invasion of cooperators in lattice populations: linear and...
TRANSCRIPT
Invasion of cooperators in lattice populations:
linear and non-linear public good games
Zsoka Vasarhelyia,∗ and Istvan Scheuringb
a. Department of Plant Systematics, Ecology and Theoretical Biology, EotvosLorand University, Budapest, H-1117, Pazmany Peter setany, 1/c, Hungary,
* corresponding authorb. Department of Plant Systematics, Ecology and Theoretical Biology, Research
Group in Theoretical Biology and Evolutionary Ecology, Eotvos Lorand Universityand the Hungarian Academy of Sciences, Budapest, H-1117, Pazmany Peter
setany, 1/c, Hungary
e-mail: [email protected]: [email protected]
Abstract
A generalized version of the N-person Volunteer’s Dilemma Game (NVD) hasbeen suggested recently for illustrating the problem of N-person social dilemmas.Using standard replicator dynamics it can be shown that coexistence of cooperatorsand defectors is typical in this model. However, the question of how a rare mutantcooperator could invade a population of defectors is still open.
Here we examined the dynamics of individual based stochastic models of theNVD. We analyze the dynamics in well-mixed and viscous populations. We show inboth cases that coexistence between cooperators and defectors is possible; moreover,spatial aggregation of types in viscous populations can easily lead to pure coopera-tion. Furthermore we analyze the invasion of cooperators in populations consistingpredominantly of defectors. In accordance with analytical results, in determinis-tic systems, we found the invasion of cooperators successful in the well-mixed caseonly if their initial concentration was higher than a critical threshold, defined bythe replicator dynamics of the NVD. In the viscous case, however, not the initialconcentration but the initial number determines the success of invasion. We showthat even a single mutant cooperator can invade with a high probability, becausethe local density of aggregated cooperators exceeds the threshold defined by thegame. Comparing the results to models using different benefit functions (linear orsigmoid), we show that the role of the benefit function is much more important inthe well-mixed than in the viscous case.
Key words: social dilemma, stochastic dynamics, probability of invasion, publicgoods, Volunteer’s Dilemma Game
Preprint submitted to Elsevier Science April 30, 2013
1 Introduction
Social dilemmas, that is situations in which collective and private interest con-flict, are frequently experienced in nature and human societies. Well-knownexamples include cooperation of unicellular organisms, hunting in groups, taxpaying, the open source software movement, etc. (Chuang et al., 2009; John-son, 2002; Kollock, 1998; MacLean et al., 2010; Packer et al., 1990; Wilson,2011). Concerning these dilemmas, two key questions emerge:
(1) What kind of mechanism maintains cooperative or altruistic behavior insituations where selfishness seems more beneficial?
And an even more challenging question:
(2) How can cooperation invade populations where selfishness is dominant?
In the literature of theoretical work treating social dilemmas, there is a weak-ening but still apparent dominance of 2-person game theory models with twostrategies, cooperation (C) and defection (D). The most popular ones, thePrisoner’s Dilemma (PD) (Hamilton, 1971; Axelrod and Hamilton, 1981) andthe Snowdrift Game (SD) (Sugden, 1986; Doebeli and Hauert, 2005) are sim-ple models that capture some key attributes of the problem. Due to decades ofresearch on the PD, it is common knowledge now that well-mixed populationsare dominated by defection, but in the case of structured, spatial, or sociallyinhomogenous populations, the stable coexistence of cooperation and defec-tion or even the dominance of cooperation can occur. (Nowak, 2006; Nowakand May, 1992; Perc and Szolnoki, 2010; Szabo and Fath, 2007). However, inmost social conflicts, like the abovementioned (and many others), there aremany more than N=2 players simultaneously active. Thus, using N-persongames is a much more desirable way of modelling social dilemmas.
Indeed, a rapidly growing body of experimental research (Chuang et al., 2009;Damore and Gore, 2012; Lee et al., 2008; MacLean et al., 2010; Packer et al.,1990; Rainey and Rainey, 2003; Wilson, 2011; Yip et al., 2008) has recentlyinvigorated interest in game-theoretical models of N-person social dilemmas(Archetti, 2009; Archetti and Scheuring, 2011, 2012; Chen et al., 2012a,b;Damore and Gore, 2012; Hauert et al., 2006a; Motro, 1991; Pacheco et al.,2009; Pena, 2012; Perc et al., 2013; Szolnoki and Perc, 2010; Souza et al.,2009; Zheng et al., 2007). Although the models have moved from 2- to thebiologically more relevant N-object interactions, the key questions, that ishow stability and invasion of cooperative behavior can be explained, have notaltered.
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Let us outline briefly the main results obtained from the simplest and mostfrequently used models of N-person social dilemmas. Consider an infinite well-mixed population with strategies C and D, where selection is studied by repli-cator dynamics (Hofbauer and Sigmung, 1998). In the model framework forN-person SD games (NSD), the coexistence of cooperators and defectors is theonly stable fixed point of replicator dynamics. However, the frequency of co-operators decreases with 1/N at equilibrium (Souza et al., 2009; Zheng et al.,2007). In the N-person PD game (NPD), which builds on the 2-person PDgame, cooperation disappears if c/b > 1/N where c is the cost and b is thebenefit of the cooperative act (for details see Appendix). It is generally arguedthat this relation is valid for most biologically reasonable situations (Archettiand Scheuring, 2012; Hauert et al., 2006a). To summarize, the frequency ofcooperators is marginally low or zero in large well-mixed populations whereindividuals play the NSD or the NPD games in larger groups. These resultssharply contradict the numerous observations of cooperation in groups of bac-teria, animals, and in human communities (Kollock, 1998; Lee et al., 2008;Rainey and Rainey, 2003; Yip et al., 2008).
Naturally, the condition of a large well-mixed population does not necessarilyhold for real situations, because, in many cases, there is some kind of positiveassortment among cooperators (e.g. colonies of bacteria or groups of coop-erative animals or humans, etc.). Positive assortment or relatedness, givingopportunity for kin-selection to act, enables cooperators to outcompete or co-exist stably with defectors even in NPD-like situations (Hamilton, 1964, 1971;van Baalen and Rand, 1998; Nowak, 2006; Nowak and May, 1992; Perc et al.,2013). Thus, the literature gives us the impression that positive assortment(or relatedness) is necessary for cooperation to persist in NPD/NSD-like situ-ations. However, in addition to positive assortment, there is another candidatemechanism that can explain how the production of public goods exists stablyin well-mixed, natural systems, which is to apply an alternative and more ad-equate N-person game to the NSD or the NPD (Archetti, 2009; Archetti andScheuring, 2012; Hauert et al., 2006a; Motro, 1991; Pacheco et al., 2009). Oneof these alternatives is the N-person Volunteer’s Dilemma Game (NVD) (orThreshold Public Goods Game) (Bach et al., 2006; Diekmann, 1985; Archetti,2009; Archetti and Scheuring, 2011).
The crucial difference between the NPD and the NVD games is the shape ofthe public goods function they use. In the NPD, the public good increaseslinearly with the number of cooperators i (Fig. 1). However, in real biologicalsystems, this linearity is at most a rare exception, and certainly not the rule.The well-documented case of cooperative hunting in groups is a good example,in which per capita success and benefit change non-linearly with the numberof participants (Bednarz, 1988; Creel and Creel, 1995; Packer et al., 1990;Stander, 1992; Yip et al., 2008). Similarly, cooperative nesting and breed-ing of vertebrates (Rabenold, 1984) and the fruiting body formation of social
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amoebas (Bonner, 2008) are examples of non-linear public goods games. In mi-crobial communities, where the public good is based on specific molecules likereplication enzymes (Turner and Chao, 2003), adhesive polymers produced byviruses (Rainey and Rainey, 2003), antibiotic resistance in bacteria (Lee et al.,2008) or invertase enzyme in yeast (Gore et al., 2009), the effect of enzymeproduction, and thus the amount of obtainable common good must generallybe a saturating function of the molecule concentration (Eungdamrong andIyengar, 2004; Hemker and Hemker, 1969; Mendes, 1997; Ricard and Noet,1986). Following this train of thought, the public goods function (B(i)) usedwhen modelling these kind of dilemmas should be a monotonously saturatingcurve with an inflection point somewhere, i.e. a sigmoid one (Fig. 1). It is quitea challenge to analyze a model with this benefit function instead of the linearNPD. In ordeer to simplify the problem, specific B(i) functions are defined(Archetti, 2009; Bach et al., 2006; Hauert et al., 2006b; Motro, 1991).
k cr N0
1
Num ber of cooperators
Pu
blic
go
od
s
Figure 1. Schematic picture of characteristically different public goods as a func-tion of number of cooperators. A general saturating non-linear B(i) function withinflection point at kcr (dashed line), the linear N-person Prisoner’s Dilemma game(dotted line) and the N-person Volunteer’s Dilemma game with threshold at kcr(continuous line).
It is biologically realistic to assume that the transition phase from acceleratingto discounting is steep at the inflection point, and, thus, as a specific model,we can use a step function for B(i) (Archetti, 2009):
B(i) =
0 if i < kcr ≤ N
b otherwise,
which defines the generalized N-person Volunteer’s Dilemma Game (Archetti,2009; Diekmann, 1985). That is, at least kcr cooperators (volunteers) are
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needed for the high benefit, otherwise cooperation does not achieve net benefit(in the classical NVD game kcr = 1). This model is more tractable, and itsdynamical behavior remains qualitatively the same as it would be for smoothbut similar functions of B(i) (Fig. 1) (Archetti and Scheuring, 2012). We notehere that the problem of a collective action in public goods game with non-linear benefit functions is a classical one in sociology and political science(Hardin, 1982; Maxwell and Oliver, 1993; Oliver, 1993; Ostrom, 2003, e.g.).However, studying this problem in the framework of evolutionary game theoryhas become an important issue only recently.
Assuming that N interacting individuals are selected randomly from an in-finitely large well-mixed population and using the replicator dynamics for theNVD game, it can be shown that if c/b > 1/r∗, then complete defection (x∗ = 0if x denotes the frequency of cooperators) is the only stable state of the dy-namics, while if c/b < 1/r∗, then x∗ = 0 and x∗ = xs < 1 are the stable fixedpoints, and x∗ = xu < xs and x∗ = 1 are the unstable fixed points of (A.4)(Archetti and Scheuring, 2011, 2012) (for details, see Appendix). Naturally,if the population structure allows positive assortment for cooperators and thesocial dilemma is described by the NVD, cooperators coexist or dominatestructured populations even more easily (Boza and Szamado, 2010; Szolnokiand Perc, 2010; Perc et al., 2013).
It is important that r∗ (which is a complex function of b/c and N) can bemuch smaller than N for the NVD and for similar games with smooth buthighly nonlinear benefit functions, so stable coexistence of D and C strategiesat xs is a typical solution of these non-linear public goods games (Archetti andScheuring, 2011, 2012). We have to note here, that the cooperative state isevolutionary unstable if investment (that is, c and b = rc) is an evolvable pa-rameter of the NVD, but remains stable for smooth sigmoid benefit functions(Deng and Chu, 2011). Since the strategies and consequently the investmentsof the presented model are fixed, this problem does not present itself here.The validity of this assumption is supported by numerous microbial systemswhere only a few discrete types are isolated, typically a cooperator and acheater strain (Chuang et al., 2009; Gore et al., 2009; Greig and Travisano,2004; Velicer, 2003; West et al., 2002, e.g.). Since x∗ = 0 remains an alterna-tive stable fixed point, rare C strategies never invade the population of D (seequestion (2) above). Stable coexistence of C and D strategies can be attainedonly if the initial frequency of cooperators is higher than xu (Fig. 2).
The deterministic model framework assumes an infinitely large population,but in real finite populations, where births and deaths are stochastic events,invasibility has to be measured by the probability of a successful invasion(spread or fixation) of an initially rare strategy (Gokhale and Traulsen, 2010;Kurokawa and Ihara, 2009; Nowak et al., 2004). In this paper, we are con-cerned with the likelihood of rare cooperators invading populations dominated
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by defection, that is, the second key question addressed above. To comparedifferent situations, we use models of cellular automata in which individualsplay stochastic NPD and NVD games or a general saturating non-linear N-person game with their neighbors, while dispersal can vary from low to veryhigh for modelling cases ranging from highly viscous and spatially structuredpopulations to well-mixed populations.
0 0.2 0.4 0.6 0.8 1-0.2
0
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Figure 2. Coexistence of cooperators and defectors in the analytic NVD model. Thefitness of cooperators (gray) and defectors (black) are depicted in function of thefrequency of cooperators. Filled circles denote stable, open circles denote unstablefixed points of the dynamics (A.4). Arrows show the direction of motion, thus xuand xs are the nontrivial unstable and the stable fixed points respectively. c/b = 0.2,kcr = 4, N = 9.
2 Models and Methods
To study invasion probabilities we consider a toroidal square lattice grid of L2
(generally 100x100) nodes with at most one individual per grid point. Everyindividual interacts with its eight nearest neighbors (Moore neighborhood),thus N = 9. Individuals differ only in their strategies: cooperators (C) alwayscooperate with cost c and defectors (D) never do. Payoffs are determined byinteractions with neighbors. Thus, the fitnesses of cooperators and defectorsare
WD =W0 +B(i) (1)
WC =W0 +B(i)− c,
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where B(i) is the benefit function of the focal individual having i number ofcooperators in the neighborhood, and W0 denotes the basic payoff, indepen-dent of the individual’s strategy (in most simulations, W0 = 0.8). The strengthof selection can be estimated by the maximal possible payoff difference, thatis, by [(W0 + b) − (W0 − c)]/W0 = (b + c)/W0. In most cases, this value was0.11/0.8=0.1375 (b = 1, c = 0.1). During simulations, we examined the effectof three different benefit functions discussed below.
In case of the NVD game, B(i) = b if the number of cooperators (i) is equal orhigher than the critical threshold, kcr in the neighborhood. If i < kcr, the focalindividual remains to have the basic payoff. Consequently, using the notation+ for higher and ◦ for lower B(i) values, payoffs are defined by the followingequations:
W+D =W0 + b and W+
C = W0 + b− c (if k ≥ kcr) (2)
W ◦
D =W0 and W ◦
C = W0 − c (if k < kcr)
The linear benefit function (NPD game) is defined by B(i) = αi, where B(i)is the value of the benefit, which depends on the number of cooperators i,and α is used to regulate the maximal benefit to be b (B(9) = b, α = b/9).
The sigmoid benefit function is defined by B(i) = b β(i)−β(0)β(9)−β(0)
where β(i) =[
(1 + e−σ(i−kcr))−1]
. Thus B(i) depends on the number of cooperators i, kcr isthe threshold, σ regulates the steepness of the function at the inflection point,and the benefit varies between 0 and b; that is, B(0) = 0, B(9) = b. Duringthe simulation we used σ = 1.5.
At the beginning, we place n0 < L2 cooperators randomly on the grid, whilethe rest of the grid points (L2 − n0) are occupied by defectors. First we com-pute the payoff of every individual with the method discussed above withinone Monte Carlo cycle (MC). After this, individuals die with a constant δprobability, independent of the strategies (in most of the cases δ = 0.2), leav-ing empty sites. Empty sites can be occupied again by the offspring of livingindividuals in their neighborhood. Every neighbor of an empty site (ie, je) pro-duces progeny with probabilityWf/
∑
ijǫNeWij (Nowak and May, 1992; Nowak
et al., 1994), where Wf is the payoff of the focal neighbor of the empty siteand
∑
ijǫNeWij is the sum of the payoffs in neighborhood Ne of (ie, je). Thus
neighbors place progeny to the empty site with probability proportional totheir relative payoff. (We note that other update rules also could be used here(Szabo and Toke, 1998). However, the benefit of the selected rule is its clearbiological interpretation: progenies are in local competition, and their successis proportional to their local relative fitness.) The next MC starts when allempty sites, which had at least one non-empty neighboring site are occupied
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again. Although it is possible that at the beginning of a new MC there remainempty sites on the grid (that is if empty patches of at least 3×3 grid pointsemerge after individuals die), it is realized with very low probability (δ9) atδ = 0.2, so presence of empty sites at the beginning of a new MC has only amarginal effect on the system, if any. We did not find characteristically dif-ferent results when varying δ between 0 and 0.4, the only change appeared inthe speed of the dynamics.
Using the algorithm explained above, we examined both extreme and inter-mediate population structures:
(1) The highly viscous model, in which the population is not mixed in thesimulation (apart from the weak mixing generated by reproduction toneighboring sites). We call this the ’viscous model’.
(2) The well-mixed model, in which each MC cycle is followed by intensemixing, such that every individual exchanges its position with a randomlychosen other individual.
(3) The variable mixing model, i.e. variable viscosity, in which each MC cycleis followed by a given number (denoted by m) of mixing steps. One stepmeans the exchange of the position of a randomly chosen individual withits randomly chosen neighbor. (On average, m = L2 = 104 mixing stepsare needed for each position to be chosen.)
Our basic parameter set consisted of the kcr = 3, c/b = 0.1, δ = 0.2 andW0 = 0.8 values. This seemingly arbitrary decision has relevance as long as itrepresents parameters in the middle of the relevant area and can be used as astarting-point of further and extended studies. Consequently, we also examinedthe system with parameters kcr = 2, 4 and c/b = 0.15, 0.2. When varying c/bwe also varied the value of W0 to hold the maximal payoff differences and thusthe strength of selection ((b + c)/W0) constant across simulations. We alsovaried the strength of selection by using parameters W0 = 0.1, 3.2 along withthe original c/b = 0.1.
To estimate the relative abundance of cooperators in the stable polymorphpopulation, we ran simulations with an initial frequency of cooperators x0 =n0/L
2 = 0.5. (For kcr ≤ 7 the unstable equilibrium (xu) of the analyticalmodel is below this value, so we assume that cooperators can reach stable fre-quency similar to xs from this initial value instead of being eliminated at ourparameter set.) Simulations were run for 3× 104 MC cycles. In each case, wecounted the average ratio of cooperators from the last thousand generations,and then we counted the mean of 100 independent averages of the same pa-rameter sets. To make comparisons we also evaluated the stable and unstablefixed points of the analytic model (Archetti and Scheuring, 2011).
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In the invasion experiments, we modified the initial ratio of cooperators (x0)on a continuum from 1/L2 to 0.4. We considered invasion to be successfulwhen the ratio of cooperators exceeded a critical threshold fraction, xcr. Sincein the previous experiment we observed that cooperators always remain inpolymorph equilibrium with defectors (or go to fixation) if the initial ratio, x0
was 0.5, we simply chose xcr = 0.5 as the threshold for indicating successfulinvasion of cooperators. Obviously, invasion was considered being unsuccessfulif invading cooperators were eliminated. The invasion probability ρ(x0) is theratio of successful experiments divided by all experiments. Although it is truethat if we had let the simulations run a very long time, one of the strategieswould have surely gone extinct, it can be shown that if a polymorphic equi-librium is stable in an infinite population, it remains at the correspondingmetastable state for a very long time in a stochastic finite population as well(the average fixation time scales with exp(L2) (Antal and Scheuring, 2006)).In our cases, simulation time was generally observed to be much smaller thanthe average fixation time from the metastable polymorphic state. We notea situation in the following section when this was not the case. To computeρ(x0), we used a different number of independent simulations between 5× 102
and 2× 105, depending on the parameter sets.
Finally, we examined the role of the population size, using alternative gridswith sizes varying between 50x50 and 250x250 grid points. For these simula-tions, we used the highly viscous and the well-mixed models. In the well-mixedmodel with larger grid sizes (L2 ≥ 200x200) we had to set the xcr criticalthreshold closer to the stable equilibrium (thus xcr = 0.45 in these cases) notto let the simulation last for too long.
3 Results
Using the analytic model (Archetti and Scheuring, 2011) we evaluated thenontrivial fixed points of the NVD and we estimated the stable equilibria in thewell-mixed and viscous models defined above (Fig. 3). Results show that whileour well-mixed and the analytic model behave very similarly, the viscous modeldiffers notably from them: the number of cooperators in the stable equilibriumis much higher for every kcr than in the well-mixed models. Further, in theviscous model, the population soon reaches pure cooperation except for lowvalues of kcr. Not having enough time and computational capacity, we couldnot estimate the unstable fixed points.
The probability of invasion by the cooperator strategy was measured as afunction of the initial number of cooperators. In the well-mixed case, the in-vasion probability ρ(x0) follows a sigmoid curve (Fig. 4). If the initial numberof cooperators, x0 is below the unstable fixed point, the invasion probabil-
9
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Figure 3. Ratio of cooperators as it varies with kcr at the stable and unstablefixed points of the NVD game, according to the analytic model and simulations.Filled triangle denotes the stable fixed points, filled circle denotes the unstable onescalculated by the analytical model, empty upside-down triangle denotes the stablefixed points estimated by the simulations in the well-mixed model, and empty squaredenotes the ones estimated in the viscous models. c/b = 0.1; for other parameterssee the main text.
ity is lower than it would be in a neutral case. We consider neutrality, whenthere is no selection and thus every individual has equal probability of fix-ing, i.e. ρ(x0) = n0/(L
2 xcr), where n0 is the initial number of cooperatorsand xcr is the critical threshold proportion over which we considered invasionsuccessful (Nowak et al., 2004). Varying parameters kcr and c/b in a givenparameter range, the shape of the ρ(x0) curve remains characteristically thesame, only it moves from left to right as kcr or c/b increases (Fig. 4). Varyingthe strength of selection, the ρ(x0) curve remains qualitatively the same. Inthe viscous model, the ρ(x0) function follows a concave saturating curve. Thecurve is above the neutral case in all measurements, even if x0 = 10−4, i.e. ifthere is only one cooperator in the population at the beginning (Fig. 4 B, D).Varying parameters kcr, c/b or the strength of selection in a given parameterrange, the shape of the ρ(x0) curve remains qualitatively the same (Fig. 4).In the well-mixed case, the number of cooperators fluctuates more stronglyaround xs, than in the viscous case (Fig. 5). Consequently, if the polymorphicstable state is at a low cooperator concentration, the cooperator strategy getseliminated by random drift more easily in the well-mixed populations than inthe viscous ones (fixation does occur within the simulation time scale). Thatis why there are no data for c/b = 0.2, kcr = 4 on Fig. 4 A. There, cooperatorsalways disappeared within some hundreds of generations, independent of theirinitial frequency.
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Figure 4. Invasion probability (ρ(x0)) of the cooperator strategy, as a function ofthe initial ratio of cooperators (x0) in the well-mixed (A and C) and the viscous(B and D) models of the NVD. Data differ in parameters c/b (A and B) and kcr(C and D). Above: the triangle denotes c/b = 0.1, the circle denotes c/b = 0.15,and the square denotes c/b = 0.2. Below: the triangle, circle and square denote thekcr = 2, kcr = 3 and kcr = 4 cases, respectively. Curves are fitted for visualization.The straight line denotes the hypothetical neutral case, when there is no selection.
When using smaller and larger grids we found that grid size (population size)did not alter the results qualitatively, that is, the shapes of the invasion prob-ability curves remained the same. However, these simulations revealed thatwhile in the well-mixed case the probability of invasion depends on the intitialfraction of cooperators, in the viscous model it depends on the initial number
of cooperators (Fig. 6.). We also found that ρ(x0) of the well-mixed modelconverges to a step function as the population size (grid size) increases (seeinset on Fig. 6.).
In the variable mixing model, we found that with increasing mixing, ρ(x0)decreases monotonically (Fig. 7). 4 × 104 mixing steps/MC returns a goodapproximation of the well-mixed case and, obviously, if the number of mixingsteps tends to zero, we recover the viscous case. The probability of fixationdecreases continuously as mixing increases, thus a high invasion probabilityof cooperators still occurs when mixing is rather intense, though not perfect
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0 5000 10 0000.5
0.6
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x
0 5000 10 0000
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x0 5000 10 000
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1.
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x
0 5000 10 0000
0.1
0.2
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Generations
x
Figure 5. The number of cooperators in the first 104 generations of the well-mixed(on the right) and viscous (on the left) models of the NVD. At the beginning, thefractions of cooperators and defectors are equal. c/b = 0.1, kcr = 3, other parametersare defined in the text.
(Fig. 7).
When using the linear benefit function (NPD game) we found that the invasionprobability of C is higher than that of a neutral strategy in the viscous case(similar to the basic model using the step function, although there, the differ-ence is more pronounced). However, invasion probability is constantly zero inthe well-mixed case (Fig. 8). Using the sigmoid benefit function, the shape ofρ(x0) is generally much more similar to that of the step function model thanto the linear model. However, the ρ(x0) values of the sigmoid function modelare always lower than those of the step function model.
This gives us the impression that sigmoid functions are generally closer to thestep extreme than to the linear extreme in behavior. Naturally, an infinitenumber of sigmoid benefit functions can be defined, and their steepnesses atthe inflection points determine which extremes (NVD or NPD) they are closerto (Archetti and Scheuring, 2012). Repeating the simulations at different gridsizes, we found that invasion probability depends on the initial number ofcooperators for NPD and sigmoid benefit functions as well (results not shown).
To better understand the surprising success of the invading cooperator strat-egy in the viscous case, we examined in more detail what happens at the
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Figure 6. Invasion probability of the cooperator strategy with varying grid sizes(L2) in the viscous (left) and the well-mixed (right) cases. In the viscous case, theinvasion probability depends on the initial number (n0) of cooperators. The triangledenotes grid size L2 = 100x100, the circle denotes L2 =200x200, and the squaredenotes L2 = 50x50. In the well-mixed case, the invasion probability depends onthe initial fraction (x0) of cooperators. Darker points represent larger grids. The usedand plotted grid sizes are 50x50 (lightest gray), 70x70, 100x100, 150x150, 250x250(black). The inset figure on the right shows steepnesses of fitted curves at theirinflection points (St) as functions of the grid size (L2) on logarithmic scales. Dataare obtained from simulations using the above mentioned and two other (120x120and 200x200) grid sizes. The line is fitted for visualisation.
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óó ó
ç
ç
ç
ç
çç ç ç
áá
á
á
á
á
ááá á á
õ
õ
õ
õ
õ
õõõ
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
x0
ΡHx
0L
Figure 7. Invasion probability (ρ(x0)) as function of mixing (m/MC). Left: Inva-sion probability (ρ(x0)) at varying mixing and fixed initial fractions of cooperators.Triangle denotes the x0 = 0.05, circle denotes the x0 = 0.1 case. Right: Invasionprobability as function of the inital fraction (x0) of cooperators at different lev-els of mixing. Triangle denotes the well-mixed case, square denotes the case withm = 2 × 104/MC, circle denotes the one with m = 104/MC, upside-down triangledenotes the viscous simulations. Curves are fitted for visualization, the straight linedenotes the hypothetical neutral case. c/b = 0.1, kcr = 3.
beginning of invasion if we put a single cooperator amongst defectors. We usethe term elementary states to indicate when there are 1, 2, 3, etc. cooper-ators in the population. These states are denoted by e1, e2, e3, etc. Duringthe simulations, we estimated the probabilities of transitions between thesestates (i.e. the probability of the number of cooperators increasing by one) as
13
ó ó
ó
ó
ó
ó
ç ç ç
ç
ç
ç
ç
ç
á á á á á á
0 0.1 0.2 0.3 0.40
0.2
0.4
0.6
0.8
1
x0
ΡHx
0L
ó
ó
ó
óó ó
ç
ç
ç
çç
ç
á
á
á
á
áá
0 0.002 0.004 0.006 0.008 0.010
0.2
0.4
0.6
0.8
1
x0
ΡHx
0L
Figure 8. Invasion probability ρ(x0) with different benefit functions in case ofwell-mixed (on the left) and viscous (on the right) population structures. Triangledenotes the step, circle denotes the sigmoid and square denotes the linear functions.Curves are fitted for visualization, the straight line represents the hypothetical neu-tral case. c/b = 0.1, kcr = 3, other parameters are in the text.
p{ei → ei+1} = p{ei+1} for small i-s, as follows. Initially we put i cooperatorson the grid. In the viscous case these cooperators had at least one neighborwith the same strategy (for i > 1), a connected patch of cooperators. For smallnumbers of cooperators this setting can be a good approximation of a patchof cooperators with one common ancestor, the initial single invader. In thewell-mixed case, cooperators were placed randomly on the grid. After placingthe i cooperators, there could be two outcome of the dynamics: either thenumber of cooperators increased by one after some time (ei → ... → ei+1), ordecreased to zero (ei → ... → e0), where the expression (→ ... →) can meanany number of steps (for example ei → ei−1 → ei−2 → ei−1 → ei → ei+1
for i > 2). Thus, the probability that the number of cooperators reaches thefollowing elementary state, i.e. increases by one, is
p{ei+1} =
∑
(ei → ... → ei+1)∑
ei, (3)
where∑
ei denotes all (ei → ... → ei+1) and (ei → ... → e0) cases and∑
(ei → ... → ei+1) denotes those ones, where the number of cooperatorsincreases by one.
Computing p{ei+1}, i = 1, 2, ..5 by simulation in the viscous model, we foundthat the first elementary transitions (that is e1 → e2, e2 → e3,..) happenedwith a lower probability than in the neutral case (denoted by p◦{ei+1}), but ifthe number of cooperators locally exceeded the threshold kcr, the probabilitiesof the elementary transitions soon exceeded the neutral probabilities (Fig. 9).It also can be seen that the difference between these probabilities does notnecessarily increase with increasing number of cooperators. The reason of thisphenomenon could be that the growing patch of cooperators starts to split tosmaller patches, which behave just as the initial patch did. Thus, the key stepsfor a rare cooperator of invading a population of defectors are the transitions
14
at the threshold kcr. For comparison, at the well-mixed case, the probabilityof these transitions stays below the neutral (Fig. 9). We calculated analyti-cally the probabilities of some transitions in the case of the viscous model forkcr = 2 (for details see Appendix), and the calculated trends are consistentwith the simulations: p{e2} < p◦{e2}, but p{e3} > p◦{e3}.
ó
ó
ó
óó
ç ç
ç
çç
á
á
á á á
2 3 4 5 6
0
0.01
0.02
e i+1
p8e
i+1<-
p°8
ei+
1<
ó
óó ó
óç
çç ç
çá á á
á á
2 3 4 5 6
0
0.01
0.02
ei+1
p8e i+
1<-
p°8e
i+1<
Figure 9. Differences between the probabilities of elementary transitions of the NVDand neutral games (p{ei+1}−p◦{ei+1}). Population is well-mixed on the right paneland viscous on the left. Square denotes the kcr = 2, circle denotes the kcr = 3 andtriangle the kcr = 4 cases, other parameters are the same as before.
15
4 Discussion
Examining stochastic individual based models of the N-person Volunteer’sDilemma we showed that while a rare mutant cooperator has very weak chanceof spreading in a well-mixed system, its success in a viscous population is quitehigh. Thus, not only is the stable coexistence of cooperators and defectors tobe expected in the NVD game, as has already been shown by models of wellmixed populations (Archetti and Scheuring, 2011), but the invasion of raremutant cooperators is also highly supported in a viscous population that hasstochastic selection dynamics. The invasion probability as a function of theinitial concentration of cooperators follows a sigmoid curve in the well-mixedcase and a concave saturating curve in the viscous case. The robustness of ourresult is supported by the fact that shapes of the curves remain character-istically the same despite varying parameters, such as the critical threshold,kcr or the cost/benefit ratio, c/b (Fig. 4). Similarly, cooperators spread suc-cessfully even if mixing is more intense (Fig. 7) and even if we use a smoothsigmoid benefit function instead of the step function (Fig. 8). These resultscomplement somehow the studies using similar spatial systems (Chen et al.,2012b; Szolnoki and Perc, 2010). These papers showed that the equilibriumfraction of cooperators reaches maximum at an intermediate threshold in theNVD game (Szolnoki and Perc, 2010) and at an intermediate inflection pointand steepness with the sigmoid benefit function (Chen et al., 2012b). Whatthey had not studied are the problem of invasion and the effect of mixing onthe invasion.
In a viscous population, the invasion of cooperators is more probable as theirnumber in a neighborhood exceeds the threshold kcr. These cooperators be-have as nuclei of further invasion, and these nuclei grow further with a higherprobability than collapse (Fig. 9). In this respect, the NVD and the NPD aresimilar models, because cooperators in the NPD also need some kind of pos-itive assortment to spread (for the two person PD game see (Langer et al.,2008)).
However, there is still an important difference between the two models: in theNPD, not only is the invasion of cooperation impossible without some kind ofassortment, but also the coexistence of cooperators and defectors is (Hauertet al., 2006b), whereas in the NVD, coexistence is possible without any kindof assortment. Our numerical results using different benefit functions supportthis statement. With sigmoid benefit functions, cooperation is able to spreadin both the well-mixed and the viscous cases, just as in the step function.When the benefit function is linear (NPD game), cooperation spreads suc-cessfully only in the viscous case. In the well-mixed case, there was no initialfraction of cooperators that would result in any successful invasion inside thereasonable range (x0 ∈ [0; 0.3]), which is in accordance with analytical and
16
numerical results for two person PD games (Traulsen et al., 2006). Anotherimportant difference between the well-mixed and the viscous populations isthat in the well-mixed case, the number of cooperators fluctuates strongly, soviscosity can help cooperators by lowering the magnitude of stochastic effectsand consequently the probability of random drift driven extinction (Fig. 5).We cannot state, however, that viscosity generally helps the invasion of coop-eration, since limited dispersal can increase not only the relatedness (or spatialcorrelation) among socially interacting individuals, but it may increase relat-edness among potential competitors and intensity of competition, which mightdiminish cooperation (van Baalen and Rand, 1998; Hamilton, 1971; Platt andBever, 2009; Taylor, 1992; West et al., 2002, e.g.). Our lattice model is specificin this sense, since viscosity does not increase the intensity of competition,thus positive effect of local relatedness is not overbalanced by more intensecompetition. Similarly, it is known that overlapping generations can have apositive or negative effect on the invasion of the altruist strategy (Lion et al.,2011; Platt and Bever, 2009; Taylor and Irwin, 2000). In our model, where gen-erations overlap, vacant grid points emerge separately after death phase if thedeath rate is not too large, thus, the intensity of competition is independentof the survival of individuals (1 − δ). Further, the reproductive success of in-dividuals is also independent of the death rate, thus, as numerical simulationssupport, overlapping generations have no effect on the results.
Our numerical studies and calculations of the transition probabilities at the ini-tial invasion support our intuition that the invasion of cooperators is successfulin viscous populations if the number of cooperators exceeds the threshold ina local patch. This patch (nucleus) will continue to grow with a high proba-bility. This is in accordance with the observation that the invasion probabilityfunction depends on the initial number of cooperators in the viscous case in-stead of the initial fraction. Initial aggregation naturally helps the spread ofcooperators even in the NPD game, but the effect is much less pronounced.
Many conflicts in nature can be described by N-person non-linear games (Bed-narz, 1988; Creel and Creel, 1995; Gore et al., 2009; Lee et al., 2008; Packeret al., 1990; Yip et al., 2008). Thus, reevaluating experimental and theoret-ical studies in the light of recent advances in the theoretical understandingof these games could move us closer to the understanding of social dilemmasin general. To give an example, our models have much in common with uni-cellular systems: the social dilemma has N participants, the medium can beviscous, and the benefit function is probably sigmoid and highly non-linear.In light of the results given by our model, we can expect to observe the stablecoexistence of cooperators and defectors in experimental populations with asimilar structure. There are well-known empirical examples in the literature ofpublic goods situations in unicellular populations (Gore et al., 2009; MacLeanet al., 2010, e.g.), but these experimental works follow the tradition (alongwith many other) and apply the 2-person Snowdrift game instead of the more
17
appropriate N-person non-linear game as a model framework to explain theexperimental results. Both model systems predict the coexistence of cooper-ators and defectors in concordance with the experimental results, but in anN-person non-linear game (NVD or sigmoid payoff), cooperators can invadeonly viscous populations easily, while in the 2-person Snowdrift game, coop-erators easily invade well-mixed populations too. Based on this difference, theapplicability of these two model frameworks can be tested experimentally.
5 Acknowledgements
This work is supported by the Hungarian Scientific Foundation (OTKA) noK 100299. We thank T. Antal for helping in calculations presented in theappendix and we also thank M. Archetti, D. Yu and the anonymous refereesfor their useful comments on the manuscript.
A Appendix
NPD and NVD games in infinite well-mixed populations
In the NPD, similar to the 2-person PD game, individuals follow either acooperator (C) or a selfish defector (D) strategy in the NPD game too. Coop-erators take on a costly act (with cost c) to provide a common good (b) whichis equally divided among all group members, even those who did not pay acost (defectors). We assume that the population is infinitely large and that allindividuals form random groups of size N . The payoff of defectors (PD) andcooperators (PC) in groups having i number of cooperators are
PD =bi
N
PC =bi
N− c = PD − c (A.1)
respectively. Thus the average fitness of the strategies D and C are
WD(x) =N−1∑
i=0
fi,N−1(x)bi
N
WC(x) =N−1∑
i=0
fi,N−1(x)
(
b(i+ 1)
N− c
)
, (A.2)
18
where
fi,N−1(x) =
(
N − 1
i
)
xi(1− x)N−1−i (A.3)
is the probability of having i cooperators among the other N − 1 group mem-bers if individuals are selected randomly from a population containing coop-erators in fraction x. It is easy to see that WC(x) = WD(x) + b/N − c. Thusif c/b > 1/N , defection is always better off. Naturally if c/b < 1/N , the oppo-site is valid. The selection process can be described by the replicator equation(Taylor and Jonker, 1978)
x = x(1− x) [WC(x)−WD(x)] . (A.4)
In case of c/b > 1/N , for every xǫ[0, 1] WC(x) < WD(x), and therefore x∗ = 0is the only stable fixed point of equation (A.4). If c/b < 1/N , x∗ = 1 is the onlystable fixed point of the dynamics. There is no social dilemma in this case sinceC is the winner of the selection. However, it can be argued that usually b/c issmaller than the typical number of interacting individuals N (i.e. c/b > 1/N),and thus cooperation in general will go extinct (Hauert et al., 2006a).
Most of the biological examples of social dilemmas have a common feature:the benefit function increases monotonously in a non-linear and saturatingmanner. To model properly these situations, the public goods function in use,B(i), must meet the following criteria:
(1) The public good increases monotonically with the number of cooperatorsin the group.
(2) B(i) saturates (tends to a maximal value b as the number of cooperatorsi tends to the group size, i → N).
(3) At low i, the increase in public goods production is accelerating, and afteran inflection point (kcr), the increase in public goods production becomesdiscounting (see Fig. 1 in the main text). (If kcr → 0, the benefit increasesin a completely discounting manner, while if kcr → N , the benefit iscompletely accelerating)
Thus, the average fitness of the strategies D and C are
WD(x) =N−1∑
i=0
fi,N−1(x)B(i)
WC(x) =N−1∑
i=0
fi,N−1(x) (B(i+ 1)− c) , (A.5)
19
where B(i) is a general public goods function satisfying the conditions listedabove. It is biologically realistic to assume that the transition phase from ac-celerating to discounting is steep at the inflection point, and, thus, as a specificand easily tractable model, we can use a step function forB(i) (Archetti, 2009),which defines the generalized N-person Volunteer’s Dilemma Game (Archetti,2009; Diekmann, 1985).
Calculations concerning the initial steps of invasion by a rare mutant
We considered a fraction of the grid with 5× 5 points with one cooperator inthe middle and defectors all around. If one of its neighbors dies, this coopera-tor can leave an offspring on the empty node with a probability proportionalto its relative payoff. The probabilities of these events is calculated accordingto (3). We assumed that in every MC at most one of the 5×5 individuals dies,which is a good approximation if δ is small (in the corresponding simulationsδ = 0.01). Following this logic, one can calculate the probabilities of all pos-sible transitions even for different ”shapes” of cooperator patches within this5 × 5 square (Fig. A.1). Since the number of different possible patch shapesincreases in a highly accelerating manner as the number of cooperators form-ing the patch increases, we did it only for the first three states (which meanstransitions e1 → e2, e1 → e0, e2 → e3, e2 → e1).
According to the law of total probability, one can calculate the probabilitiesin question as follows:
p{1 → ... → 3} = p{1 → 2}p{2 → ... → 3}
p{2 → ... → 3} = p{2 → 1}p{1 → ... → 3}+ p{2 → 3}
⇒
p{1 → ... → 3} = (p{1 → 2}p{2 → 3})/ (1− p{1 → 2}p{2 → 1}), (A.6)
where the expression p{i → ... → i+1}means a transition with any number ofintermediate states before reaching state ei, while p{i → i+1} means a tran-sition with no intermediate state. To compute p{i → i+ 1} we have to studytransition probabilities for all topologically different cases and compute theweighted sum of these probabilities (see Fig. A.1 for topologies and notations).Here we only show the calculation of one probability value (p{e2 → e3C}), butall the others can be calculated similarly.
The two topologically different e2 states (e2A and e2B) develop from e1 withequal probability, but this is not true in case of the five different e3 states.For example e3C (in the middle of the bottom of Fig. A.1) can develop fromboth e2 states, but with different probabilities. These probabilities depend on
20
Figure A.1. Possible topologies of states e1, e2 and e3. Black squares are cooperatorsand grey squares are positions where these cooperators can leave offspring to. Thewidth of the arrows represents the number of possible ways how an ei state developsinto a given ei+1 state (thin line one way, thick line two ways).
several parameters:
• the number of those neighboring empty grid points, from wich the givenconfiguration can evolve (for example the e2B → e3C transition can nothappen if a diagonal neighbor dies)
• the number of cooperators neighboring each of these points• the fitness values of all neighbors of these points
Let us denote the probabilities of transitions between these topologically differ-ent ei states by pe. Using notations defined in (2), the probability of transitione2 → e3C (denoted by pe{2 → 3C}) is
pe{2 → 3C} =1
2
(
4
25
2W+C
2W+C +W+
D + 5W ◦
D
+2
25
2W+C
2W+C +W+
D + 5W ◦
D
)
,
(A.7)
where the 12is needed because of the two different e2 states,
425
and 225
are theprobabilities of having those empty neighboring positions occupying which e3C
would evolve from e2A and e2B respectively and the two2W+
C
2W+
C+W+
D+5W ◦
D
expres-
sions are the probabilities of leaving an offspring with cooperator strategy tothese empty places (see also Models and Methods). (Note that these ratios areusually not equal.)
Following the logic of (A.7), one can calculate the probabilities of all {eim →
21
ejn} transitions, where i = {1; 2}, j = {2; 3}, m ∈ M := {A;B} and n ∈N := {A;B;C;D;E}. The {ei → ej} probabilities emerge as the sum of thesevalues. For example pe{2 → 3} =
∑
mǫM
∑
nǫN p{2m → 3n}
The last step to get the probabilities p{1 → ... → 3}, p{2 → ... → 3} andp{1 → ... → 3} from equation A.6 is to calculate the p{i → i+1} probabilitiesusing the pe values. For example
p{i → i+ 1} =pe{i → i+ 1}
pe{i → i+ 1}+ pe{i → i− 1}.
Using numerical values counted the way we presented above, we get the fol-lowing probabilities for the basic parameter set with kcr = 2:
p{1 → 2} = 0.497246
p{1 → 0} = 0.502754
p{2 → 3} = 0.519639
p{2 → 1} = 0.480361
p{1 → ... → 3} = 0.317291
p{2 → ... → 3} = 0.682709
Note that in neutral case, the probabilities of all of the p{i → j} transitionswould be 0.5. In the neutral case, p{1 → ... → 3} = 1/3 and p{2 → ... →3} = 2/3.
22
References
Antal, T., Scheuring, I., 2006. Fixation of strategies for an evolutionary gamein finite populations. Bull. Math. Biol. 68, 1923–1944.
Archetti, M., 2009. The volunteer’s dilemma and the optimal size of a socialgroup. J. Theor. Biol. 261, 475–480.
Archetti, M., Scheuring, I., 2011. Coexistence of cooperation and defection inpublic goods games, from the prisoner’s dilemma to the volunteer’s dilemma.Evolution 65, 1140–1148.
Archetti, M., Scheuring, I., 2012. Review: Game theory of public goods inone-shot social dilemmas without assortment. J. Theor. Biol. 299, 9–20.
Axelrod, R., Hamilton, W. D., 1981. The evolution of cooperation. Science211, 1390–1396.
Bach, L. A., Helvik, T., Christiansen, F. B., 2006. The evolution of n-playercooperation-threshold games and ESS bifurcations. J. Theor. Biol. 238, 426–434.
Bednarz, J. C., 1988. Cooperative hunting in Harris’ hawks (Parabuto unicinc-tus). Science 239, 1525–1527.
Bonner, J. T., 2008. The Social Amoeba. Princeton University Press, Prince-ton.
Boza, G., Szamado, S., 2010. Beneficial laggards: multilevel selection, cooper-ative polymorphism and division of labour in threshold public good games.BMC Evol. Biol. 10:336.
Chen, X., Liu, Y., Zhou, Y., Wang, L., Perc, M., 2012a. Adaptive and boundedinvestment returns promote cooperation in spatial public goods games.PLoS ONE 7, e36895.
Chen, X., Szolnoki, A., Perc, M., Wang, L., 2012b. Impact of generalizedbenefit functions on the evolution of cooperation in spatial public goodsgames with continuous strategies. Phys. Rev. E. 85, 066133.
Chuang, J. S., Rivoire, O., Leibler, S., 2009. Simpson’s paradox in a syntheticmicrobial system. Science 323, 272–275.
Creel, S., Creel, N. M., 1995. Communal hunting and pack size in African wilddogs, Lycaon pictus. Anim. Behav. 50, 1325–1339.
Damore, J. A., Gore, J., 2012. Understanding microbial cooperation. J. Theor.Biol. 299, 31–41.
Deng, K., Chu, T., 2011. Adaptive evolution of cooperation through darwiniandynamics in public goods games. PLoS ONE 6, e25496.
Diekmann, A., 1985. Volunteer’s dilemma. J. Confl. Res. 29, 605–610.Doebeli, M., Hauert, C., 2005. Models of cooperation based on the prisoner’sdilemma and the snowdrift game. Ecol. Lett. 8, 748–766.
Eungdamrong, N. J., Iyengar, R., 2004. Modeling cell signaling networks. Biol.Cell 96, 355–362.
Gokhale, C. S., Traulsen, A., 2010. Evolutionary games in the multiverse.Proc. Natl. Acad. Sci. 107, 5500–5504.
Gore, J., Youk, H., Oudenaarden, A., 2009. Snowdrift game dynamics and
23
facultative cheating in yeast. Nature 459, 253–256.Greig, D., Travisano, M., 2004. The prisoner’s dilemma and polymorphism inyeast SUC genes. Proc. Roy. Soc. B 271, 25–26.
Hamilton, W. D., 1964. The genetical evolution of social behaviour I. J. Theor.Biol. 7, 1–16.
Hamilton, W. D., 1971. Selection of selfish and altruistic behavior in some ex-treme models In: Man and Beast: Comparative Social Behavior. (Eisenberg,J. F., and W. S. Dillon, eds.). Smithsonian, Washington.
Hardin, R., 1982. Collective Action. Baltimore, MD: Johns Hopkins UniversityPress for Resources for the Future.
Hauert, C., Holmes, M., Doebeli, M., 2006a. Evolutionary games and popu-lation dynamics: maintenance of cooperation in public goods games. Proc.Roy. Soc. B. 273, 2565–2571.
Hauert, C., Michor, F., Nowak, M. A., Doebeli, M., 2006b. Synergy and dis-counting of cooperation in social dilemmas. J. Theor. Biol. 239, 195–202.
Hemker, H. C., Hemker, P. W., 1969. General kinetics of enzyme cascades.Proc. Roy. Soc. B. 173, 411–420.
Hofbauer, J., Sigmung, K., 1998. Evolutionary games and replicator dynamics.Cambridge Univ. Press.
Johnson, J. P., 2002. Open source software: Private provision of a public good.J. Econ. and Man. Strat. 11, 637–662.
Kollock, P., 1998. Social dilemmas: The anatomy of cooperation. Ann. Rev.Sociol. 24, 183–214.
Kurokawa, S., Ihara, Y., 2009. Emergence of cooperation in public goodsgames. Proc. Roy. Soc. B. 276, 1379–1384.
Langer, P., Nowak, M. A., Hauert, C., 2008. Spatial invasion of cooperation.J. Theor. Biol. 250, 634–641.
Lee, H. H., Molla, M. N., Cantor, C. R., Collins, J. J., 2008. Bacterial charityworks leads to population-wide resistance. Nature 467, 82–86.
Lion, S., Jansen, V. A. A., Day, T., 2011. Evolution in structured populations:beyond the kin versus group debate. Trends. Ecol. Evol. 26, 193–201.
MacLean, R. C., Fuentes-Hernandez, A., Greig, D., Hurst, L. D., Gudelj, I.,2010. A mixture of cheats and co-operators can enable maximal group ben-efit. PLoS Biol. 8.
Maxwell, G., Oliver, P. E., 1993. The Critical Mass in Collective Action: AMicro-Social Theory. Cambridge Univ. Press, Cambridge.
Mendes, P., 1997. Biochemistry by numbers: simulation of biochemical path-ways with Gepasi 3. Trends. Biochem. Sci. 22, 361–363.
Motro, U., 1991. Co-operation and defection: playing the field and the ESS.J. Theor. Biol. 151, 145–154.
Nowak, M. A., 2006. Five rules for the evolution of cooperation. Science 314,1560–1563.
Nowak, M. A., Bonhoeffer, S., May, R. M., 1994. More spatial games. Int. J.Bifurc. and Chaos 4, 33–56.
Nowak, M. A., May, R. M., 1992. Evolutionary games and spatial chaos. Na-
24
ture 359, 826–829.Nowak, M. A., Sasaki, A., Taylor, C., Fudenberg, D., 2004. Emergence ofcooperation and evolutionary stability in finite populations. Nature 428,646–650.
Oliver, P. E., 1993. Formal models of collective action. Ann. Rev. Sociol. 19,271–300.
Ostrom, E., 2003. How types of goods and property rights jointly affect col-lective action. J. Theor. Pol. 15, 239–270.
Pacheco, J. M., Santos, F. C., Souza, M. O., Skyrms, B., 2009. Evolutionarydynamics of collective action in N-person stag hunt dilemmas. Proc. Roy.Soc. B. 276, 315–321.
Packer, C., Scheel, D., Pusey, A. E., 1990. Why lions form groups: Food is notenough. Am. Nat. 136, 1–19.
Pena, J., 2012. Group-size diversity in public goods games. Evolution 66, 623–636.
Perc, M., Gomez-Gardenes, J., Szolnoki, A., Florıa, L. M., Moreno, Y., 2013.Evolutionary dynamics of group interactions on structured populations: areview. J. Roy. Soc. Int. 10, 20120997.
Perc, M., Szolnoki, A., 2010. Coevolutionary games – A mini review. BioSys-tems 99, 109–125.
Platt, T. G., Bever, J. D., 2009. Kin competition and the evolution of coop-eration. Trends. Ecol. Evol. 24, 370–377.
Rabenold, K. N., 1984. Cooperative enhancement of reproductive success intropical wren societies. Ecology 65, 871–885.
Rainey, P. B., Rainey, K., 2003. Evolution of cooperation and conflict in ex-perimental bacterial populations. Nature 425, 72–74.
Ricard, J., Noet, G., 1986. Catalytic efficiency, kinetic co-operativity of oligo-metric enzymes and evolution. J. Theor. Biol. 123, 431–451.
Souza, M. O., Pacheco, J. M., Santos, F. C., 2009. Evolution of cooperationunder N-person snowdrift games. J. Theor. Biol. 260, 581–588.
Stander, P. E., 1992. Hunting in lions: The role of the individual. Behav. Ecol.and Sociobiol. 29, 445–454.
Sugden, R., 1986. The economics of rights, cooperation and welfare. Blackwell,Oxford.
Szabo, G., Fath, G., 2007. Evolutionary games on graphs. Phys. Rep. 446,97–216.
Szabo, G., Toke, C., 1998. Evolutionary prisoner’s dilemma game on a squarelattice. Phys. Rev. E. 58, 69–73.
Szolnoki, A., Perc, M., 2010. Impact of critical mass on the evolution of coop-eration in spatial public goods games. Phys. Rev. E. 81, 057101.
Taylor, P. D., 1992. Altruism in viscous populations - an inclusive fitnessmodel. Evol. Ecol. 6, 352–356.
Taylor, P. D., Irwin, A. J., 2000. Overlapping generations can promote altru-istic behavior. Evolution 54, 1135–1141.
Taylor, P. D., Jonker, L. B., 1978. Evolutionarily stable strategies and game
25
dynamics. Math. Biosci. 40, 145–156.Traulsen, A., Nowak, M. A., Pacheco, J. M., 2006. Stochastic dynamics ofinvasion and fixation. Phys. Rev. E. 74, 011909.
Turner, P. E., Chao, L., 2003. Escape from prisoner’s dilemma in RNA phageφ 6. Am. Nat. 161, 497–505.
van Baalen, M., Rand, D., 1998. J. Theor. Biol. 193, 631–648.Velicer, G. J., 2003. Social strife in the microbial world. Trends Microbiol. 11,330–337.
West, S. A., Pen, I., Griffin, A. S., 2002. Cooperation and competition betweenrelatives. Science 296, 72–75.
Wilson, C., 2011. Cheatobiotics: Send in the subversive superbugs. New Sci-entist, 43–45.
Yip, E. C., Powers, K., Aviles, L., 2008. Cooperative capture of large preysolves scaling challenge faced by spider societies. Proc. Natl. Acad. Sci. 105,11818–11822.
Zheng, D. F., Yin, H. P., Chan, C. H., Hui, P. M., 2007. Cooperative behav-ior in a model of evolutionary snowdrift games with N-person interactions.Europhys. Lett. 80, 18002.
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