Multi-Level Selection and the Explanatory Value of Mathematical Decompositions Christopher Clarke British Journal for the Philosophy of Science∗ Abstract Domulti-level selection explanations of the evolutionof social traits deepen the understanding provided by single-level explanations? Central tomultilevel explanations is a mathematical theorem, the multi-level Price decomposition. I build a framework through which to understand the explanatory role of such non-empirical decompositions in scienti c practice. Applying this general framework to the present case places two tasks on the agenda. e rst task is to distinguish the variousways bywhich onemight suppresswithin-collective variation in tness, or indeedbetween-collective variation in tness. I distinguish ve such ways: increasing retaliatory capacity; homogenising assortment; collapsing either tness structure or character distribution to a mean value; and boosting tness uniformly within collectives. I then evaluate the biological interest of each of these hypothetical interventions. e second task is to discover whether one of the right-hand terms of the Price decomposition measures the effect of any of these interventions. On this basis I argue that themulti-level Price decomposition has explanatory value primarily when the sharing-out of collective resources is 'subtractable'. us its value is more circumscribed than its champions Sober and Wilson ([1998]) suppose. 1 Single-Level andMulti-Level Selection 2 ree Conditions on Explanatory Decompositions 3 e Multi-Level Price Decomposition 4 e Biological Interest Problem for Sober andWilson ∗Forthcoming 2014/2015. ismanuscriptwas created July 23, 2014- e content is identical to the copy submitted for nal proo ng. 1 5 Explanatory DepthWhenever Resources are Subtractable 6 Other Alterations to Within-Collective Variation 7 Alterations to Between-Collective Variation 8 Alternative Approaches to Explanatory Depth 9 Conclusion 1 Single-Level andMulti-Level Selection One of the key variables in evolutionary theory is character– tness covariance; the degree to which those organisms that possess a given character are statistically more likely to be tter than those organisms that don't possess the character. Take for example a lion's inclination to hunt socially rather than on its own. Suppose that the tness of each lion in a population is given by Table 1. So it's determined by whether or not that lion has this inclination to hunt socially, and by whether or not the lions that it interacts with have this inclination. Making some simple assumptions one can calculate that the covariance between character and tness in this case is f0(1 − f0)(4f0 − 1); where f0 is the proportion of the lion population who are presently social hunters.1 Consider the case in which the population is evenly divided at present between social hunters and lone hunters; in other words f0 = 12 . In these circumstances it follows that there is a positive covariance between social hunting and tness, namely of 1 4 . is fact about covariance is key because it can provide a simple explanation of why the frequency of social hunters increased from the present generation of lions to the next generation. Explanation: lions inclined to hunt socially were-in the circumstances above-more likely to be tter and this caused such lions to have relatively more offspring, most of whom inherited this inclination. And so the frequency of social hunters increased. who interacts with who interacts with social hunters lone hunters Fitness of social hunter 4 0 Fitness of lone hunter 1 1 Table 1: Example Fitness Matrix For reasons that will soon become clear I will call such explanations 'singlelevel selection' explanations. Such explanations are underwritten by the Robertson–Price identity. is equation describes how the covariance of character and 1Assume that lions form pairs of lions completely at random. 2 tness determines the increasedprevalence of a character in apopulation (Robertson [1966]; Price [1970]). is equation follows deductively from some common simplifying assumptions: that there is no migration into or out of the population; that the character in question is heritable and inherited without 'transmission bias'; and that there are no stochastic effects at work (Price [1972]; Sober [1984]; Okasha [2006]). In the wake of Darwin's On the Origin of Species single-level selection explanations have become so commonplace in evolutionary biology as to be unremarkable: It would be advantageous to the Melipona [bee], if she were to make her cells closer together, and more regular in every way than at present; for then, as we have seen, the spherical surfaces would wholly disappear, and would all be replaced by plane surfaces; and the Melipona would make a comb as perfect as that of the hive-bee. . . . us, as I believe, the most wonderful of all known instincts, that of the hive-bee, can be explained by natural selection . . . (Darwin [1859], pp. 174–75) Moving from explanations of concrete biological cases over to abstractmathematical models, this 'single-level' emphasis upon character– tness covariance remains commonplace.2 For example in textbook treatments of evolutionary theory one sees tness matrices (such as Table 1) being used to identify the circumstances under which this character– tness covariance will be positive, negative or zero (McElreath and Boyd [2007], p. 203). In the lion hunting case, for example, this depends upon the initial frequency f0 of social hunters in the population. Indeed one could describe the search in evolutionary game theory for so-called evolutionary stable states or strategies roughly as the search for the conditions under which character– tness covariance is zero: f0 = 0, or 14 , or 1 in this example.3 is illustrates how the covariance of character with tness across the whole population is a central explanatory variable. Now in its multi-level form4 the socalled Price equation decomposes this central variable into the sum of two other variables (Okasha [2006]). To put it brie y, one of these variables is supposed to relate in some sense to 'selection at the level of individual lions' and the other to 'selection at the level of groups of lions'. I will say much more about these two variables in Section 3. For now it will suffice to say that both these variables 2McElreath and Boyd ([2007], §5.1) call the use of single-level explanations the 'personal tness approach' to evolution. 3Note that this is a necessary but not sufficient condition for a distribution of characters across a population to constitute an evolutionarily stable distribution. 4 e multi-level Price equation is a variation of the Price equation (Price [1972]), which itself is a more general form of the Robertson–Price identity (Robertson [1966]; Price [1970]). 3 are statistical functions of the distribution of character and tness among lion groups. Consider for example those cases in which selection for social hunting at the level of lion groups outweighed selection against social hunting at the level of individual lions. (Again, muchmore on this in Section 3.) In such cases themultilevel Price decomposition suggests a controversial explanation for the increase in the prevalence of social hunters from one generation to the next: group-level selection for social hunting outweighed individual-level selection against social hunting. As a consequence explanations that employ these two variables from themulti-level Price decomposition are o en called 'multi-level selection' explanations. e main focus of this paper will be the contrast between multi-level selection explanations and single-level selection explanations. And this will leave no time to say anything about the explanations afforded by sel sh-gene theory (Dawkins [1976], [1982]) or inclusive tness theory (Hamilton [1964]; Frank [1998]). Moreover considerations of space preventme fromdiscussing the alternative form of multi-level selection theory based on contextual analysis (Heisler and Damuth [1987]; Goodnight et al. [1992]) rather than the multi-level Price decomposition. In contrasting the multi-level explanatory framework with the single-level framework I do not mean to imply that these frameworks offer competing explanations. As I de ne the concept, two explanations of the same case compete exactly when it is highly implausible, if not impossible, that they both be correct. Take for instance the explanation that the CIA shot Kennedy, and the explanation that Soviet agents shot Kennedy. In fact I'm happy to accept the so-called pluralist idea that multi-level explanations and single-level explanations-and for that matter sel sh-gene and inclusivetness explanations-o en posit the sameprocess (Kerr andGodfrey-Smith [2002]) and so each framework canplausibly provide a correct explanation of the same case. Instead, by contrastingmulti-level explanationswith single-level explanations, what I aim to do is to address the issue of explanatory depth. For example an explanation of why a car accelerated that speci es the car's mechanics or the psychology of its driver provides a deeper explanation than merely citing the fact that the accelerator pedal was pressed. is shows how an explanation can be deeper than another without competing with it. On the one hand Sober and Wilson ([1998]) think that explanations of the evolution of social characters that employ the multi-level Price decomposition are deeper than single-level explanations. But on the other hand there are those who disagree. Maynard-Smith disagrees because he ndsmulti-level explanations altogether dubious;5 whereas 5See Okasha ([2005], pp. 1000, 1004) for references and discussion of the complexities of 4 Dugatkin and Reeve ([1994], pp. 121, 124) disagree because they think multilevel explanations are fully equivalent to single-level explanations.6 e distinctive strategy of this paper will be to separate this issue of explanatory depth from the other issues in the 'levels of selection' literaturewithwhich it is entangled. In addressing it I will draw instead upon the general philosophical literature on explanation. us I will not discuss what it means for selection to 'act at a particular level' such as that of the group (Lloyd [1986], [2000]; Okasha [2006]), norwhat it takes for something such as a group of organisms to count as a 'biological individual' (Clarke [Forthcoming]), nor whether groups can be vehicles inDawkin's ([1982]) sense, or interactors inHull's ([1981]) sense. Indeed one could perhaps think that there is no fact of thematter about such questions,7 questions concerning vehicles or interactors say, but still think that there is a fact of the matter about the topic of this paper, namely the depth of the multi-level selection framework. is focus on the explanatory depth of the multi-level Price decomposition will also raise wider philosophical questions. For the decomposition is a mathematical theorem: it's truth isn't contingent on what the world happens to be like; and one doesn't need any scienti c evidence to know that it's true. Consequently one might wonder how such 'non-empirical' propositions could play a genuine role in scienti c explanation (Pincock [2007]; Baker [2009]; Batterman [2010]). As Lange and Rosenberg ([2011], p. 593) point out in response to Sober ([2011]), it is 'difficult to see how [propositions in evolutionary theory that are knowable a priori] could gure in causal explanations'. So I will look beyond the philosophy of biology literature to explore how non-empirical decompositions such as the multi-level Price theorem can play an explanatory role. e suggestion will be-to put it somewhat laconically-that such decompositions highlight those constitutive relationships that help glue different factors in our explanatory reasoning together. Applying this suggestion to the multi-level Price decomposition shows that this decomposition has explanatory value, I will argue, primarily in cases in which the sharing-out of resources is 'subtractable'. us the range of cases across which the decomposition provides deep explanations is more modest than its champions suppose. Maynard Smith's views. 6 ings are not quite as clear cut as this. See Dugatkin and Reeve ([1994], p. 123). What is clear is thatmuch confusion has been generated in contrastingmulti-level selection explanations with their 'individualist' rivals, but not making clear what rivals one has in mind. 7See Sterelny ([1996]), Okasha ([2004a]), and Sarkar ([2008]) for discussion of this sort of pluralism. 5 2 ree Conditions on Explanatory Decompositions What does one need to know in order to explain a phenomenon? In the philosophical literature a very popular suggestion is that one needs to know what would happen under certain 'hypothetical alterations' to the system in question. Would the phenomenon still have occurred if certain things had gone differently (Lewis [1986]; Woodward [2003])?8 To explain why the economy shrank in 2008 for example it helps to know that the size of the economy would have been greater if banks had been more tightly regulated. So I am going to follow Lewis and Woodward in assuming that to explain is to answer important whatif-things-had-been-different questions. Accordingly the depth of an explanation is in proportion, roughly speaking, to the number of what-if questions it allows one to answer concerning important hypothetical alterations to the system in question. is measure of explanatory depth is by no means uncontroversial, but I will wait until Section 8 to examine it in further detail. One quali cation: to explain why the economy shrank it does not help to know that the size of the economy would have been greater if extra-terrestrials had landed from outer space and donated a billion barrels of oil to the treasury. e what-if question about bank regulation is therefore different to the question about extra-terrestrial oil donation in that answering the former has explanatory value, but answering the latter does not. I will assume that the standard account of such differences is correct: we just happen to be more interested in hypothetical alterations to bank regulation than in far-fetched questions about extraterrestrial oil donations.9 e importance of a hypothetical alterationdepends in this respect upon our personal interests; and thereby so does explanatory depth according to my measure.10 (Accordingly, the notion of what is interesting to biologists will play a central role later in this paper.) I will now use the Lewis–Woodward approach to explanation in order to build a toy model of how a non-empirical decomposition can play a modest role in explanation. Consider the following decomposition: the number of guests booked into a hotel is equal to the number of guests who are on holiday to ski plus the number of guests who are not on holiday to ski. is decomposition is non-empirical, guaranteed by the logical truth that everyone is either a skier or a non-skier. Compare this decomposition for example to a second decomposition, a decomposition of the guests into those guests with blond hair and 8Lewis ([1986]) doesn't put it in quite these terms. He says that to explain is to cite a cause; but for Lewis to cite a cause is just to say what could have gone differently such that the phenomenon wouldn't have occurred. 9But see Hart and Honore ([1965]) for an alternative account. 10For an account that emphasises interests but notwhat-if questions see vanFraassen ([1977]) and Achinstein ([1983]). 6 whose name begin with a 'K', on the one hand, and those guests who don't have both attributes on the other hand. e question I want to ask is this: when will a non-empirical decomposition (for example the rst decomposition) be more explanatorily valuable than any of the in nitely many other non-empirical decompositions that one might think of (such as the second decomposition)? To explore this question let us consider how the rst decomposition ts into the following story. (1) is winter has been unusually warm and so the average depth of snow on the Brixental ski slopes has been half a meter, in contrast to last winter's three meters. As a result (2a) there are two hundred skiers booked into the Brixental hotel, in contrast to last winter's nine hundred. (2b) And, like last year, there were one hundred non-skiers also booked into the Brixental hotel. Most of these non-skiers were there for the annual Wittgenstein conference. So applying our decomposition to (2a) and (2b) we see can see that (2) the hotel has had under ve hundred guests rather than over ve hundred as they did last winter. As a result the hotel has gone bankrupt. Note that the low number of guests (factor 2) on its own provides a simple explanation of the bankruptcy. And this explanation is made deeper by adding the point about the lack of snow (factor 1). But to have a really satisfying explanation of the bankruptcy one needs also to be able to answer what-if questions of the following form: (Z) what if xmeters of snow had fallen, and other factors like the Wittgenstein conference been arranged in such-and-such a way?11 To answer such what-if questions one will typically reason as follows. 'In this hypothetical what-if scenario there would be ga skiing guests on account of the snow; and there would be gb non-skiing guests on account of the other factors such as the Wittgenstein conference. According to our decomposition this constitutes there being g guests in total. ere would therefore be under ve hundred guests, and so the hotel would be bankrupt. Alternatively: there would be over ve hundred guests, and so the hotel would not be bankrupt.' Let's be fully explicit about how this works. To know our decomposition is to know a constitutive relationship X : g is constituted by ga and gb. And knowing this constitutive decomposition X is in practice how we come to know the following causal determination relationshipsY : an interesting rst factor (snowfall) combines with other factors (such as theWittgenstein conference) to determine a second factor (total guests) which in turn determines the to-be-explained phenomenon (bankruptcy). And knowing these causal determination relationshipsY in turn allowsus to answer some importantwhat-if questionsZ. us this knowledge deepens our simple 'undecomposed' explanation (of the bankruptcy in terms of the total number of guests alone). In short our decomposition high11Arranged, for example, such that there were gb non-skiing guests. 7 lights a constitutive relationship that helps us to glue together the relevant factors in our explanatory reasoning. In principle, of course, one could know these causal determination relationships Y without knowing the constitutive decomposition X. So the explanatory role of our decomposition is what one might call an 'ancillary' one. It is dispensable in principle, but not in practice. It will be important for later to abstract three crucial aspects from this toy example concerning the guests at the Brixental hotel. Independence aspect: the value of a term on the right-hand of the decomposition (gb non-skiing guests) is independent of the rst factor (snowfall). In other words its value is preserved by some hypothetical alteration to that rst factor (eliminating snowfall). Observe that this aspect of the Brixental case is crucial in that, without it, knowledge of the constitutive relationship X would be of no real help in calculating the causal dependencies Y. Later on, I will repeatedly draw upon the observation that this independence aspect of the Brixental case is equivalent to the following condition. Independence aspect (alternate rendering): the effect upon the value of the le hand term (g total guests) of this alteration (eliminating snowfall) is measured by the attendant change to the value of a right-hand term in the decomposition (ga skiing guests). A er all, the other right-hand term (gb) is a residual term that measures the effect of other factors only (such as the Wittgenstein conference). Interestingness aspect: this hypothetical alteration (eliminating snowfall) is interesting. is aspect of the Brixental case is crucial in that, without it, the what-if question Z would not be an important one. Hence answering this question would be of no explanatory value according to the Lewis–Woodward thesis about explanation; just as in my extraterrestrial oil donation example. Knowledge aspect: one knows how the value of the le -hand term (g total guests) determines the to-be-explained phenomenon (bankruptcy) in the circumstances. is aspect of the Brixental case is crucial in that, without it, one could not use causal decomposition X to answer what-if question Z. My conclusion is this: the Lewis–Woodward approach to explanation issues in three criteria that are in general individually necessary and jointly sufficient for a non-empirical decomposition to provide explanatory value in the above manner. at is, to issue in an explanation of greater depth than an explanation (of the bankruptcy) in terms of only the le -hand term of a decomposition (the total number of guests). I note in passing that the decomposition involving guests with blonde hair and names beginning with 'K' would in normal circumstances fail both the independence criterion and the interestingness criterion. Some non-empirical decompositions are evidently more explanatorily valuable than others. 8 I emphasise that the above are criteria only for the explanatory value of nonempirical decompositions, not empirical ones. To extend them to the case of empirical decompositions would be mistaken. For example the ideal gas law ln(P ) = ln(V ) + ln(T ) has clear explanatory value. But it fails my independence criterion: when a gas is heated in an expandable chamber both the value of the ln(V ) term and of the ln(T ) term are altered as a result. So my rst criterion is not necessary as regards the explanatory depth of empirical decompositions, as opposed to non-empirical ones. Conversely the length of Edward Heath's premiership is equal to the length of Romano Prodi's premiership plus the length of John F. Kennedy's. is equation may well meet all my criteria, but it is too accidental to have any explanatory value.12 So my three criteria are also not jointly sufficient as regards the explanatory value of empirical decompositions, as opposed to non-empirical ones. At any rate the explanatory role played by the toy decomposition involving hotel guests, I will suggest, is the same explanatory role that many non-empirical decompositions play in the actual practice of science; inparticular themulti-level Price decomposition in evolutionary biology. 3 eMulti-Level Price Decomposition To spell out themulti-level Price decomposition letme introduce some standard formalism. Consider a population of individuals, be it a population of genes, cells, organisms or social groups; although the most intuitive case is when one takes individuals to be individual organisms. Take an arbitrary individual i. Let ωi denote that individual's (relative) tness.13 Let zi denote the degree to which individual i possesses a particular character in which one is interested. is character of interest will conventionally be a 'pro-social' character such as a lion's being inclined to hunt cooperatively or a vampire bat's being inclined to donate blood to other vampire bats who are in need. e multi-level Price decomposition states that:14 Cov(ω, z) = Cov[Expg(ω), Expg(z)] + Exp[Covg(ω, z)] (1) What do these three terms mean? e le -hand term Cov(ω, z) denotes the covariance of character with tness across the whole population: to what extent do individuals who score high on character z tend statistically to be tter than 12It certainly isn't invariant under interventions (Woodward [2003]). In contrast note that non-empirical decompositions are by de nition maximally invariant under interventions. 13Relative tness is de ned to be an individual's absolute tness divided by the mean tness of all individuals in the population. I shall henceforth use ' tness' to mean relative tness. 14See Price ([1972]) andHamilton ([1975]) for a seminal formulation. SeeOkasha ([2006]) for a very clear commentary. 9 individuals in the populationwho score lowon z? For example are grouphunters tter on average than other lions? Now imagine that our population of individuals is partitioned into collectives; so each individual is a member of exactly one collective. (I will leave it entirely open what it is for an individual to be a member of a collective.) So Covg(ω, z) denotes the covariance of character with tness within collective g, rather than across the whole population: to what extent do individuals in collective g who score high on character z tend statistically to be tter than those in the same collective who score low on character z? us the third term of the decomposition Exp[Covg(ω, z)] is an average of this measure across the whole population: on average do group hunters tend statistically to be tter than those in the same collective who hunt alone? Finally the second term. Expg(ω) is the average tness of the members of collective g. Let's call this the collective's tness. Similarly Expg(z) is the average character of themembers of collective g. Let's call this the collective's character.15 So the second termof themulti-level Price decompositionCov[Expg(ω), Expg(z)] is the covariance between these two variables: to what extent do collectives who score high on character z tend statistically to be tter than collectives who score low on character z?16 Putting this less technically and more intuitively: the second term of the decomposition measures the association between collectives of (collective) tness with (collective) character, whereas the third term measures the association of (individual) tness with (individual) character within collectives. Importantly the multi-level Price decomposition is a mathematical theorem, guaranteed by the logic of covariance and of expectation. It is worth noting at this point that my third criterion for a mathematical decomposition to have explanatory value-the knowledge criterion-just requires that we know how the value of the le -hand term determines our to-beexplained phenomenon in the circumstances. And one does in this case. For one knows the Robertson–Price identity discussed in Section 1, which formally underwrites the intuition that the tter character z is, so to speak, the more it will increase in frequency. So one knows how the value of the le -hand term (the degree of character– tness covariance in the whole population) determines our to-be-explained phenomenon, the evolution of character z. So the knowledge criterion is satis ed. Consequently, this paper will focus on the circumstances under which themulti-level Price decomposition satis es the independence and the interestingness criteria. 15 us I am focusing on what Damuth and Heisler ([1988]) call multi-level selection type one, rather than type two. 16Strictly speaking the summation Cov[] is over individuals in the population not collectives. So strictly speaking: to what extent do individuals that are part of collectives who score high on character z tend to be members of t collectives? 10 4 e Biological Interest Problem for Sober andWilson One suggested explanatory role for the multi-level Price decomposition emphasises the factor of within-collective variation (Sober and Wilson [1998]). And by this I strongly suspect that Sober and Wilson mean variation in tness rather than variation in character.17 Sober and Wilson's key claim is that the third termof the decompositionmeasures the effect ofwithin-collective variation (pp. 32–33, 73–75). (Sober andWilson also claim that the second termof themultilevel Price decomposition measures the effect of between-collective variation. I will set the examination of this claim aside until Section 7.) e general framework developed in Section 2 shows why Sober and Wilson's key claim bears upon the explanatory value of the multi-level Price decomposition. For this key claim is more or less an application of my independence criterion for explanatory value. Imagine eliminating within-collective variation in tness. Let ε denote the attendant effect upon character– tness covariance across the whole population; that is, the effect on the value of the le -hand term of the multi-level Price decomposition. Independence criterion (the alternate rendering): this effect ε is measured by the attendant change in the value of a right-hand term in the decomposition; for example the third term. So Sober and Wilson's key claim is more or less an application of the rst of my three criteria for the multi-level Price decomposition to have explanatory value. Unfortunately Sober andWilson do not provide an argument for this key claim. What follows is the most plausible way of developing such an argument in my view. Take a population of individuals in an environment and consider the ' tness structure' generated by that environment. is tness structure is the mapping which speci es how an individual's tness is determined by her character and by the characters of the individuals with whom she interacts. Take for illustration the function ωi = 2Expg(z) − 12zi. Now consider a hypothetical alteration to this tness structure such that each individual in any given collective g will now enjoy the same tness as the other individuals in collective g. More precisely the tness an individual is to enjoy under this alteration is identical to the mean tness-prior to this alteration-of the individuals in her collective. Sticking with the above illustration, ωi becomes equal to 2Expg(z)− 12Expg(z). In other words it's equal to 3 2 Expg(z). Call such alterations 'Structural Collapse to the Mean' (SCM) alterations. is alteration is one straightforwardway of eliminating any 17See pp. 54, 66–67, 80–91, 115, 139 of Sober and Wilson ([1998]) for textual evidence; indeed see Sober ([1984]). At any rate my criticism of Sober and Wilson' idea as reconstructed in Sections 6 and 7 will work just as well if you substitute ' tness' for 'character' and 'character' for ' tness'. is is because covariance is symmetric: Cov(ω, z) = Cov(z, ω). So themathematical reasoning inmy criticismwill hold even if Sober andWilsonmean 'variation in character' rather than 'variation in tness'. 11 within-collective variation in individual tness. Notehowever that theSCMalterationpreserves themean tness of themembers of each collective, and thus preserves collective tness. But individual character is also preserved; so collective character is preserved. us the SCM alteration preserves the covariance of collective tness with collective character. In other words SCMpreserves the value of the second term of themulti-level Price decomposition. And this is equivalent to saying that the independence criterion for explanatory value, on its original rendering, is satis ed here.18 Incidentally, let ε denote the effect of SCMupon character– tness covariance across the whole population; that is, its effect on the value of the le -hand term in the decomposition. SCMhaving preserved the value of the second term, it follows that this effect ε is measured by the attendant change in the value of the third term in the decomposition. And this is equivalent to saying that the independence criterion for explanatory value, on the alternate rendering, is satis ed here. Having established that my rst criterion for explanatory value is satis ed with respect to hypothetical SCM alterations, can we now establish my second criterion, the interestingness criterion? Is theSCMeliminationofwithin-collective variation in tness especially interesting to biologists? I will now argue that are some cases in which the answer is no. Recall the example in which ωi = 2Expg(z) − 12zi which we can rewrite as 2Expg(z) − zi − 12(−zi). Let's imagine that this describes the tness structure for the Polistes fuscatus wasp in a given environment. Wasps with high z scores are hard workers. And wasps enjoy tness bene ts when they are in a collective whose members are hard working; hence the 2Expg(z) term. But working hard requires a costly expenditure of energy; hence the−zi term. But those lazywasps who do not work hard run the risk of being stung by the queen, and indeed the risk of other forms of retaliation from the queen (Gamboa et al. [1990]); hence the−1 2 (−zi) term. In the case of the Polistes wasp there is indeed a highly interesting way of altering the tness structure that eliminates within-collective variation in tness. One imagines an increases in retaliatory capacity: queens are better able to identify the lazy workers, or the queens increase the severity of the punishment for those who are so identi ed. In particular it will be interesting to know what would happen were the 1 2 coefficient-the retaliation parameter so to speak-to be altered such that each individual in a collective enjoys the same tness; withincollective variation thus being eliminated. One can calculate that the answer is that the coefficient becomes 1 and that ωi becomes 2Expg(z). It is crucial to note however that this highly interesting hypothetical alter18Moreover one can easily show that SCM alters the value of the third term to zero. So the magnitude of this attendant change in the third term is given by the unaltered third term itself. 12 ation to tness structure is distinct from the Structural Collapse to the Mean alteration I considered above. A er all, recall that the SCM alteration has it instead that ωi becomes equal to 32Expg(z); not to 2Expg(z). In contrast, there is nothing of especial biological interest I contend in the SCM alteration applied to our wasp population. Such alterations have no greater interest than hypothetical alterations that eliminate within-collective variation by letting ωi become 7 13 Expg(z), or to lnExpg(z), or that collapse individual tness to the collectivemedian or the collective mode, or so on. is illustrates how the Structural Collapse to the Mean alteration is not biologically interesting across every case in general. In other words SCM does not in general satisfy my second criterion for explanatory value. But I've been considering hypothetical SCM alterations in an attempt to develop Sober and Wilson's analysis into an argument that establishes a general explanatory role for the multi-level Price equation. And one can now see that this attempt has failed. I emphasise that my intention here is not to criticise the application of the multi-level Price theorem to the Polistes wasp case. A er all, the theorem is just a mathematical truth. Rather I am urging a more sanguine assessment of its explanatory value in this case. A er all, nothing that I've said so far establishes that the decomposition adds any explanatory depth. ere will, of course, be some theorists who will resist my conclusion here by objecting to my relatively narrow conception of what is biologically interesting. I cannot hope to fully persuade such objectors. But I do hope to persuade them of a somewhatmoremodest point: the SCMalteration in the wasp case is just as interesting as the in nity of other hypothetical alterations to the distribution of tnesses-such as those that let ωi become 713Expg(z), or lnExpg(z), or so on. It follows that, in the case of the Polistes wasp, we have not established that the explanatory value of the multi-level Price decomposition will be any greater than the in nity of other mathematical decompositions of character– tness covariance. We've not identi ed any special explanatory value for the decomposition in the case of the Polistes wasp. 5 Explanatory DepthWhenever Resources are Subtractable One question naturally arises from the last section. Can one appeal to SCM alterations in order to establish the explanatory value of themulti-level Price equation in a more limited class of cases, rather than across all cases in general? is section will identify a class of cases in which Structural Collapse to the Mean alterations are biologically interesting. In other words I identify a class of cases that satisfy my second criterion (interestingness) for explanatory value. ese cases are, namely, those cases in which the sharing-out of resources amongst the 13 individuals in a collective is, in the parlance of economics, subtractable. But I've already shown in Section 3 that my third criterion (knowledge) is satis ed by the multi-level Price decomposition. And I've just shown in Section 4 that my rst criterion (independence) is satis edwith respect tohypothetical SCMalterations. So all my three conditions are satis ed here. us this section establishes the explanatory value for the multi-level Price decomposition in a limited class of cases, namely those in which the sharing-out of resources is subtractable. Before getting down to business, I will need to invest a substantial amount of time carefully illustrating what I mean by subtractability. An excellent illustration of the subtractability of resources in a biological context is found in the literature on social or cooperative foraging (Giraldeau and Caraco [2000]). To see this, note that many social foraging models can be thought of as having two parts. Consider the amount of food that a collective of foragers will gather. e resource acquisition part of themodel describes how this amount depends upon the cooperative behaviour of the members of the collective, and upon the environment. e resource sharing-out part of themodel describes how this amount is divided amongst the individual members of the collective. Now, to talk of resources being genuinely 'shared out' here presupposes the following: there is an 'analytic separation' of the allocation of resources into a mechanism whereby a collective acquires its resources, and a mechanism whereby these resources are shared out amongst the individual members of the collective. By this very stipulation, resource sharing-out is subtractable only if these mechanisms are analytically separable. is is the rst of my two individually necessary and jointly sufficient conditions for subtractability. Let me be clear about analytic separation. I don't intendmy de nition of analytic separation to turn upon any substantial notion of 'mechanism'. Similarly I allow that two analytically separable mechanisms may operate simultaneously, that they may interact, and that they may have overlapping parts. Instead, what I mean by 'analytic separation' is that there is a biologically interesting alteration to the manner in which resources are divided out amongst individuals, an alteration which leaves unaltered the manner in which resources are collectively acquired. Tomake this intuitive, consider for example those 'scroungers' who have 'cheated' by refusing to cooperate during foraging. Inmany cases it is biologically interesting to ask what would occur if it became more difficult for scroungers to gain access to the food that the collective has foraged. What if, in the extreme, scroungers were excluded from these resources altogether? My second condition on subtractability is also rather intuitive. Rough and informal version: whenever one individual consumes a resource it must reduce the quantity of the resource available for other users to consume. To spell out the second condition formally I will make the simplifying assumption that one 14 can use a single variable Rg to quantify the resources that a collective g has acquired. In a simple foraging case this is just the quantity of food that the collective has foraged. Furthermore I will assume that Rg is entirely determined by the 'pro-social' character of each member of collective g, characters which one might represent by the vector zg. (In a simple foraging case this pro-social character might measure how much energy the individual in question chooses to invest in the group hunt.) To emphasise this point I will o en write collective resources Rg as Rg(zg) highlighting that it is a function of zg, and indeed of zg alone. Now consider the sum total of the tnesses of the members of a collective g; in formal terms ∑ g ωi. e sharing-out of collective resources is subtractable I stipulate only if this total tness is entirely determined by collective resources Rg(zg); more speci cally just in case this total tness is an increasing function of collective resources. Choose the right scale on which to measure resources and this becomes the requirement that the tness structure is characterised by:∑ g ωi = Rg(zg) (2) Why is this requirement a tting formalization of the rough and informal condition on subtractability that I gave above? Notice that were any individual to be tter than they actually are-but collective resources to remain as they actually are-then Equation 2 requires that some other individual or individuals would be less t than they actually are, and by an equal amount. In the foraging case, holding xed the amount of food collectively foraged, one individual's gain in tness is precisely counterbalanced by another's loss. It is of crucial importance to emphasise that thepresent requirement-concerning whatwouldhappenwere collective resources to remain as they actually are-obviously does not entail that collective resources must remain as they actually are. erefore there will be many subtractable tness structures for which collective resources vary according to the distribution of individual characters within the collective. In the foraging case for example the amount of food foraged Rg(zg) can vary depending on how the individuals are inclined to cooperate during the hunt, asmeasured by zg. So I emphasise that subtractability of resources does not entail that individuals are playing a zero-sum game that precludes them from cooperating to increase collective resources. A similar point: subtractability does not entail that the tness structure in play is additive. In other words it does not entail that the tness structure be given by ωi = λzi + μExpg(z). In summary, I stipulate that the sharing-out of resources is subtractable just in case (i) one can analytically separate resource allocation into a mechanism of resource acquisition and into a mechanism of resource division, and (ii) Equation 2 characterises the tness structure in play. 15 A second illustration of the subtractability of resources comes from simple diploid genetics models. An AB genotype causes the organism in which it is instantiated to exemplify a corresponding phenotype, and this organism interacts with the environment and has a number of offspring. And these offspring by extension are counted as the offspring of the AB genotype itself. Call this process the acquisition of the AB genotype's reproductive resources. (I'm happy to be fairly liberal about what counts as a resource.) Consider next that during meiosis the A allele in the AB genotype will be copied to a certain number of gametes and so will enjoy a particular chance of being represented in each of the aforementioned organism's offspring. e same goes for the B allele. Call this the sharing-out of the AB genotype's reproductive resources amongst its two alleles, A and B. Again one can analytically separate resource allocation into collective resource acquisition and the sharing-out of these resources between individuals. For it is biologically interesting to ask what would occur if meiosis were to unfold differently: what if segregation distortion (Lyttle [1991]) occurred and the A-allele in the AB genotype enjoyed more than its y-percent share of reproductive resources (Maynard Smith and Szathmáry [1995], §10)? So my rst condition for subtractability is satis ed here. Equally my second condition for subtractability is also satis ed here: holding the AB genotype's resources xed, an increased chance of the A-allele of being represented amongst the organism's offspring would be precisely counterbalanced by a decreased chance for the Ballele. Finally, an example in which resources are, in contrast, not shared-out subtractably is that of the Polistes wasp. In this case a worker's tness is sensitive to whether he is stung by the Queen. In virtue of this, avoiding being stung by the Queen is a key resource. But it would be absurd to attempt to analytically separate the allocation of this sting-avoidance resource into a mechanism whereby the wasp collective acquires sting-avoidance, and a mechanism in which stingavoidance is then shared out amongst individual wasps. So this resource is, by my de nition, not 'shared out'. A second example in which resources are not shared-out subtractably is that of beavers building a channel from their dam to the river bank. I concede that one can analytically separate resource acquisition and resource sharing-out here. But one beaver's using this channel does not exclude other beavers from doing likewise. So this sharing-out is not subtractable. Almost there. I want now to make Equation 2 easier to work with mathematically. Consider the following constraint on the tness ωi of each individual i in collective g: ωi = ( 1 n − α[zi − Expg(z)])Rg(zg) (3) Letmeunpack this equation. Expg(z) is just the average character of themembers of collective g. So [zi − Expg(z)] denotes the degree to which our individual 16 i scores especially highly on pro-social character z. In other words whenever an individual has a perfectly average character then this becomes zero and the overall expression reduces to 1 n Rg(zg). In other words, whenever this is so, this individual's tness is equal to collective resourcesRg(zg) divided by the number of members of the collective n. So whenever an individual is perfectly average she receives her 'fair share' of collective resources. Similarly note that whenever an individual scores especially highly for prosocial character z then the −α[zi − Expg(z)] term will be negative; assuming α is positive. So she will enjoy a lesser proportion of the collective's resources and thus she will be less t. Conversely whenever an individual scores especially low on z-in other words she has an especially 'anti-social' character-then this expression will be positive. And so she will enjoy a greater proportion of collective resources and thus will be more t. So the α parameter denotes the degree to which anti-social individuals can command an unfair share of the resources that the collective has acquired. us parameter α measures an important feature of the sharing-out of resources between individuals, as opposed to a feature of collective resource acquisition itself. It is a feature of the tness-structure generated by the environment. (Table 2 illustrates the tness structure that Equation 3 requires in a simple case; namely in the case of two-membered collectives, and inwhich an individual either has character z fully or not at all. In formal terms z = 0 or z = 1.) who interact with who interact with a Z individual a non-Z individual Fitness of Z individuals 1 2 R (1 2 − 1 2 α)R′ Fitness of non-Z individuals (1 2 + 1 2 α)R′ 1 2 R′′ Table 2: Fitness of each individual in the subtractability case Take the expression in round brackets in Equation 3 and sum it over all individuals in the collective. Since this necessarily sums to one it is evident that Equation 3 entails Equation 2. But I don't believe that to assume subtractability in the speci c form of Equation 3 rather than more generally in the form of Equation 2 amounts to a signi cant loss in generality.19 So from now on I will workwithEquation 3 as part ofmyde nition of subtractability, rather thanwith Equation 2. Having carefully illustrated what I mean by subtractability, one can now get down to business. I will now show that the multi-level Price decomposition has the ancillary role of answering questions about how character z would evolve if 19Frank's ([1995]) model however satis es Equation 2 but not Equation 3. 17 anti-socially inclined individualswerenotpermittedunfair access to subtractable resources. Suppose that the sharing-out of resources amongst individuals is subtractable. Hence it can be characterised by a parameter α which measures the degree to which the tness-structure in play permits anti-socially inclined individuals to access more than their fair share of collective resources. So intuitively, and as Equation 3 con rms, altering α to become zero will reduce within-collective variation in tness to zero. In these circumstances all individuals will receive an equal share of tness, namelyRg(zg) divided by n. (One example of this is an alteration of the visual environment such that would-be cheaters can be spotted, and thereby prevented from stealing extra resources.) But this hypothetical alteration of α is evidently a Structural Collapse to the Mean alteration. And I've already shown in Section 4 that all SCM alterations satisfy the independence criterion for explanatory value: the effect ε of this SCM alteration will be measured by the attendant change to the value of the third term in the multi-level Price decomposition.20 My second criterion (interestingness) for explanatory value requires that this alteration to α be of interest to biologists. Note, however, that the genuine sharing-out of resources-as I've de ned it-entails that one can analytically separate resource allocation into the acquisition of resources by the collective and the sharing-out of these resources amongst individuals. is in turn entails-againbymyde nition-that there is an interesting alteration to themechanism of sharing-out resources amongst individuals, an alteration that does not alter how these resources were acquired by the collective. erefore all cases of subtractable sharing-out will be cases in which alterations to α are biologically interesting. So my interestingness criterion for explanatory value is, by de nition, satis ed in cases in which resources are genuinely shared out. Here are two such cases; just to illustrate that such cases plausibly exist. Case one: α measures the degree to which visual environment is such that cheating foragers can go undetected, and therefore can steal resources rather than being excluded from them. Case two: in the population genetics example, αmeasures the degree of so-called segregation distortion, the extent to which the meiotic 20Moreover one can show that the relationship between the third term of the Price equation and α is a linear one. For observe that it follows from Equation 3 that Covg(ω, z) = Covg([ 1 n − αz + αExpg(z)]Rg, z) = αRg(zg)Varg(z) (4) But one can substitute this intoExp[Covg(ω, z)], the third termof themulti-level Price decomposition, to yield Exp[αRg(zg)Varg(z)]. And this yields αExp[Rg(zg)Varg(z)]. For, being a feature of the environment, α doesn't vary from collective to collective. So the third term of the Price decomposition depends linearly upon α. 18 environment allows sel sh alleles to enjoymore than their fair share of representation in the offspring organisms. ese are just two examples of a biologically interesting α parameter. So my interestingness criterion for explanatory value is satis ed non-trivially. But I've already shown in Section 3 that the third criterion (knowledge) for explanatory value is in general satis ed by the multi-level Price decomposition. So all three of my criteria are satis ed. us this section has established an explanatory role for the multi-level Price decomposition in a limited class of cases; namely cases in which the sharing-out of resources is subtractable. In such cases themulti-level Price decomposition deepens single-level explanations of the evolutionof character z basedonpopulation-level character– tness covariance alone. To put it intuitively, it has the ancillary role of answering questions about what would happen if anti-socially inclined individuals could no longer gain unfair access to subtractable resources. Recall that section 4 showed that appealing to SCMalterations cannot establish everything that Sober and Wilson want to establish. For it cannot establish the explanatory value of the multi-level Price decomposition across all cases in general; for example the case of retaliation in wasps. Instead the present section has shown how appealing to SCM alterations establishes the explanatory value of the decomposition in the special case in which the sharing-out of resources is more or less subtractable. Unfortunately, I contend, there are no other obvious cases in which SCM alterations have any biological interest. (See my discussion in Section 4.) So it's likely that appealing to SCM alterations can only establish the explanatory value of the multi-level Price decomposition in cases in which resources are more or less subtractable. 6 Other Alterations toWithin-Collective Variation ere are hypothetical alterations other than SCM, however, which eliminate within-collective variation. is naturally raises the following question: can one appeal to any of these other alterations in order to establish a further explanatory role for the multi-level Price decomposition? Perhaps the decomposition does indeed have a general explanatory role, or at very least a role in some cases in which resources are not subtractably shared-out. As I will illustrate momentarily, however, I can't nd any such alterations which obviously satisfy the independence and interestingness criteria for explanatory value simultaneously; even for a limited range of cases. erefore it's likely that appealing to (alterations to) within-collective variation can establish no more than Section 5 did: the multilevel Price decomposition is explanatorily valuable in cases in which resources are more or less subtractable. 19 is sectionwill supportmy claimhere by examining three alternatives to the Structural Collapse to theMean alteration: the 'Increased Retaliatory Capacity' alteration, the 'Homogenizing Assortment' alteration, and the 'Character Collapse to the Mean' alteration as I will label them. Character Collapse to the Mean. Consider a collective of vampire bats composed of a few very t members and many very un t ones. Imagine for example a ve-member collective containing individuals with tnesses ω = 1, 1, 1, 2, and 10. Imagine altering the character of every member in the collective, and in turn their tnesses, such that they are allmoderately t. Imagine in particular that this yields tnesses of ω = 3, 3, 3, 3, and 3. us by altering character, tnesses have been collapsed to the collective mean. So within-collective variation in tness has been eliminated. Note that this Character Collapse to the Mean (CCM) alteration differs from the Structural Collapse to the Mean alteration in that it does not alter tness via altering tness structure; instead it does so by altering the frequency of the character in the population. To see an immediate problem for appealing to CCM alterations, calculate the values of the second term in the multi-level Price decomposition for the example given in Table 3: the term is originally 90 but falls to 84 under the CCM alteration. SoCCMdoesn't just alter the value of the third termof the decomposition;21 it also alters the value of the second term. In other words, with respect to the CCM alteration in this case, the independence criterion for explanatory value is not satis ed. erefore one cannot appeal to the CCM alteration to identify an explanatory role for the multi-level Price decomposition for all cases in general. Original z Original ω CCM z CCM ω 3 1 24 2 24 2 24 2 81 3 24 2 - - - 81 3 192 4 192 4 192 4 375 5 192 4 Table 3: Character Collapse to the Mean for two three-membered collectives and with ωi = 13 3 √ zi But this raises the following question: might appeals to CCM establish the explanatory value of themulti-level Price decomposition in amore limited range 21Which it alters to zero; see Section 4. 20 of cases, rather than across all cases in general? Take for instance cases in which collective character maps one-to-one onto collective tness. One can show that the hypothetical CCM alteration does satisfy my independence criterion for explanatory value in such cases. is is because the CCM alteration will preserve collective tness. And so, given the one-to-one mapping, it will preserve collective character. And so it will in turn preserve the covariance of collective tness and collective character. In other words CCM will not alter the second term of the multi-level Price decomposition in this case. So the independence criterion for explanatory value is met. What about the interestingness criterion however? I certainly do not want to claim that cases of one-to-one mapping are uninteresting as such. Indeed this range of cases includes as a subset an important range of cases, namely those in which individual tness is 'additive'.22 Additive cases are those in which tness is a linear function of individual character and collective character: ωi = λzi + μExpg(z). us collective character maps one-to-one onto collective tness: Expg(ω) = (λ+ μ)Expg(z).23 Instead what I want to question is the biological interest of the Character Collapse to theMean alteration itself. A er all, the problems I identi ed in Section 4 with respect to the Structural Collapse to the Mean alteration can all be extended to Character Collapse to the Mean. For there is no range of cases-at least obviously-forwhichhypothetical collapses to the erstwhilemean aremore biologically interesting than collapses to any other value (Section 4). us it is unlikely that CCM alterations ever satisfy the interestingess criterion explanatory value; even in a more limited range of cases. Homogenizing Assortment. One biologically interesting alteration is the alteration to the mechanism of 'assortment', the mechanism that determines which individuals in a population join themselves into collectives with which other individuals. For example one might imagine that the mechanism of assortment is altered such that individuals only interact with individuals of a similar character. In the extreme case then assortment will be fully homogenous: within-collective variation in character will be zero. And therefore within-collective variation in tness will be zero. us the Homogenizing Assortment (HA) alteration differs from the CCM alteration in that it does not alter the overall composition of characters in the population, merely how individuals are assorted into collectives. It is clear that this HA alteration is in general biologically interesting. In other words it satis es my second criterion for explanatory value. 22See Birch ([2014]) for a discussion of assumptions similar to this additivity assumption but in a slightly different context. 23I note incidentally that cases of one-to-one mapping exclude any form of synergism. In other words, it precludes individuals coordinating their activities so that the bene t to the collective is greater than the sum of each individual's own efforts. 21 Unfortunately, with respect to theHomogenizingAssortment alteration,my independence criterion for explanatory value is not satis ed; except perhaps in a gerrymandered range of cases. To see this, note that HA only alters how individuals in the whole population are grouped into collectives. It preserves the overall compositionof characters in thepopulation. But take the very simple case in which an individual's tness only depends upon her own character. It follows that HA preserves each individual's tness here. In summary, it preserves the joint distribution of character and tness in the overall population.24 In such cases thereforeHA does not affect character– tness covariance across the whole population. In formal terms, the effect of HA on the value of le -hand term of the multi-level Price decomposition is zero. But the attendant change to the third term will be non-zero.25 It follows that HA also affects the value of the second term. In other words, with respect to the Homogenizing Assortment alteration, my independence criterion for explanatory value is not satis ed in this very simple case. And there is noobvious range ofmore complex cases, I contend, for which one might expect HA not to alter the second term; as well as altering the third term. Or at least not for any non-gerrymandered range of cases. erefore, I contend, it is unlikely that HA alterations ever satisfy the independence criterion for explanatory value; even in a more limited range of cases. Increasing Retaliatory Capacity. Recall the Polisteswasp example in which tness was given by 2Expg(z)−zi− 12(−zi). is is a special case of themore general tness structure ωi = f(zg)− p(−zi); where p is the parameter that measures retaliatory capacity (Section 4). Consider the hypothetical alteration inwhich this parameter is increased by ∆p: queen wasps can for example more easily punish lazy workers, or punish them more severely. One can easily show that this Increasing Retaliatory Capacity (IRC) alteration increases the value of the second term of the multi-level Price decomposition, namely by Var[Expg(z)]∆p. Ruling out the trivial case in which there is no variation in collective character, this expression will be non-zero. In other words IRC doesn't just alter the value of the third term of the decomposition,26 but also the value of the second term. So the IRC alteration fails the independence criterion for the explanatory value of the decomposition in all non-trivial cases. 24I am most grateful to Cedric Patternotte for spotting, prior to publication, a subtle but egregious error at this point. 25Homogenizing Assortment will eliminate the variation within any collective. So it will eliminate the character– tness covariance within any collective. So it ensures that the value of the third term of the multi-level Price decomposition Exp[Covg(ω, z)] will become zero. Setting aside the trivial case in which within-collective variation was already zero, this demonstrates that the attendant change to the value of the third term is non-zero. 26 e attendant change to the third term is, one can show: Exp[Varg(z)]∆p. And this is only zero when there is no within-collective variation in individual character. 22 Totake stock, this sectionhas considered three alterations towithin-collective variation: IncreasingRetaliatoryCapacity,HomogenizingAssortment, andCharacter Collapse to the Mean. And I've shown decisively that one cannot appeal to the IRC alterations to identify any explanatory role for the multi-level Price decomposition at all. And I've shown decisively that one cannot appeal to the CCMorHA alterations to identify a general explanatory role for the decomposition in all cases. Moreover it's unlikely that we can nd an explanatory role by appealing toCCMorHA in even amore limited range of cases; excluding gerrymandered ranges of cases. So an appeal to any of these three alterations-CCM, IRC, or HA-to establish any explanatory role for the multi-level Price decomposition is unlikely to be successful. erefore, the SCM alteration from Sections 4 and 5 is the only alteration of within-collective variation to which one might successfully appeal. e tentative conclusion is that appealing to (alterations to) within-collective variation can establish no more than Section 5 did: the multi-level Price decomposition is explanatorily valuable in cases in which resources are more or less subtractable. 7 Alterations to Between-Collective Variation Sections 4 to 6 askedwhether appealing to (alterations to)within-collective variation can establish the explanatory value of themulti-level Price decomposition. is was prompted by Sober and Wilson's suggestion that the third term of the decompositionmeasures the effects ofwithin-collective variation. But Sober and Wilson, I've already noted, also place a lot of weight upon an idea that is symmetrical to this one: the second termof themulti-level Price decompositionmeasures the effects of between-collective variation. If this symmetrical idea is true, then we have an additional strategy for vindicating the decomposition: appeal to alterations to between-collective variation in tness. Unfortunately, it turns out that it is very difficult to construct a plausible argument that favours Sober andWilson's symmetrical idea. e following is my best attempt, but one which ultimately fails. Take a ve-member collective with individual tnesses of ω = 1, 3, 6, 6, and 9; and thus of average tness 5. Consider a hypothetical alteration that changes the character of each member such that their tness is 'boosted' by one unit, resulting in a ve-member collective with tnesses of ω = 2, 4, 7, 7, 10, and thus of average tness 6. Note that it's a mathematical fact that this alteration won't alter within-collective variation in tness. Consider also a second ve-member collective with individual tnesses of ω = 1, 6, 8, 10, and 10; and thus of average tness 7. But this time consider a 'boost' of minus one unit, so that this second collective now also has an average tness of 6. us all collectives are altered to 23 have the same collective tness, in this case6, thus eliminatingbetween-collective variation in collective tness. Consequently this Uniform Boosting alteration reduces to zero any covariance of collective tness with other factors. erefore Cov[Expg(ω), Expg(z)], the second term of the multi-level Price decomposition, will become zero. Calculate, however, the values of the third term in the multi-level Price decomposition for the example given inTable 4: the term is originally 62but falls to 56 under the Uniform Boosting alteration. In other words, with respect to UniformBoosting, themulti-level Price decomposition doesn't in general satisfy the independence criterion for explanatory value. Moreover, let ε denote the effect of thisUniformBoosting alteration upon character– tness covariance across the whole population; that is, upon the value of the le -hand term in themulti-level Price decomposition. Uniform Boosting having altered the value of the third term, it follows that this effect ε is not measured by the attendant change to the second term of the multi-level Price decomposition. Sober and Wilson's symmetrical idea does not in general hold for all cases. Original z Original ω Boost z Boost ω 3 1 24 2 24 2 81 3 81 3 192 4 - - - 81 3 24 2 192 4 81 3 375 5 192 4 Table 4: Uniform Boosting for two three-membered collectives and with ωi = 1 3 3 √ zi One response might be to insist that nevertheless the attendant change in the second term measures effect ε in a limited but non-gerrymandered class of cases. Take for example those cases in which an individual's tness is a linear functionof that individual's owncharacter alone; put in formal termsωi = mzi+ c. Whenever the tness of each member of a collective is uniformly boosted by k, then each member's character will have been uniformly boosted by k m , given this linear relationship. But the logic of covariance has it that Covg(ω + k, z + k m ) = Covg(ω, z). So Uniform Boosting preserves the value of the third term in this case. It follows that this effect ε is measured by the attendant change to the second term of the multi-level Price decomposition. Unfortunately this class of cases is a completely irrelevant class for present purposes. For there's an intuitive sense in which there is no selection at the level 24 of the collective at all in such cases. A er all, in such cases individual tness is not in uenced by the collective at all. And I have no doubt that Sober and Wilson would agree with this point. is is because, applying their own 1998 de nition of 'trait groups', there are no genuine collectives in this special case. And hence there is no genuine collective-level selection. So theproblemremains: consider this effect εof eliminatingbetween-collective variation viaUniformBoosting, that is, the effect upon the value of the le -hand term of the multi-level Price decomposition. I contend that there is no obvious non-gerrymandered class of relevant cases for which this effect ε is measured by the attendant change to the second term of the decomposition. So, with respect to the Uniform Boosting alteration, it is unlikely that there are any cases for which the independence criterion for explanatory value holds. So appeals to Uniform Boosting are unlikely to establish any explanatory value for the multilevel Price decomposition. But there are no other obvious, biologically interesting ways-I contend-to alter between-collective variation. I conclude that appeals to (alterations of ) between-collective variation are unlikely to establish any explanatory value for the decomposition. 8 Alternative Approaches to Explanatory Depth is paper has taken for granted that the depth of an explanation is in proportion, roughly speaking, to the number of important what-if questions that it allows one to answer. But why should one accept this? I cannot offer a full defence of this view, although interested evolutionary biologists might consult Woodward ([2003]), which has quickly become a philosophical classic. Instead this sectionwill brie y examine the prospects for an alternative approach to explanatory depth, one that draws upon alternative accounts of explanation. e rst thing to note is that the philosophical literature contains scarcely any alternatives to the what-if account of explanatory depth. Why, for example, did the patient die? Hempel's Deductive Nomological approach might say that the following was a correct explanation: the patient ingested a large dose of digitalis, and it's a law that all people who ingest that dose will die soon a erwards (Hempel and Oppenheim [1948]). But Hempel's account is not an account of explanatory depth. For it does not offer us a criterion according to which this explanation counts as less deep than an explanation that includes details about how digitalis is metabolised and how it affects the heart. Hempel's approach is an account of explanatory correctness, not an account of the depth of a correct explanation. Next considerKitcher's ([1981]; [1989]) uni cationist approach to explanation. Kitcher provides a criterion for what one might call explanatory promise, 25 the ability of a candidate explanation to deepen one's understanding of what one already knows. And famously Kitcher's approach is a 'winner takes all' account. Indeed it cannot bemodi ed to admit degrees of explanatory promise on pain of admitting some embarrassing counter-examples (Woodward [2003], p. 368).27 So-even if one were willing to equate explanatory promise with explanatory depth-Kitcher's approach doesn't delineate degrees of explanatory depth. Kitcher's approach shouldnotbe confusedwith themoremodest-and thereby more plausible-idea that there are at least two virtues with respect to which an explanatory framework such as the multi-level selection framework can be assessed. e rst virtue is what I've called depth, which I've urged is to be cashed out in terms of what-if questions. e second virtue is cashed out in terms of the framework's scope of correct application: the broader the range of cases that can be correctly explainedwithin that framework, themore 'unifying' the framework.28 But it is evident that anyone tempted by this more modest uni cationist idea will have no complaints with the assumptions that this paper has made about explanatory depth. All that the modest uni cationist insists upon is that one also acknowledge the existence of an additional dimension to explanatory frameworks, uni cation qua broad scope of correct application. I'm happy to do so. Admittedly I've said very little about the relative scope of application of the single-level selection and multi-level selection frameworks. But this is because the answer is trivial: the multi-level selection framework has a narrower scope. A er all, it embodies an extra restriction, namely that one's population be partitioned into collectives. So, for this trivial reason, the present consideration concerning breadth of scope is not probative. It does not provide a sense inwhichmulti-level selection explanations add value over and above singlelevel selection explanations. Finally let's consider the causal approach to explanation. Why have I been talking about the explanatorydepthof themulti-level Price decomposition, rather than asOkasha ([2004b]; [2004c]) does ofwhether thedecomposition is 'causally adequate' or 'causally inadequate'? Mymain reason is that the notion of a decomposition's being causally adequate is incredibly tricky (Okasha [Forthcoming]). at is why I have le the discussion in this paper incomplete as a far as causal questions are concerned. But one might worry that in 'ignoring' causation the discussion in this paper is in danger of being not just incomplete but also unsound. I will now address this worry. I've taken for granted through-out this paper that thedepthof an explanation is, roughly speaking, in proportion to thenumber of importantwhat-if questions 27Indeed see Woodward ([2003], §8) for what I take to be decisive counter-examples to the view overall. 28Birch ([2014], §5) proposes this more modest approach, although he seems to suggest that there is a sensible way of aggregating these two virtues into one overall score. 26 that it helps to answer. And I've noted that the importance of a what-if question is in part determined by our personal interests. But philosophers who favour the causal approach to explanation might wish to place an additional restriction on what counts as an important what-if question. e causal restriction: a what-if question is only important if the correct answer to it cites a cause of the to-beexplained event. I have no doubt that Lewis ([1986]), Lipton ([1991]), Ruben ([1990]), and Woodward ([2003]) amongst others would endorse this restriction.29 Adding this restriction, however, makes no difference to the soundness of the arguments of this paper. Firstly my criticism of Sober and Wilson in Sections 4 and 6 relied primarily on the fact that certainwhat-if questions are uninteresting. And so my criticism required only that interestingness be a necessary condition for a what-if question to be important. It did not require that interestingness constitute the only necessary condition on importance. Secondly my positive point in Section 5 relied primarily on the importance of questions about what would happen were parameter α to be different. What happens to my argument if we add the requirement that α has to be a cause of the evolution of social character z in order for such questions to count as important? Nothing. For there is no reason to think that α-an interesting feature of the environment that determines how much command anti-social individuals have over resources-cannot be a cause of the evolution of character z. So endorsing a causal approach to explanation does not generate a reason to resist the conclusions of this paper. is concludes my defence of the measure of the depth of an explanation as, roughly, the number of important what-if questions that it helps to answer. 9 Conclusion Sections 2 and 8 built and defended a general framework through which to understand the explanatory role of non-empirical decompositions such as themultilevel Price decomposition. Such decompositions have the ancillary role of describing the constitutive relationships that help glue different factors in our explanatory reasoning together. And I provided three individually necessary and jointly sufficient criteria for a non-empirical decomposition to play this role. is motivated a search to nd a hypothetical intervention that simultaneously meets my independence criterion and my interestingness criterion. Taking my lead from Sober and Wilson, I assume that any such intervention would either be one that (i) eliminates between-collective variation in tness, or (ii) 29But note that, given Lewis' and Woodward's views of the nature of causation, this restriction is a trivial one: roughly speaking, all answers to (the right sort of ) what if things had been different questions cite causes. 27 eliminates within-collective variation in tness. And this paper considered ve interventions in total: (i) UniformBoosting (Section 7); (ii) Increasing Retaliatory Capacity (Sections 4 and 6), Structural Collapse to the Mean (Sections 4 and 5), Homogenizing Assortment (Section 6), and Character Collapse to the Mean (Section 6). Only some of these hypothetical alterations turn out tomeet my interestingness criterion: HA and IRC are in general interesting; and SCM is interesting whenever resources are subtractable. In contrast, CCM is of dubious interest. Similarly, only some of these hypothetical alterationsmeetmy independence criterion. at is, only some of these alterations have their effects measured by a right-hand term of the multi-level Price decomposition: the SCM alteration in all cases, and theCCMalteration in cases of one-to-onemapping of character to tness. All the other interventions likely fail this criterion in all cases, excluding gerry-mandered ones. In summary, none of these ve alterations meet both criteria simultaneously in all cases. Indeed there isn't even a more limited range of cases for which the IRC, HA, CCM or the UB alteration meet both criteria simultaneously. However, in the limited case in which resources are subtractable, the Structural Collapse to the Mean alteration does satisfy both criteria. But I assume that these ve alterations are the only ones to which one might obviously appeal to in order to establish the explanatory value of the multi-level Price decomposition. My conclusion is that the decomposition has explanatory value, most likely, primarily when collective resources are more or less subtractable. Its value is more circumscribed than its champions Sober and Wilson ([1998]) believe. Let me put the main thrust of the paper in intuitive form. What would happen if environmental conditions made it more difficult for anti-socially inclined individuals to access an unfair proportion of the subtractable resources acquired by their collective? I have argued that the explanatory value of the multi-level Price decomposition is that it helps us to answer such questions; questions about what would happen were the 'policing' of subtractable resources strengthened. 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