How to make an inclusive-fitness model

Social behaviours are typically modelled using neighbour-modulated fitness, which focuses on individuals having their fitness altered by neighbours. However, these models are either interpreted using inclusive fitness, which focuses on individuals altering the fitness of neighbours, or not interpreted at all. This disconnect leads to interpretational mistakes and obscures the adaptive significance of behaviour. We bridge this gap by presenting a systematic methodology for constructing inclusive-fitness models. We find a behaviour's 'inclusive-fitness effect' by summing primary and secondary deviations in reproductive value. Primary deviations are the immediate result of a social interaction; for example, the cost and benefit of an altruistic act. Secondary deviations are compensatory effects that arise because the total reproductive value of the population is fixed; for example, the increased competition that follows an altruistic act. Compared to neighbour-modulated fitness methodologies, our approach is often simpler and reveals the model's inclusive-fitness narrative clearly. We implement our methodology first in a homogeneous population, with supplementary examples of help under synergy, help in a viscous population and Creel's paradox. We then implement our methodology in a class-structured population, where the advantages of our approach are most evident, with supplementary examples of altruism between age classes, and sex-ratio evolution.

Kin discrimination, negative relatedness, and how to distinguish between selfishness and spite

Spiteful behaviors occur when an actor harms its own fitness to inflict harm on the fitness of others. Several papers have predicted that spite can be favored in sufficiently small populations, even when the harming behavior is directed indiscriminately at others. However, it is not clear that truly spiteful behavior could be favored without the harm being directed at a subset of social partners with relatively low genetic similarity to the actor (kin discrimination, causing a negative relatedness between actor and harmed recipient). Using mathematical models, we show that (1) the evolution of spite requires kin discrimination; (2) previous models suggesting indiscriminate spite involve scenarios where the actor gains a direct feedback benefit from harming others, and so the harming is selfish rather than spiteful; (3) extreme selfishness can be favored in small populations (or, more generally, under local competition) because this is where the direct feedback benefit of harming is greatest.

Structural symmetry in evolutionary games

In evolutionary game theory, an important measure of a mutant trait (strategy) is its ability to invade and take over an otherwise-monomorphic population. Typically, one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population. However, in a structured population, this fixation probability may depend on where the mutant arises. Moreover, the fixation probability is just one quantity by which one can measure the success of a mutant; fixation time, for instance, is another. We define a notion of homogeneity for evolutionary games that captures what it means for two single-mutant states, i.e. two configurations of a single mutant in an otherwise-monomorphic population, to be 'evolutionarily equivalent' in the sense that all measures of evolutionary success are the same for both configurations. Using asymmetric games, we argue that the term 'homogeneous' should apply to the evolutionary process as a whole rather than to just the population structure. For evolutionary matrix games in graph-structured populations, we give precise conditions under which the resulting process is homogeneous. Finally, we show that asymmetric matrix games can be reduced to symmetric games if the population structure possesses a sufficient degree of symmetry.

Collective action problem in heterogeneous groups

I review the theoretical and experimental literature on the collective action problem in groups whose members differ in various characteristics affecting individual costs, benefits and preferences in collective actions. I focus on evolutionary models that predict how individual efforts and fitnesses, group efforts and the amount of produced collective goods depend on the group's size and heterogeneity, as well as on the benefit and cost functions and parameters. I consider collective actions that aim to overcome the challenges from nature or win competition with neighbouring groups of co-specifics. I show that the largest contributors towards production of collective goods will typically be group members with the highest stake in it or for whom the effort is least costly, or those who have the largest capability or initial endowment. Under some conditions, such group members end up with smaller net pay-offs than the rest of the group. That is, they effectively behave as altruists. With weak nonlinearity in benefit and cost functions, the group effort typically decreases with group size and increases with within-group heterogeneity. With strong nonlinearity in benefit and cost functions, these patterns are reversed. I discuss the implications of theoretical results for animal behaviour, human origins and psychology.

Structure coefficients and strategy selection in multiplayer games

Evolutionary processes based on two-player games such as the Prisoner's Dilemma or Snowdrift Game are abundant in evolutionary game theory. These processes, including those based on games with more than two strategies, have been studied extensively under the assumption that selection is weak. However, games involving more than two players have not received the same level of attention. To address this issue, and to relate two-player games to multiplayer games, we introduce a notion of reducibility for multiplayer games that captures what it means to break down a multiplayer game into a sequence of interactions with fewer players. We discuss the role of reducibility in structured populations, and we give examples of games that are irreducible in any population structure. Since the known conditions for strategy selection, otherwise known as [Formula: see text]-rules, have been established only for two-player games with multiple strategies and for multiplayer games with two strategies, we extend these rules to multiplayer games with many strategies to account for irreducible games that cannot be reduced to those simpler types of games. In particular, we show that the number of structure coefficients required for a symmetric game with [Formula: see text]-player interactions and [Formula: see text] strategies grows in [Formula: see text] like [Formula: see text]. Our results also cover a type of ecologically asymmetric game based on payoff values that are derived not only from the strategies of the players, but also from their spatial positions within the population.

Evolution of global cooperation driven by risks

Globalization facilitates our communication with each other, while it magnifies problems such as overharvesting of natural resources and human-induced climate change. Thus people all over the world are involved in a global social dilemma which calls for worldwide cooperation to reduce the risks of these extreme events and disasters. A collective target (threshold) is required to prevent such events. Everyone may lose their wealth once their total individual contributions fail to reach the threshold. To this end, we establish a model of threshold public goods games in a group-structured population and investigate its evolutionary process. We study multilevel public goods games with defectors, local cooperators, and global cooperators and are primarily concerned with how the global cooperative behavior evolves. We find that, compared with the standard public goods games, the strategy of global cooperation accounts for a bigger proportion in the stationary distribution of threshold public goods games. On the other hand, the fixation time of the global cooperation strategy is greatly shortened with increase of the probability of disaster striking. Therefore, global risks induced by the threshold can effectively promote global cooperation in environmental investment and the reduction of greenhouse gas emissions.

Evolutionary games in deme structured, finite populations

We describe a fairly general model for the evolutionary dynamics in a sub-divided (or deme structured) population with migration and mutation. The number and size of demes are finite and fixed. The fitness of each individual is determined by pairwise interactions with other members of the same deme. The dynamics within demes can be modeled according to a broad range of evolutionary processes. With a probability proportional to fitness, individuals migrate to another deme. Mutations occur randomly. In the limit of few migrations and even rarer mutations we derive a simple analytic condition for selection to favor one strategic type over another. In particular, we show that the Pareto efficient type is favored when competition within demes is sufficiently weak. We then apply the general results to the prisoner's dilemma game and discuss selected dynamics and the conditions for cooperation to prevail.

Diffusion approximations for one-locus multi-allele kin selection, mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size.

Evolution of altruism in stepping-stone populations with overlapping generations

We study the evolution of altruism in one- and two-dimensional stepping-stone populations with discrete overlapping generations. We find that increasing survival probability facilitates the evolution of altruism, in agreement with recent results for a patch-structured population. We allow the altruistic behaviour to affect either fecundity or survival probability. In the first case, altruism is favoured compared to a randomly interacting population, but in the second case, altruism is less likely to evolve. We consider the iterated prisoner's dilemma as a description of an altruistic interaction and compare our results with recent simulations of lattice populations.