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# The mathematics of kindness

By on 23/11/2016

Charles Darwin’s theory of evolution by natural selection is one of the most profound scientific theories to have ever been developed. However, there were several questions about evolution that Darwin himself could not answer.

An article by Wim Hordijk

Originally published on Plus Magazine on November 1, 2016 and here partially reproduced by courtesy of the Plus Magazine site and the author

Worker bees give up their chance to reproduce to serve the hive.

Charles Darwin’s theory of evolution by natural selection is one of the most profound scientific theories to have ever been developed. However, there were several questions about evolution that Darwin himself could not answer. Not that he wasn’t smart enough (in fact, his intuition often pointed in the right direction), but the answers to those questions required sophisticated mathematical insights that were not developed far enough, or even available yet, in Darwin’s time. One such problem was the evolution of altruism. In biology, altruism is defined as an organism (or individual) performing an action which is at a cost to itself, but which benefits (directly or indirectly) another individual, often without the expectation of reciprocity or compensation. Altruistic behaviour seems abundant in nature: a mother bear protecting her cubs, possibly at the risk of injury; worker bees giving up their reproductive capacity entirely, effectively reducing their fitness to zero; a bird giving out warning signals to others, thereby revealing its own presence to an approaching predator; human beings going to war to defend their country, knowing very well they might die on the battle field; and so on.

However, if evolution by natural selection is all about competition and survival of the fittest, how can altruistic behaviour (which, by definition, lowers the altruist’s fitness and increases the receiver’s fitness) ever evolve? As Darwin himself wrote: “Natural selection will never produce in a being anything injurious to itself, for natural selection acts solely by and for the good of each.” (Origin of Species, 1859, Ch. 6.) Pondering the riddle of altruism, Darwin later suggested that if natural selection would sometimes act at a level higher than the individual, then altruistic behavior, for the good of the community, could indeed evolve. This intuitive idea already reflected what is now referred to as group selection, a notion to which I will return below.

### Hamilton’s rule

As some of the above examples indicate, one particular situation in which altruistic behaviour is often observed is when it involves close family members, or kin. A mother bear cares for and protects her own cubs (but not others!), because they are closely related to her. In a bee colony, all worker bees are sisters born from the same mother (the queen bee). And even humans are generally more likely to perform “selfless acts of kindness” towards closely related family members than to complete strangers (although not always).

A bear will defend her own cubs, but not others.

The idea of kinship can be made mathematically more precise by calculating a coefficient of relationship, call it $r$, which is defined as the probability that two individuals share a common gene (technically this should be stated in terms of alleles, or gene values, but as in most other descriptions, I’ll use the term gene here, for simplicity). When you were conceived, you inherited (roughly) half of your genes from your mother, and the other half from your father. In general this is a (mostly) random process, with no particular preference for which genes are inherited from which parent. So, the coefficient of relationship between you and either one of your parents is $r=0.5$. Now, if you have a sibling (brother or sister), they also inherited half of their genes from your mother and half from your father, but they may not all be the same genes as the ones you inherited. In fact, because the process of inheritance is random, you and your sibling (on average) share only half of the half of the genes that each of you inherited from your mother (and similarly for the half you inherited from your father). So, the coefficient of relationship between you and your sibling is

$$r=0.5^2+0.5^2 = 0.5$$

In a similar way, you can calculate your coefficient of relationship with other family members. For example, for you and a (first) cousin, it is $r = 0.125$ (I’ll leave the calculation as an exercise).

So how can this notion of genetic relatedness explain the evolution of altruism? It was the evolutionary biologist Bill Hamilton, born in Egypt from New Zealand parents who then settled in England, who formulated the answer in a precise mathematical way. Hamilton argued that if the benefit ($B$) of an altruistic act, devalued by the coefficient of relationship ($r$) between the two individuals involved, is greater than the cost ($C$), then (genes for) altruistic behaviour can evolve. In mathematical terms, if

$rB>C,$

then altruism is worth it. This is now known as Hamilton’s rule.

To give a simple example, if you sacrifice your own life to save two or more siblings, then for every gene that is lost with your own death, at least one copy can be expected to be saved. After all, each of your siblings has a probability of $0.5$ to share a given gene with you. Mathematically, the coefficient of relationship is $r=0.5$ (between you and your siblings) and the cost is $C = 1$  (you). So, according to Hamilton’s rule, if the benefit is at least $B = 2$  (your siblings), you’re OK (or rather, your genes are).

A bird’s warning call benefits all other birds around.

The founders of population genetics, the trio Ronald Fisher, J. B. S. Haldane, and Sewall Wright, apparently had been intuitively aware of this general idea. Fisher had already published a table calculating genetic distances between kin in 1918, and Wright formally introduced the coefficient of relationship in 1922. For some reason, though, they never related this to the problem of altruism. However, according to legend, Haldane had first expressed the logic behind Hamilton’s rule when he announced that he was prepared to lay down his life for eight cousins or two brothers. But in the end it was Hamilton who generalised the idea and formalised it mathematically in 1964.

So, clearly, altruistic behavior is associated with kinship. Or at least it is in many cases. However, as some of the above examples indicate, it need not always be. Warning signals from one bird are received by all birds that happen to be nearby, whether they are genetically related to the altruist or not. And people going to war to defend their country don’t only fight to protect their own immediate family. Even though one could argue that in these cases there is a good chance that there will be a sufficiently large number of close kin among the receivers of the altruistic act to make it worthwhile, there seems to be something more general going on.

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