The Snowdrift Game, also known as the Hawk-Dove Game, is a social dilemma in which an individual can participate (cooperate) or not (defect) in producing a public good. It is relevant to a number of collective action problems in biology. In a population of individuals playing this game, traditional evolutionary models, in which the dynamics are continuous and deterministic, predict a stable, interior equilibrium frequency of cooperators. Here, we examine how finite population size and multilevel selection affect the evolution of cooperation in this game using a two-level Moran process, which involves discrete, stochastic dynamics. Our analysis has two main results. First, we find that multilevel selection in this model can yield significantly higher levels of cooperation than one finds in traditional models. Second, we identify a threshold effect for the payoff matrix in the Snowdrift Game, such that below (above) a determinate cost-to-benefit ratio, cooperation will almost surely fix (go extinct) in the population. This second result calls into question the explanatory reach of traditional continuous models and suggests a possible alternative explanation for high levels of cooperative behavior in nature.
Evolutionary game theory (Hofbauer and Sigmund 1998 [
The Prisoner’s Dilemma, in which
The Snowdrift Game (Sugden 1986 [
In what we might think of as a “standard" model in evolutionary game theory, there is an infinitely large, well-mixed population within which individuals either cooperate or defect in one-time, pairwise games with each other. One generally assumes that individuals reproduce asexually and that the number of offspring each individual has is proportional to its fitness. This makes it easy to track how the frequencies of cooperators and defectors change in the population over time. The rate of change of a strategy is given by the replicator dynamics (Hofbauer and Sigmund 1998 [
If a population of individuals is playing this game, then traditional evolutionary models, in which the dynamics are continuous and deterministic, predict a stable, interior equilibrium frequency of cooperators (Doebeli and Hauert 2005 [
However, all biological communities are finite, and many are small and organized into groups that together form a metapopulation (Gilpin 2012 [
Here we explore a discretization of the standard model of the Snowdrift Game. We consider a metapopulation composed of a finite set of discrete, nonintermixing groups, which are themselves composed of a finite set of discrete individuals who either cooperate or defect in the Snowdrift Game. The evolutionary dynamics both between and within groups are governed by a discrete time Moran process (Moran 1958 [
Our analysis has two main results. First, we show that the combination of within-group stochasticity and group selection can promote the evolution of cooperation in the metapopulation and can even result in cooperation’s fixation. This would be an impossible result were the evolutionary dynamics deterministic.
Second, we describe a phase transition for the fixation and extinction probabilities of cooperation in any finite group of a constant size whose members play the Snowdrift Game. Letting
As we discuss, our results provide insight into the evolution of cooperation, particularly from a multilevel perspective (Luo 2014 [
Here, we describe the within- and between-group Moran processes in our model.
Payoff Matrix 2 provides the payoff matrix for the two-player Snowdrift Game we will assume throughout this paper. Let
Note that “cooperation” in this context refers to the shoveling snow behavior—that is, strategy
Suppose there is a metapopulation composed of a finite number
Since we are interested in modeling interactions that generate public goods that can be used by all, we will assume individuals play the Snowdrift Game with all of the members of their respective groups, including themselves, simultaneously. This is equivalent to using the expected fitness of random pairwise interactions among members of the group allowing for self-interaction.
Formally, we can represent the fitness of cooperators and defectors in a group
The composition of a group can change in one of two ways: a cooperator can replace a defector, or a defector can replace a cooperator. Letting
Since no other changes within a group are possible, the probability that the state of the group will not change is given by,
Were each group well-mixed and infinitely large, each group
However, because group size is finite in our model and there are no mutations, the only truly stable states of a group are the two absorbing states, in which cooperators fix
Our model also involves a discrete time Moran process that occurs between groups. Within the metapopulation, a “parent" group is chosen to replicate, thereby producing a “daughter" group, which replaces uniformly at random some group in the metapopulation, perhaps its parent, but not itself. A daughter group of some group
The probability that a given group is chosen to reproduce is proportional to the average individual fitness of its members relative to the average individual fitness of the members of all the groups in the metapopulation. Taking the first and second derivatives of (
In our model, a group’s fitness is equal to the probability it will give rise to a daughter group at the next time step. Hence, a group’s fitness increases along with its frequency of cooperators. (One can of course consider other configurations of the relationship between individual fitness and group fitness—e.g., where a group’s fitness decreases as its frequency of cooperators increases—but we do not pursue such extensions here.)
In our model, we assume individual-level birth-death events occur more frequently than group-level reproduction-extinction events. Again, the abstract structure of the model itself does not require this. Presumably, the relative rates of the individual- and group-level events in a model should be governed by the target system one is modeling.
There is a straightforward way to calculate the probability that a given group
The probability that there will be no change in the metapopulation is given by
We ran two classes of simulations. In the first class, we were concerned exclusively with the Moran process within a group. In the second class, we were concerned with our “complete” model, in which Moran processes occur both within and between groups. Both classes of simulations were implemented in R version 3.2.3 (R Core Team 2016 [
In our first class of simulations, we sought to explore how the within-group evolutionary dynamics were affected by the size of a group and the group’s
In our second class of simulations, we simulated the complete version of our model, in which there are discrete time Moran processes that occur both within groups and between groups.
In our first set of simulations within this class, we considered a scenario in which the groups in the metapopulation do not vary in their
In our second set of simulations within this class, we considered a scenario in which the groups in the metapopulation initially varied in their
These simulations were “truncated” in that they could stop before the metapopulation became monomorphic. We conducted truncated simulations because we were interested in documenting the evolutionary dynamics in the short and medium term, not only as one takes the limit of time. As our results show, the groups in the metapopulation, and the metapopulation itself, often spend a good deal of time at a polymorphic state. This is because the fixation times can be long, which underscores the need to not only consider the end state of the system. Eventually, this polymorphism will disappear, and so the simulations are an inaccurate representation of the dynamics of the system for an arbitrarily long sequence of events.
Following Traulsen et al. (2005 [
Equation (
The precise connections between the Moran process and the corresponding SDE and the Moran process and the corresponding ODE are given by the following lemma. A rigorous proof follows from standard stochastic analysis (Durrett 1996 [
Let where for any sequence
Convergence (
Convergence (
Lemma
In evolutionary biology, one is often interested in the probabilities of fixation and extinction. Let
This probability can be found in standard textbooks on the subject (e.g., Otto and Day 2007 [
Let
Through an application of Laplace’s method to the explicit formula for the fixation probability obtained in Lemma
The plot of
The plot of
The plot of
There exists a threshold number
The endpoints are given by
Asymptotic analysis (see, e.g., Murray 1984 [
The results of our purely within-group simulations are presented in Figure
In accord with analytic predictions, over a given number of generations, relatively smaller groups are more susceptible to larger fluctuations in group composition. Moreover, given a sufficient amount of time, and large enough groups, the within-group dynamics tend to follow a given pattern: in the short term, the frequency of cooperators in a group migrates toward the equilibrium frequency of cooperators given by (
Theorem
When groups’
The results of our metapopulation simulations are presented in Figures
There is a good deal of work that explores how stochasticity affects the evolutionary dynamics of a population of individuals playing a game in general (Fudenberg et al. 2006 [
There is, in fact, a considerable amount of work that explores how stochasticity affects the evolution of cooperation in a single, unstructured population of individuals playing the Snowdrift Game (Ficici and Pollack 2000 [
It is worth commenting on the general approach we have used to derive the value of this threshold quantity. In many traditional analyses in evolutionary game theory, one takes the limit of population size (
Within the context of the Snowdrift Game, if the population is large and one is concerned with a short- or medium-term time horizon, then the evolutionary trajectory of the system may be adequately represented by a “deterministic approximation” of the microscopic dynamics (Sandholm 2011 [
Our results provide insight into when within-group stochasticity and group selection do (and do not) result in evolutionary dynamics that are different from those in our two null models.
For instance, our results show that when the members of each group in the metapopulation play precisely the same parameterization of the Snowdrift Game, and when group size is small (
Why does this occur? Within each group, the frequency of cooperators is oscillating. Because the probability that a group will replicate at the next group replication event is proportional to its frequency of cooperators at the time of the group replication event, those groups with a relatively higher frequency of cooperators during the group replication event are more likely to replicate. When this occurs, the frequency of cooperators in the metapopulation will “jump” upward, as in a jump diffusion process. Indeed, within-group stochasticity can result in cooperators fixing in a group, which results in that group having the highest possible group fitness, making it possible that this group will then overtake the entire metapopulation. It is for this reason that we say within-group stochasticity in combination with group selection can promote the evolution of cooperation in the metapopulation, even its fixation, for a range of parameter values. (Note, as the number of groups in the metapopulation becomes increasingly large, the between-group selection dynamics become increasingly deterministic; thus, that there are a finite, rather small number of groups in our metapopulation simulations in fact dampens the efficacy of group selection; this modeling choice was intentional. We wanted to, if anything, load the die against the evolution of cooperation.)
However, our results also show that as the size of each group increases (
When we allow the groups in the metapopulation to initially vary in their
Why have we considered a case in which we allow groups to vary in the parameter values of the Snowdrift Game their members play? First, in a biological context, if a metapopulation is composed of groups whose members are engaged in collective action that is
Second, by allowing group payoff matrices to vary in our model, we learn something about the evolution of games themselves. Note that a central change brought about by selection and replacement of groups is a change to the payoff matrix of the underlying game. That is, as groups reproduce and go extinct, the payoffs of the game evolve. The possibility of an evolving game has been explored in other contexts, especially that of the Prisoners Dilemma, where it tends to promote cooperation in some contexts (Akçay and Roughgarden 2011 [
Our model and results connect in illuminating ways to other work on multilevel selection in general, as well as to applications of multilevel selection to the Snowdrift Game in particular. Others have used a multilevel Moran model, or close variants, to explore the evolution of social traits (Hauert and Imhof 2012 [
Ours is a different model than Traulsen and Nowak’s in that we consider a “strict” two-level Moran process, wherein both group size and metapopulation size are held constant. While two-level Moran models have been discussed before, and close variants of such a model have been described within the context of the Snowdrift Game (SI Traulsen and Nowak 2006 [
In fact, the dynamics of a strict two-level Moran model can differ in nontrivial ways from more general two-level Markov chain models. For instance, both Traulsen and Nowak (2006 [
Finally, there has been other work that explores how group structure can affect the evolution of cooperation in the Snowdrift Game, where it has been shown to sometimes promote the evolution of cooperation, though not always (Hauert and Doebeli 2005 [
In contrast, our analysis shows that quite a different kind of population structure—namely, in which the groups are discrete units that are spatially isolated within a metapopulation and replicate as a function of their internal composition—can, in certain cases, facilitate the evolution of cooperation, rather than hinder it. Whether a lattice structure or the form of population structure we consider above is a better representation of a given biological system of course depends on the details of that system.
This article presented a model of discrete individuals packaged into discrete groups, where the individual- and group-level dynamics were stochastic. We showed that within-group stochasticity and group selection promote the evolution of cooperation in the metapopulation when group size is small, and also when group size is large and groups are allowed to vary in the payoff matrices their members play. We further showed that the long-term fate of cooperation as one takes the limit of group size is determined by the cost-to-benefit ratio of cooperating in the Snowdrift Game and that this ratio effectively determines the long-term fate of cooperation even when group size is fairly small (e.g.,
All biological populations are finite (though they can be quite large), and they are often nested within a larger metapopulation. When these factors are incorporated into a model that would otherwise assume deterministic dynamics and no group structure (or group structure but no group selection), the differences in the predicted levels of cooperation can be substantial.
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This study was led by Laurence Loewe, Brian McLoone, and Wai-Tong Louis Fan, all three of whom conceptualized the project and developed the model. The simulations were implemented and analyzed in
The authors would like to thank Caitlin Pepperell and Tracy Smith for discussion and for detailed comments on earlier versions of this paper and Neil Van Lysel and Christina Koch at the Center for High Throughput Computing at UW-Madison for help in conducting their simulations. Shishi Luo, Burton Simon, and William Sandholm provided valuable comments on the manuscript. John Yin and David Schwartz raised important questions on the project in conversation. Brian McLoone acknowledges support from NSF Career Award 1149123 to Laurence Loewe and from an NHGRI training grant to the Genomic Sciences Training Program 5T32HG002760. Wai-Tong Louis Fan is supported by NSF Career Award DMS-1149312 to Sebastien Roch and by the Wisconsin Institute for Discovery at UW-Madison. Laurence Loewe is supported by NSF Career Award 1149123 and by the Wisconsin Institute for Discovery at UW-Madison.