# Reformulation of Hamilton's rule

Who (and in which article) was the first to reformulate Hamilton's rule using the letters $B$ and $C$?. See below comments on this reformulation.

Hamilton, in his 1964's article gave a mathematical formulation to explain the direction of social traits. Below is its formulation. Note: Hamilton used quite a complicated formulation, I hope I am not misunderstanding its meaning. Please let me know if it is the case.

$$R\cdot\frac{dw(x,y,z)}{dx}>\frac{dw(x,y,z)}{dy}$$ where $R$ is the coefficient of relatedness which can itself be expressed as a correlation between the variables $x$ and $y$. $w(x,y,z)$ is the fitness function of a focal individual expressing trait $x$ and interacting with an individual draw 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).

Most often this formula is expressed in the following form:

$$RB>C$$

$B$ and $C$ are called cost and benefit of the social trait. I believe that $B$ and $C$ are often not understood as a consequence of a change of the social trait but rather as a cost and benefit of carrying the social trait (compare to not carrying it).

• Interesting question but since B and C basically strip the idea of a lot of mathematical content, isn't it possible that the first use was, say, in a graduate text for non-mathematicians to give a sort of quick idea? That would be hard to track down I bet. – daniel Mar 23 '14 at 23:34
• @daniel Yes that's possible indeed! One thing that got me interested into this question is that $B$ and $C$ are usually conceptually understood a bit differently than $\frac{dw(x,y,z)}{dx}$ and $\frac{dw(x,y,z)}{dy}$ and I am wondering how the first guy has somehow argued for this new understanding of Hamilton's rule. – Remi.b Mar 24 '14 at 9:00

Tracing it backwards this was the earliest reference I found via google scholar, it's from 1988 and uses the $r B C$ notation in the format we are used to.

Hamilton's rule states that for a social action to be favored under natural selection, rb - c > 0, where c is the cost to the actor in terms of the effect on (usually a reduction in) his expected number of future offspring, bis the benefit to the recipient in terms of the effect on (usually an increase in) his expected number of future offspring, and r is the relatedness of the actor to the recipient

However, I've traced back a bit further by going through the references and found this Michod paper from 1978 where they defined terms $b_i$ and $c_i$ as the

"fitness effect of genotype $i$'s behaviour on others"

and,

"fitness effect of genotype $i$'s behaviour on self"

respectively, which was the first such notation I could find which could be interpreted as $b$ = benefit via the increased fitness of others and $c$ as a cost to self.

As far as I can tell this leaves a 14 year gap between Hamilton's long version and the beginnings of the current notation by Michod in 1978, with the first "rB - c > 0" type of notation appearing in 1988. Obviously my search is not exhaustive - many papers from this period will be hard to find and text search (e.g. using "rB" as a search term) - but it's a good start. If I find an older reference any time I will update.

• (I will try to examine the papers further to try to find both "r = relatedness, b = benefit to recipient, c = cost to provider" and "altruism will evolve if rB>C" types of statements, but that will have to wait - It's quite laborious, even this much took over an hour, and I have a long list of jobs to do by Friday!) – rg255 Dec 9 '14 at 20:14
• Nice work. Thanks a lot Robert! Michod in his abstract talk about kin selection theory as the cost-benefit rule as if it was obvious to other people that kin selection theory is a cost-benefit rule. I should check Hamilton's paper to see if he himself used the word cost and benefit. – Remi.b Dec 10 '14 at 5:22