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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:


$B$ 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).

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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 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 at 9:00
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