aa r X i v : . [ phy s i c s . s o c - ph ] F e b Evolution of cooperation in costly institutes
Mohammad Salahshour ∗ Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103, Leipzig,Germany
Abstract
We show that in a situation where individuals have a choice between a costly institute and afree institute to perform a collective action task, the existence of a participation cost promotescooperation in the costly institute. Despite paying for a participation cost, costly cooperators,who join the costly institute and cooperate, can out-perform defectors, who predominantly join afree institute. This, not only promotes cooperation in the costly institute but also facilitates theevolution of cooperation in the free institute. A costly institute out-performs a free institute whenthe profitability of the collective action is low. On the other hand, a free institute performs betterwhen the collective action’s profitability is high. Furthermore, we show that in a structured popu-lation, when individuals have a choice between different institutes, a mutualistic relation betweencooperators with different institute preferences emerges and helps the evolution of cooperation.
Introduction
In many biological contexts, individuals in a group need to work collectively to solve collective actionproblems. Examples range from cooperation in resource acquisition in bacteria [1, 2], to cooperativedefending or foraging [2–4], and cooperative breeding [5] in animal populations, and maintaining thecommons such as fisheries [6], or environmental preservation [6, 7] in human societies. Successfulsolution of such collective action problems requires biological populations to find ways to suppressfree-riding among the individuals. Past researches suggest formal and informal social institutions playan important role in solving such collective action problems. Informal social institutions can be atwork, for instance, in the form of social norms [8–14], reputation and gossip [9, 15–17], or even humanlanguage [17, 18]. Such institutions can promote cooperation by channeling the benefit of cooperationtowards cooperators. Human institutions, formal or informal, can promote cooperation through othermechanisms as well, such as reward [19–23], and punishment [24–30]. Such institutions can help tosolve collective action problems by rewarding social behavior or punishing anti-social behavior.Recently, it is also shown having a choice between different institutions to perform a collectiveaction task can promote cooperation in the absence of any of such enforcement mechanisms [31].Here, we bring attention to a relatively simple effect that can promote cooperation in costly institutes,when a choice between different institutes exist. We show that in a situation where individuals have achoice between different institutes, the existence of an entrance cost to enter an institute can keep free-riders away. In such a context, free-riders predominantly join the free institute. On the other hand,Cooperators, by working cooperatively in a costly institute, can obtain a higher profit despite payingan entrance cost that has no direct benefit. As our analysis shows, in both mixed and structured ∗ [email protected] . The Model
We consider a population of N individuals. At each time step, groups of g individuals are drawn atrandom from the population pool. Individuals in each group can choose between two public resources:A costly institute, which we call the resource 1, and a free institute, which we call resource 2. Thecostly institute has a participation fee. The participation fee is paid by all the individuals who choosethis resource. Individuals gather payoff by playing a public goods game in their institute. In thisgame, individuals can either cooperate or defect. Cooperators pay a cost c to invest the same amountin the public resource. Defectors pay no cost and do not invest. All the investments in a publicresource i is multiplied by an enhancement factor r i and is divided equally among the individuals inthat resource. In addition to the public goods game, we assume individuals receive a base payoff, π ,from other activities not related to the public goods game.After playing the games, individuals reproduce with a probability proportional to their payoff.In the reproduction stage, the whole population is updated such that the population size remainsconstant. That is, for each individual in the next generation, an individual is chosen as a parentwith a probability proportional to its payoff. The offspring inherit the game preference (which canbe institute 1 or 2) and the game strategy (which can be cooperation C or defection D ) of its parent,subject to mutations. Mutation in the game preference and strategy occurs independently, each withprobability ν . In case a mutation occurs, the value of the corresponding variable is changed to itsopposite value (for example, 1 to 2 for the game preference and C to D for the strategy).In addition to a mixed population, we consider a structured population, in which the individualsreside on a network. We consider a first nearest neighbor square lattice with von Neumann connectivityand periodic boundaries for the population network. Each individual participates in five groups, eachcentered around itself or one of its neighbors, to perform a collective action task. Individuals in eachgroup enter and play a public goods game in their preferred institute. The payoff of the individualsfrom the public goods game is defined as their average payoff from all the games that the individualsparticipates in. In addition to the payoff from the public goods games, individuals receive a basepayoff π . After deriving their payoffs, the whole population is updated. For the reproduction, weconsider an imitation or death-birth process in which each individual imitates the strategy of one ofthe individuals in its extended neighborhood, chosen with a probability proportional to its payoff. Weassume mutations can occur as well. After imitation, the individual’s strategy and game preferencemutate independently and each with probability ν .2igure 1: The density of different strategies in the c g − r plane. The densities of different strategies inthe c g − r plane are color plotted. The phase diagram of the model is superimposed. For both smalland large enhancement factors, r , the dynamics settle in a fixed point, denoted by FP. In between,the dynamics settle in a periodic orbit, denoted by PO. White lines show the boundary of the cyclicphase. For a small cost, the model is bistable for medium values of r . Green circles show the lowerboundary of the bistable region, above which the cooperative orbit becomes stable. The red squaresshow the upper boundary of the bistable region above which the dynamics settle in the cooperativeperiodic orbit starting from all the initial conditions. The filled green circle shows the point wherethe transition between the two periodic orbits becomes a continuous transition. Parameter values: g = 5, nu = 10 − , π = 2. The replicator dynamic is solved for 8000 time steps and time average aretaken over the last 2000 steps. Results
Mixed population
As shown in the Methods, the model can be solved analytically in terms of the replicator-mutatordynamics in a mixed population. We begin, by setting r = r = r , and color plot the densities ofdifferent strategies in the c g − r plane, in Fig. 1. Here, the replicator dynamic is solved starting froma uniform initial condition in which all the strategies’ initial density is equal. The phase diagramof the model is superimposed as well. The results of simulations in finite populations are in goodagreement with the replicator dynamics results (see Fig. S1). We also plot the time average densitiesof different strategies as a function of r , for three different values of c g , in Fig. 2. Blue dots andred squares represent the solutions of the replicator dynamics for two different initial conditions, andorange circles show the results of simulations starting from a uniform initial condition in which theindividuals’ strategies are assigned at random. Throughout this manuscript we fix c = 1.For small enhancement factors r , the dynamics settle in a fixed point where only defectors inthe free institute survive. As r increases beyond a threshold, the advantage of the costly institutebecomes apparent: Cooperators in the costly institute, but not in the free institute, start to appearin the system. This shows paying a participation cost works as an incentive for the individuals tocooperate. However, as the enhancement factor increases, public goods’ return can increase well abovethe participation cost. This gives an incentive for free-riding: In this region, free-riders’ density inthe costly institute increases slowly by increasing r .For large enough costs, as r increases, at a transition line indicated by the dashed white line inFig. 1, temporal fluctuations sets in. These fluctuations are derived from the cyclic dominance ofdifferent strategies. The model shows two qualitatively different periodic orbits. In the first one,which occurs for smaller values of r , cooperation in the costly institute, but not in the free instituteevolves. We call this periodic orbit the defective periodic orbit. For larger values of r , cooperationin both the costly and the free institutes evolves. We call this the cooperative periodic orbit, owing3 (a) c g = 0.18 (c) c g = 0.18 (b) c g = 0.18 (d) c g = 0.18 c g = 0.398 c g = 0.398 c g = 0.398 c g = 0.398 c g = 0.5 c g = 0.5 c g = 0.5 c g = 0.5 Figure 2: Density of different strategies as a function of r for three different values of cost. Thereplicator dynamics is solved for two different initial conditions, a cooperation favoring initial conditionin which all the individuals are cooperators and prefer the costly institute (blue dots), and a uniforminitial condition in which strategy and game preference of the individuals are assigned at random (redsquares). The result of simulations in a population of size N = 10000 starting from a random initialcondition is shown by orange circles. The system shows two different cooperative phases. For smallenhancement factors, cooperation in the costly institute, but not in the free institute, evolves. Forlarger enhancement factors, cooperation in both costly and free institutes evolves. While for a smallcost, the transition between the two cooperative phases is discontinuous and shows bistability (a),for high cost, there is a cross-over between the two phases by increasing the enhancement factor (c).Parameter values: g = 5, nu = 10 − , π = 2. The replicator dynamic is solved for 9000 time steps,and the time averages are taken over the last 2000 time steps. The simulation is performed for 6000time steps, and the averages are taken for the last 3000 time steps. (a) (b) (c) (d)
200 400 600 800 10000.20.40.60.8 0.20.40.60.8 200 400 600 800 100000.20.40.60.8 00.20.40.60.8 200 400 600 800 10000.20.40.60.8 0.20.40.60.8 200 400 600 800 100000.20.40.60.8 00.20.40.60.8
Figure 3: Examples of the time evolution of the system. Examples of the time evolution of the systemfor the defective periodic orbit, (a) and (b), and the cooperative periodic orbit (c) and (d). The toppanels show the replicator dynamics results, and the bottom panels show the result of a simulationin a population of size N = 40000 individuals. While in the defective periodic orbit only in thecostly institute, cooperation evolves, in the cooperative periodic orbit, cooperation in both institutesevolves. Parameter values: g = 5, nu = 10 − , π = 2, and c g = 0 .
18. In (a) and (b) r = 2 .
2, and in(c) and (d) r = 2 . r increases beyonda final threshold (white dashed line for large values of r ), the system settles in a cooperative fixedpoint, where most of the individuals prefer the free institute and cooperate.For small costs, the system possesses a bistable region where both periodic orbits are stable. Green4igure 4: Evolution of cooperation. (a): Time average total density of cooperators, ρ C = ρ C + ρ C inthe r − c g plane. (b): Time average difference between the probability that an individual in the costlyinstitute is cooperator from the probability that an individual in the free institute is a cooperator, γ = ρ C / ( ρ C + ρ D ) − ρ C / ( ρ C + ρ D ). Individuals are always more likely to be cooperator in the costlyinstitute. (c) and (d): The time average total density of cooperators, ρ C = ρ C + ρ C in the r − r plane for c g = 0 . c g = 0 . g = 5, nu = 10 − , and π = 2. Thereplicator dynamics is solved for 8000 time steps and time average are taken over the last 2000 steps.circles show the lower boundary of the bistable region, below which the cooperative orbit is unstable.Its upper boundary, above which the defective periodic orbit becomes unstable, is plotted by redsquares, in Fig. 1.The bistability of the dynamics for small costs can be seen in Fig. 2 (top panels), as well. Here, theblue dots and red squares show the replicator dynamics’ stationary state starting from two differentinitial conditions. While for both small and large r the dynamic is mono-stable and the attractors ofthe dynamic for different initial conditions coincide, the situation is different for medium values of r :For r between around 2 and 3, the system shows a bistable region where the two initial conditionsresult in different attractors. On the other hand, for large costs (bottom panels), no bistabilityis observed, and the stationary state of the dynamics is independent of the initial condition. Inthis region, by increasing r , the system shows a cross-over from the defective periodic orbit to thecooperative periodic orbit without passing any singularity. In between, the transition between thetwo periodic orbits becomes continuous at a single critical point (middle panel).An example of the defective periodic orbit, in smaller values of r , is presented in Figs. 3(a)and 3(b). Here, r = 2 . c g = 0 .
18. The top panels show the replicator dynamics results, andthe bottom panels show the result of a simulation in a population of N = 40000 individuals. D experiences an advantage over C , and proliferates when the density of C increases. Being exploitedby D , the density of C declines when D increases in the population. This decreases the return inthe costly institute below its entrance cost. At this point, due to not paying an entrance cost, D performs better than D and increases in density.The dynamic of the model is different in the cooperative periodic orbit. This can be seen in Fig.3(c) and 3(d), where an example of the cooperative periodic orbit is presented. Here, r = 2 .
35 and c g = 0 .
18. The top panels show the replicator dynamics results, and the bottom panels show the resultof a simulation in a population of N = 40000 individuals. In this case, cooperation in both institutesevolves. As cooperators’ density in an institute i increases, defectors in this institute start to increasein number. This decreases the profitability of the institute i , and thus, cooperators in the competinginstitute increase in number. This, in turn, motivates free-riding in the competing institute. In thisway, the cyclic dominance of the four possible strategies, derives periodic fluctuations in the timeevolution of the system.An interesting question is how the population’s cooperation depends on the participation cost and5igure 5: The density of different strategies in the r − r plane. The densities of different strategiesin the r − r plane, for two different costs, c g = 0 . c g = 0 . c g = 0 .
1, top), for both smalland large enhancement factor values, r , the dynamics settle in a fixed point, denoted by FP. Inbetween, the dynamics settle in a periodic orbit, denoted by PO. White lines show the boundaryof the cyclic phases. For a small cost, the model is bistable for medium values of r . Green circlesshow the lower boundary of the bistable region, above which the cooperative periodic orbit becomesstable. The red squares show the upper boundary of the bistable region above which the dynamicssettle in the cooperative periodic orbit starting from all the initial conditions. Blue triangles show thephase boundary, resulting from a uniform initial condition. For a large cost ( c g = 0 .
4, bottom), thebistability is lost, and the periodic phase’s domain increases. Parameter values: g = 5, nu = 10 − ,and π = 2. The replicator dynamic is solved for 8000 time steps and time average are taken over thelast 2000 steps.the enhancement factors. To see this, in Fig. 4(a) we plot the density of cooperators in the population, ρ C = ρ C + ρ C . For large r , increasing the entrance cost of the costly institute has a detrimental effecton cooperation in both institutes. This is because, although the density of defectors in the costlyinstitute remains close to zero, fewer individuals are willing to choose a costly institute with a highcost. This decreases the density of costly cooperators in the population. Due to its inability to attractindividuals, the costly institute plays a less viable role as an alternative to a free institute. As havingsuch a choice between different institutes promotes cooperation [31], the limitation imposed on theindividuals’ choice by a high participation cost can deteriorate cooperation in the free institute, aswell.On the other hand, having a participation cost can be beneficial for small enhancement factors.This is because an entrance cost deters defectors from entering the costly institute. To more clearlysee this is the case, we consider the difference between the probabilities that an individual in the costlyinstitute is a cooperator and the probability that an individual in the free institute is a cooperator, γ = ρ C / ( ρ C + ρ D ) − ρ C / ( ρ C + ρ D ). This is contour plotted in Fig. 4(b), where it can be seen it isalways positive. This fact increases a costly institute’s profitability and allows individuals to reach ahigher payoff by entering the costly institute.So far, we have assumed that the costly institute and the free institute have the same quality. In6 a) C1 t (b)
D1 t (c)
C2 t (d)
D2 t
Figure 6: The density of different strategies in the c g − r plane in a structured population. The densitiesof different strategies in the c g − r plane are color plotted. Parameter values: g = 5, nu = 10 − , and π = 2. The population resides on a 200 ×
200 first nearest neighbor square lattice with von Neumannconnectivity and periodic boundaries. The simulation is performed for 5000 time steps starting froman initial condition in which all the individuals are defectors and prefer one of the two institutes atrandom. The time average is taken over the last 2000 steps.general, the quality of the two resources may be different. To study this case, in Figs. 4(c) and 4(b)we plot the density of cooperators in the population, ρ C = ρ C + ρ C , in the r − r plane, for twodifferent costs. The phase diagram of the model is superimposed as well. The density of differentstrategies in these cases are plotted in Fig. 5 (See Figs. S2 and S3 in the Supplemental Information forcomparison with simulations). In the case of a small cost, c g = 0 .
1, the model settles in a fixed pointwith a low level of cooperation for small enhancement factors. As the enhancement factors increase,the model becomes bistable: In addition to the fixed point, a periodic orbit in which cooperatorssurvive and cyclically dominate the population emerges. Green circles plot the lower boundary of thebistable region, and its upper boundary, above which the fixed point becomes unstable, is plotted byred squares. The phase boundary can be defined as the boundary where a transition between the twophases occurs starting from a uniform initial condition in which the initial density of all the strategiesare equal [32]. This is marked by blue triangles. Further increasing the enhancement factors, abovea final transition line plotted by the dashed white line, the dynamics settle in a fixed point wherecooperators dominate the population.The situation is different for large cost ( c g = 0 . r − r plane. In addition, even for small r ( r = 1), cooperation in the costly institutecan evolve as long as r is larger than a rather small threshold (around 1 . r <
1: As individuals in the free institute defectfor too small r and the free institute yields a zeros outcome, decreasing the value of r below 1 doesnot affect the evolution of cooperation in the costly institute. This shows that the existence of a viablealternative is not necessary for the evolution of cooperation in a costly institute. Rather, having anentrance cost acts as a deterrence mechanism that can control free-riding in the institute, as longas participation in the public goods is voluntary. This contrasts the situation in the free institute,where, for too small r , cooperation does not evolve even for large values of r . This, in turn, shows,maintenance of cooperation in a free institute is contingent on the presence of a viable alternative,which guarantees individuals to have a choice between different viable institutes. Structured population
In contrast to the mixed population, the model shows no bistability in a structured population, andthe fate of the dynamics is independent of the initial condition. To see why this is the case, wenote that in a mixed population, a situation where all the individuals are defectors, and randomly7igure 7: The density of different strategies in the r − r plane in a structured population. Thedensities of different strategies in a structured population for two different costs, c g = 0 . c g = 0 . r − r plane. The population resides on a 200 ×
200 firstnearest neighbor square lattice with von Neumann connectivity and periodic boundaries. Parametervalues: g = 5, nu = 10 − , and π = 2. The simulation is performed for 5000 time steps starting froman initial condition in which all the individuals are defectors and prefer one of the two institutes atrandom. The time average is taken over the last 2000 steps.prefer one of the two institutes, is the worst case for the evolution of cooperation, as in this case,mutant cooperators are in a disadvantage in both institutes. However, in a structured population,starting from such an initial condition, due to spatial fluctuations blocks of defectors, the majority ofwhom prefer the same institute form. A mutant cooperator who prefers the minority institute in itsneighborhood obtains a high payoff and proliferates. This removes the bistability of the dynamics ina structured population.To study the model’s behavior in a structured population, we perform simulations starting froman initial condition in which all the individuals are defectors and prefer one of the two institutes atrandom. The model shows similar behavior in a structured population to that in a mixed population.This can be seen in Fig. 6 where the densities of different strategies are color plotted in the c g − r plane. As was the case in a mixed population, cooperation does not evolve for too small values of r . As r increases beyond a threshold, cooperation does evolve in the costly institute, but not in thefree institute. In this region, for a fixed enhancement factor, an optimal cost exists, which optimizesthe cooperation level in the population. On the other hand, cooperation in both the costly and thefree institutes evolves for large enhancement factors. In this region, increasing the cost can slightlyincreases defection in the free institute and have a detrimental effect on the evolution of cooperation,but not as much as it does in a mixed population.To look at the model’s behavior in a structured population when the two resources have differentqualities, in Fig. 7 we plot the densities of different strategies in the r − r plane for two differentparticipation costs. In the top panels c g = 0 .
1, and in the bottom panels c g = 0 .
4. As mentionedbefore, contrary to a mixed population, the model does not show bistability in a structured population.Consequently, the behavior of the model is qualitatively similar for small and large costs. For too8mall r , for all the values of r , only non-costly defectors, D survive. On the other hand, for r larger than a small value (approximately r = 1 .
5) costly cooperators survive even for r equal orsmaller than 1. This shows, similarly to the case of a mixed population, having a choice betweendifferent resources is not necessary to promote cooperation in a costly institute. Rather, cooperationevolves in a costly institute even for relatively small enhancement factors as long as participation isoptional. Increasing r however increases cooperation level in the costly institute (as long as r < r ).This results from the beneficial effect of freedom of choice between different public resources for theevolution of cooperation. Comparing the results for a small and a high participation cost shows thathigher participation cost improves cooperation level in the costly institute for small r , i.e., when aviable alternative does not exist.On a spatial structure, the model’s dynamic is governed by the cyclic dominance of differentstrategies, which can give rise to traveling waves. To take a deeper look into the dynamics of thesystem, in Fig. 8 we present snapshots of the population’s stationary state in different phases. Inthis figure, we consider a model in which individuals reproduce with a probability proportional to theexponential of their payoff, π , times a selection parameter, β , exp( βπ ) (see the Supplementary Infor-mation S.1), with β = 5. The situation in the model where individuals reproduce with a probabilityproportional to their payoff is similar. In Fig. 8(a), we have set r = r = 1 .
7. This phase correspondsto the defective periodic orbit in the mixed population case. Here the majority of the population arenon-costly defectors. Costly cooperators experience advantage over the former and can proliferate inthe sea of non-costly defectors. However, costly cooperators are at a disadvantage in comparison toboth costly defectors and non-costly cooperators. The former can only survive in small bands aroundcostly cooperators, as they rapidly get replaced by non-costly defectors once they eliminate costlycooperators. This phenomenon shows that competition between defectors with differing institutepreferences can positively impact the evolution of cooperation. Non-costly cooperators, in turn, cansurvive by forming compact domains where they reap the benefit of cooperation among themselves.However, as in this region, the effect of network reciprocity is too small to promote cooperation,non-costly cooperators get eliminated by non-costly defectors once costly cooperators are out of thepicture. Consequently, the dynamic of the system is governed by traveling waves of costly cooperatorsfollowed by small trails of costly defectors and non-costly cooperators in a sea of non-costly defectors(see the Supplementary Video, SV.1, for an illustration of the dynamics).Fig. 8(b), shows a snapshot of the population for r = r = 3 .
5. In this region, non-costlycooperators dominate the population. However, non-costly defectors can survive in small bands inthe sea of non-costly cooperators. While at a disadvantage in the sea of non-costly cooperators, costlycooperators beat non-costly defectors. Consequently, small blocks of costly cooperators are formedwithin the bands of non-costly defectors. These blocks of costly cooperators move along the bands ofnon-costly defectors and purge the population from non-costly defectors. In this way, although costlycooperators exist only in small density, they play a constructive role in helping non-costly cooperatorsto dominate the population.Finally, a similar phenomenon can also occur in the opposite regime of r > , where costlycooperators dominate the population. An example of this situation is plotted in Fig. 8(c). Here, r =3, r = 1 . c g = 0 .
6. In this case, costly cooperators dominate the population. Costly defectors cansurvive in the sea of costly cooperators in small bands. Non-costly cooperators, while at a disadvantagein the sea of costly cooperators, can grow within the small bands of costly defectors. A similarphenomenon holds for non-costly defectors who, due to not paying the participation cost, receive ahigher payoff than costly defectors once costly cooperators are eliminated in their neighborhood. Boththese latter types can form and grow in domains of costly defectors. This, in turn, helps the evolutionof cooperation in the costly institute.Analysis of the spatial patterns at work in the evolution of the system reveals competition or syn-9 a)
50 100 150 200 250 300 350 400 x y D D C C (b)
50 100 150 200 250 300 350 400 x D D C C (c)
50 100 150 200 250 300 350 400 x D D C C Figure 8: Snapshots of the population in the stationary state for different parameter values. Differentstrategies are color codded (legend). In (a), r = 1 .
7, 1 = 1 .
7, in (b), r = 3 . r = 3 . r = 3, and r = 1 .
8. In all the cases c g = 0 .
6. Here, individuals reproduce with aprobability proportional to the exponential of their payoff with a selection parameter equal to β = 5.The population resides on a 400 ×
400 square lattice with von Neumann connectivity and periodicboundaries. Parameter values: g = 5 and ν = 10 − ergistic relation between individuals with different institute preference plays an essential role in theevolution of cooperation in the system. Defectors with different institute preferences always appear ascompetitors who compete over space. By eliminating each other, they play a surprisingly constructiverole in the evolution of cooperation (this can be seen, for instance, in Fig. 8(c)). Cooperators, on theother hand, while having a direct competition over scarce sites, can also act synergistically and helpthe evolution of cooperation in their opposite institute. In a neighborhood of defectors who prefer thesame institute, a mutant cooperator who prefers the opposite institute performs better than the neigh-boring defectors and can grow. In this way, by purging defectors with an opposite game preference,cooperators help fellow cooperators with an opposite game preference. Consequently, cooperatorswith different game preferences, despite having direct competition for scarce spatial resources, canengage in a mutualistic relation and help each other to overcome defectors. Discussion
The question that how participation cost affects the evolution of cooperation had been consideredin the context of binary interactions (prisoner’s dilemma), for instance, in the context of indirectreciprocity [33], structured populations [34–36], and in repeated group interactions [37]. Some previouswork show when interactions are obligatory, participation cost can have a detrimental effect on theevolution of cooperation [33, 34]. As we have shown, the advantage of a costly institute becomesapparent when it is accompanied by an alternative free institute. In such contexts, while defectorspredominantly join a free institute, cooperators can reach a higher payoff by working cooperatively in acostly institute, despite paying a participation cost with no direct benefit. Furthermore, the existenceof a costly institute has a positive impact on the evolution of cooperation in the free institute. In amixed population, this is brought about by the fact that non-costly defectors, those who join a freeinstitute and free ride, can be eliminated by costly cooperators, those who join a costly institute andcooperate. While this mechanism can be at work to promote cooperation in a mixed population, itis further strengthened in the presence of population structure. In a structured population, costly10ooperators, while at a disadvantage in the presence of non-costly cooperators or costly defectors, canbeat non-costly defectors. Consequently, they can grow in domains of non-costly defectors, and bypurging the population from non-costly defectors, they help the evolution of non-costly cooperators.Analysis of the model in a structured population reveals interesting insights about how having achoice between different resources can help the evolution of cooperation. In a structured population,defectors with different game preferences are in direct competition over scarce spatial resources. Byundermining each other, they help the evolution of cooperation in their opposite institute. Cooperatorswith opposite game preferences are in direct competition as well. However, a mutualistic relationemerges between these two as well. This comes about because cooperators can beat defectors withthe opposite game preference and grow in their domain. By eliminating free-riders in the oppositeinstitute, they help their rival cooperators with a different game preference to flourish. Consequently,a mutualistic relation between cooperators with opposite game preference emerges, which helps theevolution of cooperation in the population. We note that such a dynamic is at work to help theevolution of cooperation in a structured population in a more general context where individuals havea choice between different institutes, irrespective of the existence of a participation cost.Empirical examples of a situation where individuals have a choice between different public resourcesto perform a collective action task appear to abound in human societies. Some examples include com-petition between political parties, firms, and associations, team incentives within firms [38]. A choicebetween geographic locations such as cities [40, 41], or having a choice between different collectiveaction tasks such as communal hunting or communal agriculture [42] provide other examples. Asour analysis suggests, the existence of a costly alternative can positively impact the evolution of co-operation both in the costly resource and its competing resources. In the case of large scale humaninstitutes, such a participation cost can be costly signaling of a common goal, for example, in the formof a donation to charity [43], or transaction cost associated with the search, bargaining, or monitoringto maintaining the commons [44]. Formation and maintenance of costly relations [34, 45, 46], whensome heterogeneity in formation and maintenance costs exists, can be another example at work ina situation where individuals can form or join different groups with different participation costs, toperform a collective action task.Examples of a situation where individuals have a choice between different resources appear to existin other biological populations as well. An example is the fission-fusion dynamics observed in differentanimal groups [3, 5, 47], or even hunter-gatherer human societies [39]. In many such cases, sub-groupsare formed within a group to perform a collective action task, such a cooperative hunting [3, 4], orcooperative defending [3]. Another example is resource competition in microorganisms [1, 2]. Ouranalysis predicts cooperation is easier to evolve when joining a group or embarking on a collectiveaction task is more costly than others. Future empirical research can shed more light on the extentto which such a mechanism is at work in different biological populations.
Methods
The replicator dynamics
The model can be described in terms of the replicator-mutators equation [48]. This equation readsas follows: ρ ix ( t + 1) = X y,j ν x,iy,j ρ jy ( t ) π jy ( t ) P z,l ρ lz ( t ) π lz ( t ) . (1)Here, x , y , and z refer to strategies and can be either C or D , and i , j , and l refer to the publicresources which can be 1 or 2. ν x,iy,j is the mutation rate from a strategy profile that prefers public11esource j and plays strategy y to a strategy combination that prefers public resource i and playsstrategy x . These can be written in terms of mutation rates as follows: ν x,iy,j = 1 − ν + ν , if ( i = j and x = y ) ν x,iy,j = ν − ν , if ( i = j and x = y ) or ( i = j and x = y ) ν x,iy,j = ν if ( i = j and x = y ) (2)In eq. (1), π jy is the expected payoff of an individual who prefers public resource j and plays strategy y . These terms can be written by averaging a focal individual’s payoff with game preference j (for j = 1 and 2) in a group composed of n jC cooperators and n jD defectors who prefer public resource j ,over all possible group configurations. In this way, we have the following equations for the payoffs: π C = g − − n C X n D =0 g − X n C =0 cr n C n C + n D (1 − ρ C − ρ D ) g − − n C − n D ρ Dn D ρ C n C (cid:18) g − n C , n D , g − − n C − n D (cid:19) − c − c g + π ,π D = g − − n C X n D =0 g − X n C =0 cr n C n C + n D (1 − ρ C − ρ D ) g − − n C − n D ρ Dn D ρ C n C (cid:18) g − n C , n D , g − − n C − n D (cid:19) − c g + π .π C = g − − n C X n D =0 g − X n C =0 cr n C n C + n D (1 − ρ C − ρ D ) g − − n C − n D ρ Dn D ρ C n C (cid:18) g − n C , n D , g − − n C − n D (cid:19) − c + π ,π D = g − − n C X n D =0 g − X n C =0 cr n C n C + n D (1 − ρ C − ρ D ) g − − n C − n D ρ Dn D ρ C n C (cid:18) g − n C , n D , g − − n C − n D (cid:19) + π . (3)In this equation, cr n C n C + n D − c in the first equation is the payoff of a cooperator who prefers thepublic resource 1, and cr n C n C + n D in the second equation is the payoff of a defector who prefers publicresource 1. (1 − ρ C − ρ D ) g − − n C − n D ρ Dn D ρ jC n C (cid:0) g − n c ,n D ,g − − n C − n D (cid:1) , is the probability that such a groupcomposition occurs. Here, (cid:0) g − n c ,n D ,g − − n C − n D (cid:1) = ( g − n C ! n D !( g − − n C − n D )! is the multinomial coefficientand is the number of ways that n C cooperators and n D defectors who prefer game 1 can be chosenamong g − c g . Finally, a base payoff of b is added toall the payoffs.It is easy to derive the expected payoff of those who prefer public resource 2, using a similarargument. 12 olution of the replicator dynamics and simulations Figs. 1, 4, and Fig. 5 result from numerical solution of the replicator dynamics. The replicatordynamics is solved for T = 8000 time steps, and an average over the last 2000 time steps is taken.The initial condition is a uniform initial condition in which all the strategies’ initial densities are equal( ρ C = ρ D = ρ D = ρ D = 0 . Acknowledgments
The author acknowledges funding from Alexander von Humboldt Foundation in the framework of theSofja Kovalevskaja Award endowed by the German Federal Ministry of Education and Research.