Strategy evolution on higher-order networks

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

A geometric approach to the evolution of altruism

Fisher's geometric model provides a powerful tool for making predictions about key properties of Darwinian adaptation. Here, I apply the geometric model to predict differences between the evolution of altruistic versus nonsocial phenotypes. I recover Kimura's prediction that probability of fixation is greater for mutations of intermediate size, but I find that the effect size that maximises probability of fixation is relatively small in the context of altruism and relatively large in the context of nonsocial phenotypes, and that the overall probability of fixation is lower for altruism and is higher for nonsocial phenotypes. Accordingly, the first selective substitution is expected to be smaller, and to take longer, in the context of the evolution of altruism. These results strengthen the justification for employing streamlined social evolutionary methodologies that assume adaptations are underpinned by many genes of small effect.

Evaluating the structure-coefficient theorem of evolutionary game theory

In order to accommodate the empirical fact that population structures are rarely simple, modern studies of evolutionary dynamics allow for complicated and highly heterogeneous spatial structures. As a result, one of the most difficult obstacles lies in making analytical deductions, either qualitative or quantitative, about the long-term outcomes of evolution. The "structure-coefficient" theorem is a well-known approach to this problem for mutation-selection processes under weak selection, but a general method of evaluating the terms it comprises is lacking. Here, we provide such a method for populations of fixed (but arbitrary) size and structure, using easily interpretable demographic measures. This method encompasses a large family of evolutionary update mechanisms and extends the theorem to allow for asymmetric contests to provide a better understanding of the mutation-selection balance under more realistic circumstances. We apply the method to study social goods produced and distributed among individuals in spatially heterogeneous populations, where asymmetric interactions emerge naturally and the outcome of selection varies dramatically, depending on the nature of the social good, the spatial topology, and the frequency with which mutations arise.

Conformity and content-biased cultural transmission in the evolution of altruism

The evolution of altruism has been extensively modeled under the assumption of genetic transmission, whereas the dynamics under cultural transmission are less well understood. Previous research has shown that cultural transmission can facilitate the evolution of altruism by increasing 1) the probability of adopting the altruistic phenotype, and 2) assortment between altruists. We incorporate vertical and oblique transmission, which can be conformist or anti-conformist, into models of parental care, sibling altruism, and altruism between individuals that meet assortatively. If oblique transmission is conformist, it becomes easier for altruism to invade a population of non-altruists as the probability of vertical transmission increases. If oblique transmission is anti-conformist, decreasing vertical transmission facilitates invasion by altruism in the assortative meeting model, whereas in other models, there is a trade-off: greater vertical transmission produces greater assortment among genetically related altruists, but lowers the probability of adopting altruism via anti-conformity. Compared to conditions for invasion under genetic transmission, e.g., Hamilton's rule, we show that invasion can be easier with sufficiently strong anti-conformity, and in some models, with sufficiently high assortment even if oblique transmission is conformist. We also explore invasion by an allele A that increases individuals' content bias for altruism, in the absence of other forms of cultural transmission. If costs and benefits combine additively, A invades under previously known conditions. If costs and benefits combine multiplicatively, invasion by A and by altruism become more difficult than in the corresponding additive models.

Properties of network structures, structure coefficients, and benefit-to-cost ratios

In structured populations the spatial arrangement of cooperators and defectors on the interaction graph together with the structure of the graph itself determines the game dynamics and particularly whether or not fixation of cooperation (or defection) is favored. For networks described by regular graphs and for a single cooperator (and a single defector) the question of fixation can be addressed by a single parameter, the structure coefficient. This quantity is invariant with respect to the location of the cooperator on the graph and also does not vary over different networks. We may therefore consider it to be generic for regular graphs and call it the generic structure coefficient. For two and more cooperators (or several defectors) fixation properties can also be assigned by structure coefficients. These structure coefficients, however, depend on the arrangement of cooperators and defectors which we may interpret as a configuration of the game. Moreover, the coefficients are specific for a given interaction network modeled as a regular graph, which is why we may call them specific structure coefficients. In this paper, we study how specific structure coefficients vary over interaction graphs and analyze how spectral properties of interaction networks relate to specific structure coefficients. We also discuss implications for the benefit-to-cost ratios of donation games.

Fixation properties of multiple cooperator configurations on regular graphs

Whether or not cooperation is favored in evolutionary games on graphs depends on the population structure and spatial properties of the interaction network. The population structure can be expressed as configurations. Such configurations extend scenarios with a single cooperator among defectors to any number of cooperators and any arrangement of cooperators and defectors on the network. For interaction networks modeled as regular graphs and for weak selection, the emergence of cooperation can be assessed by structure coefficients, which can be specified for each configuration and each regular graph. Thus, as a single cooperator can be interpreted as a lone mutant, the configuration-based structure coefficients also describe fixation properties of multiple mutants. We analyze the structure coefficients and particularly show that under certain conditions, the coefficients strongly correlate to the average shortest path length between cooperators on the evolutionary graph. Thus, for multiple cooperators fixation properties on regular evolutionary graphs can be linked to cooperator path lengths.

Antisocial rewarding in structured populations

Cooperation in collective action dilemmas usually breaks down in the absence of additional incentive mechanisms. This tragedy can be escaped if cooperators have the possibility to invest in reward funds that are shared exclusively among cooperators (prosocial rewarding). Yet, the presence of defectors who do not contribute to the public good but do reward themselves (antisocial rewarding) deters cooperation in the absence of additional countermeasures. A recent simulation study suggests that spatial structure is sufficient to prevent antisocial rewarding from deterring cooperation. Here we reinvestigate this issue assuming mixed strategies and weak selection on a game-theoretic model of social interactions, which we also validate using individual-based simulations. We show that increasing reward funds facilitates the maintenance of prosocial rewarding but prevents its invasion, and that spatial structure can sometimes select against the evolution of prosocial rewarding. Our results suggest that, even in spatially structured populations, additional mechanisms are required to prevent antisocial rewarding from deterring cooperation in public goods dilemmas.

Fixation Probabilities for Any Configuration of Two Strategies on Regular Graphs

Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of rare mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.

Models of social evolution: can we do better to predict 'who helps whom to achieve what'?

Models of social evolution and the evolution of helping have been classified in numerous ways. Two categorical differences have, however, escaped attention in the field. Models tend not to justify why they use a particular assumption structure about who helps whom: a large number of authors model peer-to-peer cooperation of essentially identical individuals, probably for reasons of mathematical convenience; others are inspired by particular cooperatively breeding species, and tend to assume unidirectional help where subordinates help a dominant breed more efficiently. Choices regarding what the help achieves (i.e. which life-history trait of the helped individual is improved) are similarly made without much comment: fecundity benefits are much more commonly modelled than survival enhancements, despite evidence that these may interact when the helped individual can perform life-history reallocations (load-lightening and related phenomena). We review our current theoretical understanding of effects revealed when explicitly asking 'who helps whom to achieve what', from models of mutual aid in partnerships to the very few models that explicitly contrast the strength of selection to help enhance another individual's fecundity or survival. As a result of idiosyncratic modelling choices in contemporary literature, including the varying degree to which demographic consequences are made explicit, there is surprisingly little agreement on what types of help are predicted to evolve most easily. We outline promising future directions to fill this gap.

Evolutionary Stability of Jointly Evolving Traits in Subdivided Populations

The evolutionary stability of quantitative traits depends on whether a population can resist invasion by any mutant. While uninvadability is well understood in well-mixed populations, it is much less so in subdivided populations when multiple traits evolve jointly. Here, we investigate whether a spatially subdivided population at a monomorphic equilibrium for multiple traits can withstand invasion by any mutant or is subject to diversifying selection. Our model also explores the correlations among traits arising from diversifying selection and how they depend on relatedness due to limited dispersal. We find that selection tends to favor a positive (negative) correlation between two traits when the selective effects of one trait on relatedness is positively (negatively) correlated to the indirect fitness effects of the other trait. We study the evolution of traits for which this matters: dispersal that decreases relatedness and helping that has positive indirect fitness effects. We find that when dispersal cost is low and the benefits of helping accelerate faster than its costs, selection leads to the coexistence of mobile defectors and sessile helpers. Otherwise, the population evolves to a monomorphic state with intermediate helping and dispersal. Overall, our results highlight the effects of population subdivision for evolutionary stability and correlations among traits.

When is bigger better? The effects of group size on the evolution of helping behaviours

Understanding the evolution of sociality in humans and other species requires understanding how selection on social behaviour varies with group size. However, the effects of group size are frequently obscured in the theoretical literature, which often makes assumptions that are at odds with empirical findings. In particular, mechanisms are suggested as supporting large-scale cooperation when they would in fact rapidly become ineffective with increasing group size. Here we review the literature on the evolution of helping behaviours (cooperation and altruism), and frame it using a simple synthetic model that allows us to delineate how the three main components of the selection pressure on helping must vary with increasing group size. The first component is the marginal benefit of helping to group members, which determines both direct fitness benefits to the actor and indirect fitness benefits to recipients. While this is often assumed to be independent of group size, marginal benefits are in practice likely to be maximal at intermediate group sizes for many types of collective action problems, and will eventually become very small in large groups due to the law of decreasing marginal returns. The second component is the response of social partners on the past play of an actor, which underlies conditional behaviour under repeated social interactions. We argue that under realistic conditions on the transmission of information in a population, this response on past play decreases rapidly with increasing group size so that reciprocity alone (whether direct, indirect, or generalised) cannot sustain cooperation in very large groups. The final component is the relatedness between actor and recipient, which, according to the rules of inheritance, again decreases rapidly with increasing group size. These results explain why helping behaviours in very large social groups are limited to cases where the number of reproducing individuals is small, as in social insects, or where there are social institutions that can promote (possibly through sanctioning) large-scale cooperation, as in human societies. Finally, we discuss how individually devised institutions can foster the transition from small-scale to large-scale cooperative groups in human evolution.

Does evolution lead to maximizing behavior?

A long-standing question in biology and economics is whether individual organisms evolve to behave as if they were striving to maximize some goal function. We here formalize this "as if" question in a patch-structured population in which individuals obtain material payoffs from (perhaps very complex multimove) social interactions. These material payoffs determine personal fitness and, ultimately, invasion fitness. We ask whether individuals in uninvadable population states will appear to be maximizing conventional goal functions (with population-structure coefficients exogenous to the individual's behavior), when what is really being maximized is invasion fitness at the genetic level. We reach two broad conclusions. First, no simple and general individual-centered goal function emerges from the analysis. This stems from the fact that invasion fitness is a gene-centered multigenerational measure of evolutionary success. Second, when selection is weak, all multigenerational effects of selection can be summarized in a neutral type-distribution quantifying identity-by-descent between individuals within patches. Individuals then behave as if they were striving to maximize a weighted sum of material payoffs (own and others). At an uninvadable state it is as if individuals would freely choose their actions and play a Nash equilibrium of a game with a goal function that combines self-interest (own material payoff), group interest (group material payoff if everyone does the same), and local rivalry (material payoff differences).

Evolutionary dynamics of fairness on graphs with migration

Individual migration plays a crucial role in evolutionary dynamics of population on networks. In this paper, we generalize the networked ultimatum game by diluting population structures as well as endowing individuals with migration ability, and investigate evolutionary dynamics of fairness on graphs with migration in the ultimatum game. We first revisit the impact of node degree on the evolution of fairness. Interestingly, numerical simulations reveal that there exists an optimal value of node degree resulting in the maximal offer level of populations. Then we explore the effects of dilution and migration on the evolution of fairness, and find that both the dilution of population structures and the endowment of migration ability to individuals would lead to the drop of offer level, while the rise of acceptance level of populations. Notably, natural selection even favors the evolution of self-incompatible strategies, when either vacancy rate or migration rate exceeds a critical threshold. To confirm our simulation results, we also propose an analytical method to study the evolutionary dynamics of fairness on graphs with migration. This method can be applied to explore any games governed by pairwise interactions in finite populations.

Social evolution and genetic interactions in the short and long term

The evolution of social traits remains one of the most fascinating and feisty topics in evolutionary biology even after half a century of theoretical research. W.D. Hamilton shaped much of the field initially with his 1964 papers that laid out the foundation for understanding the effect of genetic relatedness on the evolution of social behavior. Early theoretical investigations revealed two critical assumptions required for Hamilton's rule to hold in dynamical models: weak selection and additive genetic interactions. However, only recently have analytical approaches from population genetics and evolutionary game theory developed sufficiently so that social evolution can be studied under the joint action of selection, mutation, and genetic drift. We review how these approaches suggest two timescales for evolution under weak mutation: (i) a short-term timescale where evolution occurs between a finite set of alleles, and (ii) a long-term timescale where a continuum of alleles are possible and populations evolve continuously from one monomorphic trait to another. We show how Hamilton's rule emerges from the short-term analysis under additivity and how non-additive genetic interactions can be accounted for more generally. This short-term approach reproduces, synthesizes, and generalizes many previous results including the one-third law from evolutionary game theory and risk dominance from economic game theory. Using the long-term approach, we illustrate how trait evolution can be described with a diffusion equation that is a stochastic analogue of the canonical equation of adaptive dynamics. Peaks in the stationary distribution of the diffusion capture classic notions of convergence stability from evolutionary game theory and generally depend on the additive genetic interactions inherent in Hamilton's rule. Surprisingly, the peaks of the long-term stationary distribution can predict the effects of simple kinds of non-additive interactions. Additionally, the peaks capture both weak and strong effects of social payoffs in a manner difficult to replicate with the short-term approach. Together, the results from the short and long-term approaches suggest both how Hamilton's insight may be robust in unexpected ways and how current analytical approaches can expand our understanding of social evolution far beyond Hamilton's original work.