## Abstract

Reconciling the evolution of altruism with Darwinian natural selection is frequently presented as a fundamental problem in biology. In addition to an exponentially increasing literature on specific mechanisms that can permit altruism to evolve, there has been a recent trend to establish general principles to explain altruism in populations undergoing natural selection. This paper reviews and extends one approach to understanding the ultimate causes underlying the evolution of altruism and mechanisms that can realise them, based on the Price equation. From the Price equation, we can see that such ultimate causes equate to the different ways in which the frequency of an altruistic allele in a population can increase. Under this approach, the ultimate causes underlying the evolution of altruism, given some positive fitness costs and benefits, are positive assortment of altruistic alleles with the altruistic behaviour of others, positive deviations from additive fitness effects when multiple altruists interact or bias in the inheritance of altruistic traits. In some cases, one cause can be interpreted in terms of another. The ultimate causes thus identified can be realised by a number of different mechanisms, and to demonstrate its general applicability, I use the Price equation approach to analyse a number of classical mechanisms known to support the evolution of altruism (or cooperation): repeated interaction, ‘greenbeard’ traits, games played on graphs and payoff synergism. I also briefly comment on other important points for the evolution of altruism, such as the ongoing debate over the predominant status of inclusive fitness as the best way to understand its evolution. I conclude by arguing that analysing the evolution of altruism in terms of its ultimate causes is the logical way to approach the problem and that, despite some of its technical limitations, the Price equation approach is a particularly powerful way of doing so.

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## Notes

Although by the 6th edition (p. 86) ‘each [...] profits’ had changed to ‘the community [...] profits’; see Sober (2009).

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## Acknowledgements

I thank S. Okasha for discussions on causal decomposition and comments on Eq. 5, H. Ohtsuki for clarification of his results concerning the evolution of altruism on graphs, R. Boyd for discussions on his model of cultural evolution in groups, K. Foster and two anonymous reviewers for comments on the manuscript.

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## Additional information

Communicated by Guest Editor J. McNamara

This contribution is part of the Special Issue ‘Mathematical Models in Ecology and Evolution: Darwin 200’ (see Marshall et al. 2010).

## Appendices

### Appendix 1: Covariances for repeated interaction model

The expectations we need to calculate our required covariances are:

and

These expectations give the following covariances:

and

### Appendix 2: Expectations for Greenbeard model

The required expectations for the greenbeard model can be calculated as:

and

### Appendix 3: Expectations for graph model

To calculate the required expectations for the ‘death-birth’ model of altruism on graphs, we begin by calculating the expected genotype of the actors competing for reproduction, and the unconditioned expected genotype of their neighbours,

To calculate the joint expectation E(*GG*′) required to determine Cov(*G*,*G*′) it would seem we should be able to directly apply Eq. 16, however this would be incorrect, as we must take account of the fact that all competitors interacted with a common neighbour in determining their payoff, the site being replaced. This focal site is an altruist with probability *g*, otherwise it is a defector. Thus, we modify Eq. 16 to calculate the joint expectation of *G* and *G*′ as

where the first term is the expected genotype of the site being replaced, and the second term is the conditional expectation of Eq. 16, weighted by the remaining proportion of neighbours (*k* − 1)/*k*. The variance and covariance presented in the main text follow directly from the above results.

### Appendix 4: Analysis of synergism model

The expectations required for the synergism model are:

and

As mentioned in the main text, it is of interest to briefly note that frequency-dependent social dilemmas such as the stag hunt are often referred to as having ‘risk dominant’ and ‘payoff dominant’ equilibria (Harsanyi and Selten 1988). These refinements of the Nash equilibrium concept arose in classical game theory. In evolutionary game theory, under the replicator dynamics for example, equilibrium selection is frequency-dependent, and a ‘risk dominant’ equilibrium can be defined as that having the greatest basin of attraction (Samuelson 1998), but the covariance approach naturally captures uncertainty and thus might provide a different and interesting perspective. For example a partial simplification from Eq. 20 to Eq. 24 could be written

where *G* is the random variable corresponding to the population frequency of stag hunters, *p*. Thus, the risk-dominant stag hunting equilibrium can be thought of as being favoured when the variance (i.e. the uncertainty (Zidek and van Eeden 2003)) in the synergistic benefit from both individuals cooperating, exceeds the magnitude of the cost of altruism multiplied by the expected frequency of non-cooperators.

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Marshall, J.A.R. Ultimate causes and the evolution of altruism .
*Behav Ecol Sociobiol* **65, **503–512 (2011). https://doi.org/10.1007/s00265-010-1110-1

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DOI: https://doi.org/10.1007/s00265-010-1110-1

### Keywords

- Assortment
- Synergy
- Transmission bias
- Evolution
- Altruism
- Cooperation
- Price equation
- Hamilton’s rule