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 battlefield; and so on.

However, if evolution by natural selection is all about competition and survival of the fittest, how can altruistic behavior (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 behavior 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).

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 = 0.52 + 0.52 = 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 behavior 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).

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 generalized the idea and formalized 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.

The Price Equation

Enter George Price, an American physical chemist living in London on his savings, reading, and writing on evolutionary biology. After applying for a grant and obtaining a research position in mathematical genetics at University College London, and reading Hamilton’s 1964 papers on kin selection, Price derived an equation (in the early 1970s) that generalizes Hamilton’s rule, and provides a formal method for the hierarchical analysis of the effects of natural selection.


The Price Equation, as it is now known, consists of two terms. Let’s start with considering only the first term, also known as the covariance equation (which, incidentally, was discovered independently, and unbeknownst to Price, a few years earlier by two other researchers, Robertson (1966) and Li (1967)). Suppose we have a quantitative trait, such as human height, or the propensity for altruistic behaviour. Let’s write z for this trait — so z is a variable that takes a numerical value measuring the trait (e.g. height in centimeters). Given a population of people, each comes with his or her value for z and we write z1, z2, z3, etc, for all the different values that occur in the population, and  for the population average. Now each value of zi (for i = 1, 2, 3, …) comes with a fitness value wi (e.g. being tall might come with a high fitness value because it enables you to reach more of the fruit on a tree than a small person). Let’s again write  for the population average. 

There is a statistical quantity called the covariance between the trait values and their respective fitness values. It’s a measure of how the wand zi vary together and denoted by Cov(wi, zi(see the box above for a definition). Roughly speaking, a positive value of the covariance indicates that the zand ware related, with wincreasing when zdoes and vice versa (with a high positive value indicating a strong relationship). A negative value indicates there’s an inverse relationship between the two, with wdecreasing when zincreases and vice versa (with a low negative value indicating a strong inverse relationship). Finally, a 0 value of the covariance means that there is no relationship between the two.

The covariance equation states that the change in average trait value from one generation to the next (denoted by Δz̄) is proportional to the covariance between the trait values and their respective fitness values:

Δz̄ = 1Cov(wi, zi)

Here is a graphical illustration of the meaning of the covariance equation for selection.

If the covariance between trait values and fitness values is positive (so a higher trait value means higher fitness), the average trait value in the current population (, dashed blue line) will move upward in the next generation (z̄’, dashed red line) due to selection.

This may seem a trivial statement, but it actually represents a significant insight in at least two ways. First, it provides a formal, quantitative description of how selection works, which can be used to analyze any kind of selective process, not only in biology but also in economics or learning, for example. And second, it generalizes Hamilton’s rule by showing that what really matters is the statistical association (i.e., covariance), not just genetic relatedness (which is only one particular source of statistical association, although a very important one in biological evolution).


George Price (1922-1975)

Covariance is the proper way to think about the role of genetic relatedness in evolution. After all, selection acts directly on traits, and only indirectly on genes (through those traits the genes are responsible for). This crucial insight had been missed (or at least under-appreciated) by everyone else: Darwin himself, the founding trio of population genetics Fisher, Haldane, and Wright, and the originator of kin selection Hamilton. However, George Price was the first to realize its importance, which subsequently led Hamilton to reformulate his theory of kin selection in terms of covariance. Indeed, it can be shown mathematically that Hamilton’s rule is a specific instance (given appropriate assumptions) of the covariance equation.

In the examples above, this means that birds (or people) who interact in an altruistic way do not necessarily need to be genetically related. It could be sufficient for the altruistic individuals to belong to a clearly distinct group, such as all birds nesting in a particular patch of trees, or all people living in a particular country, which provides the required statistical association. This, then, brings us to the full Price equation and the notion of group selection.

Group selection

The full Price equation consists of the original covariance equation plus an additional term:

Δz̄ = 1Cov(wi, zi) + 1E(wiΔzi),

where wiΔzi measures the change in character values between ancestor and descendant, weighted by the fitness wi, and E(wiΔzidenotes the expected (or mean) value of wiΔzi.

One interpretation of the full equation is that it provides a natural and hierarchical decomposition of selection within and between groups. To see this, let’s assume the population consists of several groups (e.g. birds nesting in different parts of a forest) and write zfor the average trait value within a group g. The corresponding average fitness within the group is wg. We can now rewrite the equation, replacing individuals by groups:

Δz̄ = 1Cov(wg, zg) + 1E(wgΔzg),

where  and  are the corresponding averages over all groups g. Now, note that the Δzg in the expectation term (in bold in the equation above), which is the change in average trait value of a given group g, can itself be written as a full Price equation (in bold in the equation below) in terms of the individuals i that make up this group g:

\[ \Delta \bar{z}=\frac{1}{\bar{w}}Cov(w_ g,z_ g) + \frac{1}{\bar{w}}E\left(w_ g\mathbf{\left[\frac{1}{w_ g}}Cov(w_{g,i},z_{g,i}) + \frac{1}{w_ g}E(w_{g,i}\Delta z_{g,i})\right]\right). \]

This simplifies to

\[ \Delta \bar{z} = \frac{1}{\bar{w}}Cov(w_ g,z_ g)+\frac{1}{\bar{w}}E\left(Cov(w_{g,i},z_{g,i})+E(w_{g,i}\Delta z_{g,i})\right). \]

This recursive expansion of the Price equation can be repeated for yet another level of subgroups, replacing Δzg,i in the expectation term by yet another full Price equation, and so on.

However, what is important here is that this recursive expansion provides additional insight into the possible evolution of altruism. Recall that, by definition, altruistic behavior decreases an individual’s fitness, but increases its group’s fitness (relative to other groups). In other words, the covariance between an individual’s altruistic trait Δzg,i and that individual’s fitness wg,i is negative, Cov(wg,i, zg,i) < 0, but the covariance between the group average of an altruistic trait zand that group’s average fitness wis positive, Cov(wg, zg) < 0. So, according to the Price equation, altruistic traits can only evolve in those situations where the positive between-group covariance Cov(wg, zgis large enough to make up for the negative within-group covariance Cov(wg,i, zg,i). The Price equation thus provides a mathematical formulation (and verification) of Darwin’s original intuition that if natural selection acts at a level higher than the individual (in other words, group selection), then altruistic behavior can evolve (within a group, for the good of the community).

Note that in such a group selection scenario, there is reciprocity at the group level, but not necessarily directly between the same two individuals. In other words, if I do you a favor, I do not necessarily need to expect you personally to return the favor to me, as long as I can expect it from the group as a whole. Altruism based on direct reciprocity, i.e., between the same two individuals, can also be explained using game theory (see this article to find out more). Incidentally, it was also Price who introduced this idea in evolutionary biology, together with John Maynard Smith, one of the leading evolutionary biologists during the second half of the 20th century (and a former student of Haldane). Alternatively, direct reciprocity can also be viewed in a group selection setting, where the group size is just two.

What had motivated Price to develop his equation was a quest to find true selfless kindness. However, what his mathematics seemed to tell him is that altruism, as a product of evolution, in the end still serves a selfish purpose, whether it is at the level of the gene, the individual, or the group as a whole. Even within groups, altruism only evolves when there is competition between groups. Perhaps disillusioned by what may have looked to him like a failure in his quest, George Price took his own life in January of 1975. At that time probably only a handful of evolutionary biologists (including Hamilton) truly understood the significance of his equation.

Further reading

The complete and highly intriguing story of the life and work of George Price is beautifully told in the book The Price of Altruism: George Price and the Search for the Origins of Kindness by Oren Harman, professor of the history of science at Bar Ilan University, Tel Aviv, Israel. The current article is largely based on that book, together with the more technical overview of George Price’s contributions to evolutionary genetics by Steven A. Frank, professor of ecology and evolutionary biology, University of California, Irvine, CA, USA. I also thank my colleague Mike Steel, professor of mathematics and statistics at the University of Canterbury, Christchurch, New Zealand, for gifting me a copy of Harman’s book, which inspired me to learn more about the Price Equation and “the mathematics of kindness”.


This article originally appeared at Plus Magazine.

Published On: April 6, 2018

Wim Hordijk

Wim Hordijk

Wim Hordijk is an independent and interdisciplinary scientist interested in the origin and evolution of life. After spending several formative years as a graduate student at the Santa Fe Institute, and earning a PhD in computer science from the University of New Mexico, USA, Wim has worked on many research and computing projects at different universities and research institutes all over the world. A wandering scientist by choice, he has collaborated with biologists, physicists, mathematicians, computer scientists, chemists, and also an archaeologist. He currently holds a senior fellowship at the Konrad Lorenz Institute for Evolution and Cognition Research in Klosterneuburg, Austria.

Wim is also an enthusiastic popular science writer, with contributions to (among others) The Scientist, Plus magazine, and TVOL. You can follow his research and writings on Twitter (@WanderingWim) or on his personal website at www.WorldWideWanderings.net


  • It would help if you provided a worked out example. You could use a simple data set and readers could work along on a spreadsheet. But as it is I still do not know how to use the Price equation.

  • Steve Davis says:

    A very good article, but hindered somewhat by this;
    “Even within groups, altruism only evolves when there is competition between groups.”
    If the mathematics does indeed lead to this conclusion, then the math is wrong.

    • Charles Goodnight says:

      The math does lead to that conclusion, and the math is not wrong. I have run into this many times. If the math and your intuition disagree, it is almost always the case that your intuition is wrong.

      • Steve Davis says:

        Charles, I’m not basing my opinion on intuition, I’m basing it on the fact that the sentence makes no sense.
        It is a mistake to make sweeping statements about a concept as fluid as altruism, as was done here.
        There are no laws in biology, as Ernst Mayr said, and certainly no rules when it comes to behaviours.

  • Steve Davis says:

    It’s a pity that the revered figures from the history of evolution are not revered a little more.
    Just as Darwin’s conclusion that the social instincts dominate individual instincts has been ignored, even treated with contempt by those calling themselves Darwinists, so also Ernst Mayr’s rebuttal of gene-centrism in which he declared “There are no laws in biology” has also been ignored.
    And so we see would-be-biologists tying themselves in knots explaining behaviours, (that is, activities influenced by factors impossible to anticipate,) by way of mathematical formulae of mind-numbing complexity. To come up with conclusions such as this above “in such a group selection scenario, there is reciprocity at the group level, but not necessarily directly between the same two individuals” which can be worked out by 5 minutes of logical thought. And despite the complex calculations they still get it wrong as in the post above.
    Why do they get it wrong?
    Because there are no laws in biology.

  • Helga Vierich says:

    Yes, the idea that you need to have competitive intergroup situations in order to develop altruism is very puzzling.

    I think it is more likely that it is not so much competition, but rather the differential efficiency of caloric returns that make a difference to the long term overtaking of one population by another, not in a necessarily competitive or hostile way, but just in sheer numbers, that is sufficient. A tendency to share food and to defend each other – even among those who are not closely related – can more efficiently make use of any returns to calories expended in procuring food. Less is wasted, and the overall niche of the population within any local ecosystem is snug – less disruptive of plant and animal components and their interactions.

    Indeed, when such sharing and other co-nurture is extended beyond the local group, something else is gained, a kind of intergroup signalling of willingness to help. This ought not be underestimated: the climatic turmoil of climate and geomorphological events (volcanoes, earth quakes, floods, asteroids..) during the Pleistocene made mutual help – even from more distant populations – a distinct advantage to any hominin community.

    The other aspect is that for human ancestors at some point cultural behaviours, especially subsistence economic and material cultural behaviours, were ramped up i a feedback with behaviours with more biological potentiality. Just as the variation available in the genome is the raw material that natural selection draws upon to drive species through bottlenecks, so too cultural variability is the raw material of cultural evolutionary adaptation. If inventive and playful experimental genius supplies the mutations of cultural practice and insight, so too information exchange between populations supplies the vital boost of variation similar to that by which gene flow between groups prevents inbreeding depression. Cultural stagnation has likely been just as deadly at times, and both have likely contributed more to the stunning loses of species and subspecies with the genus Homo than any “competition”.

    I suspect that, if anything, the more open-hearted, hospitable, and sexually adventurous representatives of Homo not only became our main ancestral line, but it was the best fo them, individually and collectively, that also rescued and embraced every other “archaic” groups they encountered as they expanded their range – over and over again, not just in Africa and across the whole world.

  • Steve Davis says:

    Helga, thank you for a lengthy and thoughtful comment.
    I emphasise “thoughtful” because just a little thoughtful reflection is enough to demolish just about all of the gene-centrism that flowed from Hamilton’s work, and with no need for mathematical crutches.
    I agree with all your points.

  • Wim Hordijk says:

    Why the hostility against mathematics? Let me remind you of some wonderful quotes from two of the great pioneers in evolutionary biology mentioned in this article 😉

    First, one by J.B.S. Haldane: “An ounce of algebra is worth a ton of verbal argument.”

    And then one by Darwin himself: “I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.”

    George Price had a deep understanding of the process of selection, not only in evolution, but in general. That understanding unfortunately still seems to be under-appreciated.

    • Steve Davis says:

      Hi Wim,
      A mathematical approach is only as good as the assumptions on which it is based.
      Numerous biologists have made careers out of inclusive fitness, but the calculations are based on an assumption for which no evidence is given; the kinship requirement for altruism.
      There is no kinship requirement; there is a proximity requirement.

  • JoseAngel says:

    E. O. Wilson used to be a major advocate of inclusive fitness, following Hamilton. However in recent years he has changed his mind and has written in favour of a more extended understanding of group selection. Most specifically, and mathematically, here:
    Allen, B., M. A. Nowak, and E. O. Wilson. “Limitations of Inclusive Fitness.” Proceedings of the National Academy of Sciences of the United States of America 110 (2013): 20135–20139.
    —Would this refutation of inclusive fitness and kin selection extend to the mathematical argument of George Price’s theory? What do you think of Wilson’s strictures against the mathematics of inclusive fitness?

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