Effects of group size and population size on the evolutionary stability of cooperation
Abstract
Evolutionary game theory has classically been developed under the implicit assumption of an infinite population. Exact analytical results for finite populations are rare, and those that exist apply to situations in which strategy sets are discrete. We rigorously analyse a standard model for the evolution of cooperation (the multi-player continuous-strategy snowdrift game) and show that in many situations in which there is a cooperative evolutionarily stable strategy (ESS) if the population is infinite, there is no cooperative ESS if the population is finite (no matter how large). In these cases, contributing nothing is a globally convergently stable finite-population ESS, implying that apparent evolution of cooperation in such games is an artefact of the infinite population approximation. The key issue is that if the size of groups that play the game exceeds a critical proportion of the population then the infinite-population approximation predicts the wrong evolutionary outcome (in addition, the critical proportion itself depends on the population size). Our results are robust to the underlying selection process.
1. Introduction
Many evolutionary games assume—for mathematical convenience—that populations are infinitely large (e.g. [1–7]). This assumption is sometimes justified on the grounds that ‘[p]opulations which stay numerically small quickly go extinct by chance fluctuations’ [8, §2.1]. Of course, all real populations are finite, and important differences in evolutionary dynamics between finite and infinite populations have been demonstrated [9–15].
In spite of the technical challenges of working with finite populations, some exact analytical results have been obtained for games with discrete strategy sets [9,12,14–16], notably also in structured populations [17–24]. However, most existing finite-population results rely on approximate analytical methods and simulations [11,15,25–29]. For example, a diffusion approximation is often employed for its analytical convenience [1,27,30]; such approximations are useful, but have limitations (e.g. some real populations may be better-described using non-diffusive processes [31]).
Almost all existing finite-population results involve discrete strategy sets, such as when individuals must choose between making a fixed positive contribution to a public good, or nothing at all (e.g. [9,12,14–16]). Yet, many traits are better described on a continuum (e.g. allocation of time or effort to a communal task); consequently, evolutionary games involving continuous strategy sets are widely applicable, and they have been extensively studied using infinite-population models [32]. Moreover, to our knowledge, almost all existing results for finite populations assume a particular selection process—almost always the Moran process, and occasionally the Wright–Fisher process [33,34] (with few exceptions, e.g. [35,36]).
Here, we present mathematically rigorous results for finite-population evolutionary games with continuous strategy sets. We consider a standard model for exploring the evolution of cooperation—the continuous multi-player snowdrift game [3]—which has previously been studied in infinite populations using exact analysis and simulations [3,7,37–39] and in finite populations using approximations and simulations [11,29,40,41]. We focus on a subclass of the snowdrift game, which we refer to as the natural snowdrift game (NSG) because its definition excludes parameter regions that are not biologically sensible. For the NSG in finite populations, we characterize all evolutionarily stable strategies (ESSs) and expose the roles of group size and population size in determining the evolutionary stability of cooperation.
An important outcome of our analysis is that we rigorously identify critical differences in the predictions of evolutionary games in finite and infinite populations. The NSG always has at least one ESS, the non-cooperative (defection) ESS, regardless of whether the population is finite or infinite. When played in an infinite population, the NSG always has a second (cooperative) ESS [42], but we find conditions under which there is no cooperative ESS when it is played in finite populations. This qualitative difference in predictions for finite and infinite populations can occur no matter how large the finite population is, and is universal in the sense that it is independent of the selection process (a notion we clarify at the end of §2 and define rigorously in [43,44]). To our knowledge, there are no other examples in the literature of qualitatively different dynamics in finite and infinite populations that persist for arbitrarily large populations and are independent of the selection process; other studies that demonstrate such differences (e.g. [45]) are restricted to particular selection processes. Our results underline the need to model real populations as explicitly finite in size.
The results we present are supported by formal mathematical theorems, which we state in §3 and prove in §5 and appendices A and B.
2. Terminology
The snowdrift game is an abstraction of the situation in which a group of individuals encounters a snowdrift that blocks their path. We suppose that players are drawn from a population of self-interested individuals ( is the group size), and that each player chooses how much to contribute to a public good—e.g. snow cleared off the path—from which all group members benefit. A focal individual contributing incurs a cost that depends only on its own contribution, whereas its benefit depends on the total good contributed by the group as a whole. The focal individual’s payoff—which is interpreted as a change in fitness—is then
To avoid mathematical complexities that are not relevant to the biological issues that concern us, and to ensure that the fitness function in equation (2.1) is biologically sensible, we impose a few natural conditions on the cost and benefit functions and refer to the natural snowdrift game (NSG; see §5a). The NSG was introduced in [42], where it was shown that—when played in infinite populations—the game always has a cooperative ESS. Cost, benefit and fitness functions for an NSG example are shown in figure 1.
Figure 1. Example cost, benefit and fitness functions for a natural snowdrift game (NSG, defined in §5a). (a) The cost function is simply . (b) The benefit function is given in §5 equation (5.5); parameter values are , , , . (c) Fitness is shown for three situations involving groups of individuals. (i) Residents cooperate and contribute the (light green, ), (ii) Residents cooperate but contribute less than the (medium green, ). (iii) Residents defect, i.e. contribute nothing (dark green, ). Resident strategies are indicated by dotted vertical lines in the same colour as the associated fitness function. In the case of defecting residents, a focal individual’s fitness function does not depend on the group size () and has a local maximum at the maximizing total good (, thin grey vertical line).
Traditionally, an evolutionarily stable strategy (ESS) is one such that, when adopted by the entire population, a single mutant individual playing a different strategy cannot invade the population [46]. Because the phenotypic changes caused by mutations are often small, local ESSs are of particular interest: a population of individuals playing a local ESS is resistant to the invasion of a single individual playing a slightly different strategy. A strategy is convergently stable if a population playing a different strategy evolves toward it [47]; convergence can be either global or local.
In infinite populations, the theory of adaptive dynamics [2,8,48] identifies a singular strategy as one at which the selection gradient, , vanishes [49, table 1]; for an NSG, this reduces to
The definition of singular strategies can be extended to finite populations: the defining feature of a singular strategy is that when it is played by a resident population, directional selection (equation (A 4a)) vanishes2; for an NSG, this condition reduces to
Fixation probabilities depend on the selection process [43], i.e. the stochastic process by which differences in fitnesses of individuals playing different strategies generate changes in the frequencies of strategies in the population over time.3 As a result, the strategies that are evolutionarily stable in finite populations depend on the selection process. Variants of the Moran and Wright–Fisher processes [33,51,52] are commonly assumed, but are idealizations that do not exactly describe realistic populations (e.g. [31]). We are spared this complication in this paper because, for the games we consider, every ESSN is a universal ESSN [44, §5], that is, all ESSNs are evolutionarily stable irrespective of the selection process [43]. Consequently, we need not specify the population-genetic processes underlying selection, and we obtain general results about evolutionary stability. We use the term universal more generally to indicate that a property or statement holds for any selection process.
3. Results
(a) ESSs in infinite populations
As we have previously shown [42], if an NSG (§5a) is played in an infinite population then there are always two (and only two) ESSs:
defect: contribute nothing (), or
cooperate: make a positive contribution that is inversely proportional to the group size ().
(b) ESSs in finite populations
In a finite population, NSGs do not necessarily have a cooperative , and when they do it is not necessarily possible to find an explicit formula for evolutionarily stable cooperation levels in terms of the parameters of an NSG (nevertheless, cooperative s are always easy to find numerically within the interval (3.3) identified in the following theorem).
Theorem 3.1. (Existence and universality of stable cooperation levels in the natural snowdrift game)
Consider a finite population (of individuals) that is subject to selection resulting from groups of individuals playing an NSG (defined in §5a). A strategy is singular if and only if
Necessary condition for : Any cooperative () satisfies both equation (3.2) and
Sufficient condition for universal : If satisfies equation (3.2) and
s in large populations: If and the group size is either fixed, or satisfies , then for any sufficiently large population size , there is a universal satisfying equation (3.5). Moreover, as .
While the evolutionarily stable cooperation levels in finite and infinite populations are never exactly the same, theorem 3.1 shows that the difference is negligible in sufficiently large populations if as the population size , groups become a vanishingly small proportion of the population (cf. figure 2). However, if group size is not sufficiently small relative to the total population size then evolutionary predictions from finite population models differ qualitatively from the predictions for infinite ones: it may actually be impossible for cooperation to evolve at all. This is formalized in the next theorem.
Figure 2. ESSs in the NSG (§5a), with the sigmoidal benefit function shown in figure 1. For several group sizes (), the infinite population ESS (, equation (3.1)) is shown as a horizontal line, and the finite population () is shown with dots as a function of population size . The vertical line segments indicate the critical population size threshold (, (3.10)). A cooperative exists if and only if .
Theorem 3.2. (s of the natural snowdrift game)
Consider a finite population (of individuals) that is subject to selection resulting from groups of individuals playing an NSG (defined in §5a with fitness defined by equation (5.2)). Let denote the maximal marginal fitness, i.e.
This theorem predicts qualitatively different evolutionary outcomes, depending on the maximal marginal fitness (): equation (3.7) gives the critical maximal marginal fitness above which a cooperative exists, and below which defection is the only . Theorem 3.2 thus connects the maximal marginal fitness—a property of the fitness function that relates investments in the communal task to fitness benefits—with properties of the population of interacting agents: the population size (), the number of players in a group () and the number of groups ().
Equation (3.7) expresses the critical maximal marginal fitness in terms of a given population size and given group size. To clarify the roles of group size and number of groups in the evolution of cooperation, it is useful to think instead of the maximal marginal fitness () as given (i.e. as a fixed property of the strategic interaction) and one of or as also fixed. Then, in the inequality (see (3.8a)), we can replace by the expression on the right-hand size of equation (3.7), and solve for a critical number of groups () or critical group size ().
(f) ESS conditions in relation to the number of groups () with group size () fixed
Condition (3.8a) can be expressed equivalently as
(g) ESS conditions in relation to group size () with the number groups () fixed
Rearranging condition (3.8a) again, we can write

Figure 3. ESSs in the NSG (§5a), with the sigmoidal benefit function given in §5 equation (5.5); parameter values are , , , . For several numbers of groups (), the infinite population ESS (, equation (3.1)) is shown as a black curve, and the finite population () is shown with blue dots as a function of population size . For each number of groups, the minimum population size considered is . The red vertical line segments indicate the critical population size threshold (, (3.12)), below which a cooperative exists (in contrast to the situation in which the group size is fixed and an exists only above a critical population size; cf. figure 2).
(h) Lack of for any population size
It is even possible that there is a cooperative ESS if the population is infinite, but no cooperative for any finite population size . This is easy to verify for an NSG as follows. As noted above, an NSG always has an infinite-population cooperative ESS (3.1). An exists if and only if (3.8a) (or (3.11) or (3.9)) is satisfied. Rearranging inequality (3.9) (or equation (3.7)), we can write, equivalently,
Above, we have considered populations divided into a given number of groups. Alternatively, we could consider groups of a given size (), and ask whether it is possible for a public goods game to have a cooperative ESS if the population is infinite but no cooperative for any finite population size. As we show elsewhere, NSGs do not have this property, but there are snowdrift games that do have it [44].
(i) Confirmation with both selection and mutation
Lastly, in figure 4 we complement our rigorous analyses with individual-based simulations of finite populations in which individuals undergo both selection and mutation (see §5b for details). Simulations such as these confirm that rigorous game-theoretical analyses—which are based on selection acting with only two types in the population—correctly predict evolutionary outcomes in realistic populations in which each individual can, in principle, be playing a different strategy.
Figure 4. Individual-based simulations (details in §5b) of populations playing an NSG with cost and benefit functions as in figure 1 and group size , for population sizes (red), 165 (black) and 120 (grey). The values of the additional parameters required to simulate using algorithm 1 (§5b) were , , , , , , , . The horizontal axis is the number of generations elapsed, and the vertical axis is the strategy (contribution level) of each individual in the population. The strategies present in the population in each generation are plotted on a vertical line intersecting the horizontal axis at the corresponding point. For , defecting is the unique, globally convergently stable ; for , a cooperative (B 21) is predicted, and specifically and (marked with a horizontal yellow line). The ESS for an infinite population playing this game is . In these simulations, the mutation rate is high enough—i.e. the probability is large enough—that populations contain more than two strategies at any given generation (in contrast to our rigorous mathematical analysis of dimorphic populations). Nonetheless, for population sizes , for which a cooperative is predicted, we see evidence for its existence. With the width () of the truncated Normal distribution of mutation effect sizes used here (cf. algorithm 1, §5b), the simulation with the largest population size () would eventually leave the vicinity of the cooperative and settle at the non-cooperative . A narrower distribution of mutation step sizes (smaller ) would increase the probability that any population with size would remain in the basin of attraction of the cooperative for a longer period.
4. Discussion
We have seen that the evolutionary dynamics of the class of natural snowdrift games (NSGs, defined in §5a) are different when played in finite versus infinite populations. Since all real populations are finite, it is important to understand how inferences based on infinite-population analyses of the multi-player snowdrift game (e.g. [3,42,53]) might be affected. More generally, under what circumstances are infinite-population analyses of the evolution of cooperation likely to lead to invalid inferences about real populations?
We have shown that there are situations in which cooperation in the snowdrift game can evolve in an infinite population but not in any finite population (no matter how large). This extreme possibility emphasizes that inferences drawn from infinite population analyses should be regarded cautiously when applied to groups that are relatively large compared with the population size. Other models may or may not have parameter regimes in which cooperation can evolve only if the population is infinite, but it is important to be aware of the possibility that the infinite-population approximation might predict incorrect evolutionary outcomes if the number of individuals playing the game (the group size, ) is substantial relative to the total population size (). Exactly what ‘substantial’ means will depend on the game in question and the population size8; we have specified this threshold precisely for NSGs in (3.11). Evolutionary predictions derived from infinite population analyses can be incorrect for finite populations of any size (figure 2 and theorem 3.2). The origin of such erroneous inferences is that finite groups (no matter how large) are always negligible in size compared to an infinite underlying population, but not compared to a finite underlying population. This highlights the fact that, when evaluating whether an infinite-population approximation is appropriate, it is important to consider whether and how the group size changes as the (finite) population size is increased.
Intuition for how different predictions arise in finite and infinite populations can be developed by considering a thought experiment in which the population (of size ) is simultaneously divided into groups that play the game. If a single mutant invades the resident population, the probability that a randomly chosen group contains the mutant is . If the population size were then increased by adding more and more groups of the same size (, keeping fixed), then the effect of the mutant on the residents would be ‘infinitely diluted’ (the mutant would have a negligible effect on residents’ fitnesses as ). This example is illustrative of a more general difference between well-mixed finite and infinite populations: it can be shown that in a well-mixed population containing both mutants and residents, on average, mutants interact with fewer mutants, and residents interact with more mutants in a finite population than in an infinite one. This difference between the environments experienced by—and thus payoff functions of—residents in finite and infinite populations gives rise to the difference in evolutionary outcomes. If, instead, the population size were increased by adding individuals to the existing groups (without increasing the number of groups) then the probability that a randomly selected group contains the mutant would not change; however, in this version of the thought experiment, the limit entails the size of each group also becoming infinitely large.
Adaptive dynamics, which has been extensively used in the study of evolutionary dynamics (e.g. [3,53,54], as well as [55] and references therein), relies on an infinite-population approximation [8]. Previous work has presented reasonable arguments to justify this approximation (e.g. [48]) and reported general agreement between adaptive dynamics and stochastic simulations of finite populations (see [56] for a review). In addition, specific agreement has been noted [15] between the finite- and infinite-population evolutionary dynamics of the multi-player snowdrift game with discrete strategies. These results appear to contrast with those presented here, though [15] did observe that defectors prevail when the group size approaches the population size (even in situations in which cooperators and defectors can coexist in an infinite population). In other work, there has been a focus on situations in which the group size is much smaller than the population size, which reduces the chance of discovering discrepancies between finite and infinite population evolutionary predictions.
Our analysis of the class of natural snowdrift games is rigorous (theorems 3.1 and 3.2), and our conditions for existence of a cooperative are universal (in the sense of being entirely independent of the selection process [43]). These exact results for the finite-population NSG, together with exact results for the infinite-population NSG [42], make it easy to identify differences in predictions when the game is played in finite versus infinite populations. For the NSG, we have found that the infinite-population approximation yields the wrong evolutionary outcome for group sizes that are substantial relative to the population size. More broadly, our results indicate that approximating large populations by infinite ones (as in the classical adaptive dynamics framework [8]) has the potential to generate misleading conclusions. There is a general need to reevaluate the theoretical justification for approximating large populations by infinite ones, and to derive clear conditions for when such approximations are valid.
Finally, the recently introduced concept of the Social Efficiency Deficit (SED) [57,58] captures the ‘opportunity cost’ that an evolving population experiences, in comparison to what players could attain at the social optimum. Because evolutionary outcomes can differ between finite and infinite populations, an interesting direction for future inquiry would be to compare the different SEDs experienced in these two settings.
5. Methods
(a) The natural snowdrift game
This biologically motivated version of the continuous snowdrift game (§2) was introduced in [42].
We consider a population of individuals that are identical except (possibly) with respect to the strategy (contribution level) adopted when playing the snowdrift game. In particular, there is no age, spatial, social or other structure in the population. Evolution affects only the contribution levels of individuals, so at any time the population is completely characterized by the set of strategies present in the population and the numbers of individuals (or population proportions) playing each strategy. Fitnesses are determined entirely by payoffs from the continuous snowdrift game played in groups of individuals. We say that this population plays a natural snowdrift game (NSG) if, in addition, the cost and benefit functions have the following properties (which are satisfied by the example shown in figure 1):
(a) | The cost to the focal individual of a contribution is measured in units of its impact on this individual’s fitness, that is, 5.1 Thus, the focal individual’s fitness is
5.2 where is the total contribution in the focal individual’s group. | ||||
(b) | The benefit is a smooth function of the total contribution (more precisely, exists for all ). | ||||
(c) | There exist total contribution levels and () such that decreases for and and increases for . (See [42, 2] for the biological motivation for this assumption, the key aspect of which is that the marginal cost of an increase in contribution eventually outweighs its benefit.) Consequently, given condition (a), if only one member of a group contributes anything then that individual’s fitness [take in equation (5.2)] is locally minimized (maximized) if its contribution is (). | ||||
(d) | There is a net fitness cost to an individual who contributes when all other group members contribute nothing, 5.3 but there is a net incremental fitness benefit for contributing if other group members contribute that amount,
5.4 |
In an infinite population, condition (c) implies that and 0 are the only local ESSs [42]. Adding condition (d) guarantees that they are both global ESSs (0 via (5.3) and via (5.4); see [42]).
(i) Benefit function used for numerical examples
For the purpose of making example graphs and running simulations, we have used sigmoidal benefit functions. The biological motivation for this is that one would expect a nonlinear increase in the ease of passing the barrier as more snow is cleared, but eventually there can be no further benefit from additional work because all the snow has been cleared.
Specifically, for any integer and real numbers , and , consider the benefit function
Figure 1 shows the benefit function (5.5) for particular values of , , and , together with the corresponding fitness function (5.2) that results if residents defect, or—in groups of two individuals—if residents play the infinite population ESS (equation (3.1)). Based on equation (5.5), in appendix B we derive explicit formulae for , , and and (in terms of , , and ).
The class of sigmoids based on generalized error functions is much more flexible than the more common ‘logistic’ sigmoid used by Molina & Earn [42] and Cornforth et al. [53] (which is based on shifting, and horizontally and vertically stretching, the hyperbolic tangent function, ). Whereas the maximum slope, horizontal asymptote and position of the inflection point uniquely determine the ‘width’ of a logistic sigmoid, the generalized error function allows the width to be set independently via the parameter (see equation (B 14)).
(b) Individual-based simulations
The three individual-based simulations shown in figure 4 (for population sizes (grey), 165 (black) and 225 (red)) were run using algorithm 1 (§5b), which we implemented in an R [60] package. In the following description, we denote the normal distribution truncated to the interval by . It is assumed that values of the following parameters have been set:
— | Parameters (, , and ) of the benefit function (5.5). | ||||
— | Group size () and population size (), such that is an integer. | ||||
— | Number of repetitions of the NSG between reproductive events (). | ||||
— | Maximum number of generations to evolve (). | ||||
— | Upper bound for contribution level (). | ||||
— | Mean () and standard deviation () of an underlying Normal distribution of strategies; the initial strategies (, ) are to be sampled from . | ||||
— | Mutation probability () per individual per generation. | ||||
— | Standard deviation () of an underlying Normal distribution of the strategy changes caused by mutations, and upper and lower bounds on mutation sizes, ; when an individual playing strategy mutates, its new strategy is sampled from , so that the mutation is within the interval and the mutated strategy is in . |
Data accessibility
Code used to produce the figures in this manuscript (in particular, the evolvr R package) is attached as ESM. The latest version of the evolvr R package can be found from the GitHub repository: https://github.com/davidearn/evolvr [61].
Version 0.0.3 is provided in electronic supplementary material [62].
Declaration of AI use
We have not used AI-assisted technologies in creating this article.
Authors' contributions
C.M.: conceptualization, formal analysis, investigation, methodology, software, visualization, writing—original draft, writing—review and editing; D.J.D.E.: conceptualization, formal analysis, investigation, methodology, software, supervision, visualization, writing—original draft, writing—review and editing.
Both authors gave final approval for publication and agreed to be held accountable for the work performed therein.
Conflict of interest declaration
We declare we have no competing interests.
Funding
D.J.D.E. was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). C.M. was supported by the Ontario Trillium Foundation, the United States Defense Advanced Research Project Agency NGS2 program (grant no. D17AC00005), and the Army Research Office (grant no. W911NF1810325).
Acknowledgements
We are grateful to Sigal Balshine, Ben Bolker, Michael Doebeli, Jonathan Dushoff, Gil Henriques and Paul Higgs for valuable discussions and comments.
Footnotes
2 See definition 4.3.5 and appendix 4.C of [50] for a more general exposition of evolutionarily singular strategies, from which it follows that the derivative in equation (A 4a) must vanish.
3 In [43], we rigorously define and study selection processes. These are Markov processes describing the evolution of populations in which there are two types of individuals, there are no mutations, and at any time, the number of individuals of the type that has a higher mean fitness is expected to increase. The Moran and Wright–Fisher processes are particular examples of selection processes.
4 In (3.8a), ‘generically’ means excluding the unlikely possibility of singular strategies also being inflection points of ; in (3.8b), it excludes the possibility of the marginal benefit being constant in a neighbourhood of .
5 Condition (c) in the definition of the NSG (§5a) implies that , so is always well defined in (3.10).
6 Note that is always finite for a given population size, but when the number of groups is fixed and larger than , then there is an for any number of players .
7 Note, however, that the group size as the number of groups , and while the cooperative equilibrium exists in the infinite population limit for any finite number of groups , it tends to 0 as .
8 In fact, to underscore the dependence of ‘substantial’ on the game, note that it is possible to construct snowdrift games for which ‘substantial’ can be any desired quantity. More precisely, for any given group and population sizes it is possible to choose quadratic cost and benefit functions (as used in, e.g. [3]) such that the infinite population approximation yields the wrong evolutionary outcome [44].
9 More precisely, for a given maximal marginal fitness () and horizontal asymptote (), controls the distance between the benefit function’s inflection point () and the total contribution at which the marginal benefit is half of its maximum.
10 Equation (A 2) remains valid if individual fitnesses are obtained by averaging payoffs from an arbitrary (either fixed or random) number of rounds of the NSG, as long as groups are selected independently in each round.
Appendix A. Proofs
(a) Analysis of the natural snowdrift game (NSG, §5a) in a finite population
Our main results are stated in theorems 3.1 and 3.2 (§3). Before developing the proofs in detail, it is useful to note that:
— | (where is defined in assumption (c) of the definition of the NSG, §5a). To see this, suppose that . Then assumption (c) implies that , contradicting assumption (d). | ||||
— | The benefit function is twice-differentiable [assumption (b) in the definition of the NSG (§5a)]. | ||||
— | , for , and otherwise [these properties of follow from assumption (c)]. Consequently, [cf. equation (3.6)] and . |
(i) The mean fitness difference between mutants and residents
Consider a population of individuals, comprised of mutants who play and residents who play , and denote the proportion of mutants in the population by . Suppose that groups of individuals are randomly sampled from this population without replacement, which implies that the number of mutants in each such group is hypergeometrically distributed with parameters , and [50,63]; thus, the probability that the number of mutants in a randomly sampled group of individuals is is given by
— | is independent of , and | ||||
— | is linear in . |
(ii) Evolutionary and convergent stability of defection
Lemma A.1. (Evolutionary stability of defection)
If the NSG (§5a) is played in a finite population then not contributing () is a locally convergently stable for any selection process. Moreover, if the population and group sizes are the same (, so the entire population plays the game together) then defecting is the unique and is globally evolutionarily and convergently stable.
Proof.
because decreases for , so using equation (A 4a),
Now suppose groups constitute the entire population, i.e. . Then, for any resident strategy and any number of mutants , mutants contributing less than residents to the public good () have a higher payoff than residents; hence defection is the unique and is globally convergently stable. Defection is also globally evolutionarily stable because for any mutant strategy and any number of mutants (), residents obtain a higher payoff than mutants (because they receive the same benefit without paying a cost).
(iii) Proof of theorem 3.1
Inserting equation (A 4a) into the definition of an evolutionarily singular strategy (definition 4.3.5 of [50]) implies that cooperative singular strategies are characterized by equation (3.2). Any solution of equation (3.2) must satisfy , because the right-hand side of equation (3.2) is greater than 1 and, as noted above, if then .
Necessary condition for : Suppose that solves equation (3.2) but . Plugging equation (3.2) into equation (A 4a) gives . Rearranging equation (A 4b), we have
Sufficient condition for universal : The sufficient condition for local universal evolutionary and convergent stability follows immediately from theorem 4.D.1 of [50] and equation (A 4).
s in large populations: Suppose that and consider the equation
(iv) Proof of theorem 3.2
First, note that is always a locally convergently stable (lemma A.1). From corollary 4.3.8 of [50], selection opposes invasion of a cooperative resident strategy by sufficiently similar mutant strategies only if is singular, which (using equation (A 4a)) occurs iff satisfies
Case . | Because and is a continuous function of on the interval , it follows from the intermediate value theorem [64] that equation (3.2) has a solution in this interval. Let be the set of singular strategies, i.e. solutions of equation (3.2), A 12 Note that from theorem 3.1, . Denote the largest solution of equation (A 11) by , i.e.
A 13 (this maximum exists because the continuity of on a closed interval implies ). | ||||
Generically,11 . We claim that . To see this, suppose, in order to derive a contradiction, that . Then, increases in a neighbourhood of , so there exists such that and A 14 From the intermediate value theorem, there exists such that
A 15 a contradiction. | |||||
Thus and , so theorems 4.D.1 and 4.3.9 of [50] imply that is a local and is locally convergently stable. | |||||
Case . | Suppose, in order to derive a contradiction, that is a cooperative . From theorem 3.1, must solve equation (A 11) so, from the definition of in equation (3.7), A 16 Suppose further that does not contain an interval (i.e. the marginal benefit is not maximal for an interval of total contributions ), which happens generically. Then, any total contribution in is a local maximum of . It follows that if and is sufficiently close to , then
A 17 and therefore from equation (A 3),
A 18 which, together with the identity [50, eqn (4.63), p. 138],
A 19 implies that . Hence, similar to an argument in the proof of theorem 3.1, since , we must have for any sufficiently close to (and this is true for any number of mutants ). Consequently, selection favours the invasion and replacement of by any such , so is not evolutionarily stable. | ||||
To see that defection is globally evolutionarily stable, substitute in equation (A 3) to get A 20 Noting that for all , , we have
A 21 where we have used equations (3.7) and (A 19) in the last equality. Thus, is non-decreasing in . Moreover, if , then for all , so similarly, equations (A 19) and (A 20) imply that . Because , it follows that for all (regardless of the proportion of mutants in the population). Thus, from [43, corollary 5.4], when residents defect, selection opposes invasion and fixation of any mutants. | |||||
Case . | In this case, equation (3.2) has no solution, and no cooperative exists. | ||||
To see that defection () is globally evolutionarily and convergently stable, observe first that implies A 22 Then, using equations (A 3), (A 22) and equation (4.63) on p.138 of [50], it follows that
so decreases with for any . Thus, from [43, corollary 5.4], defection () is a globally evolutionarily and convergently stable strategy. |
Appendix B. Analysis of the benefit function used for numerical examples
In this appendix, we define the class of sigmoidal benefit functions that we have used to illustrate our results, and derive a variety of analytical formulae that we have found useful when working with these functions.
(a) Sigmoids using generalized error functions
For any integer and real , and , consider the benefit function
Expressing generalized error functions using gamma functions: It is sometimes convenient to express in terms of gamma functions. For , the transformation ( and ) gives
Parameter meanings: Because equation (B 3) implies
From the integral definition of the generalized error function (equation (B 2))
The minimizing and maximizing total goods: Since is monotonic on each of the intervals, and and is even, for any , we can find two real values of for which (although one of these values may be negative and therefore biologically irrelevant, because total contributions to the public good cannot be negative). To find these values of total contribution , we set in equation (B 8a), and get
The infinite-population cooperative ESS: Equation (3.1) then gives
Using and equation (B 13b) in equation (B 8b), we have
Using equation (B 1) and the fact that is odd,
Singular and evolutionarily stable cooperative strategies in finite populations: In a finite population of size , a singular strategy of the NSG is a solution of equation (3.2), that is,
The curvature of the benefit function at the : Similar to equation (B 16), we have
Condition for the fitness difference having a minimum when a single mutant defects and residents play the ESS: To guarantee that when a single mutant invades a population playing the ESS, the fitness difference has both a minimum and a maximum (as a function of the mutant strategy), we need the mutant contribution that minimizes fitness to be positive; equivalently, the total contribution of the non-focal individuals—all of whom are residents—must be less than the minimizing total good . Thus,
The payoff extrema difference: We now calculate the payoff extrema difference (PED), , that is, the difference between a mutant’s local minimum and maximum fitnesses when residents contribute the infinite-population ESS.
The mean fitness slope: To choose parameter values that generate a fitness difference with a distinct peak at the ESS (when residents play the ESS), we would like to find the mean fitness slope between the extrema, i.e. the ratio of the PED, , and the distance between the fitness extrema as a function of our parameters. To that end, using equation (B 24), the distance between the fitness extrema is
In addition, equation (B 3) implies that for any ,
The ratio of ESSs in infinite and finite populations: Using equations (B 14), (B 15) and (B 21),
(b) Sigmoid using standard error-function
In the special case (i.e. ), since , equation (B 1) reduces to