Figures and data in The general version of Hamilton’s rule

Figures
Tables
Additional files

8 figures, 3 tables and 1 additional file

Figures

Figure 1

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Two Price-like equations and two models.

In Model A, the number of offspring follows a binomial distribution with an expected number of offspring of $α + β_{1} p_{i}$ . This means that the model can be summarized as $w_{i} = α + β_{1} p_{i} + ε_{i}$ (see Detailed Calculations 1.6 at the end of Appendix 1 for details). For the transition depicted in panels a and c, we generated an offspring generation using Model A, with $α = 1$ and $β_{1} = 1$ . Combining the Generalized Price equation in regression form with Model A includes choosing $\hat{α}$ and ${\hat{β}}_{1}$ so that they minimize the sum of squared differences between $w_{i}$ and $\hat{α} + {\hat{β}}_{1} p_{i}$ . In Model B, the number of offspring follows a binomial distribution with an expected number of offspring of $α + β_{1} p_{i} + β_{2} {p_{i}}^{2}$ , which means that $w_{i} = α + β_{1} p_{i} + β_{2} {p_{i}}^{2} + ε_{i}$ . For the transition depicted in panels b and d, we generated an offspring generation using Model B, with $α = 1$ , $β_{1} = 0$ , and $β_{2} = 1$ . Combining the Generalized Price equation in regression form with Model B includes choosing $\hat{α}$ , ${\hat{β}}_{1}$ , and ${\hat{β}}_{2}$ so that they minimize the sum of squared differences between $w_{i}$ and $\hat{α} + {\hat{β}}_{1} p_{i} + {\hat{β}}_{2} p_{i}^{2}$ . Reproduction is asexual, so parents and their offspring are always identical, and $E (w Δ p) = 0$ by definition. For both of these models, we started with a population consisting of 2500 parents for each p-score, ranging from 0 to 1 in increments of 0.1. The four panels represent the four combinations of the two Price-like equations and the two datasets. The red lines in all panels represent the estimated fitnesses as a function of the p-score, as implied by the respective Price-like equations. The aim of this example is not to show that the estimated fitnesses from Price-like equation A match the data generated by Model A, and the estimated fitnesses from Price-like equation B match the data generated by Model B, but the estimated fitnesses from Price equation A do not match the data generated by Model B; that part is obvious. The purpose of this example, instead, is to illustrate that both Price-like equations remain identities, also when they are overspecified (panel c) or underspecified (panel b) with respect to the transition between parent and offspring population they are applied to. With underspecification, it is also visually clear that the estimated fitnesses do *not* match the data, even though Price-like equation A remains an identity, also when combined with data generated by Model B.

Figure 1—source data 1 Simulation data generated by a simple computer algorithm.: https://cdn.elifesciences.org/articles/105065/elife-105065-fig1-data1-v1.xlsx
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Appendix 1—figure 1

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Dependence on the parent population.

We assume that the population is infinitely large, so that the average fitness matches the expected values. On the left, a parent population, all members of which either have a p-score of 0 or a p-score of $\frac{1}{2}$ . With expected fitnesses belonging to these p-scores of 0 and $\frac{1}{2}$ , respectively, that results in a $β$ of 1. On the right, a parent population, all members of which either have a p-score of $\frac{1}{2}$ , or a p-score of 1. With expected fitnesses belonging to these p-scores of $\frac{1}{2}$ and 2, respectively, that results in a $β$ of 3. In the literature, the dependence of the $β$ on the parent population is sometimes referred to as *dynamical insufficiency*. That, however, is not what this is in this case, as here, this is really a symptom of misspecification. If the model that generated the data would have been linear (model A), then, absent the noise, the $β$ would have been the same, regardless of the composition of the parent population. Dynamic (in)sufficiency is discussed in Section 6 of this appendix.

Appendix 1—figure 2

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A graphical representation of the life cycle for the equilibrium state of the population.

*Step 1: differential survival*. All individuals in the parent generation with a p-score of 0 die, and half of those with a p-score of 1 do. None of the parents with a p-score of $\frac{1}{2}$ die, so the heterozygote has the highest fitness. *Step 2: mothers and fathers*. Half of all surviving parents are female, and half are male. *Step 3 and 4: random matching*. The parents are randomly matched. *Step 5: fair meiosis*. In each pair, the offspring inherits one allele from either parent. *Step 6: scaling up*. Parent pairs have on average three kids.

Appendix 1—figure 3

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Nested models and their Price-like equations.

There are different models, and each model has its own Price-like equation. These Price-like equations are general, in the sense that they can be written for *any* dataset, regardless of the underlying data generating process or, in a theory context, for *any* model. The terms in it, however, only have a meaningful interpretation if the data are generated by a model in the set that the Price-like equation belongs to. In line with the setup in this appendix, set A would represent models that are linear in the p-score ( $w_{i} = α + β_{1, 0} p_{i}$ ). This is the set of models for which the regression coefficient in the original Price equation in regression form has a meaningful interpretation. Set B could consist of models that are quadratic in the p-score ( $w_{i} = α + β_{1, 0} p_{i} + β_{2, 0} {p_{i}}^{2}$ ), set C could consist of models that also include a coefficient for the p-score to the power 3 ( $w_{i} = α + β_{1, 0} p_{i} + + β_{2, 0} {p_{i}}^{2} + β_{3, 0} {p_{i}}^{3}$ ), set D models that are linear in the p-score and the q-score ( $w_{i} = α + β_{1, 0} p_{i} + β_{0, 1} q_{i}$ ), and set E models that include an interaction term between the p-score and the q-score ( $w_{i} = α + β_{1, 0} p_{i} + β_{0, 1} q_{i} + β_{1, 1} p_{i} q_{i}$ ). The Price-like equations for these different models are all different.

Appendix 2—figure 1

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Symptoms of underspecification.

The red bars indicate frequencies of parents with a p-score of 0, and with 0, 1, …, 10 offspring for the example described in the text. The blue bars indicate frequencies of parents with a p-score of 1, and with 0, 1, …, 10 offspring. The composition of the parent population differs between the panels; in the top panel, 1 out of 4 parents has a p-score of 1; in the middle panel, this is 2 out of 4 parents; and in the bottom panel, it is 3 out of 4 parents. For the correctly specified model $w_{i} = α + β p_{i} + ε_{i}$ , the red and the blue dotted lines indicate the values for $α$ and for $α + β$ , if those are chosen so as to minimize the sum of squared errors relative to the correct model. The $α$ is the expected value of the number of offspring for parents with a p-score of 0, while $α + β$ is the expected value of the number of offspring for parents with a p-score of and 1. The $α$ and $β$ that minimize the sum of squared errors do not depend on the composition of the population. For the mis-specified model $w_{i} = α + ε_{i}$ , the black dotted lines indicate the $α$ ’s for which the sum of squared errors is minimized. These do move around. The sum of squared errors when using the mis-specified model is also much larger at the minimum.

Appendix 2—figure 2

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Fitnesses for Model 2.

On the left, $r = 0$ , and the likelihood of being matched with a cooperator is the same for cooperators and defectors, and equal to the frequency of cooperators. On the right, $r = \frac{1}{4}$ , where the probability of being matched with a cooperator is $\frac{1}{4} + \frac{3}{4} f_{C}$ for cooperators, and $\frac{3}{4} f_{C}$ for defectors, where $f_{C}$ is the frequency of cooperators.

Appendix 2—figure 3

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Fitnesses for Model 3.

On the left,, $r = 0$ and on the right, $r = \frac{1}{4}$ . The lines are still straight, but no longer parallel.

Appendix 2—figure 4

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Applying the Generalized Price equation for Model 2 to Model 3.

For this figure, we chose Model 3 with $δ_{1, 0} = - 2$ , $δ_{1, 1} = 0$ , and $δ_{1, 1} = 3$ . The red lines are ${\hat{γ}}_{1, 0}$ (the lower one), and ${\hat{γ}}_{0, 1}$ (the higher one), which, when applied to Model 2, would have been the effect on the individual itself and the effect on the partner. The average effect on the individual itself is the lower unbroken blue line, which is accompanied by the average effect for the defectors (the dashed line below it) and the average effect for cooperators (the dashed line immediately above it). The average effect on the partner is the higher unbroken blue line, which is accompanied by the average effect for the defectors (the dashed line immediately below it) and the average effect for cooperators (the dashed line above it).

Tables

Table 1

Three rules and three models.

This table gives all combinations of the three rules and the three models discussed in the example. All rules indicate the direction of selection correctly for all models. Yellow indicates a combination of a rule and a model, where the rule is more general than is needed for the model. This leads to one or more $b$ ’s being 0. These are relatively harmless overspecifications. Red indicates a combination of a rule and a model, where the rule is not general enough (underspecified) for the model. This leads to $b$ ’s and $c$ ’s that depend on the population state. Terms that depend on the population state are abbreviated as follows: $r = r_{0, 1} = \frac{C o v (p, q)}{V a r (p)}$ , $r_{1, 1} = \frac{C o v (p, p q)}{V a r (p)}$ , $s_{b} = \frac{C o v (p, q) C o v (p, p q) - C o v (q, p q) V a r (p)}{{(C o v (p, q))}^{2} - V a r (p) V a r (q)}$ , and $s_{c} = \frac{C o v (p, p q) V a r (q) - C o v (p, q) C o v (q, p q)}{{(C o v (p, q))}^{2} - V a r (p) V a r (q)}$ (see Detailed Calculations 2.3 at the end of Appendix 2 for calculations). Rule 1 is the standard rule for non-social evolution for linear non-social traits. Rule 2 is the classical Hamilton’s rule. Rule 3 is Queller’s rule (Queller, 1985), which is a rule that allows for an interaction effect. Rule 1 is nested in Rule 2, which is nested in Rule 3, which can be nested in more general rules as well.

		Model 1:	Model 2:	Model 3:
		$α + β_{1, 0} p_{i}$	$α + β_{1, 0} p_{i} + β_{0, 1} q_{i}$	$α + β_{1, 0} p_{i} + β_{0, 1} q_{i} + β_{1, 1} p_{i} q_{i}$
Rule 1:
$0 > c$	$c =$	$- β_{1, 0}$	$- (β_{1, 0} + r_{0, 1} β_{0, 1})$	$- (δ_{1, 0} + r_{0, 1} β_{0, 1} + r_{1, 1} β_{1, 1})$
Rule 2:
$r b > c$	$c =$	$- β_{1, 0}$	$- β_{1, 0}$	$- β_{1, 0} + s_{c} β_{1, 1}$
	$b =$	$0$	$β_{0, 1}$	$β_{0, 1} + s_{b} β_{1, 1}$
Rule 3:
$r_{0, 1} b_{0, 1} + r_{1, 1} b_{1, 1} > c$	$c =$	$- β_{1, 0}$	$- β_{1, 0}$	$- β_{1, 0}$
	$b_{0, 1} =$	$0$	$β_{0, 1}$	$β_{0, 1}$
	$b_{1, 1} =$	$0$	$0$	$β_{1, 1}$

Appendix 1—table 1

Four transitions and the values for the terms in the Price equation.

This is an overview of all terms that are in the different versions of the Price equation.

	Parent 1	Parent 2	$\bar{w}$	$Δ \bar{p}$	${\hat{β}}_{1, 0}$	$C o v (p, w)$	${\hat{ϑ}}_{1, 0}$	$C o v (p, v)$
Transition 1	$0$	$1$	$\frac{1}{2}$	$\frac{1}{2}$	$1$	$\frac{1}{4}$	$1$	$\frac{1}{2}$
Transition 2	$0$	$3$	$\frac{3}{2}$	$\frac{1}{2}$	$3$	$\frac{3}{4}$	$\frac{1}{3}$	$\frac{1}{2}$
Transition 3	$2$	$1$	$\frac{3}{2}$	$- \frac{1}{6}$	$- 1$	$- \frac{1}{4}$	$- \frac{1}{9}$	$- \frac{1}{6}$
Transition 4	$2$	$3$	$\frac{5}{2}$	$\frac{1}{10}$	$1$	$\frac{1}{4}$	$\frac{1}{25}$	$\frac{1}{10}$
Average	$1$	$2$	$\frac{3}{2}$	$\frac{7}{30}$	$1$	$\frac{1}{4}$	$\frac{71}{225}$	$\frac{7}{30}$

Appendix 2—table 1

Three rules and three models.

This table shows all combinations of the three rules and the three models. All rules indicate the direction of selection correctly for all models. Yellow indicates a combination of a rule and a model, where the rule is more general than is needed for the model. This leads to one or more $b$ ’s being 0. Red indicates a combination of a rule and a model, where the rule is not general enough for the model. This leads to one or more $b$ ’s and $c$ ’s that depend on the population state. Terms that depend on the population state are abbreviated as follows: $r = r_{0, 1} = \frac{C o v (p, q)}{V a r (p)}$ , $r_{1, 1} = \frac{C o v (p, p q)}{V a r (p)}$ , $s_{b} = \frac{C o v (p, q) C o v (p, p q) - C o v (q, p q) V a r (p)}{{(C o v (p, q))}^{2} - V a r (p) V a r (q)}$ , and $s_{c} = \frac{C o v (p, p q) V a r (q) - C o v (p, q) C o v (q, p q)}{{(C o v (p, q))}^{2} - V a r (p) V a r (q)}$ . Rule 1 is the standard rule for non-social traits. Rule 2 is the classical Hamilton’s rule. Rule 3 is a rule that allows for an interaction effect. An appropriate name for this rule would be Queller’s rule (Queller, 1985). This rule can in turn be nested in a sequence of ever more general rules if we allow for p- and q-scores that are not restricted to be binary.

		Model 1:	Model 2:	Model 3:
		$α + β p_{i}$	$α + γ_{1, 0} p_{i} + γ_{0, 1} q_{i}$	$α + δ_{1, 0} p_{i} + δ_{0, 1} q_{i} + δ_{1, 1} p_{i} q_{i}$
Rule 1:
$0 > c$	$c =$	$- β$	$- (γ_{1, 0} + r_{0, 1} γ_{0, 1})$	$- (δ_{1, 0} + r_{0, 1} δ_{0, 1} + r_{1, 1} δ_{1, 1})$
Rule 2:
$r b > c$	$c =$	$- β$	$- γ_{1, 0}$	$- δ_{1, 0} + s_{c} δ_{1, 1}$
	$b =$	$0$	$γ_{0, 1}$	$δ_{0, 1} + s_{b} δ_{1, 1}$
Rule 3:
$r_{0, 1} b_{0, 1} + r_{1, 1} b_{1, 1} > c$	$c =$	$- β$	$- γ_{1, 0}$	$- δ_{1, 0}$
	$b_{0, 1} =$	$0$	$γ_{0, 1}$	$δ_{0, 1}$
	$b_{1, 1} =$	$0$	$0$	$δ_{1, 1}$