Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Effects of group size and population size on the evolutionary stability of cooperation

Chai Molina

Chai Molina

Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA

International Institute for Applied Systems Analysis, Laxenburg 2361, Austria

[email protected]

Contribution: Conceptualization, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing

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David J. D. Earn

David J. D. Earn

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Contribution: Conceptualization, Formal analysis, Investigation, Methodology, Software, Supervision, Visualization, Writing – original draft, Writing – review & editing

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Published:https://doi.org/10.1098/rspa.2023.0305

    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. [17]). 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 [915].

    In spite of the technical challenges of working with finite populations, some exact analytical results have been obtained for games with discrete strategy sets [9,12,1416], notably also in structured populations [1724]. However, most existing finite-population results rely on approximate analytical methods and simulations [11,15,2529]. 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. [9,12,1416]). 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 [3,7,3739] 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 n players are drawn from a population of self-interested individuals (n 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 x incurs a cost C(x) that depends only on its own contribution, whereas its benefit B(τ)0 depends on the total good τ contributed by the group as a whole. The focal individual’s payoff—which is interpreted as a change in fitness—is then

    W(x,τ)=B(τ)C(x).2.1
    If x is a continuous variable, as we assume here, the game is said to be continuous. Positive contributions represent cooperative strategies, and individuals who contribute nothing are said to defect. If the population is finite and contains N individuals, then for convenience we refer to the ratio G=N/n as the number of groups; however, we do not assume that the population is simultaneously subdivided into groups of n individuals (and in particular, G need not be an integer).

    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.

    Figure 1. Example cost, benefit and fitness functions for a natural snowdrift game (NSG, defined in §5a). (a) The cost function is simply C(x)=x. (b) The benefit function B(τ) is given in §5 equation (5.5); parameter values are L=10, k=1, m=1.5, τturn=15. (c) Fitness is shown for three situations involving groups of n=2 individuals. (i) Residents cooperate and contribute the ESSN (light green, Xres=9.63), (ii) Residents cooperate but contribute less than the ESSN (medium green, Xres=5). (iii) Residents defect, i.e. contribute nothing (dark green, Xres=0). 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 (n) and has a local maximum at the maximizing total good (τmax=19.3, 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, xW(x,x+(n1)X)|x=X, vanishes [49, table 1]; for an NSG, this reduces to

    B(nX)1=0.2.2
    A singular strategy for which the mutant fitness is concave near the singular strategy is a local ESS.1 Local convergent stability of singular strategies is also defined via a condition on the local fitness difference (see table 1 of [2]).

    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

    NnN1B(nX)1=0.2.3
    The finite-population extension of the concept of evolutionary stability is more involved, because it must account for the fact that selection can favour fixation of a mutant strategy, even if selection opposes its invasion [9]. Thus, the standard definition of an evolutionary stable strategy in a finite population (ESSN [9]) requires that selection oppose both invasion by, and fixation of, mutant strategies. In addition, the presence of one or more mutants in a finite population has a non-negligible effect on the fitness of residents (whereas finitely many mutants cannot affect the mean fitness of residents in an infinite population).

    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 (x=0), or

    • cooperate: make a positive contribution that is inversely proportional to the group size n (x=X>0).

    Both ESSs are global, and both are locally convergently stable (theorem 4.1 of [42]). At the cooperative ESS, everyone contributes an equal share of the amount that maximizes individual fitness given that everyone contributes equally. In terms of this maximizing total good τmax (see §5a and figure 1), the cooperative ESS is
    X=τmaxn.3.1

    (b) ESSs in finite populations

    In a finite population, NSGs do not necessarily have a cooperative ESSN, 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 ESSNs 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 N individuals) that is subject to selection resulting from groups of n individuals playing an NSG (defined in §5a). A strategy X is singular if and only if

    B(nX)=1+n1Nn,3.2
    and any such strategy X lies in the open interval
    (τminn,τmaxn),3.3
    (hence, from (3.1), X<X).

    Necessary condition for ESSN: Any cooperative ESSN (X>0) satisfies both equation (3.2) and

    B(nX)0.3.4

    Sufficient condition for universal ESSN: If X satisfies equation (3.2) and

    B(nX)<0,3.5
    then X is a universal ESSN that is (universally) locally convergently stable.

    ESSNs in large populations: If B(τmax)0 and the group size n is either fixed, or satisfies n(N)/NN0, then for any sufficiently large population size N, there is a universal ESSN X=XN satisfying equation (3.5). Moreover, XNX as N.

    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 N, 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.

    Figure 2. ESSs in the NSG (§5a), with the sigmoidal benefit function shown in figure 1. For several group sizes (n), the infinite population ESS (X, equation (3.1)) is shown as a horizontal line, and the finite population ESSN (XN) is shown with dots as a function of population size N. The vertical line segments indicate the critical population size threshold (Nmin, (3.10)). A cooperative ESSN exists if and only if N>Nmin.

    Theorem 3.2. (ESSNs of the natural snowdrift game)

    Consider a finite population (of N individuals) that is subject to selection resulting from groups of n individuals playing an NSG (defined in §5a with fitness W defined by equation (5.2)). Let m denote the maximal marginal fitness, i.e.

    mmaxτ0(Wx)=maxτ0(B(τ)1).3.6
    Then m>0 and there is a critical maximal marginal fitness threshold,
    mc=N1Nn1,3.7
    such that4
    m>mc{Generically, at least one cooperative localESSN(X=XN>0)exists that is{universal} and universallylocally convergently stable.In addition, defection is (universally) a locally convergently stable ESSN.3.8a
    m=mc{Generically, no cooperative ESSNexists.Defection is (universally) globally evolutionarilystable and locally convergently stable.3.8b
    andm<mc{No cooperative ESSNexists.Defection is (universally) globallyevolutionarily and convergently stable.3.8c

    This theorem predicts qualitatively different evolutionary outcomes, depending on the maximal marginal fitness (m): equation (3.7) gives the critical maximal marginal fitness above which a cooperative ESSN exists, and below which defection is the only ESSN. 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 (N), the number of players in a group (n) and the number of groups (G=N/n).

    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 (m) as given (i.e. as a fixed property of the strategic interaction) and one of n or G as also fixed. Then, in the inequality m>mc (see (3.8a)), we can replace mc by the expression on the right-hand size of equation (3.7), and solve for a critical number of groups (Gc) or critical group size (nc).

    (f) ESS conditions in relation to the number of groups (G) with group size (n) fixed

    Condition (3.8a) can be expressed equivalently as

    G>Gc1+1m(11n),3.9
    i.e. the number of groups G must be greater than Gc, the minimum number of groups that are needed to support stable cooperation if the group size is n. For any given number of players in a group (n), if we multiply (3.9) by n we see that cooperation cannot evolve—i.e. no cooperative ESSN exists—unless the population size is greater than a critical population size,5
    N>Nminn+n1m.3.10
    Figure 2 illustrates this result for the particular NSG specified by the benefit function shown in figure 1. Put another way, for a given group size n, if the population size N is too small then there is no cooperative ESSN, but if N is sufficiently large then there is a (universal) cooperative ESSN. For any given population size N, there are group sizes n and benefit functions B(τ) that yield Nmin>N, so a qualitative difference between the evolutionary outcomes in finite and infinite populations can occur for any population size N.

    (g) ESS conditions in relation to group size (n) with the number groups (G) fixed

    Rearranging condition (3.8a) again, we can write

    n<nc{11m(G1) if m<1(G1), otherwise,3.11
    i.e. for cooperation to evolve, the group size n must be less than nc, the maximum size of groups that support cooperation in a population divided into G groups6. Multiplying (3.11) by G and rearranging, we obtain
    N<Nmax{G1m(G1) if m<1(G1), otherwise,3.12
    i.e. if the number of groups is fixed (and smaller than 1+1/m) then in order for a cooperative ESSN to exist, the population size must be less than the threshold in (3.12), as illustrated in figure 3.
    Figure 3.

    Figure 3. ESSs in the NSG (§5a), with the sigmoidal benefit function B(τ) given in §5 equation (5.5); parameter values are L=1000, k=1, m=0.05, τturn=7. For several numbers of groups (G), the infinite population ESS (X, equation (3.1)) is shown as a black curve, and the finite population ESSN (XN) is shown with blue dots as a function of population size N. For each number of groups, the minimum population size considered is N=G+1. The red vertical line segments indicate the critical population size threshold (Nmax, (3.12)), below which a cooperative ESSN exists (in contrast to the situation in which the group size n is fixed and an ESSN exists only above a critical population size; cf. figure 2).

    (h) Lack of ESSN for any population size

    It is even possible that there is a cooperative ESS if the population is infinite, but no cooperative ESSN for any finite population size N. 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 ESSN 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,

    m>mc1(G/N)G1,3.13
    i.e. there is a cooperative ESSN if and only if the maximum marginal fitness m exceeds the threshold mc (exactly the same threshold that appears in equation (3.7), but expressed here in terms of G rather than n). Suppose now that the population is divided into a given number of groups, G2. There must be at least two individuals in each group, so N2G and hence G/N1/2. Consequently, for any possible population size N, we have
    mc12(G1).3.14
    Therefore, there is a simple, sufficient condition that precludes evolutionary stability of cooperation for an NSG played in any finite population: if the benefit function is such that
    m<12(G1)3.15
    then no cooperative ESSN exists, and defection is the only ESSN, no matter how large the population size N. When the game defined by the same cost and benefit functions is played in an infinite population, defection remains an ESS, but in addition, there is also a cooperative ESS (regardless of the group size n).7 Given G, it is easy to satisfy (3.15) since the only other constraint on m is that it must be positive; for example, equation (5.5) below can be used to construct a benefit function satisfying (3.15).

    Above, we have considered populations divided into a given number of groups. Alternatively, we could consider groups of a given size (n), and ask whether it is possible for a public goods game to have a cooperative ESS if the population is infinite but no cooperative ESSN 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.

    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 n=15, for population sizes N=225 (red), 165 (black) and 120 (grey). The values of the additional parameters required to simulate using algorithm 1 (§5b) were nRep=10, nGen=104, xmax=6, μx=5.5, σx=0.1,pmut=0.005, σ=0.1, l=0.3, u=0.3. 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 N=120, defecting is the unique, globally convergently stable ESSN; for N>155, a cooperative ESSN XN (B 21) is predicted, and specifically X165=3.53 and X225=3.54 (marked with a horizontal yellow line). The ESS for an infinite population playing this game is X=3.56. In these simulations, the mutation rate is high enough—i.e. the probability pmut 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 N>155, for which a cooperative ESSN is predicted, we see evidence for its existence. With the width (σ) of the truncated Normal distribution of mutation effect sizes used here (cf. algorithm 1, §5b), the simulation with the largest population size (N=225) would eventually leave the vicinity of the cooperative ESSN and settle at the non-cooperative ESSN. A narrower distribution of mutation step sizes (smaller σ) would increase the probability that any population with size N>155 would remain in the basin of attraction of the cooperative ESSN 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, n) is substantial relative to the total population size (N). 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 N) is simultaneously divided into G groups that play the game. If a single mutant invades the resident population, the probability that a randomly chosen group contains the mutant is 1/G. If the population size were then increased by adding more and more groups of the same size (G, keeping n 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 N). 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 N 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 ESSN 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 n 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 x is measured in units of its impact on this individual’s fitness, that is,

    C(x)=x.5.1
    Thus, the focal individual’s fitness is
    W(x,τ)=B(τ)x,5.2
    where τ is the total contribution in the focal individual’s group.

    (b)

    The benefit B(τ) is a smooth function of the total contribution τ (more precisely, B(τ) exists for all τ0).

    (c)

    There exist total contribution levels τmin and τmax (0τmin<τmax) such that B(τ)τ decreases for τ<τmin and τ>τmax and increases for τmin<τ<τmax. (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 x=τ in equation (5.2)] is locally minimized (maximized) if its contribution is x=τmin (τmax).

    (d)

    There is a net fitness cost to an individual who contributes τmax when all other group members contribute nothing,

    B(τmax)<C(τmax),5.3
    but there is a net incremental fitness benefit for contributing τmax/n if other group members contribute that amount,
    B(τmax)B((n1)τmaxn)>C(τmaxn).5.4

    In an infinite population, condition (c) implies that τmax/n and 0 are the only local ESSs [42]. Adding condition (d) guarantees that they are both global ESSs (0 via (5.3) and τmax/n 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 k>0 and real numbers m>0, L>0 and τturn0, consider the benefit function

    B(τ)=Lerf2k((m+1)Γ(1/(2k))2kL(ττturn)),τ0,5.5
    where erf(x) is the generalized error function [59] of order ,
    erf(x)=Γ(1/)0xetdt,5.6
    and Γ(x) is the gamma function (equation (B 4a)). We analyse this flexible class of sigmoidal benefit functions in appendix B, where we show that the parameters L and τturn are the horizontal asymptote and the inflection point, respectively, k controls the ‘width’ of the sigmoid9, and m+1 is the maximal marginal benefit (so that m is the maximal marginal fitness that results from this functional form, justifying our notation).

    Figure 1 shows the benefit function (5.5) for particular values of k, m, L and τturn, 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 τmin, τmax, and X and XN (in terms of m, L, τturn and k).

    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, tanh(x)). 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 k (see equation (B 14)).

    (b) Individual-based simulations

    The three individual-based simulations shown in figure 4 (for population sizes N=120 (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 (l,u) by TruncNormal(μ,σ,l,u). It is assumed that values of the following parameters have been set:

    Parameters (k, m, L and τturn) of the benefit function (5.5).

    Group size (n) and population size (N), such that G=N/n is an integer.

    Number of repetitions of the NSG between reproductive events (nRep).

    Maximum number of generations to evolve (nGen).

    Upper bound for contribution level (xmax).

    Mean (μx) and standard deviation (σx) of an underlying Normal(μx,σx) distribution of strategies; the initial strategies (xi, i=1,,N) are to be sampled from TruncNormal(μx,σx,0,xmax).

    Mutation probability (pmut) per individual per generation.

    Standard deviation (σ) of an underlying Normal (0,σ) distribution of the strategy changes caused by mutations, and upper and lower bounds on mutation sizes, (l,u); when an individual playing strategy x mutates, its new strategy is sampled from TruncNormal(x,σ,max{0,xl},min{xmax,x+u}), so that the mutation is within the interval [l,u] and the mutated strategy is in [0,xmax].

    Inline Graphic

    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

    1 We assume throughout this paper that the strategy space is one-dimensional.

    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 B(nX); in (3.8b), it excludes the possibility of the marginal benefit B(τ) being constant in a neighbourhood of argmaxB(τ).

    5 Condition (c) in the definition of the NSG (§5a) implies that m>0, so Nmin is always well defined in (3.10).

    6 Note that nc is always finite for a given population size, but when the number of groups G is fixed and larger than 1+1/m, then there is an ESSN for any number of players n.

    7 Note, however, that the group size n as the number of groups G, and while the cooperative equilibrium X exists in the infinite population limit for any finite number of groups G, it tends to 0 as G.

    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 (m) and horizontal asymptote (L), k controls the distance between the benefit function’s inflection point (τturn) 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.

    11 We need to avoid the situation in which singular strategy XN is also an inflection point of B(nx). This occurs when nXN is both a critical point and an inflection point of B(x)(N1)x/(Nn), which is generically not the case.

    12 For any positive integer k, Γ(k)=(k1)!.

    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:

    τmin>0 (where τmin is defined in assumption (c) of the definition of the NSG, §5a). To see this, suppose that τmin=0. Then assumption (c) implies that B(τmax)τmaxB(0)00,soB(τmax)τmax, contradicting assumption (d).

    The benefit function B(τ) is twice-differentiable [assumption (b) in the definition of the NSG (§5a)].

    B(τmin)=B(τmax)=1, B(τ)>1 for τmin<τ<τmax, and B(τ)1 otherwise [these properties of B(τ) follow from assumption (c)]. Consequently, m>0 [cf. equation (3.6)] and B(τmax)0.

    (i) The mean fitness difference between mutants and residents

    Consider a population of N individuals, comprised of Mp mutants who play x and NMp residents who play X, and denote the proportion of mutants in the population by ϵ=Mp/N. Suppose that groups of n 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 N, Mp and n [50,63]; thus, the probability that the number of mutants Mg in a randomly sampled group of n individuals is k is given by

    Pr(Mg=k)=(NMpnk)(Mpk)(Nn).A 1
    Suppose, moreover, that a focal individual is selected from the population by first sampling a group of n individuals, and then selecting one of the members of this group. Lastly, suppose for simplicity that individual fitnesses are given by the payoffs from a single round of the NSG played in such randomly selected groups.10 We show elsewhere [50, eqn. 4.61, p. 137] that the expected difference between the mutant and resident fitnesses is then
    δW¯ϵ(x,X)=Xx+k=0n(NMpnk)(Mpk)(N1n1)(kNMpnMp(NMp))B(kx+(nk)X).A 2
    Differentiating equation (A 2) yields
    xδW¯ϵ(x,X)=1+k=0n(NMpnk)(Mpk)(N1n1)kNMpnMp(NMp)kB(kx+(nk)X)=1+k=0n(NMpnk)(Mp1k1)(N1n1)kNMpnNMpB(kx+(nk)X),A 3
    and setting x=X, we find [50, pp. 137–138]
    xδW¯ϵ(x,X)|x=X=1+NnN1B(nX).A 4a
    Similarly, differentiating equation (A 2) with respect to x and setting x=X yields
    x2δW¯ϵ(x,X)|x=X=NnN1(N2nN2+2(n1)N2Nϵ)B(nX).A 4b
    From these expressions, we see that

    xδW¯ϵ(x,X)|x=X is independent of ϵ, and

    x2δW¯ϵ(x,X)|x=X is linear in ϵ.

    We will exploit these facts below.

    (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 (X=0) is a locally convergently stable ESSN for any selection process. Moreover, if the population and group sizes are the same (N=n, so the entire population plays the game together) then defecting is the unique ESSN and is globally evolutionarily and convergently stable.

    Proof.

    B(0)<1 because B(τ)τ decreases for 0τ<τmin, so using equation (A 4a),

    xδW¯ϵ(x,0)|x=0=NnN1B(0)1<0.A 5
    Because xδW¯ϵ(x,X)|x=X is continuous in X, for X sufficiently small,
    xδW¯ϵ(x,X)|x=X<0.A 6
    From theorem 4.3.9 in [50], it follows that X=0 (defection) is convergently stable, and selection opposes invasion of mutants contributing a sufficiently small but positive amount, x>0. To establish that X=0 is evolutionarily stable, observe that equation (A 5) implies that δW¯ϵ(x,0)<0 for sufficiently small x, so such mutants are selected against, regardless of their proportion (ϵ) in the population. Thus, corollary 5.4 of [43] implies that selection also opposes the fixation of such mutants.

    Now suppose groups constitute the entire population, i.e. N=n. Then, for any resident strategy X>0 and any number of mutants Mp{1,2,,N1}, mutants contributing less than residents to the public good (0x<X) have a higher payoff than residents; hence defection is the unique ESSN and is globally convergently stable. Defection is also globally evolutionarily stable because for any mutant strategy x>0 and any number of mutants (Mp<N), 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 nXN(τmin,τmax), because the right-hand side of equation (3.2) is greater than 1 and, as noted above, if τ(τmin,τmax) then B(τ)1.

    Necessary condition for ESSN: Suppose that X solves equation (3.2) but B(nX)>0. Plugging equation (3.2) into equation (A 4a) gives xδW¯ϵ(x,X)|x=X=0. Rearranging equation (A 4b), we have

    x2δW¯ϵ(x,X)|x=X=NnN1(1+2(n1)N2(Nϵ1))B(nX),A 7
    so x2δW¯ϵ(x,X)|x=X is increasing in ϵ and positive for any ϵ1/N (i.e. any mixed population). Thus, when mutants play x sufficiently close to X, xδW¯ϵ(x,X)|x=X is negative for x<X and positive for x>X; hence, since δW¯ϵ(X,X)=0, we must have δW¯ϵ(x,X)>0 for any x that is near but not equal to X (and this is true for any number of mutants Mp=1,,N1). Corollary 5.4 of [43] then implies that selection favours the fixation of such mutants, so X is not an ESSN, regardless of the selection process. Thus, if XN>0 is an ESSN then it cannot be that B(nX)>0, i.e. (3.4) holds.

    Sufficient condition for universal ESSN: The sufficient condition for local universal evolutionary and convergent stability follows immediately from theorem 4.D.1 of [50] and equation (A 4).

    ESSNs in large populations: Suppose that B(τmax)0 and consider the equation

    f(X,y)=B(nX)y=0.A 8
    Noting that f(τmax/n,1)=0 and that
    Xf(X,y)|(X,y)=(τmax/n,1)=B(τmax)0,A 9
    from the implicit function theorem [64, Theorem 12.40], there exists a differentiable function X(y) defined in a neighbourhood of y=1, such that X(1)=τmax/n and
    f(X(y),y)=yB(nX(y))y=0.A 10
    Now suppose that the group size n is either fixed, or varies with population size but satisfies
    n(N)NN0.
    If we define yN:=1+(n1)/(Nn) then yNN1, so for all sufficiently large population sizes N, equation (A 10) can be solved implicitly for XN:=X(yN). Such XN then solve equation (3.2), and XNNX because X(y) is continuous. Recalling that B(τmax)0 and B(τmax)0 by assumption, we have B(τmax)<0, so for sufficiently large N, B(nXN)<0. Theorem 4.D.1 of [50] then implies that for sufficiently large N, XN is a universal local ESSN and is locally convergently stable.

    (iv) Proof of theorem 3.2

    First, note that X=0 is always a locally convergently stable ESSN (lemma A.1). From corollary 4.3.8 of [50], selection opposes invasion of a cooperative resident strategy X>0 by sufficiently similar mutant strategies only if X is singular, which (using equation (A 4a)) occurs iff X satisfies

    B(nX)=1+n1Nn.A 11
    Because B(nX)>1 only if τmin/n<X<τmax/n, if a cooperative ESSN exists then it must lie in this interval.
    Case m>mc.

    Because B(τmax)=1 and B(nX) is a continuous function of X on the interval [τmin/n,τmax/n], it follows from the intermediate value theorem [64] that equation (3.2) has a solution in this interval. Let S be the set of singular strategies, i.e. solutions of equation (3.2),

    S={X|B(nX)=1+n1Nn}.A 12
    Note that from theorem 3.1, S(τmin/n,τmax/n). Denote the largest solution of equation (A 11) by XN, i.e.
    XN=maxSA 13
    (this maximum exists because the continuity of B(nX) on a closed interval implies supSS).

    Generically,11 B(nXN)0. We claim that B(nXN)<0. To see this, suppose, in order to derive a contradiction, that B(nXN)>0. Then, B(nX) increases in a neighbourhood of XN, so there exists X~ such that XN<X~<τmax/n and

    B(nX~)>B(nXN)=1+n1Nn.A 14
    From the intermediate value theorem, there exists XS such that
    X>X~>XN=maxS,A 15
    a contradiction.

    Thus B(nXN)<0 and C(X)=0, so theorems 4.D.1 and 4.3.9 of [50] imply that XN is a local ESSN and is locally convergently stable.

    Case m=mc.

    Suppose, in order to derive a contradiction, that X>0 is a cooperative ESSN. From theorem 3.1, X must solve equation (A 11) so, from the definition of mc in equation (3.7),

    B(nX)=N1Nn=mc+1.A 16
    Suppose further that argmaxB(τ) does not contain an interval (i.e. the marginal benefit B is not maximal for an interval of total contributions τ), which happens generically. Then, any total contribution in argmaxB(τ) is a local maximum of B(τ). It follows that if x<X and x is sufficiently close to X, then
    B(x+(n1)X)<N1Nn,A 17
    and therefore from equation (A 3),
    xδW¯ϵ(x,X)=1+k=0n(NMpnk)(Mp1k1)(N1n1)kNMpnNMpB(kx+(nk)X)<1+(k=0n(NMpnk)(Mp1k1)(N1n1)kNMpnNMp)N1Nn,A 18
    which, together with the identity [50, eqn (4.63), p. 138],
    k=0n(NKnk)(K1k1)(N1n1)(kNKnNK)=NnN1,A 19
    implies that xδW¯ϵ(x,X)<0. Hence, similar to an argument in the proof of theorem 3.1, since δW¯ϵ(X,X)=0, we must have δW¯ϵ(x,X)>0 for any x<X sufficiently close to X (and this is true for any number of mutants Mp=1,,N1). Consequently, selection favours the invasion and replacement of X by any such x, so X is not evolutionarily stable.

    To see that defection is globally evolutionarily stable, substitute X=0 in equation (A 3) to get

    xδW¯ϵ(x,0)=1+k=0n(NMpnk)(Mp1k1)(N1n1)kNMpnNMpB(kx).A 20
    Noting that for all x>0, B(kx)mc+1, we have
    xδW¯ϵ(x,0)1+(k=0n(NMpnk)(Mp1k1)(N1n1)kNMpnNMp)(mc+1)=0,A 21
    where we have used equations (3.7) and (A 19) in the last equality. Thus, δW¯ϵ(x,0) is non-decreasing in x. Moreover, if x<τmin/n, then B(kx)<1 for all k=0,,n, so similarly, equations (A 19) and (A 20) imply that xδW¯ϵ(x,0)<0. Because δW¯ϵ(0,0)=0, it follows that δW¯ϵ(x,0)<0 for all x>0 (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 m<mc.

    In this case, equation (3.2) has no solution, and no cooperative ESSN exists.

    To see that defection (X=0) is globally evolutionarily and convergently stable, observe first that m<mc implies

    B(τ)<1+n1Nn=N1Nn,for all τ0.A 22
    Then, using equations (A 3), (A 22) and equation (4.63) on p.138 of [50], it follows that
    xδW¯ϵ(x,X)<1+(k=0n(NMpnk)(Mp1k1)(N1n1)kNMpnNMp)N1Nn=1+(NnN1)(N1Nn)=0,
    so δW¯ϵ(x,X) decreases with x0 for any X0. Thus, from [43, corollary 5.4], defection (X=0) 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 k>0 and real m>0, L>0 and τturn0, consider the benefit function

    B(τ)=Lerf2k((m+1)Γ(1/(2k))2kL(ττturn)),τ0,B 1
    where erf(x) is the generalized error function of order ,
    erf(x)=Γ(1/)0xetdt.B 2
    This class of functions generalizes the error function, erf, which is recovered for =2 or, equivalently, k=1; see §(b) below.

    Expressing generalized error functions using gamma functions: It is sometimes convenient to express erf in terms of gamma functions. For x>0, the transformation z=t (t=z1/ and dt=z1/(1)dz/) gives

    erf(x)=1Γ(1/)0xz(1/)1ezdz=1Γ(1/)(Γ(1)Γ(1,x)),B 3
    where
    Γ(s)=0ts1etdtB 4a
    and
    Γ(s,x)=xts1etdt,B 4b
    are the gamma,12 and upper incomplete gamma functions, respectively. Note that we are only interested in generalized error functions of even order (=2k), which are odd functions of x.

    Parameter meanings: Because equation (B 3) implies

    limxerf(x)=1,B 5
    it follows that
    limxB(x)=L.B 6
    We show below that the inflection point of B (equation (B 1)) is τturn, and that the maximal marginal fitness given the benefit function B is m.

    From the integral definition of the generalized error function (equation (B 2))

    derf(x)dx=Γ(1/)exB 7a
    and
    d2erf(x)dx2=Γ(1/)x1ex,B 7b
    so
    B(τ)=πLΓ(2k)Γ(1/(2k))(m+1)Γ(1/(2k))2kL(2k)!π×exp([(m+1)Γ(1/(2k))2kL(ττturn)]2k)=(m+1)exp([(m+1)Γ(1/(2k))2kL(ττturn)]2k),B(τ)=2k[(m+1)Γ(1/(2k))2kL]2k(ττturn)2k1×(m+1)exp([(m+1)Γ(1/(2k))2kL(ττturn)]2k)B 8a
    =2k[(m+1)Γ(1/(2k))2kL]2k(ττturn)2k1B(τ).B 8b
    Consequently, τturn is the unique solution of B(τ)=0, and is thus the only inflection point. B(τ) is always positive, and hence B(τ) is monotonically increasing. However, B(τ)>0 for τ<τturn and B(τ)<0 for τ>τturn, and hence
    maxτ0B(τ)=B(τturn)=m+1,B 9
    so from equation (3.6), the maximal marginal fitness is
    maxτ0(Wx)=maxτ0B(τ)1=m.B 10

    The minimizing and maximizing total goods: Since B(τ) is monotonic on each of the intervals, (,τturn) and (τturn,) and B(τ) is even, for any bB(R0)=(0,m+1], we can find two real values of τ for which B(τ)=b (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 τ, we set B(τ)=b in equation (B 8a), and get

    logm+1b=[(m+1)Γ(1/(2k))2kL(ττturn)]2kB 11
    and
    τ=τturn±2kL(m+1)Γ(1/(2k))logm+1b2k.B 12
    To find τmax and τmin, we substitute b=B(τ)=1 in equation (B 12) and, noting that B(τ) changes sign from positive to negative at τturn, we have
    τmin=τturn2kL(m+1)Γ(1/(2k))log(m+1)2kB 13a
    and
    τmax=τturn+2kL(m+1)Γ(1/(2k))log(m+1)2k,B 13b
    and the distance between the location of the fitness minimum and maximum is
    Δτ=τmaxτmin=4kL(m+1)Γ(1/(2k))log(m+1)2k.B 14

    The infinite-population cooperative ESS: Equation (3.1) then gives

    X=τmaxn=1n(τturn+2kL(m+1)Γ(1/(2k))log(m+1)2k).B 15

    Using B(τmax)=1 and equation (B 13b) in equation (B 8b), we have

    B(τmax)=B(nX)=2k[(m+1)Γ(1/(2k))2kL]2k(τmaxτturn)2k1=2k[(m+1)Γ(1/(2k))2kL]2k(2kL(m+1)Γ(1/(2k)))2k1×(log(m+1))(2k1)/2k=Γ(12k)m+1L(log(m+1))1(1/2k).B 16

    Using equation (B 1) and the fact that erf2k is odd,

    B(τmax)B(τmin)=2Lerf2k(log(m+1)2k).B 17

    Singular and evolutionarily stable cooperative strategies in finite populations: In a finite population of size N, a singular strategy XN of the NSG is a solution of equation (3.2), that is,

    B(nXN)=1+n1Nn=N1Nn,B 18
    so equation (B 12) implies that at the ESS, the total contribution must be one of
    τ=τturn±2kL(m+1)Γ(1/(2k))log((m+1)NnN1)2k.B 19
    There are therefore two singular strategies,
    XN±=1n(τturn±2kL(m+1)Γ(1/(2k))log((m+1)NnN1)2k).B 20
    Similarly to τmin and τmax, B(nXN+)>0 and B(nXN)<0, so from theorem 3.1, the unique ESSN is
    XN=1n(τturn+2kL(m+1)Γ(1/(2k))log((m+1)NnN1)2k).B 21

    The curvature of the benefit function at the ESSN: Similar to equation (B 16), we have

    B(nXN)=2k[(m+1)Γ(1/(2k))2kL]2k(nXNτturn)2k1N1Nn=2k[(m+1)Γ(1/(2k))2kL]2k(2kL(m+1)Γ(1/(2k)))2k1×(log((m+1)NnN1))(2k1)/2kN1Nn=(m+1)N1NnΓ(1/(2k))LB 22
    ×(log((m+1)NnN1))(2k1)/2k.B 23

    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 τmin. Thus,

    τminn1nτmax>0.B 24
    Using equations (B 13b) and (B 14), this is equivalent to τmax>nΔτ, or
    n<τmaxΔτ=τturn(4kL/(m+1)Γ(1/(2k)))log(m+1)2k+12=τturnm+14kLΓ(1/(2k))log(m+1)2k+12.B 25
    Rewriting this condition in terms of the horizontal asymptote L,
    L<τturn(m+1)2k(2n1)Γ(1/(2k))log(m+1)2k.B 26

    The payoff extrema difference: We now calculate the payoff extrema difference (PED), ΔΨ, that is, the difference between a mutant’s local minimum and maximum fitnesses when residents contribute the infinite-population ESS.

    ΔΨ=[B(τmax)τmaxn][B(τmin)(τminn1nτmax)]=B(τmax)B(τmin)(τmaxτmin)=B(τmax)B(τmin)Δτ,B 27
    so using equations (B 14) and (B 17), we have
    ΔΨ=2Lerf2k(log(m+1)2k)4kL(m+1)Γ(1/(2k))log(m+1)2k.B 28

    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, ΔΨ, 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

    τmaxn(τminn1nτmax)=τmaxτmin=Δτ.B 29
    Equations (B 14) and (B 28) then yield
    ΔΨΔτ=2Lerf2k(log(m+1)2k)(4kL/(m+1)Γ(1/(2k)))log(m+1)2k1=(m+1)Γ(1/(2k))2kerf2k(log(m+1)2k)log(m+1)2k1,B 30
    which depends only on the maximal marginal fitness, m (and the order of the generalized error function, 2k). Note also that using equation (B 2) and L’Hôpital’s rule [64],
    limx0erf(x)x=limx0Γ(1/)ex=Γ(1/),B 31
    so
    limm0ΔΨΔτ=0.B 32

    In addition, equation (B 3) implies that for any x>0,

    limerf(x)=1,B 33
    (because Γ(x) as x0, and Γ(1/,x) is bounded), and
    limx0xΓ(x)=limx0Γ(x+1)=1,B 34
    so we have
    limkΔΨΔτ=m.B 35

    The ratio of ESSs in infinite and finite populations: Using equations (B 14), (B 15) and (B 21),

    XXN=τturn+(2kL/(m+1)Γ(1/(2k)))log(m+1)2kτturn+2kL(m+1)Γ(1/(2k))log((m+1)NnN1)2k=2τturn+Δτ2τturn+Δτ(log((m+1)((Nn)/(N1)))/log(m+1))2k=2τturn+Δτ2τturn+Δτ1+(log((Nn)/(N1))/log(m+1))2k.B 36
    Rewriting the population size as N=nG,
    XXN=2τturn+Δτ2τturn+Δτ1+log((G1)/(G(1/n)))/log(m+1)2k.B 37
    We see that the ratio X/XN1 as G with n fixed. However, X/XN approaches a (finite) value greater than 1 as n with G fixed (assuming XN exist for all N; see (3.12)).

    (b) Sigmoid using standard error-function

    In the special case k=1 (i.e. =2), since Γ(1/2)=π, equation (B 1) reduces to

    B(τ)=Lerf((m+1)π2L(ττturn)),τ0.B 38
    Then, setting k=1 in equation (B 13) gives the maximizing and minimizing total goods,
    τmin=τturn2Lm+1log(m+1)π,B 39a
    and
    τmax=τturn+2Lm+1log(m+1)π,B 39b
    and the distance between the location of the fitness minimum and maximum is
    Δτ=τmaxτmin=4Lm+1log(m+1)π.B 40
    Equation (B 15) then gives
    X=1n(τturn+2Lm+1log(m+1)π),B 41
    and equation (B 16) becomes
    B(τmax)=m+1Lπlog(m+1).B 42
    From equation (B 17),
    B(τmax)B(τmin)=2Lerf(log(m+1)).B 43
    Equation (B 21) gives the unique ESSN:
    XN=1n(τturn+2Lm+1log((m+1)((Nn)/(N1)))π),B 44
    and equation (B 22) becomes
    B(nXN)=m+1Lπlog((m+1)NnN1).B 45
    Condition (B 25), which guarantees 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), reduces to
    n<τmaxΔτ=τturn4m+1Lπlog(m+1)+12,B 46
    the PED, ΔΨ (equation (B 28)) becomes
    ΔΨ=2Lerf(log(m+1))4Lm+1log(m+1)π,B 47
    and the mean fitness slope (equation (B 30)) between the extrema reduces to
    ΔΨΔτ=m+12πlog(m+1)erf(log(m+1))1.B 48

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.7075470.

    Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.