A recent model shows that altruism can evolve with limited migration and variable group sizes, and the authors claim that kin selection cannot provide a sufficient explanation of their results. It is demonstrated, using a recent reformulation of Hamilton's original arguments, that the model falls squarely within the scope of inclusive fitness theory, which furthermore shows how to calculate inclusive fitness and the relevant relatedness. A distinction is drawn between inclusive fitness, which is a method of analysing social behaviour; and kin selection, a process that operates through genetic similarity brought about by common ancestry, but not by assortation by genotype or by direct assessment of genetic similarity. The recent model is analysed, and it turns out that kin selection provides a sufficient explanation to considerable quantitative accuracy, contrary to the authors' claims. A parallel analysis is possible and would be illuminating for all models of social behaviour in which individuals' effects on each other's offspring numbers combine additively.
The evolution of altruism remains an active topic in biological research, but the power and the scope of inclusive fitness theory (Hamilton 1964, 1970) remain largely unappreciated. Killingback et al. (2006) recently claim to have demonstrated a new mechanism for evolving altruism that cannot be fully accounted for by kin selection. Here, we put this claim to the test using inclusive fitness theory, developing an approach that could equally be taken for many other papers.
The test is conducted using a new formulation of inclusive fitness theory recently proposed by Grafen (2006), which fulfils Hamilton's original intentions by combining the best features of his 1964 and 1970 derivations, eliminating minor flaws, incorporating uncertainty, permitting arbitrary ploidies and arbitrary genetic architecture, and being fully explicit about optimization. Furthermore, Hamilton's original generality of interactional structure is retained; individuals may engage in different numbers of social interactions, and the interactions may all be different in nature. Most subsequent rederivations of inclusive fitness have insisted on just one kind of interaction that individuals all engage in the same number of times.
All of the inclusive fitness results used in the present paper are from Grafen (2006): the proof that the model falls within the inclusive fitness framework, the expression for inclusive fitness, the formula for relatedness and the result that inclusive fitness is maximized by selection. Other important bodies of work on inclusive fitness are by Frank (1998; see also Taylor & Frank 1996), Taylor and co-workers (e.g. Taylor 1990, 1996; Taylor et al. 2000) and Rousset and co-workers (Rousset 2004). The generality, directness and the immediate applicability of Grafen (2006) are bought at the cost of saying nothing about the link between relatedness and common ancestry, which is a central element of many other papers. The assumption of additivity is also relaxed in some of them.
An important distinction made by Hamilton (1975) is emphasized here. Inclusive fitness theory is very general and applies to genetic similarity, however caused, whether by common ancestry, assortation of genotypes or kin recognition. As Grafen (2006) shows, provided social interactions combine additively in determining offspring numbers, there is (almost always) a relatedness that can be calculated such that the direction of gene frequency change is determined by inclusive fitness. However, it is useful to reserve the term ‘kin selection’ for situations in which the relatedness arises through common ancestry. (The situation in which the relatedness cannot be defined occurs when a quantity called the ‘Hamilton residual’ cannot be rendered zero by choice of relatedness, and is discussed in detail in section 3.3 of Grafen (2006). It arises when the actors are genetically representative of the population, so that the gene frequency being studied has an exactly zero correlation with the altruistic phenotype.)
Section 2 describes the model of Killingback et al. (2006), and shows how it can be represented within the framework of Grafen (2006), and thus derives a formula for the relatedness with which inclusive fitness is maximized. The analysis of that section could be repeated with few changes for many other models. Section 3 moves on to the particularities of the current model. It defines the model more precisely, gives details of the demographic properties of the model and calculates the relatedness that would arise within a group through common ancestry alone. This relatedness allows predictions to be made about selection, and a computer calculation of the model (detailed in Appendix A) allows Section 4 to put those predictions to the test.
2. The varying group size model of Killingback et al. (2006)
Killingback et al. (2006) study a ‘public goods’ game in a grouped asexual population. Each individual j plays a value xj, which is directly a cost. xj is constrained to lie between 0 and V. Each individual receives, as a benefit, the average value of xi in the group, multiplied by an ‘interest rate’ k. Let Γj be the set of individuals in the group to which j belongs, and nj be the number in that group. We can write the net pay-off in two equivalent ways:(2.1)V is the baseline fitness in the absence of social interactions. The two forms suggest two equivalent ways forward. The upper version includes j in the recipients of her own action, as part of the group, and counts the ‘cost’ as the full xj; the lower version nets off the benefit to herself from her cost. We follow the first course for simplicity; it will affect the details of the analysis, but the outcome is obviously the same.
Killingback et al. (2006) further assume that individuals are haploid, asexual reproduction occurs proportionally to fitness, each individual has an independent chance of dispersal with probability d so that dispersers join another group at random and then population-wide mortality reduces the mean group size to the prescribed value m. Groups will thus vary in size. The total population size was fixed at 500.
The authors found that with a mean group size of 5, altruism was broadly selected when d was less than approximately 0.125 for k=2 and when d was less than approximately 0.27 for k=3. The obvious explanation is that limited dispersal (low d) brings about intra-group relatedness, which favours altruism to group members. The authors reject this explanation as insufficient. They state that in another unpublished study, they found that a different game did not evolve altruism in the same demographic circumstances. However, this is a weak argument as there may well be important differences between the two games (it would be premature to investigate these before both models have undergone peer scrutiny as part of the process of publication). They also propose that there might be a new mechanism for evolving altruism, and suggest it could be based on a combination of the fact that at very small group sizes higher x is individually advantageous, that small group sizes arise in the model, and Simpson's paradox (Blyth 1972). This proposal is rather vague: as we shall see, it is also unnecessary.
Grafen's (2006) new formulation of inclusive fitness assumes additivity of fitness interactions by representing the number of successful gametes of an individual j asbijt is the effect of individual i on the number of successful gametes of individual j, when i is acting in ‘role’ t in relation to j. Role is a crucial concept employed by Hamilton (1964) and revived more formally by Grafen (2006). A role will typically delineate a situation in which a decision needs to be made, for example, ‘parent towards offspring it is feeding’ or ‘first of two strangers to meet at a food resource towards the second’. The significance of roles is that the relatedness is defined not for individuals, but for each role. There is a special role e for ego, in which an individual affects her own number of successful gametes through non-social action.
We now show that the model of Killingback et al. (2006) falls within the scope of inclusive fitness theory by showing how to define the bijt to establish an equivalent formula for offspring number. This is simply done using equation (2.1). Let g be the role in which an individual plays the public goods game with her group members. Here are the two different ways to represent the model in inclusive fitness terms(2.2)We will pursue the analysis with the upper version. The key assumption here is just the additivity of fitness interactions. The definitions of the bijt are needed to implement the formulae of Grafen (2006) for inclusive fitness and relatedness. Non-additive situations can be handled by, for example, Taylor & Frank (1996).
This expression in terms of bijt is the crucial step: it follows from section 3.1 of Grafen (2006) that gene frequency change will favour types of individual with higher inclusive fitness. The formula for the inclusive fitness of individual i is(2.3)where the relatedness r is to all group members (including self), and we drop the superscript t because there is only one role in this simple example. The mean value of x will therefore increase if k is high enough, and specifically if rk−1>0. The critical value of k, at which the mean of x remains unaltered, is(2.4)The group size ni cancels out because each individual gives the same amount to the group as a whole, whatever its size.
But what is the value of r? The notation rIF is introduced to mean the relatedness that makes inclusive fitness work. Equation (2.3) of Grafen (2006) shows that in general(2.5)(The formula fails when the denominator equals 0—see Grafen (2006), section 3.3.) Note this depends on a particular p-score, which is either a gene frequency or a weighted linear sum of gene frequencies. The evolution of a p-score depends on inclusive fitness calculated with its value of rIF. rIF also depends on all the demographic details and the current state of the population. Thus, in line with the very general conditions for inclusive fitness, there is a great deal of complexity here, reflecting the complexity in the original model. It may be worth pointing out that if we inserted an extra dispersal stage so that the groups had the same size distribution but individuals were allocated to groups at random, it is clear that rIF=0 and so in that case inclusive fitness would equal V−xi and selection would reduce x as far as it could.
In an asexual one-locus model, the useful p-score to study is x itself. Thus, we let pj=xj, and substitute using equation (2.2), and define as the average value of x in the group to which j belongs. It is easy to show that , which could be expected from the kin selection approach of Taylor & Frank (1996), but note that the weighting by bijt has automatically ensured the appropriate weighting of different-sized groups. We further consider the situation in which there are only two genotypes with population frequencies 1−γ and γ, playing distinct values x1 and x2, and let a1 be the expected frequency of type 1 in the group of a randomly chosen individual of type 1, and mutatis mutandis for a2. It is easy to show that(2.6)This form shows that all the necessary demographic details for the polymorphic model (as defined in Appendix A) are encapsulated in a1 and a2. In particular, the effects of both the mean group size and the variability in group size are included in them.
The values of a1 and a2 will be affected by all factors influencing genetic similarity of group members. If only common ancestry is important, then we would say that kin selection explained the results. Section 3 derives an expression for the value of r that measures the genetic similarity among group members that would be expected from common ancestry alone, ranc, which is called the ancestral relatedness.
Whether rIF=ranc will be answered here by direct calculations for this particular model. The same question is considered analytically in general contexts by, for example, Wild & Taylor (2004) and Rousset (2004), though it is usually phrased as whether identity-by-descent suffices to calculate relatedness. So far as I am aware, none of the existing general results apply directly and unequivocally when there are groups of different sizes. There is a general sense that in the absence of genetic discrimination, indeed rIF=ranc. Future work will surely provide an analytic result that encompasses the present model.
It is worth noting that the analysis so far could be replicated without difficulty for any model that can be brought within the scope of inclusive fitness theory by establishing additivity of fitness interactions. From now on, the analysis becomes more particular.
3. The extent of common ancestry
In order to investigate the extent of common ancestry in groups, we need to define the model more precisely. Killingback et al. (2006) fixed their population size at 500, and obtained the number of offspring of the individuals in the parental generation from a multinomial distribution with probabilities proportional to fitnesses. Here an infinite population is assumed to ease analysis, and the appropriate limiting case of the multinomial distribution is assumed: each individual has a Poisson number of offspring with a mean proportional to its fitness, scaled so that the mean of the individual Poisson means equals 1. Parents do not survive to the next generation. Contrariwise, the original model can be understood as making the Poisson assumption, but then conditioning on the total number of offspring being 500. The size of the population is not implicated in the authors' arguments about the selective pressures at work, so it is fair to study the infinite population model to investigate their conclusions.
Two models are defined in Appendix A, which implement the assumptions above in a conceptually straightforward way. A monomorphic model with just one value of x present is used to calculate ancestral relatednesses, and a polymorphic model with two genotypes each with its own value of x is used to calculate the direction of selection at different values of the ‘interest rate’ k.
The individuals in a group whose ancestry within the group goes back to the same immigrant are called a clone, and the distribution of clone size depends on the number of generations since the immigrant's arrival. Calculations in Appendix B reveal that after G generations, the distribution of clone size in the monomorphic model haswhile the distribution of the size of a whole group has
If we pick an individual at random, she is likely to be in a larger than average clone and larger than average group. The ‘experienced clone size’ hasand the ‘experienced group size’ has
Now we turn to calculating the ancestral relatedness. We use the definition of relatedness in equation (2.5), and apply it to a p-score that indicates belonging to a very rare clone as distributed across the population. Hence, we assume that there has only ever been one arrival of that clone in any one group, and that the clone makes only a zero fraction of the population as a whole, so that p=0. We set pi=1 for clone members and zero for others, and use equation (2.2) to obtainWe can cancel kx and note that . If we let ci be the number of individuals in the group that are a member of the clone, then . This yields(3.1)
Appendix C finds an analytic expression for this quantity in the monomorphic model, which allows it to be calculated. ranc will be the same for all loci, but may differ from rIF owing to assortation or genetic discrimination at the x locus. If selection follows inclusive fitness with this relatedness, that is, if rIF=ranc, then only kin selection is at work in the model.
Selection was measured in the polymorphic model of appendix A at kanc=1/ranc, and at 1% above and 1% below kanc. The theory tells us that selection proceeds according to kcrit=1/rIF. If rIF=ranc then kcrit=kanc, and selection should be neutral at kanc, favour higher x at 1% above, and lower x at 1% below.
The analytic values of ranc from equation (3.1) were derived in Appendix C and are shown in figure 1 for the parameter values used by Killingback et al. (2006), along with the relatednesses rIF from equation (2.5) measured in the polymorphic model. Clearly, the relatedness is very high at low dispersal rates, and comes down to approximately 0.2, which is the inverse of the mean group size and reflects the fraction of the group the individual itself comprises. The virtual equality of ranc and rIF shows that common ancestry does explain the selection in the model to considerable numerical accuracy.
Figure 2 has further results from the polymorphic model. It shows how closely the ancestral relatedness predicts the direction of selection. To pursue the discrepancies would involve numerical analysis and questions of machine accuracy. There is no clear biological issue causing the minute discrepancies from the effects of common ancestry.
It is of interest to consider fig. 3 of Killingback et al. (2006) in relation to our figure 1. With k=2, the current model would predict selection to increase contributions when d is less than approximately 0.125, and to decrease it for higher d. This is fully consistent with their fig. 3(a). For k=3, theory predicts the break point to lie between 0.25 and 0.3. This is again fully consistent. The most significant discrepancy is that many of the points in their fig. 3 are intermediate between x=0 and 5, and this is probably due to their mutational scheme which will tend to push away from boundaries, relatively more strongly where selection is weak. The theory matches their findings.
The anomalous behaviour of d=0 in both parts of fig. 3 of Killingback et al. (2006) is simply explained. In the absence of any migration, their population of 500 individuals would find itself eventually all in one group for the rest of time; within any one closed group, contributions must be selected downwards.
The unavoidable conclusion is that no force other than common ancestry is required to explain the detailed quantitative pattern of selection when the public goods game is played in varying-sized groups, as modelled by Killingback et al. (2006). It is wholly unsurprising that kin selection plays some role, as it is inevitable that kinship ties build up in groups in which siblings are likely to be present together. Such ties and genetic similarity have been one of the primary objects of study in population genetics from its early days, as embodied in the F-statistics of Wright (1969). The power of the current analysis is to be able to show decisively that kin selection is the only quantitatively significant force at work. This confirms by calculation in the present case what is clearly suggested in general by the analytical work of Rousset (2004) and Wild & Taylor (2004): common ancestry is the only cause of genetic similarity in the absence of assortation or genetic discrimination.
On the particular case studied here, the previous section shows that altruism in the model of Killingback et al. (2006) is precisely explained by the building up of ties of common ancestry between group members. The authors themselves specifically assert that their mechanism is ‘clearly quite distinct from kin selection’, but the results of the previous section show the contrary. If variable group size has an effect, then it does so through affecting the extent of common ancestry among the members of a group. Whether that effect exists would naturally be investigated using models with different degrees of variability of group size. The paper's title is ‘Evolution in group-structured populations can resolve the tragedy of the commons’: this turns out to be true to the extent that limited migration increases the strength of common ancestry within groups, a point already made by Hamilton (1964, 1970, 1975).
Lehmann & Keller (2006) have appealed for the results from repeated games to be interpreted in terms of Hamilton's inclusive fitness, and present a kind of meta-model to make that task easier. The work of Grafen (2006) allows any model of social behaviour with discrete non-overlapping generations and additive fitness effects to be interpreted in terms of inclusive fitness. One advantage is conceptual clarity of biological interpretation, and another is the value for a field of having a single central theory to which everything can be referred. Developments following Wild & Taylor (2004) and Rousset (2004) are increasing the range of conditions under which we can be sure that the only cause of genetic similarity is common ancestry. It is therefore good practice to place models of social behaviour in the context of Hamilton's inclusive fitness theory, and recent theoretical advances make that increasingly straightforward in a widening set of circumstances.
Dr David Stirzaker gave timely advice on probability generating functions, and Alexis Gallagher, Mark Rendel, Dr Francisco Úbeda de Torres and Dr Marco Archetti made helpful comments on an earlier manuscript. I am particularly grateful to Prof. Peter Taylor and two anonymous referees, whose comments improved enormously the clarity and organization of the paper.