A generalization of Pontryagin's maximum principle for dynamic evolutionary games among relatives

We present two theorems that generalize Pontryagin's maximum principle to the setting of dynamic evolutionary games between genetically related individuals. The two theorems correspond to two types of interactions among individuals: patch-structured populations in which individuals locally "play the field" and pairwise interactions. These generalizations can be used in the same way that Pontryagin's maximum principle is used and they are valid for diploid organisms under a single locus, diallelic genetic model. These generalizations involve an interesting, dynamic version of Hamilton's Rule from inclusive fitness theory. We illustrate how these theoretical results can be applied by modeling the evolution of lifetime resource allocation to growth and reproduction in an annual plant when there is competition for resources among related individuals.     

Comparative statics of games between relatives

According to Hamilton's theory of kin selection, species tend to evolve behavior such that each organism appears to be attempting to maximize its inclusive fitness. In particular, two neighbors are likely to help each other if the cost of doing so is less than the benefit multiplied by r, their coefficient of relatedness. Since the latter is less than unity, mutual altruism benefits both neighbors. However, is it theoretically possible that acting so as to maximize the inclusive, rather than personal, fitness may harm both parties. This may occur in strategic symmetric pairwise interactions (more specifically, nxn games), in which the outcome depends on both sides' actions. In this case, the equilibrium outcome may be less favorable to the interactants' personal fitness than if each of them acted so as to maximize the latter. This paper shows, however, that such negative effect of relatedness on fitness is incompatible with evolutionary stability. If the symmetric equilibrium strategies are evolutionarily stable, a higher coefficient of relatedness can only entail higher personal fitness for the two neighbors. This suggests that negative comparative statics as above are not likely to occur in nature.     

The donation game with roles played between relatives

We consider a social game with two choices, played between two relatives, where roles are assigned to individuals so that the interaction is asymmetric. Behaviour in each of the two roles is determined by a separate genetic locus. Such asymmetric interactions between relatives, in which individuals occupy different behavioural contexts, may occur in nature, for example between adult parents and juvenile offspring. The social game considered is known to be equivalent to a donation game with non-additive payoffs, and has previously been analysed for the single locus case, both for discrete and continuous strategy traits. We present an inclusive fitness analysis of the discrete trait game with roles and recover equilibrium conditions including fixation of selfish or altruistic behaviour under both behavioural contexts, or fixation of selfish behaviour under one context and altruistic behaviour under the other context. These equilibrium solutions assume that the payoff matrices under each behavioural context are identical. The equilibria possible do depend crucially, however, on the deviation from payoff additivity that occurs when both interacting individuals act altruistically.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

The adaptive dynamics of altruism in spatially heterogeneous populations

Abstract.— We study the spatial adaptive dynamics of a continuous trait that measures individual investment in altruism. Our study is based on an ecological model of a spatially heterogeneous population from which we derive an appropriate measure of fitness. The analysis of this fitness measure uncovers three different selective processes controlling the evolution of altruism: the direct physiological cost, the indirect genetic benefits of cooperative interactions, and the indirect genetic costs of competition for space. In our model, habitat structure and a continuous life cycle makes the cost of competing for space with relatives negligible. Our study yields a classification of adaptive patterns of altruism according to the shape of the costs of altruism (with decelerating, linear, or accelerating dependence on the investment in altruism). The invasion of altruism occurs readily in species with accelerating costs, but large mutations are critical for altruism to evolve in selfish species with decelerating costs. Strict selfishness is maintained by natural selection only under very restricted conditions. In species with rapidly accelerating costs, adaptation leads to an evolutionarily stable rate of investment in altruism that decreases smoothly with the level of mobility. A rather different adaptive pattern emerges in species with slowly accelerating costs: high altruism evolves at low mobility, whereas a quasi-selfish state is promoted in more mobile species. The high adaptive level of altruism can be predicted solely from habitat connectedness and physiological parameters that characterize the pattern of cost. We also show that environmental changes that cause increased mobility in those highly altruistic species can beget selection-driven self-extinction, which may contribute to the rarity of social species.     

The evolution of altruism: Correlation,cost, and benefit

A simple and general criterion is derived for the evolution of altruism when individuals interact in pairs. It is argued that the treatment of this problem in kin selection theory and in game theory are special cases of this general criterion.My thanks to James Crow, Carter Denniston, Lee Dugarkin, David Wilson, and an anonymous referee of this journal for helpful discussion.     

Crystal toxins and the volunteer's dilemma in bacteria

The growth and virulence of the bacteria Bacillus thuringiensis depend on the production of Cry toxins, which are used to perforate the gut of its host. Successful invasion of the host relies on producing a threshold amount of toxin, after which there is no benefit from producing more toxin. Consequently, the production of Cry toxin appears to be a different type of social problem compared with the public goods scenarios that bacteria usually encounter. We show that selection for toxin production is a volunteer's dilemma. We make specific predictions that (a) selection for toxin production depends upon an interplay between the number of bacterial cells that each host ingests and the genetic relatedness between those cells; (b) cheats that do not produce toxin gain an advantage when at low frequencies, and at high bacterial density, allowing them to be maintained in a population alongside toxin‐producing cells. More generally, our results emphasize the diversity of the social games that bacteria play.     

Bonds of bros and brothers: Kinship and social bonding in postdispersal male macaques

Group‐living animals often maintain a few very close affiliative relationships—social bonds—that can buffer them against many of the inevitable costs of gregariousness. Kinship plays a central role in the development of such social bonds. The bulk of research on kin biases in sociality has focused on philopatric females, who typically live in deeply kin‐structured systems, with matrilineal dominance rank inheritance and life‐long familiarity between kin. Closely related males, in contrast, are usually not close in rank or familiar, which offers the opportunity to test the importance of kinship per se in the formation of social bonds. So far, however, kin biases in male social bonding have only been tested in philopatric males, where familiarity remains a confounding factor. Here, we studied bonds between male Assamese macaques, a species in which males disperse from their natal groups and in which male bonds are known to affect fitness. Combining extensive behavioural data on 43 adult males over a 10‐year period with DNA microsatellite relatedness analyses, we find that postdispersal males form stronger relationships with the few close kin available in the group than with the average nonkin. However, males form the majority of their bonds with nonkin and may choose nonkin over available close kin to bond with. Our results show that kinship facilitates bond formation, but is not a prerequisite for it, which suggests that strong bonds are not restricted to kin in male mammals and that animals cooperate for both direct and indirect fitness benefits.     

Evolutionary dynamics of biological auctions

Many scenarios in the living world, where individual organisms compete for winning positions (or resources), have properties of auctions. Here we study the evolution of bids in biological auctions. For each auction, n individuals are drawn at random from a population of size N. Each individual makes a bid which entails a cost. The winner obtains a benefit of a certain value. Costs and benefits are translated into reproductive success (fitness). Therefore, successful bidding strategies spread in the population. We compare two types of auctions. In “biological all-pay auctions”, the costs are the bid for every participating individual. In “biological second price all-pay auctions”, the cost for everyone other than the winner is the bid, but the cost for the winner is the second highest bid. Second price all-pay auctions are generalizations of the “war of attrition” introduced by Maynard Smith. We study evolutionary dynamics in both types of auctions. We calculate pairwise invasion plots and evolutionarily stable distributions over the continuous strategy space. We find that the average bid in second price all-pay auctions is higher than in all-pay auctions, but the average cost for the winner is similar in both auctions. In both cases, the average bid is a declining function of the number of participants, n. The more individuals participate in an auction the smaller is the chance of winning, and thus expensive bids must be avoided.     

The concurrent evolution of cooperation and the population structures that support it

The evolution of cooperation often depends upon population structure, yet nearly all models of cooperation implicitly assume that this structure remains static. This is a simplifying assumption, because most organisms possess genetic traits that affect their population structure to some degree. These traits, such as a group size preference, affect the relatedness of interacting individuals and hence the opportunity for kin or group selection. We argue that models that do not explicitly consider their evolution cannot provide a satisfactory account of the origin of cooperation, because they cannot explain how the prerequisite population structures arise. Here, we consider the concurrent evolution of genetic traits that affect population structure, with those that affect social behavior. We show that not only does population structure drive social evolution, as in previous models, but that the opportunity for cooperation can in turn drive the creation of population structures that support it. This occurs through the generation of linkage disequilibrium between socio-behavioral and population-structuring traits, such that direct kin selection on social behavior creates indirect selection pressure on population structure. We illustrate our argument with a model of the concurrent evolution of group size preference and social behavior.     

Stochastic evolutionary dynamics of bimatrix games

Evolutionary game dynamics of two-player asymmetric games in finite populations is studied. We consider two roles in the game, roles α and β. α-players and β-players interact and gain payoffs. The game is described by a pair of matrices, which is called bimatrix. One's payoff in the game is interpreted as its fecundity, thus strategies are subject to natural selection. In addition, strategies can randomly mutate to others. We formulate a stochastic evolutionary game dynamics of bimatrix games as a frequency-dependent Moran process with mutation. We analytically derive the stationary distribution of strategies under weak selection. Our result provides a criterion for equilibrium selection in general bimatrix games.     

To what extent does living in a group mean living with kin?

Chimpanzees live in large groups featuring remarkable levels of gregariousness and cooperation among the males. Because males stay in their natal communities their entire lives and are hence expected to be living with male relatives, cooperation is therefore assumed to occur within one large 'family' group. However, we found that the average relatedness among males within several chimpanzee groups as determined by microsatellite analysis is in fact rather low, and only rarely significantly higher than average relatedness of females in the groups or of males compared across groups. To explain these findings, mathematical predictions for average relatedness according to group size, reproductive skew and sex bias in dispersal were derived. The results show that high average relatedness among the philopatric sex is only expected in very small groups, which is confirmed by a comparison with published data. Our study therefore suggests that interactions among larger number of individuals may not be primarily driven by kin relationships.     

Evolutionary dynamics of collective action in N-person stag hunt dilemmas

In the animal world, collective action to shelter, protect and nourish requires the cooperation of group members. Among humans, many situations require the cooperation of more than two individuals simultaneously. Most of the relevant literature has focused on an extreme case, the N-person Prisoner's Dilemma. Here we introduce a model in which a threshold less than the total group is required to produce benefits, with increasing participation leading to increasing productivity. This model constitutes a generalization of the two-person stag hunt game to an N-person game. Both finite and infinite population models are studied. In infinite populations this leads to a rich dynamics that admits multiple equilibria. Scenarios of defector dominance, pure coordination or coexistence may arise simultaneously. On the other hand, whenever one takes into account that populations are finite and when their size is of the same order of magnitude as the group size, the evolutionary dynamics is profoundly affected: it may ultimately invert the direction of natural selection, compared with the infinite population limit.     

Nepotism vs Tit-For-Tat,or, why should you be nice to your rotten brother?

Summary It is well known that interactions among relatives facilitate the evolution of altruistic behaviours. Game theoretic models show, however, that guarded altruism (such as Tit-For-Tat) can evolve among non-relatives when individuals interact many times and cheating behaviours can be punished. Strangely, no one has yet asked whether the guarded altruism that evolves among non-relatives might also evolve among close relatives, supplanting unconditional altriusm. We present a series of one-locus sexual haploid models in which Tit-For-Tat, unconditional altruists and selfish individuals interact in groups of full siblings. Tit-For-Tat frequently (but not always) replaced unguarded altruism, in which case the strategic rules for interacting with kin vs non-kin are identical. Even when Tit-For-Tat is selected at a single locus, however, by withholding altruism for non-reciprocating relatives it may qualify as an outlaw from the standpoint of modifier genes at other loci.     

Evolutionary stable strategies and game dynamics for n-person games

This note contains a generalization of the definition of an evolutionary stable strategy and of the corresponding game dynamics from 2-person to n-person games. This broader framework also allows modelling of several interacting populations or of populations containing different types of individuals, for example males and females.     

Task specialization in two social spiders,Stegodyphus sarasinorum (Eresidae) and Anelosimus eximius (Theridiidae)

Understanding the social organization of group‐living organisms is crucial for the comprehension of the underlying selective mechanisms involved in the evolution of cooperation. Division of labour and caste formation is restricted to eusocial organisms, but behavioural asymmetries and reproductive skew is common in other group‐living animals. Permanently, social spiders form highly related groups with reproductive skew and communal brood care. We investigated task differentiation in nonreproductive tasks in two permanently and independently derived social spider species asking the following questions: Do individual spiders vary consistently in their propensity to engage in prey attack? Are individual spiders' propensities to engage in web maintenance behaviour influenced by their previous engagement in prey attack? Interestingly, we found that both species showed some degree of task specialization, but in distinctly different ways: Stegodyphus sarasinorum showed behavioural asymmetries at the individual level, that is, individual spiders that had attacked prey once were more likely to attack prey again, independent of their body size or hunger level. In contrast, Anelosimus eximius showed no individual specialization, but showed differentiation according to instar, where adult and subadult females were more likely to engage in prey attack than were juveniles. We found no evidence for division of labour between prey attack and web maintenance. Different solutions to achieve task differentiation in prey attack for the two species studied here suggest an adaptive value of task specialization in foraging for social spiders.     

Evolutionary dynamics of zero-sum games

Aim model in terms of differential equations is used to explain mammalian ovulation control, in particular regulation for a prescribed number of mature eggs. NIH Grant RO1 GM 32153-01GE     

SPATIAL DISTRIBUTION OF A PRIMITIVELY SOCIAL BEE: DOES GENETIC POPULATION STRUCTURE FACILITATE ALTRUISM?

Exoneura bicolor is a univoltine, facultatively social bee exhibiting a solitary/quasisocial/semisocial colony polymorphism (Schwarz, 1986, 1987). Intracolony relatedness in semisocial colonies has been previously estimated at 0.49 ± 0.06 (Schwarz, 1987), although the crucial relatedness between altruists and the brood that they rear will be about half this value. This value is unlikely to be increased by the preferential rearing of only close relatives (Schwarz, 1988a) and no known morphological specializations preclude workers from reproducing in this species. Hamilton (1972, 1975) suggested that relatedness may be increased through population subdivision, if this leads to significant inbreeding and increased between-colony genetic variance. The same process may also operate at higher levels of population structure (e.g., Wade, 1978). Population structure and intracolony relatedness in E. bicolor were investigated in seven localities in southern Victoria, Australia, to determine if inbreeding at any level of population structure was contributing to relatedness between altruists and beneficiaries within these colonies. Population structure was described using hierarchical F-statistics and an identity by descent measure, developed by Queller and Goodnight (1989), was used to estimate intracolony relatedness. It was found that inbreeding was not contributing to between-group genetic variance, at any level, in a consistent manner across localities. Therefore relatedness, considered in isolation, does not seem sufficient to account for the presence of worker behavior. It is suggested that large benefits for group living may be responsible for maintaining altruistic behavior, in part, in this species. Significant heterogeneity among localities for all F-statistics estimated in our analysis was found and this may be attributable to stochastic elements such as cofounding behavior and the low percentage of males in the brood. The possible consequences of such heterogeneity in population structure for the maintenance of altruism in E. bicolor are discussed.     

The evolution of cooperation and altruism--a general framework and a classification of models

One of the enduring puzzles in biology and the social sciences is the origin and persistence of intraspecific cooperation and altruism in humans and other species. Hundreds of theoretical models have been proposed and there is much confusion about the relationship between these models. To clarify the situation, we developed a synthetic conceptual framework that delineates the conditions necessary for the evolution of altruism and cooperation. We show that at least one of the four following conditions needs to be fulfilled: direct benefits to the focal individual performing a cooperative act; direct or indirect information allowing a better than random guess about whether a given individual will behave cooperatively in repeated reciprocal interactions; preferential interactions between related individuals; and genetic correlation between genes coding for altruism and phenotypic traits that can be identified. When one or more of these conditions are met, altruism or cooperation can evolve if the cost-to-benefit ratio of altruistic and cooperative acts is greater than a threshold value. The cost-to-benefit ratio can be altered by coercion, punishment and policing which therefore act as mechanisms facilitating the evolution of altruism and cooperation. All the models proposed so far are explicitly or implicitly built on these general principles, allowing us to classify them into four general categories.