Beneficial laggards: multilevel selection, cooperative polymorphism and division of labour in threshold public good games
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The origin and stability of cooperation is a hot topic in social and behavioural sciences. A complicated conundrum exists as defectors have an advantage over cooperators, whenever cooperation is costly so consequently, not cooperating pays off. In addition, the discovery that humans and some animal populations, such as lions, are polymorphic, where cooperators and defectors stably live together -- while defectors are not being punished--, is even more puzzling. Here we offer a novel explanation based on a Threshold Public Good Game (PGG) that includes the interaction of individual and group level selection, where individuals can contribute to multiple collective actions, in our model group hunting and group defense.
Our results show that there are polymorphic equilibria in Threshold PGGs; that multi-level selection does not select for the most cooperators per group but selects those close to the optimum number of cooperators (in terms of the Threshold PGG). In particular for medium cost values division of labour evolves within the group with regard to the two types of cooperative actions (hunting vs. defense). Moreover we show evidence that spatial population structure promotes cooperation in multiple PGGs. We also demonstrate that these results apply for a wide range of non-linear benefit function types.
We demonstrate that cooperation can be stable in Threshold PGG, even when the proportion of so called free riders is high in the population. A fundamentally new mechanism is proposed how laggards, individuals that have a high tendency to defect during one specific group action can actually contribute to the fitness of the group, by playing part in an optimal resource allocation in Threshold Public Good Games. In general, our results show that acknowledging a multilevel selection process will open up novel explanations for collective actions.
KeywordsCollective Action Free Rider Killer Whale Public Good Game Benefit Function
The intriguing phenomenon of cooperation has fascinated experimental and theoretical researchers for decades, as it is essential for understanding the complexity of life, and life itself [1, 2, 3, 4, 5]. A dilemma derives from the fact that selfish individuals have an advantage over those who act cooperatively, by which we mean a costly act that can benefit others [1, 6, 7]. Although selfish 'defectors' interacting with their own kind have a lower 'payoff' than cooperators, defectors would eventually totally replace cooperators in the population . However a wide variety cooperative behaviour can be observed in nature [7, 9]. Moreover, it is a common observation that humans [10, 11, 12] and some animal populations such as lions (Panthera leo) [13, 14] are polymorphic. Individuals that have a high tendency to cooperate live together with those that have a high tendency to defect. The proportion of these 'laggards' can be high in certain societies and they are not being punished, as in the case of lions  or in a number of human hunter-gatherer societies [11, 15]. Observations suggest that the average tendency for cooperation in these populations appears to be stable in the long term. This is a perplexing set of observations, not yet fully understood. Existing explanations rely either on the snowdrift game [16, 17] with homogeneous population structure, punishment , or spatially explicit population structure . None of these explanations can be applied perfectly here as for example lion and human populations are neither completely well-mixed, nor spatially explicit in a sense as sedentary organisms, and punishment is also not common in these instances.
Majority of the current evolutionary game theoretical studies that shed light on the mechanism behind many cooperative phenomena in biological systems concentrated on pair-wise interactions between individuals [3, 6, 7, 8, 10, 16, 20][but see for e.g. ]. Arguably numerous examples of group interactions found in nature however can be described as n-person games [22, 23, 24, 25, 21], in which more than two 'players' interact with each other at the same time in the form of group actions [26, 27, 28, 29, 30]. During group interactions individuals typically invest into common goods or common goals . In most of the cases this is available for everyone, that is the common good is non-monopolizable (or non-excludable, non-exclusive) and non-rival (synonym of joint in supply, non-diminishable) [32, 33, 34, 35]. This raises a collective action problem, where non cooperators, often termed as free riders, can reap the benefit of the common good without investing into it [24, 34, 35].
Models of Public Good Game nicely capture the main features all of the above described cooperative phenomena. But the traditionally used linear benefit return function is insufficient in capturing the threshold effect for the optimal number of individuals that is necessary to perform the given group action, as close to the threshold joining one or few more cooperators disproportionally increases the success of the group action [43, 68, 69]. So instead a nonlinear return paradigm is more appropriate [68, 70, 71, 72], such as in the n-player Threshold Public Good Game . In this version of the Public Good Game (PGG) a successful cooperative effort is achieved only if the number of cooperating individuals reaches a given threshold, just like in the above described situations. Different levels of cooperation can be evolutionary stable outcome in such games, depending on the cost of cooperation, and the proportion of initial cooperative decisions in the population .
While these situations assume that groups of individuals engage in an interaction which may or may not end in successful cooperation these groups themselves often compete with each other . For example, in one of the most studied group behavioural biological systems lioness form a pack to hunt together yet they are in direct competition with other lion packs with which they share common borders [13, 14, 47, 73, 74]. The same holds for all group hunting territorial species from hyenas through whales, African wild dogs to humans [75, 76, 77, 78].
Here we study the interaction of selection acting on the level of individuals engaged in a threshold PGG and selection acting on these groups while competing for territories. Thus, our model differs from previous models of TPGGs [71, 72] and that of multi-level selection  by explicitly integrating these two components. We do so first by giving analytical solutions for the evolutionarily stable level of cooperation for various group sizes and threshold levels at first assuming only individual selection; then by studying the interaction of individual and group level selection with a series of computer simulations validated by the numeric results. We study 3 basic setups in the computer simulations: (i) individual selection only (Figure 1.a), (ii) group selection only, (for comparison), and (iii) the combination of both, in which case first we assume that (iii/a) all individuals are obliged to participate in group defense (Figure 1.b), then we relax this assumption by allowing (iii/b) voluntary participation (Figure 1.c). Individual selection in our model allows individuals to compare their success with other individuals, or compete for resources, while group selection based on the idea that stronger groups may outcompete weaker ones by overtaking their territories. The structure of the game is described in the Methods section.
Effect of different threshold functions
Results for individual level selection
Introducing multilevel selection
Our results also indicate, in line with the general conclusion of previous models that spatial structure favours cooperation [6, 8, 19, 79], as high values of cooperation evolve for both public actions (hunting and group defense, see large red bubbles) even at high cost values (Figure 5.g and 5h). We observe division of labour in the spatially explicit model only under given circumstances (see Additional file 1 for further details), which suggests that this outcome is unstable in these cases with small group size. Note that when C(x) > 0 cooperation evolves only if the initial proportion of cooperation is not zero in the population with well-mixed structure, which indicates a separatrix in the system. However, spatial population structure promotes cooperation and it allows cooperators to invade at higher cost values with regard of both types of costs (i.e. compare 5.e with 5.g and 5.f with 5.h).
Here we show that multilevel selection in threshold PGG can maintain stable levels of polymorphism (i.e. a stable mixture of cooperators and defectors) without the need of punishment or spatially explicit population structure. We give a primary demonstration that the described dynamics holds not only for step-wise benefit function (strict threshold function), but for a wide range of sigmoid curves between the step-wise and the linear benefit functions (s-shape function). Our results further indicate that polymorphism in group hunting and defense can be adaptive in case of multilevel selection, and to our knowledge, we provide a pioneer report on the division of labour in multiple Public Good Games. We conclude that what was regarded as cheating at the individual level is in fact can play a significant part in the optimization at the group level, which optimization is provided by the described mechanisms working on behavioural polymorphism.
Many of the collective actions produce goods that are non monopolizable, meaning that no one can be excluded from benefiting [33, 34, 35], thus free riders can exploit this collective good without paying the cost of production . Threshold effects in Public Good Games can provide an often neglected, yet powerful explanation for the observed polymorphism in the population, as an alternative to punishment , spatial structure  or various payoff functions [16, 17].
Population level polymorphism is a stable outcome in Threshold Public Good Games, where cooperators and free riders stably live together in the population [71, 72], as long as the cost of cooperation doesn't exceed a limit cost value, the hysteresis point. Our model predicts that for an increased group size this hysteresis, that is a sudden drop from cooperation to defection, appears at lower cost values, compared to smaller groups (see Figure 3.b). To put it in a different way, with large groups cooperation can be maintained only if the cost of cooperation is relatively low. Also for higher cost values the instable fix points move closer to the stable fix points, which means that the invasion of cooperators in a population of defectors becomes harder, while the invasion of defectors into a cooperative population becomes more likely.
Introducing multilevel selection into the model is a logical step towards reality, as in many biological and human examples where collective hunting occurs intergroup conflicts can be also observed [35, 74, 75, 76, 77]. When group level competition introduced explicitly into the model population level polymorphism is stabilized both in well-mixed and spatially explicit populations. Thus, multilevel selection need not select for the groups with the highest number of cooperators as it is often assumed. Groups that optimize the number of hunters enjoy an advantage over those groups that hunt (cooperate) on a higher, unnecessary level. Thus, defection of a given proportion of individuals during one specific cooperative group action can be an adaptive strategy for the group depending on the cost values. Because group level performance (i.e. success in the PGG) determines both individual and group level success any deviation from the optimal group composition would cause a disadvantage in some way or other. If the number of cooperators is lower than the threshold value then the collective action (i.e. hunting) is unsuccessful, thus neither cooperators nor defectors gain anything, and these groups are being replaced by successful groups. On the other hand, if the number of cooperators is higher than what is required for providing benefit for the whole group (in our example capturing the prey), the energy loss of unnecessary effort would cause a disadvantage when the group faces a conflict (fight for territory). Hence our novel result is that laggards, who were previously seen as exploiters of the common goods provided by cooperators [13, 14], do actually contribute to the fitness of the group by keeping the group level allocation at the optimum level.
Our results also verify that if participation in two distinct collective actions which produce shared benefits is costly, such as hunting and territory defense, then selection pressure on cooperators can result division of labour to evolve, predicting that it will evolve only for a narrow range of cost values. Interestingly, behavioural polymorphism first appears at the individual level, that is all of the individuals participate in both collective actions with intermediate probabilities. However, this state is not stable and evolution drives the system towards division of labour at the population level, where individuals mostly participate only in one of the collective actions (i.e. polymorphism appears at the group level). This result is robust in our simulations at medium cost values, and turns out to be stable on the long term. Experimental support for this context dependent role specialization is poor yet [but see[56, 80]], however the idea that cooperators and 'free riders' switch roles in different contexts is not obscure [28, 35, 56, 81, 80], as long as there is a trade off situation between two energy consuming group actions .
Our model has a simplistic assumption, that individuals have only two heritable traits that describe their behavioural decisions, this two are sufficient for maintaining polymorphism in the population. Obviously regularly many genes effect behaviour, which explains higher polymorphism, still there is evidence suggesting that some may play disproportionally important role in behavioural switches . Accordingly, our model can provide a potential explanation for the observed polymorphism in lions in threshold game like situations [36, 37, 46] and the presence of laggards in the population [14, 83].
The importance of intra group conflict in human evolution seems to gain a new currency [77, 78], however since the technology of hunting humans changed a lot in the last 250,000 years (i.e. evolution of spear points, bows, arrows, etc. [84, 85]), it is more difficult to evaluate whether a threshold effect existed in hominid plio-pleistocene group hunting. The conception of hunting large preys would inevitably suggest so and the fact that hunting of medium-sized and large ungulates started long before stone-tipped and bone-tipped weapons were widely used strongly suggests that cooperation amongst hunters was essential for the capture of large games . If it did so, then our model applies and has the potential to explain the observed polymorphism in humans as well. Interesting implication of our results is that once human societies become larger and more fluid in composition this polymorphism was no longer necessarily adaptive and definitely was not looked upon as desirable. This, in turn could have triggered the evolution of cultural norms and institutions that attempts to obtain an universally high level of cooperation from all the members of the society regardless of their predispositions.
Here we present a multilevel selection model of interdependence in group living species, where cooperation is modelled as an n-player Threshold Public Good Game and where selection acts both at the individual and at the group level. We have found that the population can evolve into stable levels of cooperative polymorphism, where cooperators and free riders -laggards- can stably live together. Our results indicate: (i) that the described dynamics holds for a wide range of probabilistic sigmoid benefit functions, not only for strict deterministic step-wise function; (ii) that multilevel selection need not select for the highest number of cooperators within groups but instead it may selects for polymorphism depending on the details of the TPGG; (iii) that defectors contribute to the group fitness as much as they help the group to achieve the optimal amount of investment at the group level; (iv) that division of labour might evolve with regard of the participation in the two collective actions for medium cost values of cooperation; (v) and that spatial population structure promotes cooperation in TPGGs.
The structure of the game is as follows: individuals of a group can engage in a cooperative activity (i.e. hunting, resource purchase) where every individual can play two strategies, either to cooperate with probability x or to defect with probability 1-x. The cooperative players join in the group effort (group size is n), and thus pay a fix cost of cooperation (c). In contrast, defectors do not pay the cost of the game. If the number of cooperators is equal or above a given threshold value (T), then all of the individuals within the group can acquire the benefit (b) of cooperation regardless whether they cooperated or not. This derived from the non-rival, non-excludable features of the game [33, 34, 35], that is, no individual can be excluded from the acquired benefit of the hunt, and each consumer gains the same proportion without depleting it. However if the number of cooperators does not reach the necessary number which is required for the successful achievement of the group action then no one gets the benefit, but cooperators still pay the cost. The fitness (W) is calculated as the benefit minus the cost of the game.
While analytical solution might not be possible to calculate at large group sizes, numerical solutions can be obtained. Of course, the numerical solutions of this kind will not tell us which points are the stable and which points are the instable [but see ref. ]. To check the stability of our fixed points we used an individual based simulation of the model (details see below). The numerical results and the results of the simulations are depicted on Figure 3.
Where P is the probability that the public goal is reached, which depends on the steepness of the function (s), the number of cooperators in the group (cn) and the threshold value (T).
Individual based simulation of the model
We model the evolutionary dynamics in a finite size population consisting of N all individuals (10,000 and 12,500 individuals). We set up two different scenarios in which the population structure is modelled by the two extremes. On the one part individuals have no fixed partners in a well-mixed model. In every time step both the composition of cooperating groups and competition neighbourhoods change (n, we increase the number of cooperators up to 50 on Figure 3.b), where partners randomly chosen from the entire population, so the interaction environment of the individual is well-mixed. In the case of spatially explicit model individuals occupy the grid points of a regular lattice (N × N = N all ), with toroidal boundary conditions, and the focal individual has a constant interaction environment, its immediate neighbourhood. The neighbourhood in both cases defines both the cooperating and the competing group for the focal individual (n), which is 4 (von Neumann neighbourhood) group members in the spatially explicit model. The simulations were run for a given number of update steps with asynchronous updating. An update step consists of a game played by the group, individual competition, group competition if occurred (see Figure 1), and mutation.
We analyze multiple competition update scenarios (see Additional file 1 for details).
The chance for each group to occupy the focal site is given as described above in the competition rules, but using average group payoffs.
Next we introduce voluntary participation in both of the two group actions, and individuals have two continuous, evolving traits. Besides, x that defines the propensity to cooperate in the PGG, we introduce a, which describes the individual's propensity to participate in the territorial group defense action. The marginal values are also 0-1. If a is high, the individual will participate in the group competition with high probability, whereas if it is low or 0, it will never. As the group defense is considered as an act of cooperation here, it involves a cost C(a) for those who participated. The average group payoff is now calculated as the average of the payoffs of those group members, who participate in the group competition. Groups then compared by their average group payoffs according to the above described competition rules.
To ensure evolution, after the competition steps, mutation can occur in the traits of the focal individuals with the chance 0.01. The mutant's trait is drawn from a normal distribution, with the original trait value as a mean and with a given variance (0.01). The trait can only be between 0 and 1. After individual competition, only the focal individual's traits can mutate, after group competition in all members of the focal group has a chance for mutation to occur with the same probability.
The points on figures were calculated as the average of the last 1 M (106) update steps of multiple iterated simulation results (for Figure 4 15 and for Figure 5 3 repetitions, with insignificant variation between the repetitions). The length of the simulation was determined by preliminary simulations, that is 1,000 M, 625 M and 312.5 M update steps for Figure 3, 4 and 5 accordingly. Graphs on Figure 5.a-d were obtained by plotting the trait distribution of the population in every 100th update step for 25,000 steps.
GB worked on the model in the scope of the ESF TECT-BIOCONTRACT project (TECT-OTKA NN71700). This work was also supported by the Hungarian National Science Foundation (OTKA, Grants T049692, NK73047). The authors would like to thank Szathmáry E. and Scheuring I. for helpful comments. GB would like to thank Zachar I. for the help of the preparation of the graphs. The authors would like to thank the anonymous referees for their suggestion in improving the manuscript.
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