The Debate over Inclusive Fitness as a Debate over Methodologies University of Groningen The Debate over Inclusive Fitness as a Debate over Methodologies Rubin, Hannah Published in: Philosophy of Science DOI: 10.1086/694809 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2018 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Rubin, H. (2018). The Debate over Inclusive Fitness as a Debate over Methodologies. Philosophy of Science, 85(1), 1-30. https://doi.org/10.1086/694809 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). 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Download date: 06-04-2021 https://doi.org/10.1086/694809 https://research.rug.nl/en/publications/the-debate-over-inclusive-fitness-as-a-debate-over-methodologies(76375aeb-130b-4c06-9578-5e7f64539f3b).html https://doi.org/10.1086/694809 The Debate over Inclusive Fitness as a Debate over Methodologies Hannah Rubin*y This article analyzes the recent debate surrounding inclusive fitness and argues that cer- tain limitations ascribed to it by critics—such as requiring weak selection or providing dynamically insufficient models—are better thought of as limitations of the methodolog- ical framework most often used with inclusive fitness (quantitative genetics). In support of this, I show how inclusive fitness can be used with the replicator dynamics (of evo- lutionary game theory, a methodological framework preferred by inclusive fitness crit- ics). I conclude that much of the debate is best understood as being about the orthogonal issue of using abstract versus idealized models. 1. Introduction. The mathematical framework of inclusive fitness was first introduced by Hamilton (1963, 1964) in order to help explain the evo- lution of social traits by kin selection and has helped to give new, intuitive explanations of a variety of traits including altruism, eusociality, parental care, and genomic imprinting (Grafen 1984; Marshall 2015, and references therein). In calculating inclusive fitness, one looks at the effects an organ- ism has on other organisms’ reproductive success, rather than just looking at the organism’s own reproductive success. These effects are then weighted by the ‘relatedness’ of the organism to those organisms it affects. In recent years, there has been an extensive debate surrounding inclusive fitness. Some authors argue that inclusive fitness calculations can be wrong (van Veelen 2009), while others argue that it requires stringent assumptions *To contact the author, please write to: Department of Theoretical Philosophy, Univer- sity of Groningen, or Department of Philosophy, University of Notre Dame; e-mail: hannahmrubin@gmail.com. yI would like to thank Simon Huttegger, Brian Skyrms, Jonathan Birch, Cailin O’Connor, Kyle Stanford, Justin Bruner, three anonymous referees, members of the Social Dynamics Seminar at University of California, Irvine, and audiences at the International Society for History, Philosophy, and Social Studies of Biology 2015 meeting for helpful comments. Received July 2016; revised October 2016. Philosophy of Science, 85 (January 2018) pp. 1–30. 0031-8248/2018/8501-0001$10.00 Copyright 2018 by the Philosophy of Science Association. All rights reserved. 1 This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 2 HANNAH RUBIN All u and is less general than ‘standard’ natural selection (Nowak, Tarnita, and Wil- son 2010; Wilson 2012; Allen, Nowak, and Wilson 2013). The response is that inclusive fitness calculations are not (merely by virtue of using the math- ematical framework) susceptible to being wrong (Marshall 2011) and do not require stringent assumptions like weak selection (e.g., Abbot et al. 2011; Marshall 2015), additive payoffs (e.g., Queller 1992; Taylor and Maciejewski 2012; Birch 2014b; Birch and Okasha 2015), pairwise interactions (e.g., Tay- lor, Wild, and Gardner 2007; Abbot et al. 2011; Marshall 2011), or special population structures (e.g., Taylor and Frank 1996; Taylor et al. 2007; Abbot et al. 2011; Taylor and Maciejewski 2012; Marshall 2015). Critics of inclusive fitness often propose evolutionary game theory or pop- ulation genetics as alternatives to the inclusive fitness framework (Nowak et al. 2010, 2011; Traulsen 2010; Allen et al. 2013; Allen and Nowak 2015). Often, the comparisons are made between very simple models in quantitative genetics, which abstract away from particular details of any given population, andmorecomplexmodelsarisingout ofpopulationgenetics,whichoftentake into account more of the particular details. Here, we will look at how inclusive fitness can function in evolutionary game theory, which often makes idealiza- tionsratherthanabstractionsinordertoachievesimplemodels.Thedifference between these two modeling strategies (using abstractions vs. using idealiza- tions)andhowthisrelatestotheinclusivefitnessdebatewillbediscussedmore in sections 3.3 and 5. Looking at the way inclusive fitness can be incorporated into evolutionary game theory will help show where some of the disagree- ments about inclusive fitness arise and when inclusive fitness calculations mightbeexpectedtohavethelimitationsascribedtothembycritics.Itwillalso demonstrate how we can think of some parts of the debate as arising from dif- ferent sides emphasizing different methodologies, rather than as disagree- ments over inclusive fitness as a way of calculating fitness. First, I introduce the framework of inclusive fitness and compare it to ‘neighbor-modulated’ fitness calculations in section 2. Then, in section 3, I discuss the debate that has arisen around the inclusive fitness framework, fo- cusing on issues that can be understood as arising from the different sides of the debate emphasizing different methodologies. In section 4, I discuss how models using both neighbor-modulated and inclusive fitness are connected and provide a simple example to demonstrate these connections. Section 5 will provide a few ways to think about these connections and explain how theycanhelpusunderstandsomeissuesintheinclusivefitnessdebate.Finally, section 6 concludes. 2. Inclusive Fitness and Neighbor-Modulated Fitness 2.1. Basic Calculations. Inclusive fitness and the related concept of neighbor-modulated fitness were first proposed by Hamilton (1964). This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 3 Roughly, the neighbor-modulated fitness of an organism is calculated by adding up the number of offspring the organism is expected to have from some social interaction of interest. Inclusive fitness is an alternative math- ematical framework in which fitness calculations track the offspring caused by a particular organism, rather than tracking the offspring an organism ac- tually has. The offspring caused by the organism are then weighted accord- ing to a ‘relatedness’ parameter, which is a measure of how likely it is that the focal organism and its social partner share genetic material, relative to the rest of the population. The two types of fitness calculations provide al- ternative ways of partitioning the causal structure of social interactions. A more concrete description of the equations used in both frameworks will be provided below. The inclusive fitness framework might initially seem counterintuitive, so it is helpful to start with a basic observation: in general, a trait will increase in frequency when organisms with that trait have more offspring than the average organism in the population. To determine whether a trait of interest will increase in frequency, we want to see how many offspring organisms with that trait will have. Inclusive fitness gives us this information by telling us how many offspring are caused by an organism and how likely it is that these offspring are had by an organism with the trait of interest. We can calculate inclusive fitness for a focal organism, i, by looking at the effects from all its social interactions relevant to our trait of interest. When i interacts with other organisms, it affects its own fitness by some amount (sii) and the fitness of another organism, j, by some amount (sij). The genotype of organism i also predicts, to a certain extent, the genotype of the social partner j. This relationship is described by rij. There will be more details on calculating rij in sections 2.2 and 4, but for now we can think of it as a measure of how likely it is that i and j share genetic material. We can then calculate inclusive fitness as follows: fi 5 o j rijsij: (1) This fitness calculation gives us information about how the population will evolve. It tells us how many offspring are had by organisms with the trait of interest, and since offspring tend to be like their parents, this gives us infor- mation about how the composition of a population is expected to change. Note that, although it is sometimes described this way, inclusive fitness is not calculated by counting the number of offspring an organism has and then adding all the offspring its relatives have (weighted by relatedness). Compare the inclusive fitness approach to the neighbor-modulated fit- ness approach, where we look at an organism, i, and add up the effects of its social interactions on its own number of offspring. The neighbor- modulated fitness of organism i is then calculated as follows: This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 4 HANNAH RUBIN All u fi 5 o j sji, (2) where sji is the effect i’s social interaction with j has on i’s fitness.1 This gives us information about how many offspring i is expected to have and, since i’s offspring tend to be like i, about how the composition of a popula- tion is expected to change.2 2.2. Hamilton’s Rule, the Price Equation, and Kin Selection. Hamilton’s rule, famously associated with inclusive fitness, gives a condition for the in- crease of an altruistic behavior, where an organism performs an action that decreases its own fitness and increases the fitness of another. (An exam- ple of a model of the evolution of altruistic traits will be given in sec. 4.2.) It says simply that if the relatedness-weighted benefit of a trait exceeds its cost, then we should expect selection to favor that trait. That is, the trait is favored when bR 2 c > 0, (3) where b is the benefit to the focal organism’s social partner and c is the cost to the focal organism. Many results within the inclusive fitness framework, including Hamil- ton’s rule, are derived from the Price equation, which is a general descrip- tion of evolutionary change. Let f be the fitness of a trait in the population, relative to the average fitness in the population. Then, the Price equation describes expected evolutionary change in the following way: _E pð Þ 5 Cov f , pð Þ: (4) We can think of p as the average phenotypic value of the population, al- though p can actually represent anything a modeler might want to keep track of: phenotypic value, genetic value, frequency of a trait, and so on. 1. Note that the definition of neighbor-modulated fitness looks formally different from inclusive fitness as fitness effects are unweighted, while the fitness effects in inclusive fitness are weighted by a relatedness parameter. This apparent asymmetry disappears at the population level when we calculate the fitness of organisms with a certain trait. See sec. 4.1 for a calculation of neighbor-modulated fitness at the population level. For more information on the calculations of these two types of fitness, see Frank (1998, 48–49) and Birch (2016). 2. Technically, both inclusive fitness and neighbor-modulated fitness include a baseline nonsocial fitness component, so these calculations are the fitness effects of the social trait of interest. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 5 The change in the average value is then _E(p). The covariance term measures how fitness changes with differences in phenotype.3 When fitness effects are additive, that is, when the fitness effects on the recipient do not depend on the recipient’s genotype/phenotype and fitness effects from all an organism’s social interactions can simply be added up, we can derive equations for both inclusive fitness and neighbor-modulated fitness from the Price equation.4 These equations are discussed further in the appendix, but here we will look at Hamilton’s rule as derived from the Price equation. The inclusive fitness version of Hamilton’s rule is _E(g) > 0 when bsi2ip � Cov p, g0ð Þ Cov p, gð Þ 2 bsiip > 0: (5) When we can interpret the covariance between an organism’s phenotype and its own fitness (bsiip) as a ‘cost’ and the covariance between an organ- ism’s phenotype and its social partner’s fitness (bsi2ip) as a ‘benefit’, we have Hamilton’s rule, where R 5 Cov( p, g0)=Cov( p, g). This measure of re- latedness compares the covariance between a focal organism’s phenotype, p, and its social partner’s genotype, g0, with the covariance between the fo- cal organism’s phenotype and its own genotype, g (Orlove and Wood 1978). It is a measure of the degree to which the focal organism and its so- cial partner are genetically related, or how likely it is that the fitness effects from a trait fall on organisms with the gene(s) encoding for the trait. Section 4.1 and the appendix discuss how inclusive fitness results de- rived from the Price equation are related to the replicator dynamics, which is often used in game theoretic models, using methods drawn from Page and Nowak (2002). Section 4.2 will discuss how this definition of relatedness matches up with the definition of relatedness we will use in game theoretic models. Section 5.2 will discuss versions of Hamilton’s rule that do not rely on the assumption of additive fitness components in relation to the results discussed here. Relatedness is commonly thought of as a measure of the average kinship between interacting organisms when talking about kin selection for a trait. However, it is widely acknowledged that R, and many methods for calcu- 3. There is sometimes a second term, Ef ( _p), included that measures the fitness-weighted transmission bias, the difference between the phenotypic value of a parent and the av- erage phenotypic value of its offspring. It is often assumed that Ef ( _p) 5 0, which is gen- erally thought of as assuming there is no transmission bias. (Assuming that Ef ( _p) 5 0 is not exactly the same as assuming there is no transmission bias [van Veelen 2005], but the details of what exactly it means to assume Ef ( _p) 5 0 are not crucial here.) 4. The additivity of fitness effects requires satisfying these two conditions, which Birch (2016) refers to as actor’s control and weak additivity. Actually, only the second condi- tion is required to derive neighbor-modulated fitness, while both are required to derive inclusive fitness. See Birch (2016) for a discussion of this. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 6 HANNAH RUBIN All u lating R, can be thought of as general measures of correlation between types (Marshall 2015). In this case, R can measure how likely it is that altruists interact with other altruists regardless of whether that correlation is caused by interacting with kin or by some other mechanism, such as a green-beard effect where altruists are able to recognize and preferentially interact with other altruists. Because inclusive fitness is often used in describing traits that evolve via kin selection, the terms ‘inclusive fitness’ and ‘kin selection’ are sometimes used interchangeably. However, it is important to distinguish inclusive fit- ness from kin selection. Inclusive fitness is a method of calculating fitness, as described above. Kin selection, however, refers to the selection of a trait because of benefits falling differentially on relatives. Inclusive fitness is a mathematical framework used to describe evolution of a trait; kin selection is a mechanism by which traits can evolve (e.g., Hamilton 1975; Grafen 2007a). Some of the critiques of inclusive fitness models are aimed at showing that kin selection has been less important as an evolutionary force than many inclusive fitness theorists presume (see, e.g., Wilson 2012). Other parts of the criticism are aimed at the mathematical framework of inclusive fitness itself, such as claims that there are mathematical difficulties with the calcu- lations in inclusive fitness (Nowak et al. 2010; Traulsen 2010; Wilson 2012). This article will not discuss whether kin selection provides an adequate expla- nation of prosocial behavior. Instead, it looks at whether inclusive fitness can provide an adequate mathematical framework for use in evolutionary models. Kin selection is discussed only in considering how inclusive fit- ness can be used in models of traits evolving via kin selection. This focus will help us see which aspects of the debate are relevant to the inclusive fitness framework and which pertain to kin selection explanations of the evolution of particular traits. Section 5 will discuss this further. 3. The Debate Surrounding Methods. There are several critiques levied against the inclusive fitness framework. This article will address a couple of particularly important critiques that, as we will see, can be understood in light of an emphasis on different modeling techniques: the critiques that in- clusive fitness requires the assumption of weak selection and cannot pro- vide dynamically sufficient models. Here, I give a description of these cri- tiques and a brief motivation for thinking of them as arising from different sides of the debate emphasizing different methodologies. Section 5 pro- vides a more detailed argument for this conclusion using material that will be laid out in section 4. 3.1. Weak Selection. First, inclusive fitness has been critiqued for re- quiring the assumption of weak selection. In assuming that there is weak This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 7 selection, we assume that gene frequencies are not changing or that the changes in gene frequencies are small enough to be ignored.5 This assump- tion is used in various ways in inclusive fitness models: in employing esti- mation methods for calculating relatedness, in ignoring higher-order effects or certain types of population structure, and so on. It is easy to see why certain methods of estimating relatedness require weak selection. For example, unless very special conditions hold, estimating relat- edness using pedigrees, or family trees, requires that selection is weak. If gene frequencies are systematically changing in the population, the relatedness of an organism to its siblings, for example, will change as the genetic composi- tion of its siblings changes (Grafen 1984). However, calculating relatedness does not, in general, require weak selection, and we can calculate how relat- edness changes as gene frequencies change (Grafen 1985; Birch 2014a; Mar- shall 2015). The assumption of weak selection is also used because it allows one to ignore nonadditive fitness effects. That is, the assumption of weak selection has been used to ignore things like synergistic effects (where organisms re- ceive additional benefits from cooperation if they both cooperate) or the ef- fects of competition over resources. This is perhaps the more important use of the assumption of weak selection, as it allows one to separate the way an organism affects its own fitness (a self-effect) from the way it affects its so- cial partner’s fitness (an other effect) in cases in which the simplifying as- sumption of additive fitness components is false. Note that this critique also applies to neighbor-modulated fitness, as the fitness effects are similarly sep- arated into self- and other-effect components. At some points in the debate, it seems that critics argue against the use of inclusive fitness (and neighbor- modulated fitness) because it requires weak selection in order to achieve the separation of fitness components. That is, without the assumption of weak selection, one is restricted to a special case in which fitness effects are additive, leading to the conclusion that inclusive fitness is less general than ‘standard’ natural selection (Nowak et al. 2010). However, at some points it seems that critics want to claim that, whether or not fitness effects can be split into additive components, inclusive fitness calculations require weak selection. For instance, Nowak et al. (2010) claim that “inclusive fitness theory cannot even be defined for nonvanishing se- lection; thus the assumption of weak selection is automatic” (SI 14). It is 5. One way to achieve this in a model is to write down fitness as the sum of two com- ponents: f 5 f0 1 dfx. One of these, f0, is the ‘background’ fitness, the fitness organisms get from things that are not related to the trait of interest. This is the same for all organ- isms. The fitness the organisms get from things related to the trait of interest, fx, is then weighted by a parameter d, and as we take d to zero, we approach the limit of weak se- lection. This is what Wild and Traulsen (2007) refer to as ‘d-weak selection’. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 8 HANNAH RUBIN All u this second, stronger, claim that will addressed here. In section 4, the claim will be shown to be clearly false using modeling techniques from evolution- ary game theory, one of the preferred frameworks of critics of inclusive fit- ness. Section 5 will then discuss how, if we read the debate as about inclu- sive fitness theory as a set of methods rather than inclusive fitness theory as a framework for calculating fitness, we can make sense of this claim. 3.2. Dynamic Sufficiency. Inclusive fitness has also been criticized for not being able to provide dynamically sufficient models (Nowak et al. 2010; Wilson 2012). In a dynamically sufficient model, information about the population at any particular time is enough to make predictions about the population at all future times. So, information about a population at some starting time is enough to be able to predict how the population will evolve at all future times. In a dynamically sufficient model, one can predict whether the population will reach an equilibrium, a state at which the population is no longer evolving, and what the population composition will be at the equilib- rium should it reach one.6 Critics of inclusive fitness argue that it cannot be used to describe the evolutionary trajectories or end points of evolution (Nowak et al. 2010, SI 4). One reason this criticism might be leveled against inclusive fitness is the general reliance on the Price equation, which is not dynamically sufficient.7 More specifically, the Price equation itself is neither dynamically sufficient nor insufficient (because it merely expresses a mathematical identity), but it can be either, depending on what sort of model it is used with. When we do have a dynamically sufficient model, the Price equation will correctly de- scribe evolutionary change in the model but will not itself give any addi- tional predictions (van Veelen et al. 2012). Because many of the results in inclusive fitness theory, like Hamilton’s rule, are formulated in absence of a particular model, and because the focus is often on estimating the covariances rather than calculating them from an evolutionary model, we might not always get dynamically sufficient models within the framework. These estimations of parameters will only predict the evolutionary outcome if they do not change over time, which is not the case when selection is frequency dependent (Nowak et al. 2010; Allen et al. 2013). However, as we will see in section 4, the regression methods often 6. This article only deals with deterministic models, but stochastic models can also be dynamically sufficient. A stochastic model is dynamically sufficient when the informa- tion about the probability distribution over types at some starting time is enough to pre- dict how the probability distribution will evolve at all future times and to predict the lim- iting distribution. 7. Another reason, which will be discussed further in secs. 3.3 and 5.2, is that many of the results that do not rely on the Price equation are focused solely on equilibrium anal- ysis. See, e.g., Taylor and Frank (1996). This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 9 emphasized in inclusive fitness theory are intimately connected with the sort of dynamically sufficient models preferred by critics of inclusive fit- ness. 3.3. The Debate over Methodologies. Critics of inclusive fitness often propose population genetics or evolutionary game theory as alternative frameworks in which one can provide models that are dynamically suffi- cient and that do not require stringent assumptions like weak selection (Nowak et al. 2010, 2011; Traulsen 2010; Allen et al. 2013; Allen and Nowak 2015). It is not immediately clear how we should read this proposal, because although it is true that inclusive fitness tends to be used in quanti- tative genetics models (Frank 2013) and is seen as primarily a quantitative method in spirit (Queller 1992), it has been used in both game theoretic (e.g., Skyrms 2002; van Veelen 2009, 2011) and population genetics mod- els (e.g., Rousset 2002; Grafen 2007b; Lehmann and Rousset 2014). In fact, when Hamilton (1964) first proposed using inclusive fitness, he did so in the context of a population genetics model. The methods used in quantitative genetics are designed to handle contin- uously varying traits, such as height or weight. In models of social behavior, a continuously varying trait could be the probability of performing an altru- istic action. Models within quantitative genetics tend to emphasize simplic- ity and measurability. These models usually start with observations about phe- notypes, or other easily measurable quantities, with few assumptions about the underlying genetics of a trait. This method of modeling involves abstrac- tions, ignoring complicating details of the situation by merely leaving them out while still giving a description that is literally true (Godfrey-Smith 2009). The Price equation is often used within this approach. As mentioned in sec- tion 2.2, many of the common results within inclusive fitness theory are de- rived from the Price equation. By contrast, challenges to the inclusive fitness framework tend to come from population genetics (Frank 2013, 1153). This is an approach that tends to start with specific assumptions (such as assuming we know the underly- ing genetics of a trait, the mutation rates, etc.) and make predictions based on those assumptions. Models within this approach tend to be dynamically sufficient, meaning that information about the population at any particular time is enough to make predictions about the population at all future times. The use of simplifying assumptions also means that these models make use of idealizations rather than abstractions. That is, they talk about populations that have features we know real populations do not have (e.g., infinite pop- ulation size, no mutations) in order to provide a simple model. One way to think about models using idealizations is that they describe nonactual, fic- tional populations that we take to be similar to real populations in important ways (Godfrey-Smith 2009). As mentioned, critics propose evolutionary This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 10 HANNAH RUBIN All u game theory as an alternative to the inclusive fitness framework.8 The rep- licator dynamics is often used within this approach. This dynamics re- quires many idealizing assumptions, which will be discussed in section 4.1. The rest of this article will look more closely at the use of inclusive fit- ness in evolutionary game theory, focusing on the replicator dynamics. Since inclusive fitness is not as commonly used in evolutionary game the- ory, this will help us see the benefits and drawbacks of using inclusive fit- ness in highly idealized models. This article will also compare how inclu- sive fitness calculations can be used in evolutionary game theory with some of their uses in quantitative genetics. This comparison between the use of inclusive fitness within these two traditions for studying evolution will be helpful in understanding key issues in the debate, since they represent ex- tremes of methodologies using idealizations and abstractions: the replicator dynamics of evolutionary game theory is highly idealized, while the Price equation often employed in quantitative genetics uses only abstractions. We will see how some of the disagreement arises out of the sides of the debate emphasizing different methodologies and how this relates to arguments over the usefulness of Hamilton’s rule. It is important to note that, while this distinction between abstract models in quantitative genetics and idealized models in evolutionary game theory is illuminating for the present purposes, it does not capture the full variety of modeling techniques within the two methodological traditions. There are evolutionary game theoretic models that make the assumption of weak se- lection in order to abstract away from genetic details and fail to be dynam- ically sufficient. For instance, Taylor and Frank (1996) employ a weak se- lection assumption, allowing them to approximate regression coefficients using partial derivatives, in order to use standard maximization techniques for finding evolutionarily stable strategies (28). This method can be used to derive ‘approximate’ versions of Hamilton’s rule, which will be described further in section 5.2. This article focuses on the special case in which fitness effects are addi- tive. This is a starting point to examine how inclusive fitness can be calcu- lated in idealized evolutionary game theoretic models and to see whether there is any benefit to using inclusive fitness in this context. We will see that the assumption of weak selection is not essential to the calculation of inclu- sive fitness and that one can build dynamically sufficient models using in- clusive fitness. There is, of course, further work to be done to see whether 8. Evolutionary game theory and population genetics are sometimes seen as having dis- tinct methods, and other times they are seen as more or less continuous (Hammerstein and Selten 1994, 953). They are loosely grouped together here because they are similar in that models within both approaches tend to start with specific assumptions and be dy- namically sufficient. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 11 and how this can extend into the more complicated cases generally talked about in inclusive fitness theory. The relationship between these results and general versions of Hamilton’s rule, which do not require weak selection and do not assume additive fitness components, will be discussed in section 5.2. Note, however, that while the special case of additive fitness effects will not be applicable to many traits of interest in the real world, it is an important spe- cial case that has been studied extensively in a variety of contexts even out- side of the inclusive fitness framework (e.g., Eliashberg and Winkler 1981; Chakraborty and Harbaugh 2007; Maciejewski, Fu, and Hauert 2014). 4. Inclusive Fitness in Evolutionary Game Theory. Inclusive fitness and neighbor-modulated fitness are commonly viewed as ‘formally equivalent’ in that they yield the same predictions in terms of the direction of evolution- ary change. That is, they give the same conditions for when a social trait is favored by evolution (see Birch [2016] and references therein). This section will show that, in the special case discussed above, we can prove further that they also give the same prediction for magnitude of evolutionary change. Section 4.1 will prove that the two calculations of fitness are equivalent when used with the replicator dynamics, a standard model from evolutionary game theory. These results are then compared to more common calculations of in- clusive fitness in the appendix, which proves the equivalence between the rep- licator dynamics and both the neighbor-modulated and inclusive fitness cal- culations derived from the Price equation. Then, section 4.2 provides a simple example to illustrate the connections between these fitness calculations. 4.1. Inclusive Fitness and Neighbor-Modulated Fitness in Evolutionary Game Theory. In evolutionary game theoretic models, the replicator dy- namics is a standard model of the evolutionary process. Under this dynamic, if the fitness of a trait is greater than the average fitness of the population, the frequency of the trait will increase. The traits of interest dictate behavior in some social interaction, so a trait’s fitness is determined by how well it does against the other possible traits in the population (in addition to the population composition). If xt is the frequency of the trait of interest, and ft(x) is its fitness in a population of composition x, the replicator dynamics is governed by the following equation: _xt 5 xt ft xð Þ 2 �f xð Þ½ �, (6) where �f (x) is the average fitness in the population. There are a number of assumptions involved in using the replicator dynamics, notably that the population size is infinite and there are a finite number of traits. Since we are trying to see whether the trait of interest is favored, we can calculate the fitness of organisms that have the trait and the fitness of those This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 12 HANNAH RUBIN All u that do not in order to have a full description of evolutionary change accord- ing to the replicator dynamics. As mentioned, we will look at the case in which there are additive fitness effects. If we assume further that there are pairwise interactions, we can denote organism i’s social partner as 2i. In this case, we can write the neighbor-modulated fitness of the organ- isms with the trait of interest as ft xð Þ 5 P T2i Tij Þ sii 1 si2ið Þ 1 P N2ið jTið Þsii 5 sii 1 P T2ijTið Þsi2i, (7) where P(T2ijTi) is the probability an organism with the trait will interact with another organism that has the trait and where P(N2ijTi) is the probabil- ity an organism with the trait will interact with an organism that does not have the trait. Similarly, the neighbor-modulated fitness of organisms with- out the trait of interest is fn xð Þ 5 P T2ijNið Þsi2i, (8) where P(T2ijNi) is the probability an organism that does not have the trait will interact with another organism that does have the trait. The inclusive fitness of organisms with the trait of interest is (now using w for inclusive fitness to distinguish it from neighbor-modulated fitness, f ) wt xð Þ 5 sii 1 Rsi2i, (9) and the inclusive fitness of not having the trait is 0. The relatedness between interacting organisms, R, is defined as a difference in conditional probabil- ities (Skyrms 2002; van Veelen 2009; Okasha and Martens 2016). The re- latedness of a focal organism to its social partner is the probability that the social partner has a trait given the focal organism does minus the probability the social partner has the trait given that the focal organism does not: R 5 P T2i Tij Þ 2 P T2ið jNið Þ: (10) This is a measure of the degree to which the focal organism’s phenotype predicts its social partner’s phenotype.9 Since genotypes (to a certain extent) predict phenotypes, this can also be thought of as a measure of genetic re- latedness.10 9. Why this is the right definition to use is shown in Skyrms (2002). For a demonstration that the assortment rate from Grafen (1979) commonly used in the replicator dynamics is equivalent to a covariance definition of relatedness derived from the Price equation, see Marshall (2015, chap. 5, n. 1). 10. Note that relatedness is not just the probability that the two organisms share the al- lele of interest. It is a measure of their genetic similarity relative to the genetic compo- This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 13 If we start with the replicator dynamics with neighbor-modulated fitness as our measure of fitness, we can show that it is equivalent to using the replicator dynamics with inclusive fitness as our measure of fitness: _xt 5 xt ft xð Þ 2 �f xð Þ½ � 5 xt sii 1 P T2i Tij Þsi2i 2 xt sii 1 P T2ið jTið Þsi2ið Þ 2 xn P T2i Nij Þsi2ið Þð �½ 5 xt sii 2 xtsii 1 P T2ið jTi½ Þsi2i 2 xtP T2i Tij Þsi2i 2 1 2 xtð ÞP T2ið jNið Þsi2i� 5 xt sii 1 P T2i Tij Þ 2 P T2ið jNið Þð Þsi2i 2 xtsii 2 xt P T2i Tij Þ 2 P T2ið jNið Þð Þsi2i½ � 5 xt sii 1 Rsi2i 2 xt sii 1 Rsi2ið Þ½ � 5 xt wt xð Þ 2 �w xð Þ½ �: That is, neighbor-modulated fitness and inclusive fitness are equivalent when used with the replicator dynamics, a standard model of evolution used in evolutionary game theory.11 The appendix shows further that, given the assumptions stated above, us- ing the replicator dynamics is equivalent to the Price equation with either method of calculating fitness. That is, what follows are equivalent descrip- tions of evolutionary change: 1. The replicator dynamics used with neighbor-modulated fitness 2. The replicator dynamics used with inclusive fitness 3. The Price equation used with neighbor-modulated fitness 4. The Price equation used with inclusive fitness The equivalence between 1 and 3 is demonstrated in appendix A. The gen- eral strategy is the same as the one used in Page and Nowak (2002). First, show that the Price equation used with neighbor-modulated fitness (3) is de- scriptive of a population evolving according to the replicator dynamics used with neighbor-modulated fitness (1); then, show that when there are a finite number of types, 3 is also descriptive of a population evolving according to 1. Using the same strategy, we can show that 2 and 4 are equivalent. This is done in appendix B. Note that these four ways of modeling evolutionary change are shown to be equivalent in that they give the same prediction 11. For a discussion of the relationship between inclusive fitness and neighbor-modulated fitness in games that do not assume pairwise interactions, but with a constant relatedness, see van Veelen (2011). sition of the population as a whole. This is important because in studying altruism, for example, we want to know whether the benefits of altruistic acts fall on altruists suffi- ciently more often than they fall on nonaltruists. That is, the benefits must fall on altru- ists rather than nonaltruists with sufficient frequency to give them a reproductive advan- tage over nonaltruists. We will see an example of how R depends on the population’s genetic composition in sec. 4.2. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 14 HANNAH RUBIN All u for both the direction and magnitude of evolutionary change. This goes be- yond what is commonly meant by the claim that neighbor-modulated fitness and inclusive fitness are equivalent, which is that they give the same predic- tion for the direction of evolutionary change (see Birch [2016] and refer- ences therein). The next section provides a simple model using inclusive fitness in the context of evolutionary game theory. This simple illustrative example will let us see, in more concrete terms, the benefits and disadvantages of using inclusive fitness in such an idealized setting. Section 5 discusses how to un- derstand these equivalences in the context of the inclusive fitness debate. 4.2. A Simple Model: Altruism with Haploid Siblings. This section will provide an idealized model using haploid siblings to show how one can dynamically model relatedness within the inclusive fitness framework when selection is not weak. We will assume that these organisms either have the altruistic trait or not, which is completely determined by whether they receive a certain gene from their parent. So that the relationship be- tween this model and Hamilton’s rule is clear, we will assume that when an organism has the altruistic trait, it pays a cost c and bestows a benefit b on its social partner. When an organism lacks the altruistic trait, it does not pay the cost and does not benefit its social partner. In this model, an or- ganism’s social partner is its sibling. On the basis of these assumptions, we can calculate the inclusive fitness of altruists to be fa 5 2c 1 Rb: (11) The inclusive fitness of nonaltruists is 0 because they do not perform any action (relevant to our trait of interest) that affects their own or their social partner’s reproduction. Thus, altruism will spread when bR 2 c > 0. Since the relatedness of haploid siblings is determined by the genetic material they receive from their common parent, we can let p be the fre- quency of altruists in the parent generation and use this to calculate related- ness among the offspring. We will also account for a small mutation rate m in the calculation of relatedness. Once we rewrite the probabilities (accord- ing to the definition of conditional probability) so that they are easier to cal- culate from the assumptions of the model, we can calculate the relatedness of an altruist to its haploid sibling in the following way: R 5 P A2i Aij Þ 2 P A2ið jNið Þ 5 P A2i & Aið Þ P Aið Þ 2 P A2i & Nið Þ P Nið Þ 5 p 1 2 mð Þ21 1 2 pð Þm2 p 1 2 mð Þ 1 1 2 pð Þm 2 p 1 2 mð Þm 1 1 2 pð Þ 1 2 mð Þm pm 1 1 2 pð Þ 1 2 mð Þ : This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 15 Briefly, here is how to understand this calculation. The numerator of the first term is the probability of two haploid siblings both being altruists. Since there are two ways to get two altruistic offspring, we can calculate this as the probability the parent is an altruist (p) times the probability it has two offspring without mutations ((1 2 m)2) plus the probability the parent is a nonaltruist (1 2 p) times the probability it has two offspring that both have a mutation (m2). The denominator of the first term is then the frequency of altruists in the offspring generation. These offspring can come from an al- truist parent without mutation or from a nonaltruist parent with mutation. The second term is calculated similarly. The numerator is the probability that a focal nonaltruist will have an altruist sibling: the probability that the parent is an altruist and the focal organism mutates while its sibling does not plus the probability the parent is a nonaltruist and the focal organism does not mutate while its sibling does. This is divided by the frequency of nonaltruists in the offspring generation. Figure 1 shows how R will change when the population’s composition changes.12 In particular, it shows that relatedness decreases as the popula- tion becomes more uniform.13 To see why this is the case, it is easiest to look at the extremes of p 5 0 and p 5 1. When p 5 0, the parent population is entirely composed of nonaltruists. In the offspring generation, altruists only exist because of mutation. The probability an altruist has an altruist sibling is just m, the probability that the sibling also has a mutation. However, the probability that a nonaltruist has an altruist sibling is also m, the probability that the sibling has a mutation. So R 5 P(A2ijAi) 2 P(A2ijNi) 5 0. Similar reasoning applies when p 5 1. The parent population is composed entirely Figure 1. Relatedness graphed over the frequency of altruists in the parent popula- tion, for m 5 0:1. 12. This graph was created with a mutation rate of m 5 0:1, which is a fairly high mu- tation rate. This mutation rate was chosen in order to make the graphs more readable. Results similar to those described in this section can be obtained with much smaller mu- tation rates. 13. For a demonstration of this in a more general setting, see Rousset (2002). This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 16 HANNAH RUBIN All u of altruists, so any nonaltruists in the offspring generation arise through mutation. This means that although altruists are likely to have altruist sib- lings, nonaltruists are equally likely to have altruist siblings. So although P(A2ijAi) is high at 1 2 m, P(A2ijNi) is also 1 2 m, and R 5 0. We can also calculate relatedness in this model using covariances or re- gressions. Since phenotypes in this idealized model are completely deter- mined by genotypes (an organism with the altruistic gene is assumed to be an altruist), we can write R 5 Cov p, g0ð Þ Cov p, gð Þ 5 Cov g, g0ð Þ Cov g, gð Þ 5 bg 0g: (12) For any population composition, we can perform a regression to calculate the value of R, and it will give the same value of relatedness as the proba- bilistic definition of relatedness. Figure 2 gives a way to visualize why this is the case. In this model, an organism’s genetic value, g, is 1 if it has the gene for altruism and 0 if it does not. Thus, there are four possible places for data points on a graph of g versus g0: the four corners of the graph. Then, when we do a regression of g on g0, what matters is how many data points are in each of these locations. When the focal organisms’ genetic value is 1, its social partner’s genotype will on average be P(A2ijAi). Similarly, when the focal organisms’ genetic value is 0, its social partner’s genotype will on average be P(A2ijNi). As shown in figure 2, this is the intercept of the re- gression, and the regression coefficient is bg0g 5 P(A2ijAi) 2 P(A2ijNi). Figure 2. Illustration of a 5 P(A2ijNi) and bg0g 5 P(A2ijAi) 2 P(A2ijNi). This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 17 The inclusive fitness of altruists depends on R, so it also changes as the population composition changes. Figure 3 shows how the inclusive fitness of altruists compares with the inclusive fitness of nonaltruists over the pos- sible population compositions, for b 5 18 and c 5 10. Since relatedness drops off as the population becomes uniform, the inclusive fitness of altru- ists drops off as the population becomes more uniform. For many possible values of b, c, and m this means that altruists will have a fitness advantage for some area around p 5 :5, but their fitness will drop below the fitness of nonaltruists as the population becomes more uniform. These calculations of relatedness and inclusive fitness can be used in a dynamic model where frequencies of genotypes are changing over time; we use these calculations with an appropriate dynamics to see how the pop- ulation will evolve and to find the equilibria. For this model, we use the selection-mutation dynamics, which is just like the replicator dynamics ex- cept that there is an extra term that keeps track of mutations.14 Figure 4 shows the dynamical analysis of this model, using both inclu- sive fitness and neighbor-modulated fitness. Figures 4a and 4b show, re- spectively, how inclusive fitness and neighbor-modulated fitness change as the population composition changes. Figures 4c and 4d show the evolu- tionary trajectories in the population, in terms of the change in frequency of altruists. When this change is positive (when the solid line is above the X- axis, which is represented by the dashed line in figs. 4c and 4d), altruists will increase in frequency. Likewise when the change is negative, altruists will decrease in frequency. Information about the magnitude of selective Figure 3. Inclusive fitness graphed over the frequency of altruists in the parent pop- ulation, for m 5 0:1, b 5 18, and c 5 10. 14. With the selection-mutation dynamics, a population with two types will evolve ac- cording to the following equation: _xt 5 xt½ ft(x) 2 �f (x)� 1 m(1 2 2xt). Note that since this is the same as the replicator dynamics except for the mutation term, which does not depend on the definition of ft(x), we can prove that using neighbor-modulated fitness and inclusive fitness will be equivalent in the same way as in sec. 4.1. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 18 HANNAH RUBIN All u pressures is also represented; the further the solid line is from zero, the more selective pressure there is and the faster the population composition will change. Comparing figures 4a and 4b shows that the two methods of calculating fitness do yield different numerical values of fitness. However, in compar- ing the evolutionary trajectory found using inclusive fitness in figure 4c with the trajectory calculated using neighbor-modulated fitness in figure 4d, one can see that the choice between these fitness measures makes no difference for predicting the evolution of the population, either for the quan- titative predictions of the amount of evolutionary change over time or the qualitative predictions about the evolutionary outcomes based on the model. That is, in this simple model, inclusive fitness and neighbor-modulated fit- ness both give us the same answer when we ask how much altruists will Figure 4. Comparison of inclusive fitness and neighbor-modulated fitness, for m 5 0:1, b 5 18, and c 5 10. Comparing the calculations of inclusive fitness (a) and neighbor-modulated fitness (b) shows how the calculations of the two types of fit- nesses differ. Comparing the change in the frequency of altruists found using inclu- sive fitness (c) and neighbor-modulated fitness (d) shows that the evolutionary tra- jectories are the same regardless of which calculation of fitness is used. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 19 increase or decrease in frequency, across all possible population composi- tions. We can also use either type of fitness calculation to find when the change in altruists is zero, when frequencies are not changing and the population is at an equilibrium. With the values of m, b, and c chosen here, there are four equilibria, two of which are stable: one at about 1% altruists and one at about 75% altruists.15 5. Discussion. We can see from section 4 not only that inclusive fitness is perfectly well suited for use in evolutionary game theory but also that weak selection is not a necessary assumption for inclusive fitness calculations and that these calculations can be part of dynamically sufficient models. Some methods of calculating or estimating inclusive fitness may require stringent assumptions, but the calculations in general do not always require extra as- sumptions. How are we to understand this in the context of the debate over inclusive fitness? 5.1. Inclusive Fitness with Idealized Models. Some of the disagree- ment over inclusive fitness can be understood as arising from two sides of the debate emphasizing different methodologies. Recall from section 3.3 that inclusive fitness is seen as fundamentally within the quantitative genet- ics tradition, while critics of inclusive fitness tend to favor population genet- ics or evolutionary game theory. This means that inclusive fitness theorists tend to favor models that make use of abstractions, leaving details out while still providing literally true general claims about evolution. By contrast, evo- lutionary game theory, one of the preferred frameworks of the critics of in- clusive fitness, tends to provide highly idealized models, making many as- sumptions that we know are not true of any real population but that allow us to develop a simple model of a fictional population that we think is similar to the real world in important ways. As discussed in section 4.1, when there is an infinite population and a finite number of types, inclusive fitness calculations from quantitative genet- ics and evolutionary game theory are equivalent. Since quantitative methods are designed to handle continuously varying traits, assuming a finite number of types takes the methods out of the context in which they were developed and puts them into the context where dynamically sufficient models can be built. In doing so, we can get models with the kinds of properties valued by critics of inclusive fitness. 15. An equilibrium is stable when selective pressures will cause the population to return to the equilibrium if a small amount of drift changes gene frequencies in the population. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 20 HANNAH RUBIN All u One way to think about this is that the regression methods developed by inclusive fitness theorists do not, in themselves, provide models with the properties the critics of inclusive fitness argue evolutionary models should have. However, we can formulate idealized models that are dynamically sufficient and that incorporate selection that is not weak. Then, when we abstract away from the particular details of genetic inheritance or popula- tion structure assumed by the simplified models, we arrive at the abstract equations based on the Price equation, which are often used in inclusive fit- ness theory. Section 4.1 (and the appendix) showed how, when we make sim- plifying assumptions commonly made in evolutionary game theory, the replicator dynamics and the versions of the Price equation often used in in- clusive fitness theory are equivalent descriptions of evolutionary change. Section 4.2 gives an example of how the regression methods commonly used in inclusive fitness can be seen as abstract descriptions of models within evolutionary game theory. In this simple model, we can track how Cov( p, g0)= Cov( p, g) changes as the population evolves. Using covariances might seem a bit unnatural in this overly simplified case: because we can calculate the relatedness directly from the assumptions of the model, we do not need to estimate it using the methods of quantitative genetics. In fact, one might wonder whether there is any benefit to be gained from inclusive fitness in this sort of simplified model. One of the main perceived benefits of inclusive fitness is that it allows modelers to track changes in traits rather than the genes encoding for these traits (which are very difficult to discover empirically) while accounting for genetics by using relatedness (which is often not too difficult to estimate in real populations; Queller 1992). Because we abstract away from the mechanisms of genetic inheri- tance and how the genes encode for the trait of interest, summarizing this with a ‘relatedness’ parameter, we can develop a phenotypic model that still incorporates genetics in a way that can be empirically easy to measure. That is, one can account for genetics without knowing or making assumptions about the actual underlying genetics of a trait. When we switch to an evo- lutionary game theoretic or a population genetics model, like the replicator dynamics, we generally then must make assumptions about what these un- derlying genetics are. We no longer use relatedness to estimate genetic as- sortment; we can calculate the level of assortment directly. So, to a certain extent one might think that it is appropriate that the de- bate over inclusive fitness is a debate over methods: although we can use inclusive fitness in the highly simplified models of evolutionary game the- ory, in doing so we lose some of the main benefits of the inclusive fitness framework. The statistical methods used in inclusive fitness make the frame- work particularly useful, although these methods may require weak selection to split fitness effects into additive components and do not provide dynami- cally sufficient models. Further, if we view the debate as being about the meth- This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 21 ods commonly used in quantitative genetics, we can see where these criti- cisms come from.16 That is, since inclusive fitness has often been seen as fundamentally quantitative, and since one of the main benefits of inclusive fitness (incorporating genetics in a way that is easy to estimate in real pop- ulations) is generally tied up in the statistical methods arising out of quan- titative genetics, it makes sense that the debate over inclusive fitness will be in part a debate over methods. However, the status of inclusive fitness should not be decided by a debate over the use of methods for which inclu- sive fitness is seen as particularly beneficial. The model in section 4.2 demonstrates that there can still be some benefit to calculating inclusive fitness rather than neighbor-modulated fitness even in models that are highly idealized, where the level of assortment can be cal- culated directly. The explanation given for why the population does not evolve to a population composed entirely of altruists was that relatedness drops off as the population becomes more uniformly altruistic. This sort of in- tuitive explanation is not readily available when using neighbor-modulated fitness. Because the terms describing how the benefits of altruism fall differ- entially on altruists are split between two different fitness calculations (one for the fitness of altruists and one for the fitness of nonaltruists), there is no param- eter that systemically changes as the population composition changes that we can point to in order to explain why the fitness of altruists drops off as the population becomes more uniform.17 5.2. The Use of Hamilton’s Rule. Hamilton’s rule, the most famous re- sult arising out of inclusive fitness theory, has been criticized for not being generally true, for not having any predictive power, and for being mislead- ing in the absence of a particular model (Nowak et al. 2010). There is some truth to these claims. To get Hamilton’s rule in the form bR 2 c > 0, where c and b are interpreted as costs and benefits as described in section 2.2, one has to assume additive fitness components as we have been doing through- out this article. If fitness components are not additive, then the rule will not give a correct description of a condition for the spread of a trait. Addition- ally, if we only have enough information to estimate b, c, and R at a partic- ular point in time, we cannot predict the evolutionary outcome. Further, if 16. This is, of course, not to say that these are the only methods used in inclusive fitness theory but that the critiques of inclusive fitness are often wrapped up in critiques of the statistical methods (see, e.g., Allen et al. 2013). 17. Others argue that inclusive fitness is valuable because it allows us to maintain the analogy of organisms acting as if they are maximizing fitness (Grafen 2007b; West and Gardner 2013; Okasha, Weymark, and Bossert 2014; Okasha and Martens 2016) or more modestly that it allows us to explain the selection of social traits because of their casual contributions to fitness (Birch 2016). These benefits would hold regardless of the method one uses and so are not addressed here. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 22 HANNAH RUBIN All u bR 2 c > 0 when we estimate these parameters, we might even be misled into thinking that the population will eventually be entirely altruistic if we forget that the value of any of these parameters can change as the population evolves. However, inclusive fitness theorists will generally agree to this (see, e.g., Marshall 2015) but maintain that Hamilton’s rule has both predic- tive and explanatory power. It is not immediately clear where the disagree- ment lies.18 The distinction between idealizations and abstractions can again be help- ful in understanding part of the dispute. In particular, why should we expect Hamilton’s rule to be true in general? Results derived within population ge- netics and evolutionary game theory are never true in general, as they rely on idealizations to achieve their simplicity. By contrast, Hamilton’s rule is seen as a general result that is applicable to any real population one might wish to study. This fits well with its prominent role in quantitative genetics, relying on abstractions rather than idealizing assumptions to help provide “the general principles of social evolution theory” (Marshall 2015, xiv). In this vein, there is emphasis on providing a version of Hamilton’s rule that is generally true. Hamilton’s rule can be given in a very general form in which we do not have to assume any particular population structure or ad- ditive payoff affects (Gardner, West, and Wild 2011). Birch (2014b) and Birch and Okasha (2015) describe this in detail, but we can think of the ‘cost’ and ‘benefit’ terms in the rule as statistical associations between an organism’s fitness and its own genotype (a self-effect) and its social partner’s genotype (an other effect), respectively. This general version of Hamilton’s rule is true of any population. “In effect, this is because we are abstracting away from the complex causal details of social interaction to focus on the overarching statistical relationship between genotype and fitness” (Birch and Okasha 2015, 24). The question is then whether this version of Hamilton’s rule has any pre- dictive power. It can have predictive power if its components can be under- stood causally instead of just statistically. That is, if the self-effect and other- effect terms can be interpreted as ways in which the focal organism causally contributes to its own and its social partner’s fitness, we have a model that can be used to make predictions rather than just a statistical summary of evo- lution within a population. However, as Birch and Okasha (2015) explain, it is not entirely clear when a causal interpretation can be provided. There are, however, a variety of different rules that go under the name ‘Hamilton’s rule’, each of which follows from different assumptions about the evolutionary process. We can describe these versions of Hamilton’s rule as falling into three categories. There are ‘special’ versions of the rule (where 18. See Marshall (2015, chap. 6, n. 9) for an example of an inclusive fitness model where parameters can change as the population evolves. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 23 the b and c terms are interpreted as payoffs in a model) and ‘approximate’ versions (which provide marginal approximations of the general versions of the rule) in addition to the ‘general’ version described above (Birch and Okasha 2015). In the version of Hamilton’s rule in section 3.1, the b and c terms are in- terpreted as payoff from a game, or parameters in the model, so this can be thought of as a special version of Hamilton’s rule. The fact that we derived a condition bR 2 c > 0 for the spread of altruism depends on the particular payoff structure of the model. If there were nonadditive payoffs, we would have derived a different condition for the spread of altruism. Section 4 (and the appendix) illustrated how these general versions of Hamilton’s rule de- scribe the special versions from particular models. As mentioned in sections 3.1 and 3.3, there are also approximate versions of Hamilton’s rule that re- quire the assumption of weak selection to calculate relatedness or in order to split fitness effects into additive components. Thus, these rules abstract away from the particular payoff structure and so describe a wider range of cases than special forms of the rule. The assumption of weak selection, then, provides some restriction on the conditions under which approximate versions of Hamilton’s rule will apply but allows us to give an approximately correct condition for the spread of a social behavior for arbitrary payoff struc- tures. (See Birch and Okasha [2015] for more discussion.) Note that both general and approximate versions of Hamilton’s rule ap- ply for arbitrary payoff structures, but neither is dynamically sufficient. They instead allow us to perform a static analysis, comparing fitnesses at specific points in the evolutionary process (usually the points of interest are equilibria). Since this article has looked at how inclusive fitness is used in the replicator dynamics compared with approaches based on the Price equation, it has focused on the contrast between abstract models in quanti- tative genetics and idealized models in evolutionary game theory. However, that the critics of inclusive fitness prefer dynamic models over these static modeling techniques is perhaps the more fundamental disagreement in the debate. There is the additional issue of interpreting the R parameter in Hamil- ton’s rule. Although many inclusive fitness theorists recognize that R in in- clusive fitness calculations can be thought of as a general measure of cor- relation, Hamilton’s rule is still usually presented as a condition for the evolution of a trait by kin selection. However, this is an additional oppor- tunity for Hamilton’s rule to be misleading; a suggested biological or causal interpretation of the parameter might be unwarranted. Some criticisms seem to assume that Hamilton’s rule is only useful when R is a measure of kinship (Nowak et al. 2010). The thought behind these sort of critiques of Hamil- ton’s rule seems to be that when R does not have an intuitive biological in- terpretation, it is not clear what explanatory power is gained from forcing This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 24 HANNAH RUBIN All u terms into this particular inequality. The power of Hamilton’s rule then comes from using something like the statistical definitions of relatedness provided here and estimating relatedness using measures of kinship, like pedigrees.19 Since the statistical definitions of relatedness are historically explained and used as measures of kinship, adopting Hamilton’s rule as a starting point might seem to suggest an interpretation in terms of kin selection and may lead to theorists ignoring other mechanisms that generate assortment between types. Connecting the statistical and probabilistic definitions here is one way of emphasizing how the association between ‘relatedness’ and R is con- tingent: R just measures differences in conditional probabilities of interacting with certain types in the population. In this context, Hamilton’s rule might be thought of as a convenient mathematical description of the fact that there must be sufficient assortment between types in order for a trait such as altru- ism to evolve, a general point that has been made without the use of Ham- ilton’s rule (see, e.g., Skyrms 1996). A fully specified (but idealized) model, like the one in section 4.2, can connect R in Hamilton’s rule to kinship, giving it a meaningful biological interpretation. This is in line with one suggestion to avoid wrongly interpreting results in terms of kin selection, advanced by Taylor and Frank (1996) and Frank (2013), among others: formulate and analyze a model first, then afterward use Hamilton’s rule to give an intuitive explanation of the results if appro- priate. This allows us to set up the model with whatever mechanism of as- sortment we think is plausible, then use Hamilton’s rule if it helps illumi- nate important aspects of the causal structure. 6. Conclusion. While there can be benefits to using inclusive fitness, this does not mean that it is always beneficial to do so. Whether inclusive fitness or Hamilton’s rule should be used depends on the model or the population one is studying. Many of the issues involved in deciding whether to use these methods were not addressed here. This article has discussed the use of inclu- sive fitness in a special type of evolutionary model, in which pairwise interac- tions, additive fitness effects, and a finite number of types were assumed. In doing so, this article focused the discussion on issues surrounding the differ- ent methodologies favored by the critics and proponents of inclusive fitness theory, in absence of conceptual and mathematical complexities that can arise in more complicated scenarios. Looking at this simple case helped to illuminate several features of the mathematical framework of inclusive fit- ness and the debate surrounding it. 19. There are of course, other issues with applications of Hamilton’s rule aside from in- terpreting R in terms of kinship. Often in more biologically realistic models, in order to keep R defined in a way that is plausibly connected to relatedness, b and c become func- tions of R itself. These sorts of issues are dealt with by Frank (2013) and Birch and Okasha (2015), among others. This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 25 While there may be difficulties with partitioning fitness effects into the form demanded by inclusive fitness when interactions become more com- plicated, we have seen that the specific causal partition used in inclusive fit- ness does not prevent one from building dynamically sufficient models, nor does it require weak selection. Criticisms of inclusive fitness claiming that it requires these stringent assumptions are best thought of as criticisms of the types of quantitative methods generally used by inclusive fitness theorists. One can use inclusive fitness calculations in the sort of population genetic or evolutionary game theoretic models favored by these critics. In these models much of the advantage of using inclusive fitness, such as providing terms that can be easy to estimate empirically, disappears, but its power as an intuitive explanation of the evolution of social traits remains. Appendix A Equivalence with Neighbor-Modulated Fitness The Price Equation Describes the Replicator Dynamics. Following the definition provided in section 2.1, we can calculate the neighbor-modulated fitness of a pairwise interaction as follows: fi 5 sii 1 s2ii: (A1) Keeping track of probabilities of receiving payoffs was necessary in section 4.1 in order to show the connection between neighbor-modulated fitness and inclusive fitness, but since we are only dealing with neighbor-modulated fitness we can use this less complicated expression. In these calculations, we will track the change in g, genetic value. By definition, _E(g) 5 oigi _xi 1 oi _gixi. As mentioned, for simplicity we assume that there is no transmission bias and set oi _gixi 5 0. Then, since the replicator dynamics provides us an equation for _xi, we can plug the replicator dynamics into the Price equation: _E gð Þ 5 o i gi _xi 5 o i gixi sii 2 1 n oj sjj 1 s2ii 2 1 n oj s2jj ! 5 o i gixisii 2 o i gixi 1 n oj sjj 1 o i gixis2ii 2 o i gixi 1 n oj s2jj 5 E siigð Þ 2 E siið ÞE gð Þ 1 E s2iigð Þ 2 E s2iið ÞE gð Þ 5 Cov sii, gð Þ 1 Cov s2ii, gð Þ: (A2) This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 26 HANNAH RUBIN All u This is the Price equation with fitness partitioned into two components, the effect the focal organism has on its own fitness and the effect the social part- ner has on the focal organism’s fitness. Theorists often derive this from the original Price equation in order to use neighbor-modulated fitness calcula- tions and introduce relatedness calculations (see, e.g., Queller 1992). Hamilton’s rule can easily be derived from this equation. Since the way an organism affects the fitness of itself and others is (to a certain degree) predicted by its phenotype, we can write both fitness terms as the following regressions: sii 5 asiip 1 bsiip p 1 εsiip; (A3) s2ii 5 as2iip0 1 bs2iip0 p 0 1 εs2iip0: (A4) Since the a’s are the intercepts of the regression, they are constants and can- not covary with g. The ε’s are error terms, which do not covary with g when payoffs are additive (Queller 1992). So, plugging (A3) and (A4) into the last line of (A2), we are left with _E gð Þ 5 bsiipCov p, gð Þ 1 bs2iip0Cov p0, gð Þ: (A5) When we can interpret bsiip as a cost c and bs2iip0 as a benefit b, this gives us _E(g) > 0 when b � Cov p 0, gð Þ Cov p, gð Þ 2 c > 0, (A6) where Cov( p0, g)=Cov( p, g) is the neighbor-modulated fitness version of relatedness. The Replicator Dynamics Describes the Price Equation. When there are a finite number of types, gi can be written as an indicator function: gj 5 1 if  i 5 j 0 otherwise : ( For Page and Nowak (2002), who were considering phenotypes rather than genotypes, assuming a finite number of types was a restriction. Here, in considering genotypes, it is a natural assumption to make. We can then use this indicator function in the Price equation with two fitness components derived above and simplify: This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DEBATE OVER INCLUSIVE FITNESS 27 _E gð Þ 5 Cov sii, gð Þ 1 Cov s2ii, gð Þ 5 E siig ð Þ 2 E siið ÞE gð Þ 1 E s2iigð Þ 2 E s2iið ÞE gð Þ 5 o j gj xjsii 2 o j gj xj 1 n oj sjj 1 o j gj xjs2ii 2 o j gj xj 1 n oj s2jj 5 xisii 2 xi 1 n oj sjj 1 xis2ii 2 xi 1 n oj s2jj 5 xi fi xð Þ 2 �f½ �: (A7) Since gj 5 1 when i 5 j and 0 otherwise, ojgj xj 5 xi, and this simpli- fies to yield the replicator dynamics. Appendix B Equivalence with Inclusive Fitness The Price Equation Describes the Replicator Dynamics. This is done in the same way as appendix A, except we take into account that the genetic value of the focal organism times its relatedness to its social partner is a measure of the social partner’s genetic value: _E gð Þ 5 o i gi xi sii 2 1 n oj sjj ! 1 xiri2i si2i 2 1 n oj sj2j !" # 5 o i gixisii 2 o i gixi 1 n oj sjj 1 o i giri2ixisi2i 2 o i giri2ixi 1 n oj sj2j 5 Cov sii, gð Þ 1 Cov si2i, g0ð Þ: (B1) This is again a version of the Price equation where the fitness effect is split into two components. Here, though, fitness is split into the effect the focal organism has on its own fitness and the effect the focal organism has on its social partner’s fitness. In order to relate this to Hamilton’s rule, we can again notice that the fit- ness components are predicted by phenotype. Since in this case the focal organism causes the fitness effects, both for itself and its social partner, the phenotype of the focal organism predicts both fitness effects. So, we use the phenotype of the focal organism in both regressions: This content downloaded from 129.125.136.103 on May 31, 2018 00:28:26 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 28 HANNAH RUBIN All u sii 5 asiip 1 bsiip p 1 εsiip; (B2) s2ii 5 asi2ip 1 bsi2ip p 1 εsi2ip: (B3) We can then plug (B2) and (B3) into the last line of (B1) and rearrange to obtain the inclusive fitness version of Hamilton’s rule: _E(g) > 0 when b � Cov p, g 0ð Þ Cov p, gð Þ 2 c > 0: (B4) The Replicator Dynamics Describes the Price Equation. We can again let gi be an indicator function and write _E gð Þ 5 Cov sii, gð Þ 1 Cov s2ii, g0< i >ð Þ 5 o j gj xjsii 2 o j gj xj 1 n oj sjj 1 o j gj rj2jxjsi2i 2 o j gj rj2jxj 1 n oj sj2j 5 xisii 2 xi 1 n oj sjj 1 xiri2isi2i 2 xiri2i 1 n oj sj2j 5 xi fi xð Þ 2 �f½ �: (B5) Again, this simplifies to yield the replicator dynamics. 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