The economics of altruism and cooperation in class-structured populations: what's in a cost? What's in a benefit?

The simplicity of Hamilton's rule, as it is stated in line (1), certainly understates the tremendous scope of Hamilton's idea. It turns out that Hamilton's rule (1) can be modified so that it applies to the evolution of altruism in a wide variety of situations (Taylor, 1988, 1990). In fact, some important aspects of this variety may have been neglected by the direct fitness model presented by L&K. As we argue below, understanding the costs-to-benefit ratio in general terms can show us that the scope for the promotion of altruism (especially when altruism is mediated by coercion, punishment, or policing) might be different than that suggested by the authors. Many interesting examples of kin selection come from class-structured populations. Class structure can occur whenever it is possible to group individuals according to some shared quality. Familiar examples of class structure include sex structure (e.g. class 1 = female, class 2 = male) and age structure (e.g. class 1 = juvenile, class 2 = adult). Class structure can also arise when populations experience inbreeding depression (Denver & Taylor, 1995), or when populations occur in spatially heterogeneous environments (Leturque & Rousset, 2002). While the authors have presented some examples of class structure in their supplementary material, their work does not address class-structured models in general. Our aim was to do so in a relatively straightforward fashion. The main challenge in constructing a class-structured kin-selection model comes from the fact that individuals have different components of fitness (e.g. fitness through sons, fitness through daughters; survival, fecundity). In general, a class-structured model requires knowledge of wij, the expected number of class-i offspring produced by an individual belonging to class j (weighted by genetic contribution). We can think of wij as a function of the behaviour exhibited by one or more individuals. A deviant level of behaviour, x will change a number of different wij expressions. To determine whether a particular (positive) behavioural deviation is favoured in some population we can use either a direct fitness argument (Taylor & Frank, 1996) or an inclusive fitness argument (Taylor, 1988, 1990). In either case, recent work suggests that we should get the same mathematical expression (Taylor & Frank, 1996, P.D. Taylor, G. Wild & A. Gardner unpublished; cf. Frank, 1997). Consider a relatively simple situation in which all actors belong to the same class (e.g. adult females). In a direct fitness model, we fix attention on a particular recipient (the focal individual, FI to use the terminology of L&K), and consider how each of several actors influences the fitness of that recipient. We express the fitness of a focal recipient chosen from class j as the sum,, where the coefficients vi denote the reproductive value of class-i components of recipient fitness, thought of as the asymptotic genetic contribution of class-i individuals to a population in the distant future. Such a notion requires, of course, some assumption about the future trajectory of the population; and for this purpose we suppose that we have a pure monomorphic ecologically stable resident population. Thus reproductive value depends only on the resident behaviour, call it x*. Of course, if a mutant behaviour, x, is introduced, this assumption will no longer be valid, but for behavioural deviations of small effect (weak selection) the approximation will be reasonable (more precisely it will hold to first-order in mutant deviation x − x*). Reproductive value acts as an ‘exchange rate’ that allows us to add the wij’s and express recipient fitness as a scalar quantity. The function ΔW(x*) is often called the ‘inclusive fitness effect’, and can be derived either using a direct fitness approach (as we have done above), or using a classical inclusive fitness approach (e.g. Taylor, 1988, 1990). As long as selection is weak, the sign of ΔW(x*) tells us whether selection favours an increase or decrease in the level of behaviour. If ΔW(x*) > 0, then x* is selected to increase over time, if ΔW(x*) < 0, then x* is selected to decrease over time. When ΔW(x*) = 0 the population is said to be at ‘evolutionary equilibrium’. To establish a connection between eqns (1) and (3) we will change our notation slightly. If the derivative dwij/dxk|xk = x* > 0, we will place the triple (i, j, k) into a set called B, and we will refer to the derivative itself as a ‘benefit’ of deviant behaviour, Bijk. If the same derivative is negative we label it a ‘cost’, −Cijk and place (i, j, k) into a set called C. The quantities Bijk and Cijk might represent fecundity benefits and fecundity costs, respectively. That is to say, Bijk and Cijk could be the same ‘benefits’ and ‘costs’ described by the parameters B and C used by L&K (in fact, fecundity costs and benefits are the only kind considered by the class-structured examples presented in the supplementary material). However, Bijk and Cijk might also be used to represent other kinds of benefits or costs paid through other components of fitness (e.g. survival). It is also useful to point out that Bijk and Cijk can also describe the fitness consequences of some of the other factors considered by L&K, like repeated interactions (e.g. Irwin & Taylor, 2001). or simply ΔW(x*) = b−c, where b and c correspond to the first and second terms on the right hand side of eqn (4), respectively. Note that, in general, we expect that each term of the summand in b and c will depend on x*. Note also that our choice of notation b and c is motivated by the notation introduced by L & K (e.g. their eqn 4), but our usage differs in that our benefits and costs incorporate relatedness. Our expression (4) may well serve as a more general, and (conceptually) more straightforward framework in which to discuss the evolution of altruism. L&K concentrate their analysis on ways in which altruism between relatives can be promoted by direct manipulation of benefits Bijk and costs Cijk (e.g. via punishment, coercion). Inequality (5) provides an analogue to the ratio C/B, highlighted by L&K, but the mathematical form of eqn (4) suggests a number of alternative pathways through which the evolution of altruism might be promoted. The economics of the decision to ‘help’ or ‘not help’ a relative might also be influenced by a change in reproductive values, vi, a change in class frequencies uj, and even through a change in relatedness coefficients Rk→j. The authors have established a set of biological conditions necessary for the evolution of altruism, in particular, if altruism or cooperation has been observed then one of their four conditions must have been met. The factors they identify would presumably appear as terms in our eqn (4) and their conditions for the increase of altruistic behaviour would correspond to our condition (5). The strength of our formulation is that it points to pathways other than through direct manipulation of costs and benefits. This observation might be especially important for examples of punishment, coercion and policing among relatives. Class structure seems to be a key feature of model systems where punishment occurs (Clutton-Brock & Parker, 1995). First, it is possible that ‘punishers’ (e.g. dominant individuals) are less common than ‘punishees’ (e.g. subordinates). Secondly, being a ‘punisher’ rather than a ‘punishee’ may necessarily influence your genetic contribution to future generations (i.e. your reproductive value). Lastly, it is also possible that punishment/coercion has complicated consequences for the fitness of individuals other than those directly involved in the interaction (e.g. through reduced or increased local competition). These concerns would be naturally addressed by our general class-structured approach (4). As models of punishment, coercion and policing are still in their infancy (L&K), it seems doubly important to point out that other factors, like reproductive value and class frequency, deserve consideration alongside C and B.

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