# What if $rB=C$ in Hamilton's rule?

Hamilton's rule states that if $rB>C$ then a gene giving altruistic behaviour will increase in frequency in the population. What would happen if $rB=C$? Will an individual perform the altruistic act?

• Why has this been downvoted? Oct 7, 2013 at 7:42
• @Biogirl How have you tried to analyse the problem? Also, I rephrased your Q to better fit with a standard way of stating Hamilton's rule. Oct 7, 2013 at 8:56
• Could you edit your title so it doesn't have dollar signs in it please. Feb 4, 2017 at 20:27

I agree with @Amory in the sense that Hamilton's rule is not a rule that applies to each specific individual and explain their behavior (or other traits). The Hamilton's rule describe the direction (and not the dynamic) of how a social traits evolve. A social trait is any trait which does not only affect the fitness of its carrier but also affect the fitness of other individuals in the population. Theoretically, all (or almost) traits are actually somehow social traits.

$RB>C$ is a simplistic way of looking at Hamilton's rule which might bring you to some confusion. This formula makes more sense to me by replacing $B$ by $\frac{dw(x,y,z)}{dx}$ and $C$ by $\frac{dw(x,y,z)}{dy}$. $w(x,y,z)$ is the fitness function of a focal individual expressing trait $x$ and interacting with an individual drawn from a subpopulation expressing trait $y$ (expected value of the probability distribution of trait expression of individuals in the subpopulation) in a population expressing trait $z$ (expected value again). In the particular case where there is no population structure $y$ equals $z$. $\frac{dw(...)}{dx}$ describes the partial derivative of the fitness function according to the variable $x$.

Therefore the Hamilton's rule can be rewritten as:

$$R\cdot\frac{dw(x,y,z)}{dx}>\frac{dw(x,y,z)}{dy}$$ where $R$ is the coefficient of relatedness which can itself be expressed as a correlation between the variables $x$ and $y$.

Altruistic behavior evolves if (but not "if and only if") this rule is respected. Hope this reformulation of Hamilton's rule helps a bit understanding it.

In the special case where $R\cdot\frac{dw(x,y,z)}{dx}=\frac{dw(x,y,z)}{dy}$ then, the trait of interest is not selected (neither counter selected). Therefore the trait will evolve under genetic drift alone. While the allele frequencies are modified by drift, the Hamilton's rule might (or might not) differ from the equality and then the trait would not be neutral anymore.

"Hamilton's rule states that if rB>C then a gene giving altruistic behaviour will increase in frequency in the population."

To start here are some examples of how Hamilton's Rule works...

In a population of four individuals a pair of adults mate. A first unrelated male (r=0) is genetically coded to help raise their offspring by an allele (selfless) at the "sociality" gene. A second male, also with r=0, has a different allele (selfish) at the same locus which makes him not help with offspring care. The pair raises 4 offspring if they go it alone, and 10 if another male helps. By helping, a male sacrifices his ability to mate so produces no offspring. Going it on his own he would produce 4 offspring.

By helping, when r=0, a male passes on no copies of his genes to the next generation (fitness = 0, no offspring in the next generation have genes that are identical by descent [ibd] to his), by being selfish he passes on 4ibd copies of half of his genes (fitness = 2 = 4*0.5). Thus the male carrying the selfish allele should do better and genes promoting social behavior should not evolve in the population because 0*(10)<2 (by helping he sacrifices the fitness score of 2, this is the cost).

Now with related males, say the father's brothers. This time the males will on average be related to the offspring of the pair with a relatedness coefficient of r=0.25. With the same results of helping and selfish behavior, genes for social behavior should evolve because 0.25*10>2, the selfless male has more IBD genes in the population than the selfish male. The next generation would have many individuals in it with the selfless allele.

Hamilton's rule is used to estimate whether "selfless" type alleles will spread through a population, to do so there must be some sort of relatedness within a group otherwise r=0 so rB will never be greater than C. In populations where relatives stay together there is more chance that altruistic behavior will affect a relative, when relatives can't/don't disperse away from each other much then genes promoting altruism should evolve. In more fluid societies they should not be as common.

Hamilton's rule is not definitive though. The closer that rB and C are to each other, the weaker selection will be at driving out alleles which reduce fitness, that is because the gain of having the correct allele is so marginal. For example, if r=0.5, B=100, and C=49, then selfless alleles are 2% fitter than selfish alleles. In comparison, if we keep r=0.5 and B=100, but change C to 40, then the rB>C still holds and selfless alleles have more strongly favorable selection (they are 25% fitter) so will spread more rapidly. If C=0 the behavior is likely to evolve because r=0 is unlikely in a population (even if you only include polymorphic loci).

"What would happen if rB=C? Will an individual perform the altruistic act?"

Whether animals can use Hamilton's rule to decide behaviors plastically is different application of the equation in Hamilton's rule. It is a similar concept, but evolution does not happen with the individual; the genetic makeup of an individual will not change as a result of selection. However, ignoring the definition of how Hamilton's rule is used you could use the same equation to estimate likely behavioral outcomes when an individual is given a choice. Here is an example where an individual is given the option to give or keep food and can judge relatedness, it is a very simplified model.

Example: A locus has two alleles G and g. Individuals can then be GG, Gg, and gg and each genotype has an easily discernible unique trait. Thus when a GG individual meets a GG individual it knows it is perfectly related, r=1. For Gg r=0.5, and gg r=0. When the individuals meet our focal individual (GG) has the option to either provide something for the other one or not provide (let's say give it some food).

For our GG individual the cost of providing the food to the other is the energy it took to find the food, say 50 calories. The benefit to the recipient is the energy within the food, lets say 120 calories. Thus we can say, GG should give the food to the other individual if the other is GG (because 1*120 > 50) or Gg (0.5*120 > 50) but not when it is gg (0*120 < 50). If we reduce the value of the food to 80 calories it becomes that GG should not help Gg (0.5*80 < 120).

There are a few limitations to consider...

Firstly, how well can an individual quantify relatedness? Just using Wrights Coefficient of Relatedness we would come out with roughly accurate measures on average. But there is variability due to mutations and the second law of Mendelian inheritance; independent assortment.

Secondly, how well can an individual judge benefit of its actions? The effect on the recipient is likely to be complex, it could have serious knock-on effects such as increased fitness later in life, longer lifespan, the action might reduce predation risk for the recipient that day etc..

Thirdly and similarly, how well can the provider judge the cost it incurs? If it is giving away food, it does not know if and when it will actually find it's next food item, foraging might cause it to be predated upon, and it might lose out on a mating opportunity if a receptive mate comes by while it is foraging etc..

When rB truly equals C (unlikely due to the complex nature of r, B and C) or is approximately equal, then the individual will be less likely to make the correct choice. When it is obvious that the cost is large and benefit is small it is more likely to make the correct decision. This is analogous to the ability of selection to drive out the incorrect allele as outlined above.

This is a rewrite of a previous answer made after some thinking and I think it provides a much better answer than the previous incarnation which I have deleted

• Thanks for such a wonderful answer ! Actually, this question arose because there was this question : If an individual with an alternative of rearing its own 1 offspring or rearing its cousin's offspring,what is the minimum no. of offspring its cousin should have ? I know that 1own off. = 4 cousin off. So should the minimum no of offspring be 5 or is 4 an apt answer ? Oct 7, 2013 at 18:29
• @biogirl Well I guess you could always say that an individual might choose to help regardless of the benefit due to its lack of ability to differentiate, but that would be a risky idea! I'd suggest explaining 3-5 offspring. Such that at 5, given the individual can distinguish relatedness, costs and benefits quite accurately, that it would be altruistic. At 3 it should behave selfishly. And at 4 it theoretically could choose either strategy with no adverse effects. As long as you show your working (0.25*4=1, 0.25*3<1, and 0.25*5>1) and explain those three you should nail it on the head. Oct 7, 2013 at 21:09

As far as I understand it, Hamilton's "rule" isn't really meant to apply individually, it's meant as a way of thinking about kin selection and altruism that can be reduced to individual cases. The reality is that B and C can rarely, if ever, be easily measured or determined. If the two sides were equal then who knows? You'd have to observe it. Presumably it would split 50-50. Or maybe laziness takes over. Or maybe some species are inherently altruistic. That being said, exact equality is a near-impossible occurrence in the wild, and B and C are movable targets, so you'd have to measure it, which is, as stated, very, very difficult.