The general form of Hamilton's rule makes no predictions and cannot be tested empirically

Abstract

Hamilton's rule asserts that a trait is favored by natural selection if the benefit to others, [Formula: see text], multiplied by relatedness, [Formula: see text], exceeds the cost to self, [Formula: see text] Specifically, Hamilton's rule states that the change in average trait value in a population is proportional to [Formula: see text] This rule is commonly believed to be a natural law making important predictions in biology, and its influence has spread from evolutionary biology to other fields including the social sciences. Whereas many feel that Hamilton's rule provides valuable intuition, there is disagreement even among experts as to how the quantities [Formula: see text], [Formula: see text], and [Formula: see text] should be defined for a given system. Here, we investigate a widely endorsed formulation of Hamilton's rule, which is said to be as general as natural selection itself. We show that, in this formulation, Hamilton's rule does not make predictions and cannot be tested empirically. It turns out that the parameters [Formula: see text] and [Formula: see text] depend on the change in average trait value and therefore cannot predict that change. In this formulation, which has been called "exact and general" by its proponents, Hamilton's rule can "predict" only the data that have already been given.

Keywords: cooperation; evolution; kin selection; sociobiology.

Fig. 1.

Hamilton’s rule—general (HRG) is a relationship among slopes in multivariate linear regression. (A) A population of individuals with pairwise interactions. Blue indicates the presence of a trait, $g_i=1$, and red indicates the absence of the trait, $g_i=0$. Links imply interactions. Each individual is labeled by its fitness value, $w_i$. (B) The equivalent table representation shows the input data for fitness, $w$; trait value, $g$; and trait value of interaction partners, $h$. (C) The change in trait value, $\mathrm{\Delta }\overline{g}$, is proportional to the slope of the regression $w$ vs. $g$. (D) Slope of the regression $w$ vs. $h$. (E) There are two different relatedness values for the slopes of $h$ vs. $g$ and $g$ vs. $h$. (F) The multivariate linear regression of $w$ vs. $g$ and $h$ gives the slopes, $M_h=B$ and $M_g=-C$, that are called benefit and cost. The slopes are related as follows: $m_{wg}=M_g+M_h{m}_{hg}$ and $m_{wh}=M_h+M_g{m}_{gh}$. The first of these two equations is Hamilton’s rule. This relationship between slopes is a consequence of multivariate linear regression and has been known in statistics at least since 1897 (15).

Fig. 2.

The retrospective prediction of Hamilton’s rule is based on the numerical value of the term $BR-C$. (A and B) The numerical value of $BR-C$ does not depend (functionally) on the $h$ list, which specifies the trait values of the interaction partners. Any (generic) choice of numbers can be used for the $h$ list, for example the digits of $\pi$, and the value of $BR-C$ remains the same. The prediction of this form of Hamilton’s rule therefore does not use information about who interacts with whom or on whether interactions are between relatives or not. (C) The numerical value of $R$ depends on the $g$ and $h$ lists. The numerical values of $B$ and $C$ depend on all three lists. In particular, $B$ and $C$ also depend on the change in trait value. Therefore, they cannot be used to predict that change in any meaningful way.

Fig. 3.

The parameters $B$ and $C$ mischaracterize the underlying biology in simple examples. In all four populations of size $N=8$, there are two types of individuals: blue, which indicates presence of a trait ($g=1$), and red, which indicates absence of the trait ($g=0$). Each individual is labeled by its fitness, which is later normalized to ensure constant population size. Arrows indicate interaction partners. (A) Both blue and red have baseline fitness $3$. Blue harms blue by reducing its fitness by $2$ at no personal cost. Blue helps red by increasing its fitness by $2$ at no cost. Red does nothing to its interaction partners. The regression method yields $B\mathit{\text{,}}C>0$, misclassifying blue as altruism. (B) Blue has baseline fitness $1$, and red has baseline fitness $2$. Blue increases the fitness of blue by $3$ and decreases the fitness of red by $1$, both at no cost. Red does nothing to its interaction partners. The regression method yields $B>0$ and $C<0$, misclassifying blue as mutually beneficial. This classification is incorrect because blue harms red. (C) Blue has baseline fitness $2$, and red has baseline fitness $3$. Blue increases the fitness of blue by $1$ and decreases the fitness of red by $2$, both at no cost. Red decreases the fitness of blue by $1$ at no cost and does nothing to red. The regression method yields $B<0$ and $C>0$, misclassifying blue as spiteful. (D) Blue has baseline fitness $1$ and red has baseline fitness $2$. Blue does nothing to blue, and red increases the fitness of blue by $2$ at no cost. Red does nothing to red. The regression method yields $B,C<0$, misclassifying blue as selfish. In all four cases the regression method yields $R=7/15$. In A, C, and D, we have $BR-C<0$ (blue decreases in frequency); in B, $BR-C>0$ (blue increases in frequency).

Fig. 4.

The numerical values of the parameters $B$ (“benefit to interaction partners”) and $C$ (“cost to self”) are not robust. An infinitesimally small change in population structure or trait value can modify $B$ and $C$ from arbitrarily large positive values to arbitrarily large negative values. Blue corresponds to $g=1$, indicating the presence of a focal trait, and red corresponds to $g=0$, indicating its absence. In A, the trait values (blue/red) are held constant and the weight of a single interaction is perturbed by $\epsilon$. B shows the resulting values of $w$, $g$, and $h$. C illustrates the effects of this perturbation on $B$ and $C$ as $\epsilon \to 0$ (when $x=0.5$, $w_1=1.6$, and $w_2=w_3=w_4=0.8$). In D, all players have a trait value of $x$ (where $0<x<1$), and a single player’s trait value is slightly perturbed by $\epsilon$. E gives the table representation of this population, and in F, we see the erratic behavior of $B$ (benefit) and $C$ (cost) as $\epsilon \to 0$ (depending on $w$). In both populations, the limits of $B$ and $C$ are different from the left ($\epsilon$ small and negative) and from the right ($\epsilon$ small and positive). For $\epsilon =0$, the parameters $B$ and $C$ are undefined in both populations.