the C + D + X phase remains nonzero, but the amplitude ofoscillations increases drastically as we approach the phase transition point.Importantly, the increase in the amplitude of oscillations is not a finite size effectbecause the amplitude grows even if we increase the system size. This effect can bequantified by measuring the fluctuations of strategy X according to χ = L M M X t i =1 ( ρ X ( t i ) − ρ X ) , (2)where M denotes the number of independent values measured in the stationary state. ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma fr ac ti on s , ρ DCX
0 1 2 3 4 5 0.5 0.52 0.54 0.56 0.58 0.6 f l u c t u a ti on , χ probability, p Figure 7.
Top panel shows the cross-section of the phase diagram depicted in Fig. 6,as obtained for γ = 0 .
6. Depicted are stationary fractions of the three competingstrategies in dependence on the probability p . As the value of p increases, the absorbing D phase changes abruptly to the absorbing C , followed by the gradual emergence ofdeceitful defectors that give rise to the three-strategy C + D + X phase. The bottompanel shows the fluctuation of the vanishing density of deceitful defectors ρ X (seeEq. 2) in dependence on p , as obtained for different system sizes that are indicatedin the figure legend. The always finite value of χ indicates that the amplitude ofoscillations can always be reduced by increasing the system size, thus confirming acontinuous phase transition. As the bottom panel of Fig. 7 demonstrates, this quantity remains finite at large γ ,which means that the amplitude of oscillations can always be reduced by increasing thesystem size L . The same quantity, however, behaves very differently at small γ . Asthe bottom panel of Fig. 8 shows, the value of χ is diverging as we approach the phasetransition point, indicating that here the amplitude of oscillations cannot be loweredand that X will inevitably die out. We note that a similar type of discontinuous phasetransition was already observed in the spatial public goods game with correlated positive ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma fr ac ti on s , ρ DCX f l u c t u a ti on , χ probability, p Figure 8.
Top panel shows the cross-section of the phase diagram depicted in Fig. 6,as obtained for γ = 0 .
05. Depicted are stationary fractions of the three competingstrategies in dependence on the probability p . As the value of p increases, the absorbing D phase changes abruptly to the absorbing C , followed again by an abrupt emergenceof deceitful defectors that give rise to the three-strategy C + D + X phase. Thebottom panel shows the fluctuation of the vanishing density of deceitful defectors ρ X in dependence on p , as obtained for different system sizes that are indicated in the figurelegend. Unlike for γ = 0 . χ at the phase transitionpoint is diverging, increasing beyond bound even for very large L , and thus indicatinga fascinating discontinuous phase transition between the absorbing C phase and theheterogeneous C + D + X phase. and negative reciprocity, where cyclical dominance also emerged spontaneously betweenthe competing strategies [48]. These results thus reveal the hidden complexity behindthe evolution of deception, which appears to be commonplace in evolutionary settingswith three or more strategies in structured populations. ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma
4. Discussion
To summarize, we have shown that the introduction of conditional cooperators anddeceitful defectors to a social dilemma gives rise to many counterintuitive evolutionaryoutcomes that can only be understood as a consequence of self-organized patternformation in structured populations. Spatial systems, where players have a limitedinteraction range, allow the observation of rich behavior, including the formation ofpropagating fronts due to the spontaneous emergence of cyclic dominance. In particular,we have demonstrated the stable coexistence of all three competing strategies, as well asthe emergence of C + D and C + X two-strategy phases. Similarly to the results obtainedin well-mixed populations, in structured populations absorbing phases are also possibleat specific parameter values. Namely, if the conditional cooperators are ineffectivein identifying pure defectors, the latter dominate like under mean-field conditions.Conversely, if the conditional cooperators are sufficiently effective and if the cost ofdeception is high, then cooperative behavior dominates. Unexpectedly, our researchindicates that an imperfect ability of cooperators to properly detect defectors may bebeneficial for the evolution of cooperation, and that deceitful behavior may fare better ifit is costly. These results are rooted in the spontaneous emergence of cycling dominanceand complex spatial patterns. It is also worth emphasizing that these results are robustand remain valid if we choose other strategy updating rules or interaction networks otherthan lattices. For example, qualitatively similar evolutionary outcomes can be obtainedon random regular graphs.We have also shown that continuous and discontinuous phase transitions separatethe different stable solutions, and that changing a model parameter may have highlynontrivial and unexpected consequences. For example, the three-strategy C + D + X phase may terminate continuously or abruptly, depending solely on the cost of deception.This type of complexity is absent in traditional rock-scissors-paper or extended Lotka-Volterra-type models where the cyclic dominance is hardwired in the food web [49].Although the evolution of deception is also governed by cyclic dominance, the latteremerges spontaneously, and therefore changing even a single parameter may influencethe effective invasion rates between all three competing strategies. We have made similarobservations in the realm of the spatial ultimatum game [47] and the spatial public goodsgame with reward [50] and different types of punishment [46, 51], as well as in the spatialpublic goods game with correlated positive and negative reciprocity [48]. Taken together,this indicates that the reported evolutionary complexity is in fact much more frequentthan it might be assumed, and in fact appears to be commonplace in evolutionarysettings where three or more strategies compete in a structured population.When introducing punishing cooperators to a social dilemma, the tension betweendefectors and cooperators shifts to the tension between the cooperators that punish andthe cooperators that do not punish. Since the latter avoid the additional costs that needto be invested for sanctioning defectors, they become second-order free-riders [52]. Theevolution of deception gives rise to a conceptually similar shift in the dilemma, only that ostly hide and seek pays: Unexpected consequences of deceit in a social dilemma Acknowledgments
We thank our Referees for their insightful and constructive comments. This researchwas supported by the Hungarian National Research Fund (Grant K-101490), TAMOP-4.2.2.A-11/1/KONV-2012-0051, and the Slovenian Research Agency (Grant J1-4055).
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