Social evolution – Carbon Chemist

Tag: Social evolution

The evolution of private reputations in information-abundant landscapes

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The evolution of menopause in toothed whales

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For this study we collected, processed and analysed published data from the literature to get estimates of lifespan, reproductive lifespan, size, survival to maturity and age at maturity for as many toothed whale species as possible. We then used these estimates in analyses to understand how and why menopause evolves. All data management, analysis and plotting were performed in R with the tidyverse, rstan cmdstanr and ape packages^64,65,66,67.

Data

Lifespan data and modelling

We collected sex-specific age-structured toothed whale data from the literature and then applied a mortality model to these data to get species-sex lifespan estimates for toothed whale species. In most toothed whale species, the age of a whale can be inferred by counting tooth growth rings⁶⁸. There is a tradition in the study of toothed whales to collect age and sex data from deceased individuals, for example, after mass-stranding events. Here, we collate and use these published data to understand toothed whale mortality patterns. We collated the data by performing a systematic search of the literature through Web of Science (webofscience.com) using species name⁶⁹, alternative species name⁶⁹, common name⁶⁹ and alternative common names⁶⁹ with each of the search terms ‘life history’, ‘lifespan’ and ‘age structure’. We also repeated the search using relevant languages local to the distribution of the species in question in an attempt to reduce the English-language bias of our data⁷⁰. We repeated the search for each of the 75 species of toothed whale recognized by the International Whaling Commission⁶⁹. For each species, we also performed opportunistic searches by following references through the literature and identifying potential data sources using the relevant chapters in authoritative edited collections^71,72,73,74. From our search, we identified and extracted⁷⁵ sex-specific age data into a database⁷⁶. In the database, where possible, each dataset represents data from a single population collected at a single time; however, this is not always possible and some datasets represent data collected from several populations or over a longer timescale. We also parameterize the mortality model with dataset-specific information on population growth, sampling biases and age-estimation error each of which are based on data provided in the original publication⁷⁶. A species-sex was included in the analysis if they had at least one dataset with more than ten samples above the age of maturity and a sampling rate (number of samples/maximum observed age) of greater than 0.5. Our complete database for analysis used 269 datasets from 118 publications allowing us to estimate lifespan for 32 species of female (Fig. 1 and Supplementary Table 1) and 33 species of male toothed whale. Fisheries by-catch (n = 113) and stranding (n = 107) are the most common sources of data in our database, although most species have data from several sources. Research to date has not suggested or found evidence of systematic age differences in propensity to be taken as by-catch or stranding and when bias is suspected in a given dataset we include this bias in our modelling framework.

We developed a Bayesian modelling framework to derive the parameters, α and β, of a Gompertz mortality model for each species-sex. The framework aims to derive a distribution of mortality parameters for each species-sex most parsimonious with the data and with the potential and unknown effects of population growth, sampling bias and age-estimation error. We consider the count c of whales in age category i (where AGE is greater than age at maturity) in dataset d to be drawn from a multinomial distribution, where θ_i is a product of the probability of surviving to that age, L_i, the effect of population change, R_i, and the effect of sampling bias, S_i (equation (1)). L_i, R_i and S_i are defined on the basis of a function of the age, AGE_i, of the whales in category i. We concurrently model that the count c_d,i is the number of whales with the true age i in dataset d, where the true age of whale j, τ_j, is drawn from a normal distribution around the observed age, o_j, with a standard deviation of ε_j (equation (1)). The model is fit over all available datasets of the same species. Datasets of the sex (SEX_d) share mortality parameters, datasets from the same population (POP_d) share a population growth parameter (r_d) and each dataset has an independent sampling bias parameter (s_d). Sampling bias error is applied where AGE_d,i is within a predefined bias window W_d derived from the data source. Simulations demonstrate that this model is capable of recovering reliable estimates of lifespan under a biologically relevant range of demographic and error scenarios⁷⁶ (see also Supplementary Information 1, Equations):

$$\begin{array}{c}c\, \sim {\rm{M}}{\rm{u}}{\rm{l}}{\rm{t}}{\rm{i}}{\rm{n}}{\rm{o}}{\rm{m}}{\rm{i}}{\rm{a}}{\rm{l}}(\theta \,,N)\\ {\theta }_{d,i}=\frac{{L}_{d,i}{R}_{d,i}{S}_{d,i}}{\sum {L}_{d,i}{R}_{d,i}{S}_{d,i}}\\ {L}_{i}=\frac{{{\rm{e}}}^{-({\alpha }_{{{\rm{S}}{\rm{E}}{\rm{X}}}_{d}}/{\beta }_{{{\rm{S}}{\rm{E}}{\rm{X}}}_{d}})({{\rm{e}}}^{{\beta }_{{{\rm{S}}{\rm{E}}{\rm{X}}}_{d}}\times {{\rm{A}}{\rm{G}}{\rm{E}}}_{d,i}}-1)}}{{\sum }_{j=0}^{n}{{\rm{e}}}^{-({\alpha }_{{{\rm{S}}{\rm{E}}{\rm{X}}}_{d}}/{\beta }_{{{\rm{S}}{\rm{E}}{\rm{X}}}_{d}})({{\rm{e}}}^{{\beta }_{{{\rm{S}}{\rm{E}}{\rm{X}}}_{d}}\times {{\rm{A}}{\rm{G}}{\rm{E}}}_{d,j}}-1)}}\\ {R}_{d,i}={(1-{\rho }_{d})}^{{{\rm{A}}{\rm{G}}{\rm{E}}}_{d,i}}\\ {\rho }_{d}=\frac{{r}_{{{\rm{P}}{\rm{O}}{\rm{P}}}_{d}}}{1/2\times \,max({{\rm{o}}}_{{{\rm{D}}{\rm{S}}{\rm{E}}{\rm{T}}}_{{{\rm{S}}{\rm{E}}{\rm{X}}={\rm{S}}{\rm{E}}{\rm{X}}}_{d}}})}\\ {S}_{d,i} \sim \left\{\begin{array}{cc}{s}_{d}+1 & {{\rm{A}}{\rm{G}}{\rm{E}}}_{d,i}\in {W}_{d}\\ 1 & {{\rm{A}}{\rm{G}}{\rm{E}}}_{d,i}\notin {W}_{d}\end{array}\right.\\ {\tau }_{j} \sim {\rm{N}}{\rm{o}}{\rm{r}}{\rm{m}}{\rm{a}}{\rm{l}}({o}_{j},{{\epsilon }}_{j})\\ {{\epsilon }}_{j}=\frac{1}{20}({o}_{j}+B)\\ {c}_{d,i}=\mathop{\sum }\limits_{k=1}^{N}[\lfloor {\tau }_{k}\rceil ={{\rm{A}}{\rm{G}}{\rm{E}}}_{i},{\rm{D}}{\rm{S}}{\rm{E}}{\rm{T}}=d]\end{array}$$

(1)

We fit this using Hamiltonian Monte Carlo chains implemented in R and Stan^64,66 (Supplementary Information 2, Mortality and Corpora). For each species-sex, we then use the posterior estimates of α and β to calculate ordinary maximum lifespan (age Z): the age at which 90% of adult life years have been lived⁷⁷. This results in a distribution of ordinary maximum lifespans for each species-sex derived from 40,000 draws from the posterior of the fitted mortality model. In the following analyses, we use the mean ± s.d. of the posterior distribution of ordinary maximum lifespans as our species-sex lifespan measures.

Reproductive lifespan and modelling

We use age-specific ovarian activity to assess toothed whale reproductive lifespans. In toothed whales, after ovulation, the corpora in which the ovum develops persists in the ovary⁷⁸. A sample of corpora counts from whales of known ages can therefore be used to calculate changes in the rate of corpora deposition with age as a measure of age-specific ovarian activity⁹. In long-finned and short-finned pilot whales the rate of corpora deposition with age matches age-specific pregnancy rates⁹. We use the age at which the rate of new corpora deposition reaches 0 as our measure of reproductive lifespan.

The counts of corpora are commonly reported after dissections of deceased whales. We build on the database of age-specific corpora counts used in previous studies⁹ by adding more datasets uncovered during our systematic and opportunistic search for toothed whale life history data (Lifespan data and modelling section). Each corpora dataset consists of a sample of known-age females and the number of ovarian corpora they were found to have. We only include corpora datasets in our analysis if they have a sample size greater than or equal to 20 and a sampling rate (number of samples/maximum observed age from lifespan data) of more than 0.6. Our final sample size for this analysis is 27 datasets from 18 species (Fig. 1, Supplementary Information 2, Mortality and Corpora and Supplementary Table 2).

We apply a Bayesian model to these corpora data to find the age at which ovarian activity reaches 0. Unlike in previous studies⁹, here we directly model the process of corpora deposition. We model the count of corpora at a given adult age C_i (where i = 0 is the age at maturity) to be drawn from a Poisson distribution with mean λ_i, where λ_i is the sum of corpora (k) deposited at all previous ages, which in turn depends on the initial rate of corpora deposition α and a linear rate of decline in rate with age β (see also Supplementary Information 1, Equations):

$$\begin{array}{c}C \sim {\rm{P}}{\rm{o}}{\rm{i}}{\rm{s}}{\rm{s}}{\rm{o}}{\rm{n}}(\lambda )\\ {\lambda }_{i}=\mathop{\sum }\limits_{j=0}^{i}\Delta {k}_{j}\\ \Delta {k}_{j}=\alpha (1-\beta {{\rm{A}}{\rm{G}}{\rm{E}}}_{j})\end{array}$$

(2)

If several corpora datasets are available from the same species, they are modelled together with datasets allowed to differ in α but sharing an age-specific rate of decline β. For all species, reproductive lifespan is then calculated as 1/β + age at maturity. We calculate a distribution of reproductive lifespans from 40,000 draws from the posterior distribution of β.

Size and maturity

We use length rather than mass as our measure of species size because of the difficulty of accurately measuring mass in cetaceans⁷⁹. For each toothed whale species-sex, we consulted expert-written edited volumes^71,72,73,74 to get all available metrics of sex-specific length. For each species-sex, we collated all available measures of mean, asymptotic, minimum and maximum length, as well as standard deviation around the mean or asymptote. If metrics were not available from a sex-specific sample (n = 20) we instead used any available combined sex samples; if this occurred, the species were not included in any within-species-sex size comparisons. We then processed these metrics to get a single estimate of mean size and standard deviation around that estimated size (Supplementary Information 3, Additional Data Explanation). This error was carried through all subsequent analyses. Available estimates of mass were strongly positively correlated with length (Supplementary Information 3, Additional Data Explanation).

We use the same framework to generate estimates of age at maturity for each species-sex as we did for size (Supplementary Information 3, Additional Data Explanation). We gathered estimates of mean age of maturity and distribution from expert consensus and then processed these measures to get a single measure of mean age of sexual maturity and standard deviation around the mean for each species-sex. For simplicity, for analyses not directly testing correlates of the age of maturity we use the mean measure as the age of sexual maturity.

In analyses regressing size against age or vice versa, models take the form of a Bayesian phylogenetically controlled linear model with the required parameter and menopause as predictors. In these models, both the true size and true age at maturity are considered to be drawn from a normal distribution around the observed mean given the observed standard deviation. The effect of phylogeny is implemented as the covariance between species by means of the Ornestain–Uhlenbeck process⁸⁰.

Our kinship demography analysis requires estimates of juvenile mortality. We generate estimates of juvenile mortality for the 23 datasets with good sampling (>100) of whales under the age of maturity by applying a Gompertz mortality model with a bathtub term⁸¹ (to describe juvenile mortality) to these datasets. In these samples, we found a negative correlation between age at maturity and the probability of surviving to maturity. We used this correlation to generate a posterior distribution of the estimated proportion of whales born surviving to maturity for each species-sex (Supplementary Information 3, Additional Data Explanation).

Kinship demography analysis

We use a matrix population modelling framework to understand how many relatives of different classes a female can expect to have given her age²⁵ and hence her potential to provide intergenerational help and intergenerational harm. For each species, we build a Leslie matrix based on (1) mortality hazard at each adult age derived from the lifespan modelling, (2) mortality hazard at juvenile ages derived from the survival to maturity analysis and (3) age-specific fecundity derived from the corpora analysis β scaled by the baseline fecundity f needed to maintain a stable population (the age-distribution of a stable population is calculated by inferring the posterior distribution of mortality parameters when r = 1 for each species from our fitted mortality models). For each β draw from the corpora analysis, we calculated the baseline reproductive rate by systematically exploring the parameter space to find the value of f needed to maintain this known stable age-distribution (λ = 1) using established matrix population methods³⁴. All of these inputs are the posterior estimates from Bayesian analyses. Computational limitations mean that it was not possible to explore the entire combined posterior space, we therefore take 1,000 draws from each posterior distribution and use these to create a set of alternative Leslie matrices for each species. We apply kinship demography models²⁵—adapted to predict both sexes of offspring and grandoffspring—to each Leslie matrix to calculate a distribution of the number of offspring and grandoffspring a female can expect to have at a given age.

We quantify grandmother overlap at age x as the number of matrilineal grandoffspring below the age at maturity that a female can expect to have at that age multiplied by the probability of surviving to that age from maturity. We then sum this over all female ages. This measure gives a direct measure of the number of years a female can expect to be alive at the same time as her grandoffspring. Similarly, mother–offspring overlap is quantified as the number of years a female can expect to spend alive at the same time as her offspring and is calculated in the same way as grandmother overlap but replacing the number of grandoffspring with the number of offspring (of both sexes). We measure reproductive overlap as the cumulative proportion of a female’s reproductive life remaining when her daughter gives birth. To calculate this at each age we take the product of the expected number of grandoffspring of age 0 that a female will have and multiply this by the proportion of reproductive capacity remaining to a female of that age. This is summed over all ages to get the total reproductive overlap in the species. We use the same methodological pathway to calculate ‘relative offspring overlap’ between mothers and the offspring, determined separately for daughters and sons.

Demographic simulations

For each of the five species with menopause, we also calculated the expected grandmother years and reproductive overlap under two alternative demographic scenarios: (1) the ancestral case and (2) the slow life history case. The ancestral case is the predicted demographics of the immediate non-menopausal ancestors of the species. Under the live-long hypothesis (see the section ‘Live-long or stop-early?’), species without menopause evolve by extending their lifespan without extending their reproductive lifespan. For the ancestral case, we therefore simulate the demographic parameters of a population with the same lifespan as expected for the size of the species and the same reproductive lifespan as observed in the real data (see Supplementary Table 3 for the derivation of parameters). The slow life history case simulates a population in which, instead of evolving menopause, females continue to reproduce for their whole life. Under the slow life history case, lifespan is the same as observed for the real species but reproductive lifespan is extended to lifespan (Supplementary Table 3). We compare the ancestral case and slow life history case to the observed demographics (observed case). For this analysis, we measure reproductive lifespan as the age of the oldest known reproductive active female (Supplementary Table 4) to allow the inclusion of killer whales in the analysis, as no corpora data are available for killer whales. Unlike most of the other species in our dataset, the reproductive datasets for these five datasets are large enough to allow a relatively robust estimate of the age of the oldest reproductively active female. For each demographic scenario, we use the pipeline for the kinship demography analysis (above) to calculate distributions of values of baseline reproductive rate, relative grandmother years, relative mother years and expected reproductive overlap.

Analysis

Live-long versus stop-early

To understand if species with menopause live or reproduce longer than expected given their size and phylogenetic position we use a model of the form:

$$\begin{array}{c}\tau Z \sim {\rm{M}}{\rm{u}}{\rm{l}}{\rm{t}}{\rm{i}}{\rm{N}}{\rm{o}}{\rm{r}}{\rm{m}}{\rm{a}}{\rm{l}}(\mu ,K)\\ \tau Z \sim {\rm{N}}{\rm{o}}{\rm{r}}{\rm{m}}{\rm{a}}{\rm{l}}(\mu Z,\sigma Z)\\ \tau S \sim {\rm{N}}{\rm{o}}{\rm{r}}{\rm{m}}{\rm{a}}{\rm{l}}(\mu S,\sigma S)\\ {\mu }_{i}=\alpha +{\beta }_{{\rm{S}}{\rm{I}}{\rm{Z}}{\rm{E}}}\tau {S}_{i}+{\beta }_{{\rm{P}}{\rm{R}}}{M}_{i}\\ {K}_{i,j}={\eta }^{2}{{\rm{e}}}^{-{\rho }^{2}{D}_{i,j}}\end{array}$$

(3)

where true lifespan τZ is drawn from a multinormal distribution with a mean described by a linear model with terms for size (β_SIZE) and the effect of menopause (β_PR). True ordinary maximum lifespan τZ and log true size τS are drawn from normal distributions described by the means (μZ, μS) and standard deviations (σZ, σS) from the lifespan modelling and log observed size, respectively. In species in which menopause is present M = 1, otherwise M = 0. The dispersion of the multinormal distribution, K, is a covariance matrix derived from the phylogenetic distance matrix by the Ornestain–Uhlenbeck process (parameters η, ρ)⁸⁰. All parameters have weakly informative priors. We use 1,000 time-calibrated bootstrapped phylogenetic trees derived from a recently published cetacean phylogeny³⁵. We run the model on each of the bootstrapped phylogenies and combine the posterior estimates to get a complete posterior, which is reported. We use the same modelling structure replacing ordinary maximum lifespan Z with the age at which ovarian activity reaches 0 to understand the relationships between reproductive lifespan, size, phylogeny and the presence or absence of menopause. We confirmed the absence of an effect of menopause on reproductive lifespan by comparing the predictive power of models with and without a menopause parameter (Supplementary Table 5). There is no evidence that a model with a menopause parameter has greater predictive power than a model without a menopause parameter (elpd difference without − with = −0.8 ± 1.2 (±s.e.); Supplementary Table 6).

As the presence of menopause in beluga whales and narwhals⁵⁸ and in false killer whales⁹ has sometimes proved controversial, we repeat this and other key analyses presented in this study excluding these species. We also repeat the analyses excluding the largest and smallest whale species to confirm that they are not exerting undue influence over our interpretation. Lastly, we also repeated our analysis, coding menopause as a continuous value with error to capture potential differences between menopause species and uncertainties around menopause status. In all cases, these exclusions made no qualitative and little quantitative difference to any of our results (Supplementary Table 5).

Help and extended lifespans

We tested three predictions to investigate the role of intergenerational help in the evolution of extended lifespans (Table 1). We compared the grandmother overlap of species with and without menopause using a phylogenetically controlled Bayesian regression model (Table 1). In this model, relative grandmother overlap—calculated as grandmother overlap/age at maturity—is the response variable providing a more meaningful interspecies comparison. Posterior predictive checks demonstrate that this model captures the observed distribution of the data (Supplementary Figs. 5 and 6) and, although the sample size means the true distribution is uncertain, there is no strong evidence that the grandmother years metric is bimodal (Supplementary Fig. 6). If further data find strong evidence of bimodality it could indicate that there are two modes of life in toothed whales that might have implications for understanding the evolution of life history strategies, including menopause. We use the same analytical pathway to compare the mother overlap of species with and without menopause (Table 1). Last, we used the outputs of the demographic simulations to compare grandmother overlap in species with menopause to their non-menopause ancestor. Applying the kinship demography models to the demographic simulation ‘ancestral case’ allowed us to generate a distribution of potential relative grandmother overlap values for the proposed non-menopausal ancestor of toothed whale species with menopause.

Help and reproductive lifespans

We use two metrics to quantify the costs of offspring and to test the prediction that the offspring of species with menopause are more costly than the offspring of species without menopause (Table 1): age at maturity and size at maturity (adult size). Specifically, we test the predictions that (1) species with menopause have a later age at maturity than expected given the size of their mother, which would suggest extended maternal investment, and (2) species with menopause have larger adult size than expected given their age at maturity, which would suggest greater maternal investment during the juvenile phase. We tested these predictions separately for each sex, using a phylogenetically controlled Bayesian regression model. There are other potential costs of raising offspring not captured by these metrics, including the costs of caring for offspring beyond maturity²⁸, which could be an interesting focus of future research. In addition, we test the explicit mechanism that some models have proposed by which the costly intergenerational help of older females can benefit their younger kin. These models propose that, by ceasing reproduction, older females can invest in increasing the reproductive rate of their daughters^30,33. We test this by comparing the observed baseline fecundity in species with menopause, derived from the kinship demography models, compared to the predicted baseline reproductive rate of their non-menopausal rate. In a further analysis, we compare the baseline fecundity of toothed whale species with menopause and without menopause (Supplementary Fig. 7).

Harm and reproductive lifespans

We tested two predictions to establish the role of intergenerational harm in the evolution of reproductive lifespan (Table 1). First, we used a phylogenetically controlled Bayesian regression model to compare the predicted reproductive overlap of species with and without menopause, in which reproductive overlap was derived from the kinship demography models (Table 1). Posterior predictive checks demonstrate that this model captures the observed distribution of the data (Supplementary Fig. 5) and the metric does not show any clear evidence of bimodality (Supplementary Fig. 6). We confirm our result by comparing the predictive power of models with and without a menopause parameter (Supplementary Table 7). Similarly, neither running the model on the mean reproductive overlap estimates without uncertainty, nor replacing the true uncertainty around the reproductive overlap with the scaled uncertainty for that species from the grandmother years metric, qualitatively change our conclusions. Second, we compare observed reproductive overlap in species with menopause to the reproductive overlap in the slow life history case representing the non-menopausal analogue of the species with menopause (Table 1). The predicted reproductive overlap is derived by applying kinship demography models to the output of the demographic simulations.

Female lifespans and male longevity

We tested two predictions of the male-driven menopause hypothesis. First, we compare relative male and female lifespans in species with and without menopause (Table 1). In species for which estimates of ordinary maximum lifespan are available for both sexes (n = 30, including all five species with menopause) we compared the difference in female:male lifespan ratio (τl) in species with (M_i = 1) and without (M_i = 0) menopause using a model of the form:

$$\begin{array}{c}\delta l \sim {\rm{l}}{\rm{o}}{\rm{g}}{\rm{N}}{\rm{o}}{\rm{r}}{\rm{m}}{\rm{a}}{\rm{l}}(\mu ,\sigma )\\ {\mu }_{i}=\alpha +{\beta }_{{\rm{P}}{\rm{R}}}{M}_{i}\\ \sigma \sim {\rm{E}}{\rm{x}}{\rm{p}}(1)\\ \tau {l}_{{{\rm{f}}}_{i}} \sim {\rm{N}}{\rm{o}}{\rm{r}}{\rm{m}}{\rm{a}}{\rm{l}}(\mu {l}_{{{\rm{f}}}_{i}},\sigma {l}_{{{\rm{f}}}_{i}})\\ \tau {l}_{{{\rm{m}}}_{i}} \sim {\rm{N}}{\rm{o}}{\rm{r}}{\rm{m}}{\rm{a}}{\rm{l}}(\mu {l}_{{{\rm{m}}}_{i}},\sigma {l}_{{{\rm{m}}}_{i}})\\ \delta {l}_{i}=\frac{\tau {l}_{{{\rm{f}}}_{i}}\,}{\tau {l}_{{{\rm{m}}}_{i}}}\end{array}$$

(4)

where τl_f and τl_m are, respectively, true female and male lifespans, drawn from a normal distribution parametrized by the distribution of age Z calculated from the posterior of mortality parameters (Lifespan data and modelling section).

Second, for each of the species with menopause we also calculated the probability of each sex reaching the age of female menopause (l_M) and the expected lifespan at the age of female menopause (e_M). We use the age of last known reproduction as age of menopause to allow all killer whales to be included in the analysis (Kinship demography analysis section) but results are qualitatively similar if the reproductive lifespan is used. We calculated l_M and e_M from the mortality parameters derived from the fitted Gompertz models⁸². Morality parameters for both sexes are calculated in the same model so for each species female and male l_M and e_M can be directly compared from each posterior draw.

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Super-additive cooperation | Nature

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A sequential social dilemma with a continuous action space

In the models, pairs of players play a social dilemma. Players choose how much to cooperate from a continuous action space, and moves are sequential. The game is thus an apt description of many social dilemmas past and present. Food sharing^45,46 and alloparental care⁴⁷, for example, must be sequential social dilemmas with continuous action spaces. They are not simple prisoner’s dilemma games in which players simultaneously decide to defect fully or cooperate fully. The emphasis on continuous action spaces is not trivial. As results from the repeated interactions scenario show, intuitions honed on the analysis of reciprocal altruism in repeated prisoner’s dilemma games⁴⁸ do not extend to settings where cooperation can vary continuously. Moreover, by developing models and experiment (see below) with parallel designs, we recruit the complementary strengths of both methods in a way that renders the link between theory and empiricism transparent⁴⁹. We do not need a vague intermediate step where we extrapolate from models based on one type of social interaction to experiments involving another type of social interaction, with misleading predictions as a result⁴⁴.

With respect to the stage game in the models (supporting information section 1²⁶), each player has an endowment normalized to one. The first mover can transfer any amount up to and including her full endowment to the second mover, and the transfer is doubled. Then, the second mover can transfer any amount up to and including her full endowment to the first mover, and this transfer is also doubled. Because transfers are doubled, expected relatedness cannot explain cooperation. Given what we know about average relatedness within groups in small-scale societies²², efficiency gains would have to be much higher than this for relatedness alone to be adequate.

A one-shot interaction is one stage game. Repeated interactions consist of repeated stage games, where each repetition involves new endowments. An individual’s strategy has two parts, an initial transfer and a response function. The initial transfer specifies how much the individual transfers, if first mover, for the first interaction only. For all choices after the initial transfer, the response function specifies an individual’s current transfer as a function of her partner’s most recent transfer. Specifically, the second mover always responds to the first mover’s transfer in the same interaction. If interactions are repeated, from the second interaction onward, the first mover responds to the second mover’s transfer from the preceding interaction (supporting information sections 2.1.3, 2.2 and 2.3²⁶).

The three scenarios

The repeated interactions scenario consists of models of populations subdivided into 40 groups of 24 individuals each without any competition between groups. Individuals within groups pair off randomly to play the game. Individuals only play the social dilemma with ingroup partners, and we consider both one-shot games and repeated interactions (supporting information section 2.1.4²⁶). Because individuals only play with ingroup partners, the repeated interactions scenario isolates the effects of repeated interactions and the reputational concerns they create from the effects of intergroup competition and more generally outgroup interactions of all sorts. We ignore uncertainty about whether a game is one-shot or repeated^2,3, which maximizes the scope for repeated interactions to support cooperation when relationships actually do last a long time.

The group competition scenario also consists of models in subdivided populations. In this scenario, however, groups compete, and games are always one-shot. Groups are paired within a generation (supporting information section 2.1.5²⁶). Each individual plays both a one-shot social dilemma with a randomly selected ingroup partner and a one-shot social dilemma with a randomly selected outgroup partner from the paired group. The individual has separate strategies for ingroup versus outgroup interactions. The opportunity to cooperate with outgroup partners in our models is different from most evolutionary models of parochialism because most models limit attention to outgroup strategies that range from defection to outright aggression³⁴. Defection in these models is the most generous feasible option for an outgroup interaction.

After game play, we model the occurrence of group competitions by assuming that paired groups compete against each other with relatively low probabilities (supporting information section 2.1.7²⁶) that decrease as the groups become more similar (supporting information section 2.1.5²⁶). This approach reflects the idea that paired groups assess each other and avoid competing when they have trouble identifying the probable winner, which is consistent with both past modelling work and ethnographic evidence^4,50. We can think of a competition as a violent conflict, a competition for some limited resource, or a process where the culture of one group displaces the culture of another group³¹. In general, the group competition scenario isolates the effects of intergroup competition from the effects of repeated interactions and associated reputational concerns within groups. The joint scenario combines both repeated interactions within groups and competition between groups (supporting information section 2.1.6²⁶). It is identical to the group competition scenario with one exception; ingroup interactions are always repeated.

A framework for comprehensive variation in model structure

To develop a set of models that examine a wide range of potential ancestral conditions, we cross the six model characteristics below in all possible combinations.

The dimensionality of strategy space (all scenarios)

We vary the dimensionality of the strategy space as a way of manipulating the set of possible strategies. When a strategy is two-dimensional, it consists of an initial transfer and a second quantity controlling the slope and location of a linear response function (supporting information sections 1.1 and 2.2²⁶). Possible response functions include perfect reciprocity, escalating reciprocity, and de-escalating reciprocity. A perfectly reciprocal response function means a focal individual’s transfer is exactly the same as her partner’s most recent transfer (Fig. 1). When two perfect reciprocators interact, all transfers are identical to the initial transfer of the first mover. Escalating reciprocity means the focal player increases the degree of cooperation when possible (Fig. 1a), and unconditional full cooperation is an extreme case. When two escalators interact repeatedly, they converge on full cooperation, and in this sense escalation is a cooperative form of reciprocity. De-escalating reciprocity means the focal player decreases the degree of cooperation when possible (Fig. 1b), and unconditional full defection is an extreme case. When two de-escalators interact repeatedly, they converge on full defection, and thus de-escalation is an uncooperative form of reciprocity.

In a three-dimensional strategy space, a strategy consists of an initial transfer, as well as left and right intercepts for a linear response function (supporting information sections 1.2 and 2.1²⁶). Three dimensions allow for all the strategies feasible in two dimensions, but with a number of additional possibilities. For example, three dimensions allow for ambiguous reciprocity. Ambiguous reciprocity means the focal player has a non-negatively sloped response that escalates low transfers and de-escalates high transfers (Fig. 1c). If an ambiguous reciprocator interacts repeatedly with a partner having any positively sloped response function, the players converge on intermediate levels of cooperation (supporting information section 1.2.8²⁶). A four-dimensional strategy space adds strategies involving a wide range of non-linear response functions (supporting information section 2.3²⁶). Some of the new possibilities include non-linear analogues of ambiguous reciprocity (Fig. 1d). New possibilities also include non-linear forms of reciprocity that do the opposite of ambiguous strategies by de-escalating low transfers and escalating high transfers (Fig. 1d). Such strategies punish low transfers with even lower transfers and reward high transfers with even higher transfers.

Cancellation effects at the individual level (all scenarios)

When a population is subdivided into groups and some individuals remain in the groups where they were born, relatedness within groups is present. When individuals play the social dilemma with ingroup partners, this relatedness allows cooperators to channel the benefits of cooperation towards other cooperators. Relatedness within groups can support the evolution of ingroup cooperation as a result, but it does not necessarily do so. Life history details, demography, and local ecological conditions can offset the effects of related individuals playing the game together⁵¹. Offsetting effects of this sort are cancellation effects at the individual level. Our models vary these cancellation effects by relying on two different life cycles (supporting information section 2.1.2²⁶). In one case, the order of events in the life cycle is birth, game play, migration, group competition when relevant, and finally individual selection within groups. Game play and individual selection are decoupled. Individuals play the ingroup social dilemma with partners who are on average related to some extent. Relatedness increases the probability that cooperators end up playing together, which supports mutual cooperation. However, when individuals later compete within the group to reproduce, they compete against a different set of individuals precisely because migration occurs after game play but before individual selection. The timing of migration decouples the patterns of relatedness that hold when individuals play the social dilemma from the patterns of relatedness that hold when individuals compete to reproduce. As a result, related cooperators impose the gains from mutual cooperation as a relative advantage on others who are unrelated.

In the other case, the life cycle is birth, migration, game play, group competition when relevant, and individual selection within groups. Under this life cycle, game play and individual selection are coupled. Relatedness within groups ensures that cooperators are relatively likely to play with other cooperators. However, because migration occurs before game play, not after, cooperators who play together also end up competing against each other to reproduce. This cancels, to some extent, the degree to which relatedness supports the evolution of cooperation^24,25. In our case, this cancellation effect at the individual level does not completely offset the value of playing the social dilemma with relatives. Under both life cycles, the evolution of cooperation increases with relatedness, though the effect is weak. Playing the game with related partners thus provides some limited support for the evolution of cooperation (supporting figures 15 and 16²⁶). That said, cancellation effects at the individual level also play a role in the following precise sense. In models without group competition, the decoupled life cycle supports more cooperation than the coupled life cycle (supporting figures 15 and 16²⁶).

Importantly, in terms of the link between game play and individual selection, decoupling is a relative concept. Under the decoupled life cycle, related cooperators who play the social dilemma together might still end up competing against each other at the selection stage. This outcome is possible simply because, even when migration rates are high, some individuals remain in the natal group. Thus, two individuals who play the game together may both stay in the same group and end up competing to reproduce later. The timing of migration does not completely eliminate this possibility because not everyone migrates. Instead, the decoupled life cycle ensures that individuals who play the social dilemma together are less likely to compete against each other than they would be under the coupled life cycle.

Cancellation effects at the group level (group competition and joint scenarios)

Cancellation effects can also operate at the group level⁸, and the intuition parallels that at the individual level precisely. Imagine a competition between two groups, one group composed of cooperative individuals and the other of uncooperative individuals. The cooperative group wins and replaces the losing group with a descendant group that is also relatively cooperative. If the parent and descendant groups go on to compete with two entirely different groups in the subsequent generation, both groups are relatively likely to compete against less cooperative groups and thus win their respective competitions. This maximizes the extent to which the group-level benefits of cooperation support the evolution of cooperation via group selection. In contrast, if the parent and descendant groups go on to compete against each other, then two cooperative groups compete against each other, with neither enjoying a relative advantage. This cancels the effects of the group-level benefits that result from both groups having many cooperative individuals.

Apart from a recent and important exception⁸, multi-level selection models are like the former example. However, if ancestral human groups did not rove freely across the landscape in search of new competitions, which seems entirely plausible, ancestral conditions were at least somewhat like the latter example. To examine this distinction, we use a novel approach to manipulate cancellation effects at the group level (supporting information section 2.1.2²⁶). The 40 groups in a population constitute a population of groups. In each generation groups are paired and have a competition with positive probability. We can interpret this setting as one in which paired groups occupy adjacent territories that place the two groups in close contact. At the beginning of each generation, Ξ ∈ {0, 20, 40} groups are randomly selected to enter a pool of migrating groups that move around in space. These migrating groups are randomly redistributed to the open territories. The population of groups is well mixed when Ξ = 40. Groups move around a lot, and groups that win intergroup competitions are relatively unlikely to compete against their descendant groups in the subsequent generation. This minimizes cancellation effects at the group level. Anchoring the opposite extreme, Ξ = 0, which means groups never move. This maximizes group-level cancellation effects.

The importance of differences in aggregate resources between groups (group competition and joint scenarios)

If paired groups engage in a group competition, as explained above, the group with more resources may or may not win the competition. Specifically, the probability of winning can be more or less sensitive to the difference in total resources between the two groups. We consider four levels of sensitivity (supporting information section 2.1.5²⁶) controlled by the parameter λ ∈ {0, 10, 25, 100}. If λ = 0, which group wins is unrelated to the difference in total resources. Groups compete in this case, but outcomes are unsystematic. Therefore, group selection cannot occur, and in this sense λ = 0 is effectively like the repeated interactions scenario. As λ values increase, the group with more resources is increasingly likely to win, and the group competition and joint scenarios are increasingly different from the repeated interactions scenario.

Migration rates (all scenarios)

We vary the migration rate and by extension the relatedness within groups by allowing either 8 or 16 out of 24 individuals to migrate (m_j) per group per generation (supporting information sections 2.1.4−2.1.6 and 2.1.19²⁶).

Initial conditions (all scenarios)

In the initial generation, we seed the population with either (1) perfect reciprocators who initially transfer the full endowment, (2) unconditionally selfish individuals, or (3) individuals having random strategies drawn from a uniform distribution over the strategy space (supporting information section 2.1.8²⁶). Perfect reciprocators start by transferring the maximum possible amount, if first mover, in the first interaction. For all subsequent choices, perfect reciprocators do exactly what their partners just did. In other words, they match the most recent transfers of their partners measure for measure. Seeding the population with perfect reciprocators represents initial conditions that are favourable for the evolution of cooperation, while seeding the population with unconditionally selfish individuals represents initial conditions that are unfavourable.

Altogether, the three scenarios and six model characteristics yield 936 combinations. For each combination, we simulated 50 independent populations. In the main paper we focus on simulation results based on three-dimensional strategies. We occasionally discuss analytical results and simulation results based on two- and four-dimensional strategies. We especially do so for the repeated interactions scenario, where the dimensionality of the strategy space is decisive (supporting information section 1.2.12²⁶). The supporting information²⁶ includes additional results and analyses, including those that go beyond the core project outlined here, and we also mention these results in the main paper as appropriate.

Adding mistakes

The main paper presents results based on models that assume individuals never make mistakes. Theory based on repeated play of the standard prisoner’s dilemma suggests this may not be an innocent assumption. Without mistakes, different cooperative strategies can drift in and out of the population because the strategies in question lead to identical choices^36,52. The population eventually drifts towards some mix of cooperative strategies that is vulnerable to invasion by an uncooperative strategy, and cooperation collapses. With mistakes, however, these same cooperative strategies no longer generate identical choices. Drift accordingly plays a reduced role, and mistakes can stabilize a specific cooperative strategy from among a glut of cooperative strategies⁵².

Because of the potential importance of mistakes, we added mistakes and repeated our entire simulation study. A mistake occurs when an actual transfer deviates from the transfer stipulated by an individual’s strategy. We implemented mistakes by distributing actual transfers around the stipulated transfer (supporting information section 5²⁶). Mistakes are thus common, but they vary in magnitude. For three-dimensional and four-dimensional strategies, results remain, in effect, exactly the same. In the two-dimensional case, under repeated interactions as a stand-alone mechanism, mistakes dramatically slow down the invasion of cooperative strategies compared to otherwise identical situations without mistakes. As a result, over long but finite time scales, repeated interactions cannot support the evolution of cooperative strategies even when strategies are two-dimensional. This limitation opens the door for group competitions to interact positively with repeated interactions, which is exactly what happens (supporting information section 5.3²⁶). Mistakes thus expand the range of conditions that lead to the evolution of super-additive cooperation. Future research could vary the structure of mistakes when actions are continuous to see how robust this conclusion is.

Sequential social dilemma in Papua New Guinea

We conducted our experiment with members of Perepka and Ngenika groups, two horticultural groups in the Western Highlands of Papua New Guinea (supporting information section 6²⁶). The Western Highlands are an ideal place to evaluate evolutionary theories of human cooperation because the people who live there, relatively speaking, are beyond the reach of state institutions. Social preferences, local norms, reciprocity and group affiliation are the main forces that govern social life. These forces were probably pervasive for much of the human evolutionary past, and so they are the primary points of contention with respect to the evolution of human cooperation. By contrast, the enforceable contracts and legal institutions of contemporary large-scale societies introduce additional forces that are recent in evolutionary terms. This can confuse the interpretation of empirical findings by confounding ancestral psychologies with incentives, norms, and expectations tied to contemporary institutions.

At the time of the experiment, the Ngenika and Perepka groups inhabited territories separated by about 30 km in the Western Highlands. Although each group was aware of the other’s existence, no one had any memory of hostilities between the two groups. With adult participants, we implemented a sequential social dilemma that included both ingroup and outgroup pairings (supporting information sections 7 and 8²⁶). One author (H.B.) grew up and lived in the local area for 15 years, speaks the local language (Tok Pisin) fluently, and has a detailed knowledge of the values and cultural practices of local populations. This knowledge ensured that the experiments could be conducted in the local language and in a manner respectful of local cultures. Participants provided informed consent verbally. The Internal Review Board of the Faculty of Business, Economics and Informatics at the University of Zurich approved the study.

The players in a pair were each provided with an endowment of five Papua New Guinean Kina. This endowment was roughly half of a high daily wage for a labourer in the informal sector of the local workforce. Most participants earned less than this daily wage on average because they were not working for money on a regular basis. After receiving the endowment, the first mover in a pair transferred some amount between zero and five Kina, in increments of one Kina, to the second mover. The experimenter doubled this transfer. Before learning the amount actually transferred, the second mover specified an amount she wished to back transfer to the first mover for each of the first mover’s possible transfer levels, yielding six observations per second mover. This is the strategy method of eliciting second mover responses, and previous research has shown it to be a reliable method for measuring behavioural strategies⁵³. After eliciting the second mover’s strategy, the experimenter revealed the amount actually transferred by the first mover and implemented the appropriate back transfer. The experimenter also doubled the back transfer.

Using a between-subjects design, we implemented four treatments that differed in terms of the group affiliations of the two players. We varied affiliations in all combinations, which yielded two ingroup treatments (Ngenika/Ngenika and Perepka/Perepka) and two outgroup treatments (Ngenika/Perepka and Perepka/Ngenika). We used no statistical methods to pre-determine sample size (see Reporting Summary). All players knew the rules of the game. Each player also knew the group affiliation of her partner. The experimenter mediated all interactions in private, and so interactions were anonymous apart from information about group affiliations.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

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Tag: Social evolution

The evolution of private reputations in information-abundant landscapes

Evolution of a novel adrenal cell type that promotes parental care

Frequent disturbances enhanced the resilience of past human populations

The evolution of menopause in toothed whales

Data

Lifespan data and modelling

Reproductive lifespan and modelling

Size and maturity

Kinship demography analysis

Demographic simulations

Analysis

Live-long versus stop-early

Help and extended lifespans

Help and reproductive lifespans

Harm and reproductive lifespans

Female lifespans and male longevity

Reporting summary

Super-additive cooperation | Nature

A sequential social dilemma with a continuous action space

The three scenarios

A framework for comprehensive variation in model structure

The dimensionality of strategy space (all scenarios)

Cancellation effects at the individual level (all scenarios)

Cancellation effects at the group level (group competition and joint scenarios)

The importance of differences in aggregate resources between groups (group competition and joint scenarios)

Migration rates (all scenarios)

Initial conditions (all scenarios)

Adding mistakes

Sequential social dilemma in Papua New Guinea

Reporting summary