Altruism in a volatile world

Abstract

The evolution of altruism—costly self-sacrifice in the service of others—has puzzled biologists¹ since The Origin of Species. For half a century, attempts to understand altruism have developed around the concept that altruists may help relatives to have extra offspring in order to spread shared genes². This theory—known as inclusive fitness—is founded on a simple inequality termed Hamilton’s rule². However, explanations of altruism have typically not considered the stochasticity of natural environments, which will not necessarily favour genotypes that produce the greatest average reproductive success,^3,4. Moreover, empirical data across many taxa reveal associations between altruism and environmental stochasticity,^5–8, a pattern not predicted by standard interpretations of Hamilton’s rule. Here we derive Hamilton’s rule with explicit stochasticity, leading to new predictions about the evolution of altruism. We show that altruists can increase the long-term success of their genotype by reducing the temporal variability in the number of offspring produced by their relatives. Consequently, costly altruism can evolve even if it has a net negative effect on the average reproductive success of related recipients. The selective pressure on volatility-suppressing altruism is proportional to the coefficient of variation in population fitness, and is therefore diminished by its own success. Our results formalize the hitherto elusive link between bet-hedging and altruism^4,9–11, and reveal missing fitness effects in the evolution of animal societies.

The widespread phenomenon of organisms paying costs to help others (altruism) is a long-standing paradox in biology^1,2. Recently, variance-averse investment in stochastic environments (bet-hedging) has been suggested as an explanation for a number of major puzzles in the evolution of altruism, including the origins of sociality in birds^9,11,12, insects¹³ and rodents¹⁴, the altitudinal distribution of eusocial species⁷, and the evolution of cooperation between eusocial insect colonies¹⁵. The global distribution of animal societies is linked to environmental stochasticity⁴. In birds^6,12, mammals¹⁶, bees⁷ and wasps⁸, cooperation is more common in unpredictable or harsh environments. However, the effects of stochasticity have largely been omitted from social evolutionary theory. There are a few notable exceptions: in ref. 17 it is argued that selection will maximize expected inclusive fitness under uncertainty; ref. 18 shows that mutualism between non-relatives could counteract kin selection by dampening stochasticity; and stochastic effects on reproductive value are explored in ref. 19. However, despite speculation^11,20, the proposed link between bet-hedging and altruism⁹ has remained elusive⁴. We resolve this link by presenting a stochastic generalization of Hamilton’s rule (stochastic Hamilton’s rule), which predicts when organisms should pay a cost to influence the variance in the reproductive success of their relatives.

We allow the environmental state π to fluctuate among the possible states Π; stochasticity is the condition that states are unpredictable. We follow the established method of capturing fitness effects as regression slopes¹. Both the fitnesses w_x of individual organisms and the average fitness $\overline{w}$ in the population may vary among the states Π. We denote the kth central moment of $\overline{w}$ as ⟪^kw⟫. The joint distribution of the fitness of individual x (w_x) and $\overline{w}$ across states Π is captured by their mixed moments (covariance, k = 1; coskewness, k = 2; cokurtosis, k = 3 and so on; Supplementary Information A1). Altruists may not only alter the expected number of offspring (mean, k = 0), but also may reduce the variation in offspring number (variance, k = 1) or increase the likelihood of large numbers of offspring (skew, k = 2). We denote the effect of the actor on the expected number of offspring of the recipient as the benefit b_μ, the effect of the actor on its own expected number of offspring as the cost c_μ, and relatedness as r. Likewise, we denote the effect of the actor on the kth mixed moment defining the reproductive success of the recipient as b_k, and the effect of the actor on the kth mixed moment of its own reproductive success as c_k. The stochastic Hamilton’s rule is therefore:

$$\begin{array}{l}r.\stackrel{B}{\overbrace{\left(b_{\mu }+{\displaystyle \sum _{k=1}^{\infty }\frac{{(-1)}^k}{{𝔼}_{\pi }{\left[\overline{w}\right]}^k}}({⟪}^k\overline{w}⟫b_{\mu }+b_k)\right)}}>\\ \stackrel{C}{\overbrace{c_{\mu }+{\displaystyle \sum _{k=1}^{\infty }\frac{{(-1)}^k}{{𝔼}_{\pi }{\left[\overline{w}\right]}^k}}({⟪}^k\overline{w}⟫c_{\mu }+c_k)}}\\ \end{array}$$ (1)

Empirical tests of Hamilton’s rule have looked for benefits and costs that constitute effects on the mean reproductive success of recipients and actors, using the form rb_µ > c_µ (henceforth, means-based Hamilton’s rule)²¹. However, equation (1) reveals that b_µ is a single component of a range of potential benefits of altruism. Conclusions based on mean reproductive success (b_µ and c_µ) overlook effects on the variance of the distribution from which a recipient samples its reproductive success.

Asocial bet-hedging has been analysed extensively³, and is typically described in terms of costs and benefits: the cost is a reduction in mean reproductive success, whereas the benefit is a reduction in the variance of reproductive success³. Following speculation that these benefits and costs could be accrued by different partners^9,13—actors pay costs whereas recipients derive benefits (Fig. 1a)—we refer to decoupled benefits and costs as altruistic bet-hedging. We let b_σ and c_σ denote, respectively, the effects on the standard deviation (volatility) of the recipient and actor in reproductive success (weighted by its correlation with population average reproductive success $\overline{w}$; for details see Extended Data Table 1). We introduce the stochasticity coefficient v as the coefficient of variation in $\overline{w}$ across environmental conditions ($v=\frac{{\sigma }_{\pi }\left[\overline{w}\right]}{{𝔼}_{\pi }\left[\overline{w}\right]};$ Fig. 1b). For cases in which the actor can affect both the mean and the volatility (but not higher moments) of the reproductive success of the recipient, equation (1) simplifies (Supplementary Information A2) to:

$$r(b_{\mu }+v{b}_{\sigma })>c_{\mu }+v{c}_{\sigma }$$ (2)

Figure 1. Environmental stochasticity has been missing from models of social evolution.

In the means-based application of Hamilton’s rule (rb_μ > c_μ) to real-world organisms²¹, recipients gain an increase in average reproductive success (b_μ > 0) whereas actors suffer a decrease in average reproductive success (c_μ > 0). a, We derive an explicitly stochastic Hamilton’s rule: r(b_μ + vb_σ) > cμ + vc_σ). This shows that benefits can also arise by reducing the volatility of the reproductive success of the recipient (b_σ > 0), which depends on the magnitude of environmental stochasticity (v). An increase in the reproductive volatility of the actor (c_σ > 0) imposes a cost on the actor. Each effect represents a transformation of a probability distribution for reproductive success (bottom). Total benefits and costs (B and C) are measured in expected relative fitness¹. b, Environmental stochasticity (v) is highest when spatial patches fluctuate in sync: for instance, if drought affects a randomly chosen patch Z, it should be likely that it also affects a randomly chosen patch Y (Supplementary Information A6). Here, following ref. 3, we represent patches in a lattice connected by dispersal. Colours denote environmental condition on patches at sequential time points t. See Supplementary Information A. Image of wasp reproduced with permission from Z. Soh.

Reducing the ($\overline{w}$-correlated) volatility in the recipient’s number of offspring (b_σ > 0) confers on the recipient greater relative fitness in poor environmental states: extra offspring are disproportionately valuable when competitors produce few offspring²², underscoring the principle that the ultimate currency for benefits and costs under stochasticity is the expectation of relative fitness¹. It is straightforward to derive the established asocial bet-hedging model³ by setting r = 0 (Supplementary Information A3).

Formally, we define altruistic bet-hedging as a reduction in the reproductive volatility of a recipient (positive b_σ) that overcomes an otherwise deleterious cost to the expected reproductive success of the actor (positive c_µ). Strong benefits can arise when b_µ and b_σ are both positive, and reductions in the actor’s own reproductive volatility (c_σ < 0) diminish total costs (Fig. 2a, b). Moreover, when b_σ > c_σ, increasing stochasticity reduces the minimum relatedness (r) required for altruism to evolve (Fig. 2c). Fluctuations in relatedness (r) alter selection only if they correlate with strong fluctuations in population average reproductive success ($\overline{w}$) (Supplementary Information A4).

Figure 2. Increased stochasticity can increase the potential for selection of altruistic behaviour.

Without stochastic effects, altruism evolves when rb_μ > c_μ (shown in region ‘1’ in a and b for c_μ = 1, and r = 0.5). As stochasticity v increases, the power of b_σ:c_σ benefits increases, reducing the ratio of b_μ:c_μ needed for the evolution of altruism. a, In this scenario, altruists secure a high b_σ = 0.75, considerably increasing the scope for altruism (extending region ‘1’ to region ‘2’). Actors may also reduce the volatility of their personal fecundity (here, c_σ = −0.4), reducing the magnitude of the total cost C below c_μ and increasing the potential for altruism further (extending to region ‘3’). Altruism is always deleterious in region ‘4’. b, In this scenario, altruists secure a low b_σ = 0.1 and personal volatility reduction of c_σ = −0.1 (regions as in a). Comparing a (b_σ = 0.75) and b (b_σ = 0.1), larger reductions of recipient volatility (higher b_σ) result in larger increases in the inclusive fitness of the actor. c, The minimum relatedness required for the evolution of altruism under different c_μ values (curved lines, from c_μ = 0.05 to 0.4, when b_σ = 0.75, c_σ = 0 and b_μ = 0.2); as stochasticity (v) increases, the minimum required relatedness (r*) decreases.

We note four predictions of the stochastic Hamilton’s rule that differ from standard expectations:

• (i)
  Selection can favour altruism (C > 0) with zero increase to the expected reproductive success of the recipient (b_µ = 0). Such a seemingly paradoxical lack of benefits is observed in cases for which additional helpers appear redundant²³. Paradoxical helpers can be selected for by reducing the reproductive volatility of the recipient if:

  $$r{b}_{\sigma }>\frac{c_{\mu }}{\nu }+c_{\sigma }$$
• (ii)
  Actors may be selected to harm the expected reproductive success of their relatives (b_µ < 0, c_µ > 0). The harm is outweighed by a reduction in the reproductive volatility of the recipient (Fig. 2) if:

  $$r{b}_{\sigma }>\frac{c_{\mu }-r{b}_{\mu }}{\nu }+c_{\sigma }$$
• (iii)
  Altruists that reduce the reproductive volatility of their recipients can be favoured by selection in the absence of environmental stochasticity, but only when population size (N) is low (in extremely small populations³ or small demes with intense local competition²⁴) and b_σ² > c_σ². Effects on variance, σ², not volatility, are used here for notational convenience (Supplementary Information A5):

  $$r\left(b_{\mu }+\frac{b_{{\sigma }^2}}{N{\text{𝔼}}_{\pi }[\overline{w}]}\right)>c_{\mu }+\frac{c_{{\sigma }^2}}{N{\text{𝔼}}_{\pi }[\overline{w}]}$$ (3)
• (iv)Very strong altruistic effects (b_σ ≫ 0) can undermine the success of the altruist genotype (Extended Data Fig. 1; Supplementary Information B1–B4). Altruists that substantially reduce the reproductive volatility of their recipients spread rapidly. As successful altruists reach high frequencies, the coefficient of variation in average reproductive success $\left(v=\frac{{\sigma }_{\pi }[\overline{w}]}{{\text{𝔼}}_{\pi }[\overline{w}]}\right)$ tends towards zero (Extended Data Fig. 2). When v is small, any b_σ has a small effect (equation (2)), so altruistic bet-hedgers undermine the condition (high v) that favoured them (Extended Data Fig. 1a, b). This frequency dependence can generate a mixed population of altruists and defectors (Extended Data Fig. 1c), provided that allele frequency does not fluctuate intensively, which can otherwise destabilize the equilibrium (Extended Data Fig. 3) and lead to fixation²⁵.

Apparent reduction of the reproductive volatility of recipients (implying b_σ > 0) has been shown in starlings⁹, sociable weavers²⁶, woodpeckers¹⁰, wasps²⁷ and allodapine bees¹³. We illustrate a volatility-reduction route to sociality with two examples. First, we consider sister–sister cooperation in facultatively social insects (as in certain carpenter bees, for which a means-based Hamilton’s rule is violated²⁸). In strongly stochastic environments, altruism can evolve between haplodiploid sisters when values of mean fecundity alone would predict it to be deleterious, as predicted by equation (2) (Fig. 3a) and simulations of haplodiploid populations (Fig. 3b; Supplementary Information C1). Second, using published estimates of mean fecundity and high stochasticity in Galapagos mockingbirds (Mimus parvulus), we indicate how volatility effects could favour cooperative breeding even if helping increases the average fecundity of the recipient only as much as it reduces that of the actor (c_µ = b_µ; Fig. 3c; Supplementary Information C2).

Figure 3. Empirical studies of Hamilton’s rule may benefit from incorporating stochasticity.

a, Model of sister–sister cooperation between facultatively social insects: the means-based Hamilton’s rule (rb_μ > c_μ) is violated throughout the plot. Despite this, in the region below the dashed line (which denotes rB = C), volatility effects can favour the invasion of nonreproductive altruists. b, These predictions are matched in an individual-based haplodiploid simulation. In both a and b, good and bad years occur equally (d_π = 0.5) at random. When benefits are slight (close to the dashed line in a), chance correlated fluctuations can drive cooperators extinct. In Supplementary Information B, we discuss temporal correlation. Coordinates plot average frequency across five replicate simulations after 1,000 generations, from an initial frequency P = 0.05. c, In high-stochasticity conditions, helpers may buffer breeders from profound environmental fluctuations⁴,⁹,¹¹. We estimate rb_μ values in the Galapagos mockingbird, and show that volatility effects can, in principle, drive cooperation (above the dashed line) even when mean fecundity costs c_μ cancel out b_μ (here, b_μ = c_μ = 0.3). See Supplementary Information C. Image of bee, K. Walker (CC-BY 3.0 AU); image of mockingbird, Biodiversity Heritage Library (CC-BY 2.0).

Equation (2) reveals three core conditions for altruistic bet-hedging. First, members of the non-altruistic genotype suffer synchronous fluctuations in lifetime reproductive success driven by environmental state (high v) that can be stabilized by sociality (b_σ > 0). Second, relatedness (r) is above the threshold $r*=\frac{c_{\mu }+v{c}_{\sigma }}{b_{\mu }+v{b}_{\sigma }}.$ Third, actors either cannot predict environmental fluctuations or cannot generate phenotypes for different conditions (Fig. 4; Supplementary Information B5). If actors can obtain and utilize information at sufficiently low costs (rendering the environment predictable), plastic cooperation outcompetes constitutive cooperation (increasing b_µ and reducing c_µ).

Figure 4. The trade-off between constitutive and inducible altruism in a stochastic world depends on plasticity costs and information reliability.

We show a population fluctuating randomly between a good and a bad environmental state, comprising three alleles: ‘selfish’ (S), for which the carriers never cooperate; ‘constitutive cooperator’ (C), for which the carriers always cooperate; and ‘inducible cooperator’ (I) for which the carriers cooperate only when they believe they are in the bad (low-fecundity) state. Information reliability is set by A (actors diagnose true state with probability A). Apexes represent monomorphic populations. Without social behaviour, individuals obtain four and one offspring in good and bad states respectively. Cooperation confers on recipients 1.5 additional offspring in bad states but reduces recipient fecundity by 0.2 offspring in good states, and costs actors 0.5 offspring in all states. a, When considering only mean fecundity, the means-based Hamilton’s rule rb_μ > c_μ, commonly used empirically, mistakenly predicts that selfishness (S) will dominate. Under stochastic conditions, cooperation evolves. b, Constitutive cooperators invade (until reaching a mixture of altruists and defectors) when information is imperfect (A = 0.75) and there is a plasticity cost (0.1 offspring). c, When the reliability of information is increased (A = 1), plastic cooperators outcompete constitutive cooperators. d, Increasing plasticity costs, however (here, from 0.1 to 0.3 offspring), eliminates plasticity benefits, enabling constitutive cooperators to invade. Vectors show directions of expected changes in frequencies: these represent continuous expected trajectories when frequencies are constrained to change by small amounts per generation. Relatedness r = 0.5 in all plots. Details are provided in Supplementary Information B.

Synchronous fluctuations (high v) are generated when different patches within the population experience correlated environmental changes (Fig. 1b; Supplementary Information A6). If offspring disperse across environmentally uncorrelated patches³ but compete at a whole-population level, v decreases. Likewise, iteroparity and long generations across different environmental conditions reduce v, whereas correlated exposure to environmental conditions within lifetimes increases v. For these reasons, equation (2) suggests that the most promising avenues to detect b_σ-driven sociality may occur among social microbes, which can experience population-wide fluctuations (high v), short generations (high v), competing clones (high r), and opportunities to confer homeostasis on others (b_σ > 0), including through the construction of biofilms²⁹ and incipiently-multicellular clusters withstanding profound abiotic and biotic stress.

We have shown that altruistic effects on recipient volatility are visible to selection. Notably, Hamilton’s rule identifies ultimate payoffs by incorporating any effects of population structure¹. To make case-specific predictions, researchers should, accordingly, utilize explicit information on population structure and ecology. The empirical challenge to detect volatility-suppressing sociality in wild organisms will best be met using tailored models guided by field data for specific scenarios, led by the general framework of inclusive fitness theory^1,21,30. In summary, Hamilton’s rule reveals the action of selection under stochasticity: shielding relatives from a volatile world can drive the evolution of sociality.

Code Availability

Simulation output was generated using MATLAB code provided in section D of the Supplementary Information; this is also available from the corresponding author upon reasonable request.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Online Content

Methods, along with any additional Extended Data display items and Source Data, are available in the online version of the paper; references unique to these sections appear only in the online paper.

Extended Data

Extended Data Figure 1. The interaction between the frequency of altruists and the effectiveness of altruism.

a, The stochastic Hamilton’s rule predicts that selection on volatility-suppressing altruism with fixed costs and benefits can generate negative frequency dependence and is sensitive to mild mean-fecundity costs (c_μ). Lower values of η denote greater buffering of recipients from the environment. We evaluate a population undergoing synchronous fluctuations to identify the frequency p* at which there is no expected change in allele frequency. We illustrate the result with individual fecundities in good years (z₁) of four offspring and in bad years (z₂) of one offspring. Relatedness is r = 0.5. b, Simulated population outcomes (frequency after 100,000 generations) match predictions of the stochastic Hamilton’s rule in a. Warmer colours (pink) denote higher polymorphic frequencies of altruists. In this haploid model (Supplementary Information B1-4), 1% of breeding spots are available each year for replacement by offspring that year: with such constraints on the magnitude of the response to selection, radical stochastic shifts in allele frequency over single generations do not occur, allowing the population to settle at equilibria where all alleles have equal expected relative fitness without being continually displaced (Extended Data Fig. 3). c, Competing an altruistic allele against a defector allele reveals the action of frequency-dependent selection. Here, populations experiencing costs of c = 0.2 and η = 0.466 converge to p* = 0.359 from any initial frequency (coloured lines show five starting frequencies from 0.001 to 0.999), as predicted by the stochastic Hamilton’s rule.

Extended Data Figure 2. Stochasticity as a function of bet-hedger frequency.

Stochasticity $v=\frac{(p\eta +(1-p)){\sigma }_{00}}{{\mu }_{00}-pc}$ for the model of altruistic bet-hedging in Supplementary Information B plotted against frequency (p) and cost (c) for three different values of η. a, b, When η is small, representing high levels of volatility suppression, v declines steeply with p across the range of costs. c, When η is large, the sign of the effect of p on v depends on c. Values of other parameters: z₁ = 4, z₂ = 1, and frequency of good years d = 0.5.

Extended Data Figure 3. Weak selection negates the capacity of temporal autocorrelation to drive the frequency of altruistic bet-hedgers away from the convergence frequency.

Individual-based simulations from five different initial frequencies of an altruistic bet hedging allele (p) competing against a non-cooperator. a, The population has zero temporal autocorrelation (environmental state in each generation is random). b, The population has strong temporal autocorrelation (environmental state in the next generation has a 90% probability of remaining the same as in the current generation). Despite higher amplitude fluctuations, this population converges to the same point (from the five different starting frequencies) as the uncorrelated population (a). c, The same population is simulated with greater gene frequency changes (10% of the resident genotype frequencies are available to change each generation). The population is repeatedly carried to frequencies far from the convergence point. In this case, the utility of the stochastic Hamilton’s rule is both identifying whether a given trait is immune from invasion by competitors, and identifying the expected generational change at each frequency p. Parameters are z₁ = 4, z₂ = 1, r = 0.5.

Extended Data Table 1. Parameters of the model.

Notation	Definition	Expression
N	Population size	–
wx	Number of surviving offspring (reproductive success) of the xth individual	–
$\overline{w}$	Mean reproductive success in the population	–
π	Environmental state within the set of states Π	–
Gx	Genetic value of individual x	–
r	Relatedness	βGy,Gx
zx	Trait value of individual x	–
≪k $\overline{w}$ ≫	kth central moment of $\overline{w}$ across Π	𝔼π[($\overline{w}$ − 𝔼π[$\overline{w}$])k]
≪ wx,k $\overline{w}$ ≫	kth mixed moment of reproductive success of individual x and $\overline{w}$ across Π	𝔼π[(wx − 𝔼π[wx])($\overline{w}$ − 𝔼π[$\overline{w}$])k]
v	Stochasticity of the environment	$\frac{{\sigma }_{\pi }[\overline{w}]}{{\text{𝔼}}_{\pi }[\overline{w}]}$
ρx	Correlation between wx and $\overline{w}$ across Π	$\frac{{\text{𝔼}}_{\pi }[w_x\overline{w}]-{\text{𝔼}}_{\pi }[w_x]\cdot {\text{𝔼}}_{\pi }[\overline{w}]}{{\sigma }_{\pi }[w_x]{\sigma }_{\pi }[\overline{w}]}$
B	Total benefit in Hamilton’s rule under stochasticity	${\beta }_{{\text{𝔼}}_{\pi }\left[\frac{w_y}{\overline{w}}\right],G_x}$ Partial regression of a focal individual’s genetic value on a social partner’s expected relative fitness
C	Total cost in Hamilton’s rule under stochasticity	$-{\beta }_{{\text{𝔼}}_{\pi }\left[\frac{w_x}{\overline{w}}\right],G_x}$ Partial regression of a focal individual’s genetic value on its own expected relative fitness
bμ	Mean fecundity benefit in stochastic Hamilton’s rule	β𝔼π[wy],Gx Partial regression of a focal individual’s genetic value on a social partner’s expected number of offspring. We make use of the identity β𝔼π[wy],Gx = β𝔼π[wx],Gy in non-class-structured populations.
cμ	Mean fecundity cost in stochastic Hamilton’s rule	−β𝔼π[wy],Gx Partial regression of a focal individual’s genetic value on its own expected number of offspring
bσ	Volatility-suppressing benefit in stochastic Harmilton’s rule	−βρσπ[wy],Gx Partial regression of a focal individual’s genetic value on a partner’s standard deviation in reproductive success, where the standard deviation is weighted by its correlation with $\overline{w}$. We make use of the identity βρσπ[wy],Gx = βρσπ[wx],Gy in non-class-structured populations
cσ	Volatility-suppressing cost in stochastic Hamilton’s rule	βρσπ[wx],Gx Partial regression of a focal individual’s genetic value on a partner’s standard deviation in reproductive success, where the standard deviation is weighted by its correlation with $\overline{w}$
bk	kth moment benefit in stochastic hamilton’s rule	β≪wy,k$\overline{w}$ ≫,Gx Partial regression of a focal individual’s genetic value on the kth mixed moments of a partner’s joint distribution for reproductive success wy and population average reproductive success $\overline{w}$. We make use of the identity β≪wx,k$\overline{w}$ ≫,Gy = β≪wy,k$\overline{w}$ ≫,Gx in non-class-structured populations
ck	kth moment cost in stochastic Hamilton’s rule	−β≪wx,k$\overline{w}$≫,Gx Partial regression of a focal individual’s genetic value on the kth mixed moments of its own joint distribution for reproductive success wy and population average reproductive success $\overline{w}$.

For derivation of regression slopes, see Supplementary Information A.

Supplementary Material

Supplementary Information is available in the online version of the paper.