Queller 1985 ("Kinship, reciprocity and synergism in the evolution of social behavior") provides a generalization of Hamilton's rule that allows for non-additivity. To accomplish that, Queller writes the fitness of an individual as \begin{align} W = W_{0} - CP + BP' + DPP' \end{align} where $W_0$ is the baseline fitness; $P$ is the phenotypic value (1 for altruist or 0 for nonaltruist); $P'$ is the phenotypic value of the interacting individual; $C$ is fitness cost if $P=1$; $B$ is the fitness benefit if $P'=1$; $D$ is the deviation from additivity of $C$ and $B$ if both were altruistic (i.e., $PP' = 1$).

Queller then substitutes this equation into Price's equation $\Delta G = Cov (G, W)$, where $G$ is the frequency of altruism allele in the focal individual($0$, $1/2$ or $1$). The resulting equation is: \begin{equation} - \bar c + \bar b \frac{Cov(G,P')}{Cov(G,P)} + \bar d \frac{ Cov(G,PP') }{Cov(G,P)} > 0 \end{equation} where $\bar c$, $\bar b$, and $\bar d$ are the sample means of $C$, $B$, and $D$.

My question is: how did Queller arrive at the sample means $\bar c$, $\bar b$, and $\bar d$? I understand that $Cov(aX,Y) = a\, Cov(X,Y)$, provided that $a$ is a constant. But what Queller is doing here is different; he seems to be saying that $Cov(AX,Y) = E(A)\, Cov(X,Y)$.


1 Answer 1


It is just worded a little wierdly in my opinion. The key line in the paper is: 'Fitness components are also defined for all individuals, for example, $C$ is defined, even for a non-altruist, as the cost it would incur if were altruistic.'

Essentially, if it doesn't matter what individual you are you always pay the same cost, then $C$ is a constant. This is pretty much the definition of a constant! The expectation of a constant is just the constant, $E(C) = C$.

The reason it works is that to find whether or not the cost is realised, Queller simply multiplies by the focal individual's phenotype value $P$, which is either 0 or 1. So either the individual pays the cost or it doesn't.

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    $\begingroup$ I think Queller's discussion of his 1985 model in his 1992 paper "Quantitative Genetics, Inclusive Fitness, and Group Selection" (p. 552) is better worded. There he writes the C, B, and D as regression coefficients. $\endgroup$
    – falsum
    Commented May 30, 2015 at 16:29

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