Hamilton’s Rule: Calculating Kin Altruism

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What is Hamilton’s Rule? How does Hamilton’s Rule help quantify kin altruism?

Hamilton’s Rule is a mathematical representation of kin altruism. It compares the relatedness and benefit of helping to the cost of doing so.

Keep reading for more about kin altruism and Hamilton’s Rule.

Kin Altruism and Hamilton’s Rule: Background

Now we’ll explore how and why animals show altruistic behaviors, especially toward their family members. Genes don’t just act to protect the individual they’re inside of. We know this because animals frequently help their offspring and relatives, even at risk to themselves. This makes sense if we consider a “selfish gene” to be all copies of that gene across the entire population. Then we can assume that individuals will act to protect other individuals who are likely to have the same gene.

Before considering the quantified version of kin altruism, represented by Hamilton’s Rule, we can consider the perceptions and psychology behind these behaviors. The perceived likelihood of relatedness could come from a physical characteristic that the gene gives. For example, if there were a gene for purple hair, you might expect one purple-haired person to act altruistically toward another. However, the odds of a single gene providing both “purple hair” and “altruism toward purple hair” characteristics are astronomical—remember that genes aren’t conscious, and can’t choose what effects they have on their hosts. Granted, it’s possible that those two genes might tend to be inherited together, as they provide a mutually beneficial environment. 

However, a much easier way for genes to “recognize” themselves in others is through family relations. By definition, close family members will share a lot of the same genes. While it would be very unlikely for a gene to code both for a trait and for altruism toward that same trait, a gene that codes for altruism toward relatives would be successful all on its own. 

Therefore, it should be expected that animals will be altruistic to their family—the more closely related they are, the more altruistic they’ll be. Assuming sexual reproduction, an animal’s offspring will share half of its genes. A sibling will, on average, also share half of the same genes. 

Therefore, from a purely logical standpoint, an animal should consider its child or sibling to be worth half as much as its own life, in terms of preserving genes. That value will go down with more distant relatives. For example, a grandchild will only have 1/4 of the animal’s genes, so should only be valued at 1/4 of the creature’s life. You could calculate the “relatedness” of any animal to another by creating a family tree, and counting how many branches you have to move up and down the tree to get from one to the other. 

If g is the number of branches traveled, the relatedness would be (1/2)g

That equation assumes only one common ancestor. If there are more, multiply the result by the number of common ancestors in the most recent generation that has them.

Recognizing Kin

Naturally, the previous section raises the question of how animals recognize their relatives—or even whether they do. It’s possible that some animals recognize each other simply by appearance, and act altruistically toward animals that resemble them. Others might act altruistically toward any member of their species that happens to be nearby. In that case, we could assume that the odds of any given member of the species being a relative are good enough to make the altruism worth the risk. These behaviors arose due to common situations in nature, and they can misfire—or be made to misfire, like when farmers make nesting hens sit on eggs that aren’t their own.

An interesting evolutionary arms race can be seen between certain species of songbirds and cuckoos who lay eggs in their nests, trying to trigger just such a misfire. Cuckoos try to trick the songbirds into hatching their eggs by laying them in the nest, which takes advantage of the songbirds’ natural altruism toward eggs and small birds in their own nests.

Those songbirds who recognize the imposters and throw them out will naturally contribute more to the next generation’s gene pool, because they aren’t wasting resources raising cuckoos. Therefore, the next generation will be better at recognizing the cuckoo eggs. On the other hand, those cuckoos whose eggs more closely resemble songbird eggs will have their offspring survive and contribute their genes to the next generation. This is a perfect example of selfishness (from the cuckoos) and altruism (from the songbirds) and the struggle between them.

It’s likely that, in addition to the relatedness value, an animal will also have to consider how certain it is of that relationship. The songbirds from the previous example must be quite certain that the cuckoos are actually their offspring—of course, in that case it turns out that they’re mistaken. 

However, it’s usually much easier for most animals to recognize their own offspring than a brother or sister. Therefore, even though a sibling should have the same degree of relation as a child, parent-offspring altruism is much more common than sibling altruism in nature. By that same token, an animal is always 100% certain of its relation to itself, which can often weight decisions in favor of helping itself even over close relatives.

In reality, there are many considerations other than just relatedness and life vs. death that go into calculating how altruistic to be toward another member of the same species. For example, if an elderly animal can protect a younger relative who has a longer reproductive period still ahead of it, that might shift the decision in favor of altruism. And, naturally, not all acts of altruism result in saving the recipient’s life at the cost of the altruist’s. Risk and reward must be calculated accordingly. This tendency to act altruistically toward your relatives is sometimes called kin selection, though kin altruism may be a more accurate term.

Of course, animals don’t consciously calculate all of this before acting; it’s done instinctively and almost instantly. It’s similar to a person predicting where a ball will go so that he or she can catch it, even though the math involved in that prediction should be extremely complex. 

Hamilton’s Rule

There’s a mathematical representation of kin altruism called Hamilton’s Rule. It says that altruism is favored when rB > C, (r times B is greater than C) where r is the relatedness between the two parties, B is the benefit to the recipient, and C is the cost to the altruist. 

Because benefit is multiplied by relatedness—which is a fraction, as shown earlier—individuals will weigh benefit proportionally to how closely they’re related vs. cost to themselves. If the recipient isn’t related at all, then r = 0 and altruism will never be favored (remember that any number multiplied by zero is zero). Hamilton’s Rule also accounts for such things as an older animal risking its life to save a younger one—in that case, the benefit to the recipient is greater than the cost to the altruist.

Hamilton’s Rule: Calculating Kin Altruism

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Rina Shah

An avid reader for as long as she can remember, Rina’s love for books began with The Boxcar Children. Her penchant for always having a book nearby has never faded, though her reading tastes have since evolved. Rina reads around 100 books every year, with a fairly even split between fiction and non-fiction. Her favorite genres are memoirs, public health, and locked room mysteries. As an attorney, Rina can’t help analyzing and deconstructing arguments in any book she reads.

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