### Abstract

- Top of page
- Abstract
- Introduction
- Two steps required for inclusive fitness analysis
- Conclusions
- Acknowledgments
- References
- Appendices

A recent model studies the evolution of cooperation on a network, and concludes with a result connecting the benefits and costs of interactions and the number of neighbours. Here, an inclusive fitness analysis is conducted of the only case solved analytically, of a cycle, and the identical result is obtained. This brings the result within a biologically familiar framework. It is notable that the benefits and costs in the inclusive fitness framework need to be derived, and are not the benefits and costs that are the parameters in the original model. The relatedness is a quadratic function of position in a cycle of size *N*: an individual is related by 1 to itself, by (*N* − 5)/(*N* + 1) to an immediate neighbour, and by very close to −1/2 to the most distant individuals. The inclusive fitness analysis explains hitherto puzzling features of the results.

### Two steps required for inclusive fitness analysis

- Top of page
- Abstract
- Introduction
- Two steps required for inclusive fitness analysis
- Conclusions
- Acknowledgments
- References
- Appendices

Natural selection of social behaviour falls within the scope of inclusive fitness theory provided the fitness effects combine additively, as shown explicitly by Grafen (2006). Here, we pursue the special case of the cycle of length *N* of Ohtsuki *et al.* (2006), and we assume weak selection and so obtain additivity. Once a model falls within inclusive fitness theory, there are two steps that we take in turn. First, to calculate the costs and benefits of the possible actions. Second, to calculate the relatednesses. It is worth noting that although weak selection produces additivity in this case, it does not always do so, for example in the games of Ohtsuki & Nowak (2006).

The theory of inclusive fitness is a strong theory, in the sense that it dictates how its variables are to be measured. Ohtsuki *et al.* (2006) discuss benefits and costs that, adding together the effects of the cooperative acts towards both of an individual's neighbours, give a benefit of *b* to each neighbour and −2*c* to the individual itself. But inclusive fitness theory requires that the benefits and costs be measured in reproduction, namely the effect on the number of descendants (including a surviving self). Thus, we need to see how these ‘scores’ affect the chance of survival of individuals and the chance of their reproducing, and how these combine to affect the total expected number of descendants, which for the purposes of this paper we will call *fitness*: these all depend strongly on the demographic scheme. Ohtsuki *et al.* (2006) consider a Death–Birth scheme and a Birth–Death scheme which have very different results, and the explanation lies precisely in the process of converting scores into reproduction. It is helpful to agree to call the immediate neighbours ‘1-neighbours’, those two positions away ‘2-neighbours’, and so on.

Table 1 shows the benefits and costs as the effects on scores, and the effects on fitness under both demographic schemes. We now proceed to explain the biological significance of the benefits and costs. Consider the Death–Birth scheme first, in which each individual has an equal chance of being the one individual to die, and then one of the immediate neighbours of the vacated slot place an offspring in that slot, with probabilities proportional to their scores. An individual's own score is reduced by −2*c*, but the details of the pairwise competition mean that a given fitness effect has only a quarter of its effect in reproduction in a given competition (this calculation is in Appendix A). Thus, the −2*c* gives an expected advantage of −*c*/2 for the case where the right-hand neighbour dies, and a further expected advantage of −*c*/2 when the left-hand neighbour dies.

There is always exactly one dead individual to replace, so the fitness effect on self of −*c*/2 to one side must be exactly counter-balanced by the effect on the fitness of the rival in that competition to fill the 1-neighbour slot. The fitness effect on the 2-neighbour on that side is therefore +*c*/2. The *b* given directly to the score of a 1-neighbour has a fitness effect of *b*/4 to each side, by the same ‘quartering principle’. And the zero-sum property shows that the 3-neighbour must therefore lose out, with a fitness effect of −*b*/4. However, if *j* herself dies, both the competing neighbours have gained *b* for their score, and so the chances of them winning are unaltered by the altruism. These are all the fitness effects for the Death–Birth scheme, shown in Table 1, and the arguments just given show that they make perfect biological sense.

One of the unexplained aspects of the results of Ohtsuki *et al.* (2006) is the difference between the demographic schemes, so consider now the Birth–Death scheme, in which one individual is chosen at random to reproduce, with probabilities proportional to scores, and then the reproducing individual's offspring is placed in the slot of a neighbour chosen at random with equal probabilities. The impact of score in this case is very different. In contrast to the ‘quartering’ of the Death–Birth scheme, here an individual suffers the full −2*c* loss in fitness from her own score reduction, as it directly reduces her chance of being selected as the individual to reproduce. Further, she also suffers a −*b*/2 loss from each neighbour, for the simple reason that increasing your neighbour's score by *b* increases the chance she is selected to reproduce by *b*, and then with half a chance it is you who are selected to be replaced. The −*b*/2 for each neighbour gives the total effect on the individual's fitness as −*b* − 2*c*. Just as in the previous scheme, these effects have their equal and opposite reactions in the zero sum game in reproduction, leading to gains of *b* + *c* for each immediate neighbour, and losses of −*b*/2 for 2-neighbours.

The more biological approach of inclusive fitness theory therefore readily provides an explanation of the difference caused by the demographic scheme. To complete the analysis, the relatednesses to neighbours at varying distances are required. Using a method closely modelled on those of Taylor *et al*. (2007b) and Rousset (2004), it is shown in Appendix B that the relatedness on a cycle of size *N* to a neighbour at distance *k* is

- (1)

Table 1 shows these relatednesses, and the costs and benefits, itemized by neighbour. By multiplying and summing, we can calculate the inclusive fitness effect of a cooperator in the Death–Birth and Birth–Death schemes, respectively, as

- (2)

The benefits and costs do not depend on *N*, but the formula for the relatednesses shows that relatedness to close neighbours approaches one as *N* increases. For large *N*, eqn (2) gives the same answers as Ohtsuki *et al.* (2006), that cooperation spreads when *b*/2 − *c* > 0 in the Death–Birth scheme, and when *c* < 0 in the Birth–Death scheme. Ohtsuki & Nowak (2006) give exact results for the more general case of games on the cycle, whereas here we consider only cooperation so the effects of different individual's actions add up. But their exact results for *N*, under weak selection, are given in their equation (2.4), and they agree exactly with the results just given when additivity is indeed assumed. Thus, the analytic result of Grafen (2006) that the inclusive fitness approach gives exact results is confirmed here, and in addition it provides a biologically meaningful explanation of selection of cooperation on a cycle.

The biological interpretation can now be pressed further, to explain why the effect of the benefit is halved, compared with the cost, in the Death–Birth scheme – that is, why is the condition for the spread of altruism *b*/2 − *c* > 0? We will assume large *N*, so that according to eqn (1) the relatednesses of the focal individual *j* to her neighbours can be taken to be decreasing linearly from one in both directions. Whatever is given in fitness to *j* herself has an equal and opposite reaction on the 2-neighbours, and the net effect is the product of the effect on self with the two steps’ worth of relatedness difference. Whatever is given to a 1-neighbour, say *j* − 1, has to be considered in two halves. On half of the occasions when it matters, the dead individual is *j* herself, and the loser is *j* + 1. But *j* − 1 and *j* + 1 are equally related to *j*, and so the net effect on her inclusive fitness is zero. On the other half of occasions, it is *j* − 2 who dies, and *j* − 3 who suffers the balancing loss. The difference in relatedness to *j* between *j* − 1 and *j* − 3 is equal to the difference in relatedness to *j* between *j* and her 2-neighbours, owing to the linearity of relatedness. Thus, the only difference in the impact on the inclusive fitness of *j* between helping herself and helping her 1-neighbour is the fact that with the 1-neighbour, half of the occasions have a zero effect. This is the precise source of the factor of two by which *b* must be discounted in evaluating the inclusive fitness effect of *j*’s action.

Moving on to interpret the Birth–Death process, the disadvantage to *j* herself results in advantages of equal summed magnitude to her 1-neighbours. Thus, the net effect on inclusive fitness is the disadvantage to herself multiplied by one step's worth of relatedness difference. Next, we consider the effect of giving *b* to a 1-neighbour, say *j* − 1. This has an equal and opposite effect shared equally between *j* − 2 and *j* herself. But the linearity of relatedness means that the net effect of the disadvantages to *j* − 2 and *j* therefore exactly equals in magnitude the net effect of the advantage to *j* − 1, so far as *j*’s inclusive fitness is concerned, leaving a net effect of zero. This is the precise source of the irrelevance of *b* to the effect of natural selection.

These explanations for the extent to which the value of *b* affects the spread of altruism show that it does not derive from genetic or other kind of similarity or extent of dependence between the actor and her neighbour. The inclusive fitness analysis, therefore, does not support the quotation from Ohtsuki *et al.* (2006) earlier in this paper, which analogizes their result to Hamilton's rule. It is also worth noting that despite its very general rhetorical appeal, the quotation focusses on the Death–Birth demographic scheme and ignores the Birth–Death case.

It is worth remarking that the total inclusive fitness effects in eqn (2) are of order 1/*N* because only one individual of the *N* dies each generation, and this dependency is therefore an artefact of the schemes. There would be merit in presenting the results as the cumulative effect over *N* periods, to make one ‘generation’.

Finally, the biological significance of these models may be considered in the context of the explanation of their results. As presented by Ohtsuki *et al.* (2006), the factor of *b* in the Death–Birth scheme condition *b*/2 − *c* > 0 is presented as analogous to relatedness, but we have rejected this interpretation here. Further, the value of two is seen as a rather particular consequence of the details of the model, and it is clear that if *j* interacted so far as scores were concerned with her 2-neighbours in the Birth–Death scheme, *b* would again be irrelevant to selection. Thus, it is a fair conclusion that the model is too special and particular to be taken as having great biological significance in itself, but this is not to detract from the value of the model in assisting exploration of the complex subject of cooperation in geographically structured populations.

### Conclusions

- Top of page
- Abstract
- Introduction
- Two steps required for inclusive fitness analysis
- Conclusions
- Acknowledgments
- References
- Appendices

Network models have a great potential for exploring the evolution of social behaviour. Applying systematically the inclusive fitness methodology of Hamilton (1964, 1970) will lead to the most productive use of that resource. Indeed, after submission of the current paper, a breakthrough by Taylor *et al.* (2007a) has provided very general results about selection in viscous populations, which are further developed by Grafen & Archetti (unpublished work).

In the case considered here, the biological understanding of Ohtsuki *et al.*’s (2006) model is greatly enhanced by calculating the fitness effects on neighbours at varying distances, and the relatedness of neighbours at varying distances, and putting these together to calculate the inclusive fitness effect. The difference between the two demographic schemes is then easily understood.

The cycle is the only analytically considered case of Ohtsuki *et al.* (2006), who deal with the general case with approximate methods. The link of inclusive fitness to that general case, and to other results in social graph theory, is established in great generality by Lehmann *et al*. (2007). The present paper emphasizes the interpretative advantages and rigour of an inclusive fitness analysis in a special and simple case. By contrast, Lehmann *et al*. (2007) establish much more general results, through the use of inclusive fitness methodology, and show that all of the results of social theory on graphs can be derived that way. They therefore express their main results using the notation and concepts of graph theory.

Another interesting aspect is that an individual has, approximately, positive relatednesses to the closest 43% of the population, and negative relatednesses to the most distant 57%. The most positive relatedness is of course one to itself, and the most negative is very close to −1/2, for all population sizes. The structure of the population thus allows some individuals to have a considerable negative relatedness. On other hand, the premiss of the current model is that those individuals do not interact, so here there is no scope for spiteful behaviour.

The most important conclusion is that inclusive fitness theory is a very general and powerful theory, with great explanatory force. Models of social evolution should either be presented in terms of inclusive fitness or, if there are reasons for adopting an alternative approach, reconciled to it explicitly. Mathematical frameworks for easing this process are increasingly available (Taylor *et al.*, 2000; Rousset, 2004; Grafen, 2006; Lehmann & Keller, 2006), and are currently being used to establish retrospectively the consistency of published work with inclusive fitness theory (e.g. Grafen, 2007; Lehmann *et al.*, 2007).