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

^{a}Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138;^{b}Department of Mathematics, Harvard University, Cambridge, MA 02138;^{c}Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138;^{d}Department of Mathematics, Emmanuel College, Boston, MA 02115;^{e}Museum of Comparative Zoology, Harvard University, Cambridge, MA 02138

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Contributed by Edward O. Wilson, April 13, 2017 (sent for review February 2, 2017; reviewed by Michael Doebeli and Jan Rychtar)

## Significance

Hamilton’s rule is a well-known concept in evolutionary biology. It is usually perceived as a statement that makes predictions about natural selection in situations where interactions occur between genetic relatives. Here, we examine what has been called the “exact and general” formulation of Hamilton’s rule. We show that in this formulation, which is widely endorsed by proponents of inclusive fitness theory, Hamilton’s rule does not make any prediction and cannot be tested empirically. This formulation of Hamilton’s rule is not a consequence of natural selection and not even a statement specifically about biology. We give simple and transparent expressions for the quantities of benefit, cost, and relatedness that appear in Hamilton’s rule, which reveal that these quantities depend on the data that are to be predicted.

## Abstract

Hamilton’s rule asserts that a trait is favored by natural selection if the benefit to others,

Hamilton’s rule is a widely known concept in evolutionary biology. It has become standard textbook knowledge and is encountered in undergraduate education. For many, Hamilton’s rule expresses the intuition that cooperation evolves more easily when there are frequent interactions among relatives, because relatives are likely to share the cooperative trait. However, Hamilton’s rule goes beyond this intuition by positing a quantitative condition,

We immediately encounter the question of how the “benefit,”

A number of recent papers (7, 9, 10) have endorsed a particular formulation (4, 5) as the exact, general, and even “canonical” version of Hamilton’s rule. This formulation, called “Hamilton’s rule—general” (HRG) (12, 13), is claimed to be as general as natural selection itself (7, 14). The derivation, which we recapitulate below, is simple and contains only a few steps.

The mathematical investigation of HRG reveals three astonishing facts. First, HRG is logically incapable of making any prediction about any situation because the benefit,

The second astonishing fact of HRG is that the prediction, which exists only in retrospect, is not based on relatedness or any other aspect of population structure. A common interpretation of the terms in Hamilton’s rule is that

The third fact of HRG is that no conceivable experiment exists that could test (or invalidate) this rule. All input data, whether they come from biology or not, are formally in agreement with HRG. This agreement is not a consequence of natural selection, but a statement about a relationship between slopes in multivariate linear regression. This relationship between slopes has been known in statistics at least since 1897 (15).

## Derivation of HRG

We recapitulate the derivation of HRG given in refs. 4, 5, 7, 9, and 10. We also provide explicit algebraic formulas for

We imagine a population of

Each individual is assigned a trait value,

We will see that these two lists fully specify the numerical value of

Although the first two lists determine the value of

Some further notation is needed. The variance of a list of numbers is

Because *E*). The formula for the slope is

The derivation continues by calculating a best-fit plane to the data given in the 3Dspace *F*). The parameters

Both

The derivation is completed by calculating the term,

Any collection of numbers that is used for the **3a** and **3b** are nonzero, gives the same value of

The formal correctness of HRG is established by Eqs. **1** and **5**. Because the variance

## Slopes and Statistics

The relationship between the slopes of the various linear regressions, expressed by HRG (Eq. **5**), is not a consequence of biology and not a discovery of inclusive fitness theory. It is a fact of multivariate statistical analysis, the proof of which we recall in *Appendix*.

In a linear regression of

In three dimensions, the multivariate regression of

Inclusive fitness theorists call Eq. **6a** “Hamilton’s rule” by setting

However, it is well understood in statistics that relationships such as Eq. **6** do not themselves imply causality (18, 19). Whereas there can be causal relationships between dependent and independent variables in a linear model, these relationships cannot be deduced from linear regression alone (20, 21). Therefore, without further assumptions or information, the meanings attached to the terms in HRG have no basis in mathematics or statistics. Moreover, the derivation of HRG does not take into account any aspect of the mechanism that leads to a change in trait value and therefore cannot return a description of that mechanism (Fig. 3) (22). It merely defines quantities

## Benefit and Cost Need Not Make Sense

Although HRG does not explain or predict the change in average trait value, it could be the case that the parameters

The parameters

## Discussion

Hamilton’s rule is commonly thought to capture the idea that cooperative behaviors can be selected if the benefits go to close relatives, because these relatives are likely to share genes for cooperation. In this understanding, Hamilton’s rule is believed to make important, testable predictions for the evolution of social behavior: A trait is selected if benefit times relatedness exceeds cost. Clearly, such a simple and seemingly plausible statement has great intuitive appeal.

However, any intuition can only be as good as its mathematical or biological underpinning. The purpose of this article has been to clarify the mathematical derivation of Hamilton’s rule that has been endorsed as exact, general, and canonical by the inclusive fitness community (HRG) (4, 5, 7, 9, 10). The derivation of HRG is encapsulated in Eqs. **1**–**5**. Any collection of triples can be used as input data and will turn out to be in “agreement with Hamilton’s rule” as long as the relevant denominators are nonzero. If the denominators are zero, the quantities

The predictive power of HRG is equivalent to the following example: If you give me the shoe sizes and heights of a group of people, then I can predict the heights. My algorithm also works if you gave me the wrong shoe sizes.

That HRG has no predictive power has been previously noted by ourselves (22) and others (12, 23, 24), yet HRG is credited with making a variety of empirical predictions (7, 9, 14).

Much like the Price equation (25⇓–27), HRG provides a functional relationship between quantities that are obtained from a population at two successive points in time. Whereas the change in trait frequency,

In short, there is a startling discrepancy between the common intuitive understanding of Hamilton’s rule and the derivation of this rule that has been described as exact and general. In some cases, this discrepancy can be seen within a single paper. For example, ref. 7 uses 18 different variations of “Hamilton’s rule correctly predicts…” in reference to HRG, which makes no prediction at all.

Although HRG is the only formulation of Hamilton’s rule that is claimed to be exact and general, there are other approaches that define benefit, cost, and relatedness in different ways. For example, benefit and cost can be properties of individual phenotypes, and relatedness can be defined using common ancestry (1, 3, 12, 13, 24, 28). This approach, “Hamilton’s rule—special” (HRS) (12, 13), has the advantage of making testable predictions, because the benefit and cost of a phenotype can be determined in advance. However, it is easy to show that HRS holds only for special cases and not in general (12, 28⇓⇓–31).

The existence of these conflicting definitions makes it impossible to meaningfully test or falsify Hamilton’s rule. Any theoretical or empirical result that appears to violate Hamilton’s rule can be reanalyzed using HRG to show that the outcome is “as predicted by Hamilton’s rule.” Indeed, this pattern has been repeated many times in the literature (7, 14, 28, 32⇓–34). It appears that there are no real or hypothetical data that the inclusive fitness community would accept as a violation of Hamilton’s rule.

Some papers attempt to empirically test Hamilton’s rule (35⇓⇓⇓⇓–40). Tests of Hamilton’s rule are typically done by experimentally determining the benefits and costs of a phenotype and quantifying relatedness using genetic markers or pedigree. But such a procedure—while scientifically reasonable—tests only HRS, which is not the exact and general version of Hamilton’s rule. We are aware of only one paper (23) that attempts to apply HRG to an empirical system. They find, as we have shown here, that HRG does not predict any aspect of their system, but yields only a value of

The biological question at hand is how population structure affects the evolution of social behavior, which is a deep and important question that has been studied extensively (41⇓⇓⇓⇓⇓⇓⇓–49). The intuition that a cooperative gene can spread by preferentially conferring benefits on cobearers of this gene is correct. However, Hamilton’s rule, in its exact and general formulation, is unrelated to this biological intuition and (in general) neither predicts nor explains the evolution of social behavior.

Indeed, we should not expect that interplay of population structure and social behavior can be reduced to a simple rule with three parameters. Social interactions, which are typically multilateral (50) and nonlinear (51, 52), cannot be expressed by a single benefit and cost. Complex population structures (43, 46, 53, 54) cannot be captured by a single relatedness quantity. Assortment among relatives often has a positive effect on cooperation (41, 44⇓⇓–47), but in other cases it has a negative effect (48, 55) or no effect at all (42, 45). A good understanding of these questions, like all great problems in science, will require careful empirical observation in concert with meaningful mathematics.

## Appendix

Here, we recapitulate the derivation of HRG as a simple consequence of a well-known result in statistics.

For any collection of *i*) *ii*) *iii*) *iv*) *v*) *vi*)

### Proposition.

*These slopes satisfy the following equations*:

#### Proof.

Suppose that **7** are slopes, we may assume without a loss of generality that **9**, we obtain**9** that**10**, gives Eq. **7**.

Note that Eq. **7** can be written in matrix form as**3**.

The only situation in which we cannot solve explicitly for **7** is if

Introducing **4**.

It is interesting to note that the relationships between linear regressions with one and two explanatory variables, which are captured in Eq. **13** and give rise to HRG, appeared in the statistics literature as far back as 1897 (15).

## Acknowledgments

The authors thank Donald Rubin, Karl Sigmund, Corina Tarnita, and John Wakeley for helpful discussions.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ewilson{at}oeb.harvard.edu.

Author contributions: M.A.N., A.M., B.A., and E.O.W. designed research, performed research, analyzed data, and wrote the paper.

Reviewers: M.D., University of British Columbia; and J.R., The University of North Carolina at Greensboro.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.

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