Evolutionary stability in continuous nonlinear public goods games

Abstract

We investigate a type of public goods games played in groups of individuals who choose how much to contribute towards the production of a common good, at a cost to themselves. In these games, the common good is produced based on the sum of contributions from all group members, then equally distributed among them. In applications, the dependence of the common good on the total contribution is often nonlinear (e.g., exhibiting synergy or diminishing returns). To date, most theoretical and experimental studies have addressed scenarios in which the set of possible contributions is discrete. However, in many real-world situations, contributions are continuous (e.g., individuals volunteering their time). The “n-player snowdrift games” that we analyze involve continuously varying contributions. We establish under what conditions populations of contributing (or “cooperating”) individuals can evolve and persist. Previous work on snowdrift games, using adaptive dynamics, has found that what we term an “equally cooperative” strategy is locally convergently and evolutionarily stable. Using static evolutionary game theory, we find conditions under which this strategy is actually globally evolutionarily stable. All these results refer to stability to invasion by a single mutant. We broaden the scope of existing stability results by showing that the equally cooperative strategy is locally stable to potentially large population perturbations, i.e., allowing for the possibility that mutants make up a non-negligible proportion of the population (due, for example, to genetic drift, environmental variability or dispersal).

Appendices

Appendix 1: Motivation for assumption A3 (existence of $\eta _\mathrm{max}$)

In this appendix, we motivate assumption A3 by showing that if the focal agent’s fitness is defined by Eq. (5) and f is continuous, the following two statements are equivalent:

S1 :

For any fixed non-focal agent’s mean contribution H there exists $h^\dag (H)\ge 0$ such that the focal agent’s fitness W(h, H) decreases with its contribution h for all $h>h^\dag (H)$.

S2 :

There exists $\eta _\mathrm{max}\ge 0$ such that $f(\eta ) - \eta$ decreases with $\eta$ for all $\eta >\eta _\mathrm{max}$.

To gain some intuition, we first suppose f is a differentiable function of $\eta$ (in Appendix 1.1), and then give the general proof (in Appendix 1.2)

1.1 Appendix 1.1: Proof for differentiable f

Suppose that f is differentiable. Then, by the chain rule and Eq. (3), S1 implies that

$$\begin{aligned} \frac{\partial W}{\partial h}&= \left. \left( f'(\eta )\frac{\partial \eta }{\partial h}\right) \right| _{\eta = \eta (h,H)} -1= \left. f'(\eta )\right| _{\eta = \eta (h,H)} - 1\nonumber \\&= \left. \frac{\mathrm {d}}{\mathrm {d}\eta }\Big ( f(\eta ) - \eta \Big )\right| _{\eta = \eta (h,H)}. \end{aligned}$$

(51)

Consequently, if W(h, H) decreases with h for $h>h^\dag (H)$ then $f(\eta ) - \eta$ decreases with $\eta$ for all $\eta >\eta (h^\dag (H),H)$. Letting

$$\begin{aligned} \eta _\mathrm{max}= \min _{H\ge 0}\eta (h^\dag (H),H), \end{aligned}$$

(52)

$f(\eta ) - \eta$ decreases for $\eta >\eta _\mathrm{max}$. Thus, S1 implies S2.

Conversely, if there exists $\eta _\mathrm{max}\ge 0$ such that $f(\eta ) - \eta$ decreases for $\eta >\eta _\mathrm{max}$, then letting $h^\dag (H) = \eta _\mathrm{max}- (n-1)H$, we see that $\eta (h,H)>\eta _\mathrm{max}$ iff $h>h^\dag (H)$. It then follows from Eq. (51) that W(h, H) decreases with h for $h>h^\dag (H)$, so S2 implies S1.

1.2 Appendix 1.2: Proof for general f

Suppose S1 holds. Noting that

$$\begin{aligned} h>h^\dag (H)&\iff h + (n-1)H>h^\dag (H) + (n-1)H\nonumber \\&\iff \eta (h,H) > \eta ^\dag (H), \end{aligned}$$

(53)

where

$$\begin{aligned} \eta ^\dag (H) = \eta \big (h^\dag (H),H\big ), \end{aligned}$$

(54)

and rewriting Eq. (5) as

$$\begin{aligned} W(h,H)&= f\big (h+\left( n-1\right) H\big ) - h \nonumber \\&= f\big (\eta \left( h,H\right) \big )- \big [\eta \big (h,H\big ) - (n-1)H \big ] \nonumber \\&= f(\eta ) - \eta + (n-1)H, \end{aligned}$$

(55)

we see that S1 is equivalent to the assumption that for fixed H there exists $\eta ^\dag (H)\ge 0$ such that $f(\eta ) -\eta$ decreases for $\eta >\eta ^\dag (H)$. We define $\eta _\mathrm{max}$ to be the minimal such (non-negative) total good. Because $\eta$ can vary independently of H, it follows that $f(\eta )-\eta$ decreases for all $\eta >\eta _\mathrm{max}$, so S1 implies S2.

Conversely, if S2 is true, then Eqs. (53) and (55) imply that W(h, H) decreases for all $h>h^\dag (H) = \eta _\mathrm{max}- (n-1)H$. Thus, S2 implies S1.

Appendix 2: Boundary ESSs need not be singular strategies

In adaptive dynamics, evolutionarily singular points are singled out as candidate ESSs (e.g., Geritz et al. 1998; Doebeli et al. 2004). These are points at which there is no directional selection, since the fitness gradient D(H) vanishes.

However, when the evolving variable H is restricted to an interval (in our case $H\ge 0$), it is not necessary for the fitness gradient to vanish at an endpoint of this interval in order for it to be ES: as we have seen in Theorem 4.1, for the class of models defined in Sect. 2, the endpoint $H=0$ is globally evolutionarily stable whenever $f(\eta _\mathrm{max})<\eta _\mathrm{max}$, but the fitness gradient is negative in a right-hand neighbourhood of the endpoint $H=0$ (including at $H=0$). In fact, it is D(H) being negative near $H=0$ that ensures that $H=0$ is both locally convergently and evolutionarily stable.

The source of this issue is that the restriction to the biologically meaningful interval $H\ge 0$ is not built into the dynamical model Eq. (12), in that solutions of Eq. (12) do not necessarily remain in this interval (because the fitness gradient at the left endpoint $H=0$ points outside the interval, into $H<0$).

Note also that this cannot be easily fixed by artificially setting $D(0) = 0$, because doing so will insert a discontinuity into the fitness gradient, and adaptive dynamics assumes that the fitness gradient is at least continuous, in order to ensure the existence of solutions of Eq. (12) (see Hirsch et al. 2013) and in order to perform the local analysis leading to Table 1.

We conclude that when using adaptive dynamics to model a trait that is restricted to an interval (for biological reasons), points on the boundary of this interval may be ES, despite not being singular points. More care is thus required to examine the dynamics of such models near boundary points.

Appendix 3: The assumption that contribution is measured in units of fitness cost, $c(h)=h$

In this appendix, we comment on the biological interpretation of our assumption that the contribution of the focal agent is measured in units of the fitness cost it incurs, $c\left( h,H\right) = h$ (Eq. (2)).

Suppose, as before, that the population is engaged in an n-player public goods game, and let $h_1,\dots , h_n$ be the contributions of all the members of the focal agent’s group, including the focal agent (for example, if the index of the focal agent is $i=1$, then $h=h_1$).

Thus, substituting

$$\begin{aligned} \eta (h,H) =\sum _{i=1}^{n} h_i \end{aligned}$$

(56)

in Eq. (4), we have

$$\begin{aligned} b\left( h,H\right) = f\big (\eta (h,H)\big )=f\left( \sum _{i=1}^{n} h_i\right) . \end{aligned}$$

(57)

However, we relax our assumption in Eq. (2) and instead only assume that

$$\begin{aligned} c(h,H) = c(h), \end{aligned}$$

(58)

so that the fitness cost incurred by the focal agent is independent of the contributions of the other members in its group.

The focal agent’s fitness is then

$$\begin{aligned} W(h,H) = f\left( \sum _{i=1}^{n} h_i\right) - c(h). \end{aligned}$$

(59)

For $1\le i \le n$, let $C_i=c(h_i)$, and $C= c(h)$ be the costs incurred by the n members of the focal agent’s group, and the focal agent (respectively). Suppose that the cost function is one-to-one, so that there exists a left-inverse function $k(\cdot )$ satisfying $k\big (c(h)\big ) = h$ and $k\big (c(h_i)\big ) = h_i$ for all $1\le i \le n$. Then, Eq. (59) becomes

$$\begin{aligned} W(h,H) = f\left( \sum _{i=1}^{n} k(C_i) \right) - C. \end{aligned}$$

(60)

The benefit to the focal agent, $f\left( \sum _{i=1}^{n} k(C_i) \right)$ is then generally not a function of the sum of the group members’ fitness costs, $\sum _{i=1}^{n} C_i$.

By assuming that contributions to the public good are expressed in units of fitness cost (i.e., $c(h)=h$, as in Eq. (2)), we implicitly assumed that fitness itself is the public good. Expressed in more biological terms, we are assuming that reproductive costs are effectively transferable: each individual in a group obtains a fitness benefit $f(\eta )$ regardless of how the associated costs (which sum to $\eta$) are distributed among the group members; for example, the fitness benefit is the same if the focal agent contributes the entire cost ($h=\eta$), or if the cost is distributed equally among group members ($h_i=\eta /n$ for each i).