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Formally, Hamilton's rule states that genes should increase in frequency when
- r = the genetic relatedness of the recipient to the actor, often defined as the probability that a gene picked randomly from each at the same locus is identical by descent.
- B = the additional reproductive benefit gained by the recipient of the altruistic act,
- C = the reproductive cost to the individual performing the act.
The relatedness parameter (r) in Hamilton's rule was introduced in 1922 by Sewall Wright as a coefficient of relationship that gives the probability that at a random locus, the alleles there will be identical by descent. Subsequent authors, including Hamilton, sometimes reformulate this with a regression, which, unlike probabilities, can be negative. A regression analysis producing statistically significant negative relationships indicates that two individuals are less genetically alike than two random ones (Hamilton 1970, Nature & Grafen 1985 Oxford Surveys in Evolutionary Biology). This has been invoked to explain the evolution of spiteful behaviour consisting of acts that result in harm, or loss of fitness, to both the actor and the recipient.
There has been little empirical support for Hamilton's rule in wild animals, as it is hard to quantify the costs and benefits of different behaviours. The first study to test Hamilton's rule successfully was in 2010, involving a wild population of red squirrels in Yukon, Canada. The researchers found that surrogate mothers would adopt related orphaned squirrel pups but not unrelated orphans. The researchers calculated the cost of adoption by measuring a decrease in the survival probability of the entire litter after increasing the litter by one pup, while benefit was measured as the increased chance of survival of the orphan. The degree of relatedness of the orphan and surrogate mother for adoption to occur depended on the number of pups the surrogate mother already had in her nest, as this affected the cost of adoption. The study showed that females always adopted orphans when rB > C, but never adopted when rB < C, providing strong support for Hamilton's rule.