Optimality Explanations: Equilibrium, Idealization, and Tradeoffs

Collin Rice University of Missouri Department of Philosophy 423 Strickland Hall Columbia, MO 65211 [email protected]

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Abstract. Optimality models are widely used in evolutionary biology. A prominent approach to scientific explanation and modeling claims that for a model to provide an explanation it must accurately represent at least some of the actual causes or causal processes in the event’s causal history; e.g. at least those that made a difference to the explanandum. In this paper, I argue that many biological optimality explanations present a serious challenge to this causal approach. I contend that many optimality models in biology provide highly idealized equilibrium explanations that do not accurately represent the causes of their target system(s). Furthermore, it is often in virtue of their independence of the causes of their target system(s) that optimality models are able to provide a special kind of explanation.

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Optimality Explanations: Equilibrium, Idealization, and Tradeoffs 1. Introduction. Recently philosophers of science have begun to pay more attention to the building of idealized mathematical models (Batterman 2002, 2009; Bueno and Colyvan 2011; Weisberg 2007). An important example of idealized mathematical modeling is the widespread use of optimality models in biology. In this paper, I argue that many biological optimality explanations present a serious challenge to the causal approach to explanation and modeling. One approach to understanding the explanations of biological optimality models is as a kind of “censored causal explanation” (Elgin and Sober 2002; Orzack and Sober 1994, 1996; Potochnik 2007, 2010). A censored causal explanation is an explanation that purposely omits certain causal factors in order to focus on a modular part of the causal process that led to the explanandum. For example, Steven Orzack and Elliott Sober employ this terminology when they claim that optimality models are “censored” models, “in which the only evolutionary force is natural selection” (Orzack and Sober 1994, p. 363). Following this approach, Angela Potochnik has recently argued that their ability to provide a kind of censored causal explanation secures optimality models a permanent explanatory role within population biology (Potochnik 2007, 2010). I will call this the censored causal model approach to understanding the explanations of biological optimality models. In this paper, I first argue that many biological optimality models do not provide censored causal explanations. In response, I propose an alternative approach to understanding biological optimality explanations.1 I contend that many optimality models within biology

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I will refer to any explanation that makes essential use of an optimality model as an “optimality explanation”. In biological contexts, I will call these “biological optimality explanations”. The relationship between a model and the explanation it provides is important, but will have to be discussed in more detail elsewhere.

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provide highly idealized equilibrium explanations that do not accurately represent the differencemaking (or otherwise relevant) causes of their target system(s). Moreover, several key features of biological optimality explanations move the model in precisely the opposite direction suggested by the causal approach by eliminating many of a target system’s causally relevant details. Indeed, several of the key components of biological optimality explanations contribute to the model’s explanatory power in virtue of eliminating or distorting the various causal details of real-world biological populations. The next section presents some examples of how optimality models are used to provide explanations of biological phenomena. Then, in Section 3, I present the censored causal model approach. Section 4 argues that many biological optimality models cannot be characterized as providing censored causal explanations. Next, Section 5 argues that optimality models provide a unique kind of highly idealized equilibrium explanation, despite the fact that they often do not accurately represent the causes of any real-world system. Furthermore, I argue that for many biological explananda a model that accurately represented the target system’s causal processes would actually provide a worse (or perhaps no) explanation when compared with the explanation of an idealized optimality model. 2. Examples of Optimality Explanations in Evolutionary Biology. In this section I present three examples to demonstrate how optimality models are used to provide explanations of biological phenomena. Biological optimality models usually assume that natural selection will drive the target population towards the strategy, or mix of strategies, that optimizes the model’s criterion. Furthermore, these models commonly assume that natural selection will be able to overcome any other factors influencing the evolution of the population;

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e.g. drift. These assumptions entail that the strategy that maximizes fitness (or its proxy) in the optimality model is the equilibrium point of the evolving population.2 2.1. Foraging Lapwings. A relatively simple optimality model is described in Parker and Maynard Smith’s discussion of foraging lapwings (Parker and Maynard Smith 1990). Lapwings search for food by moving a few paces before pausing to look for their prey (insects), which they eat if they find. The available strategies in this case are the possible distances traveled in between each scan, x. The farther the lapwing moves, the less likely it will be scanning terrain already inspected (and thus more likely to find prey). However, once it has moved a distance equal to the diameter of its visual field, moving farther does not help since all of the new ground will already be outside of the previously inspected area. Furthermore, although moving farther increases the average energy intake from new prey, each step of movement costs energy. In the optimality model below, curve B represents the average caloric benefit of prey items obtained by adopting strategy x, and curve C represents the average energy cost of moving in adopting strategy x. By making the idealizing assumptions that B(x) will increase perfectly asymptotically and C(x) will increase perfectly linearly we get the following model that represents this cost-benefit tradeoff in terms of two mathematical curves:

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I should point out that my aim in this paper is to understand the kind of explanations biological optimality models are able to provide when they are satisfactory. Thus, the examples below are chosen for the clarity with which they illustrate the structure of optimality explanations, not their status as well-confirmed models.

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Figure 1: The costs and benefits of lapwing foraging strategies (Parker and Maynard Smith 1990).

The proposed optimization criterion is the average net energy gain, E, where E(x) = B(x) – C(x):

Figure 2: Model representing the average energy gain for different lapwing foraging strategies. Average energy intake is maximized at x* (Parker and Maynard Smith 1990).

The optimal strategy, x*, is that which maximizes E. The lapwing population is expected to evolve the trait represented by x* since the model makes the optimization assumptions that more average energy intake will lead to more offspring on average and, therefore, strategies with higher average energy intake will tend to increase in frequency within the population. This entails that x* is the equilibrium point of the population evolving by natural selection. The explanation provided by the model also assumes that other evolutionary factors will not deter the population on its way to this equilibrium point. This assumption is captured by the introduction of various idealizations into the model; e.g. that strategies are inherited asexually, that organisms mate randomly, and that the population is infinite in size. The optimality model provides a mathematical derivation of this locally optimal strategy given the tradeoff between the average

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caloric benefit from the prey items obtained by moving a distance x and the average energy cost of moving distance x, along with the other constraints of the particular foraging problem. Assuming that we observe actual lapwing populations exhibiting (approximately) this optimal strategy, the optimality model could provide an explanation of the adaptive behavior by showing that it is the equilibrium point of the evolving system given the constraints and tradeoffs represented within the model.

2.2. Parker’s Dung Flies. G. A. Parker utilized an optimality model in his attempts to explain why dung flies (Scatophaga stercoraria) copulate for 36 minutes on average (Parker 1978). First, Parker observed that female dung flies mate with multiple males. He then discovered, by experimentation, that when this occurs the second male fertilizes far more eggs (80%) than the first (20%). Consequently, after copulating with a female, a male dung fly spends some time guarding her before flying off in search of other mates. The total behavioral cycle time is given by summing search time, copulation time and guard time. Parker then observed that the average time spent searching plus guarding was 156 minutes. Therefore the total cycle will last 156 + c minutes, where c is the amount of time spent copulating. In Parker’s model, the x-axis represents the total time expenditure on all three tasks, and the y-axis represents the (average) number of eggs fertilized for the different strategies (i.e. for different values of c). By experiment, Parker found that increasing the copulation time increases the average number of eggs fertilized. However, there is an important tradeoff: time spent copulating with one female is time lost searching for other mates. In addition, Parker observed diminishing

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returns on time spent copulating—that is, additional copulation time brings smaller and smaller increases in the number of eggs fertilized. Parker’s observations are captured by the mathematical curve within the optimality model that represents fertilization as a function of copulation time (figure 3 below).

Figure 3 (taken from Sober, 2000): Parker’s optimality model used to investigate the copulation time of dung flies.

According to Parker’s optimization criterion, the optimal value for c is the value that maximizes the rate of eggs fertilized across several iterations of the behavioral cycle. This optimal strategy occurs at the point where a line that passes through the origin and intersects the asymptotic curve with the steepest slope (line A-B above) intersects the curve. This optimal point occurs when c is equal to 41 minutes, which is fairly close to the observed value of 36 minutes. Given this predictive accuracy, and the fact that Parker’s model was based on detailed empirical observations, the model is often thought to have captured the major constraints and tradeoffs that were involved in the evolution of this quantitatively specified behavioral trait value (Sober 2000). As a result, assuming that natural selection would maximize the rate of eggs fertilized and that other evolutionary factors would not deter the population from reaching this equilibrium state—i.e. assuming the optimization assumptions of Parker’s model are accurate—Parker’s

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model can be used to give an equilibrium explanation for why dung flies copulate for approximately 36 minutes on average. 2.3. Equilibrium Sex Ratios. Another commonly referenced group of optimality models comes from the theory of sex allocation (Charnov 1982; Fisher 1930; Maynard Smith 1982; Maynard Smith and Price 1976). In these models, the payoffs to particular strategies will often depend on the frequency of strategies within the population. Therefore, game-theoretic techniques are utilized in order to deduce an evolutionarily stable state (or strategy) (Maynard Smith and Price, 1976; Maynard Smith, 1982). For instance, R.A. Fisher (1930) asked a very general biological question, why is the sex ratio often 1:1? Fisher used a simple frequency-dependent optimality model to argue that if the sex ratio were under parental control (i.e. controlled by phenotypic strategies played by parents), the stable equilibrium sex ratio for the population would be 1:1. In rough outline, Fisher’s model claims the following. First, Fisher assumed that genes expressed in the parent determine the sex ratio they will produce and these genes are inherited perfectly asexually by offspring. In addition, Fisher assumed that organisms within the population mate randomly and that the population is infinite in size. He then reasoned that if one sex is more common in the population, there will be a fitness payoff to parents who produce the minority sex since their children will have more mating opportunities. To see this, suppose that male births are less common than female births. A newborn male thus has better mating prospects than a newborn female and is, therefore, expected to have more offspring. Therefore, parents who produce males tend to have more grandchildren on average. Consequently, the tendency to produce male offspring will spread and male births will become more common in

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the population. However, as the population approaches a 1:1 ratio, this fitness advantage fades away. The exact same reasoning applies if we start by assuming that female births are less common. The only state in which selection will not favor the production of the minority sex is when the number of males and females within the population is equal. Therefore, a 1:1 sex ratio is the stable equilibrium state of the evolving population. Fisher’s model relies on an important tradeoff between the ability to produce sons and daughters; namely, more sons means fewer daughters and vice versa. This tradeoff is described more specifically by looking at what economists refer to as the substitution cost. The substitution cost, in this case, tells us how many sons will be produced if one less daughter is produced. In Fisher’s original model this tradeoff is perfectly linear—i.e. males and females cost the same amount of resources to produce and so one fewer son means one more daughter and vice versa. This is why Fisher’s original model predicts a 1:1 sex ratio. Charnov (1982, 28-9) shows that generalizing Fisher’s model to cases where this tradeoff is not linear leads to the conclusion that the equilibrium ratio, r, for a population can be calculated by using the following formula: r = C2/(C1 + C2) Where C1 is the resource cost of one son and C2 is the resource cost of one daughter. In other words, the equilibrium sex ratio for the population is determined by the substitution cost, C2/C1; i.e., a parent can have one daughter, or C2/C1 sons. Therefore, the comparative resource cost of producing sons versus daughters—i.e. the tradeoff between producing sons and daughters—is the dominant feature of evolving systems that is the key to explaining their equilibrium sex ratio. In order for these frequency-dependent optimality models to provide an explanation for a population’s sex ratio, we must assume that within these target systems natural selection will

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optimize the criterion of these models (e.g. mating opportunities). Additionally, these models assume that other evolutionary factors (e.g. drift) will not deter the population from reaching the equilibrium favored by natural selection. In order to derive this result, these models frequently introduce idealizations, such as assuming that organisms reproduce asexually (i.e. offspring perfectly resemble their parents), mate randomly, have equal access to resources, and that the population is infinite in size, etc. As a result of idealizing away these details, the explanation provided by these models is applicable to a wide range of populations that are heterogeneous in many ways.3 This is what allows the models to provide an explanation for the kind of general biological pattern Fisher observed. 3. The Censored Causal Model Approach. One approach to understanding the explanations of biological optimality models is as a kind of “censored causal explanation”. The censored causal model approach contrasts biological optimality models with more comprehensive models that include additional causal factors; e.g. models that also represent genetic and epigenetic causes (Elgin and Sober 2002; Orzack and Sober 1994, 1996; Potochnik 2007, 2010). This approach characterizes the explanations provided by biological optimality models as causal explanations that ignore certain parts of the causal process that led to an evolutionary outcome (e.g. genetic causes) and emphasizes a modular part of that larger causal process—namely how natural selection shaped the trait in question. In other words, optimality models purposely ignore, or black box, the causal processes of genetic, epigenetic, and other evolutionary factors in favor of focusing on how the causal processes of natural selection influence the trait values in the population. For example, according to

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In addition, the male-female tradeoff remains essential even when various underlying assumptions of Fisher’s model are changed (Hamilton, 1967).

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Potochnik, “An optimality model focuses on a particular modular part of the causal process leading to the observed phenotype” (Potochnik 2007, p. 688). This causal process is modular in the sense that it is dissociable from the other causal factors involved in a trait’s evolution. Similarly, Steven Orzack and Elliott Sober claim that optimality models are “censored” models, “in which the only evolutionary force is natural selection” (Orzack & Sober, 1994, p. 363). (Orzack and Sober, 1994, p. 363). According to the censored causal model approach, an optimality model provides a special kind of causal explanation by accurately representing this dissociable component of the causal process that led to a phenotypic trait and ignoring other causal factors. In order to demonstrate that these censored causal models (i.e. optimality models) are able to provide adequate explanations of phenotypic traits, the censored causal model approach appeals to the predictive accuracy of an optimality model when compared with a model that includes additional causal factors. For example, Orzack and Sober (1994) appeal to the predictive accuracy of an optimality model in order to establish that the process of natural selection (presumably the one described by the optimality model in question) is a satisfactory explanation of a phenotypic trait. They argue that if the predictions of the optimality model fit the observations according to standard statistical criteria, then natural selection can be regarded as a sufficient explanation of the evolution of the trait. They claim, “Natural selection here provides a sufficient explanation because taking other factors into account could not significantly enhance the predictive accuracy of the optimality model” (Orzack & Sober, 1994, 363). This requirement for an optimality model to provide an adequate explanation is paralleled by Potochnik’s account (e.g. see Potochnik, 2007). Ultimately, according to the censored causal model approach, an optimality model will provide a satisfactory explanation when it accurately

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represents the causal factors of natural selection and these causes are sufficient to make the model as predictively accurate as models that include additional causal factors.

4. Against the Censored Causal Model Approach. Optimality models are distinguished by their use of Optimization Theory, which is applicable across varied scientific domains. In its most general form, Optimization Theory is just a mathematical technique that can be used to determine what values of some control variable(s) will—given a set of tradeoffs and constraints—optimize the value of some design variable(s) (Beatty 1980; Maynard Smith 1978; Seger and Stubblefield 1996). For example, if an engineer wants to construct a bridge, she may wish to optimize various design features; e.g. weight, cost, rigidity, width, etc. Not all of these can be optimized simultaneously however; certain tradeoffs (e.g. more width will mean more weight) and contextspecific limitations (e.g. limited funds) will constrain the optimal design. One way to solve this design problem is to construct and analyze an optimality model in order to deduce the set of control variables that will result in the optimization of the design variables. An optimality model describes a function, which relates each possible set of control variables (i.e. the strategies) to values of the design variable(s) to be optimized. This function and the set of available strategies are determined by the constraints and tradeoffs of the particular design problem. Once the strategy set and objective function of the optimality model are specified, one can deduce which of the available strategies will yield the optimal value of the design variables (or the single currency on to which the various design variables are mapped). In sum, optimality models identify key constraints and tradeoffs that hold within a system and then utilize those constraints and tradeoffs to determine the locally optimal (i.e. best available) solution to the design problem.

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This mathematical technique is replicated in biological optimality examples as well. For example, Parker’s copulation model provides a mathematical derivation of the locally optimal strategy given the various constraints and tradeoffs that hold within the population. Similarly, the foraging lapwing model calculates the locally optimal strategy given the population-level tradeoff between the average caloric benefit from the prey items obtained by moving a distance x and the average energy cost of moving distance x (along with the other constraints of the particular foraging problem). Indeed from the examples above we can identify some of the key features of biological optimality explanations. Namely, optimality models represent various relationships that hold between the constraints, tradeoffs and the optimal strategy of a system. In the case of evolving biological systems, it is then assumed that this optimal strategy is an equilibrium point that will be arrived at regardless of the step-by-step dynamics of the system. What biological optimality models ignore, then, is not a specific subset of the causal processes included in other dynamical evolutionary models, but rather a particular type of information— specifically all information about the step-by-step dynamics of the evolving system. This result is not surprising given that optimality models are a species of equilibrium models and it is precisely this feature that distinguishes equilibrium models from “dynamical” models (Sober 1983).4 This analysis does, however, show that the characterization of optimality models as censored causal explanations—i.e., explanations that represent a modular part of a larger causal process that led to the explanandum while omitting other causal factors—is quite misleading. For instance, Potochnik’s account suggests that optimality models differ from other evolutionary models only in which modular parts of the causal process that led to a phenotype

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I do not intend to claim that optimality models are the only kind of equilibrium models in biology. Surely they are not; e.g. many population genetic models, such as the Hardy-Weinberg law, are also equilibrium models. Optimality models do, however, provide a unique kind of equilibrium explanation by utilizing optimization theory. It is this feature that distinguishes them as optimality explanations.

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they represent; namely, optimality models represent the causal processes that are involved in selection and omit the rest (those that have to do with other evolutionary factors). However, what optimality models actually ignore is all of the step-by-step dynamics of evolving systems in favor of specifying an equilibrium point of the system that results from the structural tradeoffs and constraints present in the evolving system. In other words, optimality models are simply a different kind of evolutionary model when compared to models that represent the dynamics of causal evolutionary processes. Optimality models do not represent, “a particular modular part of the causal process leading to the observed phenotype” (Potochnik 2007, p. 688). Rather, biological optimality models explain by focusing on an entirely different set of relationships: the constraints and tradeoffs that hold between the represented variables at the level of the population and those variables’ relationship to the equilibrium point of the evolving system. Therefore, optimality models are not censored causal explanations, but instead offer a very different kind of explanation of evolutionary phenomena by showing how population-level constraints and tradeoffs entail the evolution of an equilibrium state(s).

5. A New Approach: Equilibrium, Idealization, and Tradeoffs. The previous section argued that many biological optimality models do not provide censored causal explanations. In response, I propose an alternative approach to understanding optimality explanations and argue that many biological optimality explanations present a serious challenge to the prominent causal approach to explanation and modeling. According to the causal approach, in order to explain an event a model must accurately represent the (contextually or metaphysically) salient causes of, or causal mechanisms that gave rise to, the target explanandum (Strevens, 2009; Woodward, 2003). However, the important features of biological optimality

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explanations are ones that move the model in precisely the opposite direction suggested by the causal approach by eliminating many of a target system’s causally relevant details. Furthermore, the key components of a biological optimality explanation do not contribute to the explanation in virtue of accurately representing causes. Below, I discuss three key features of biological optimality explanations. First, optimality explanations in biology are equilibrium explanations that do not reference the particular initial conditions or causal trajectories of their target system(s). Second, these models are often highly idealized such that their explanations do not accurately represent the causes of any real-world population(s). Finally, most of the explanatory work in these models is done by synchronic mathematical representations of multiply realizable constraints and tradeoffs that do not reference the causal processes of the model’s target system(s). 4.1. Equilibrium Explanation. Optimality explanations in evolutionary biology are equilibrium explanations. The optimality model explains the current state (or trajectory) of the population by assuming that natural selection will move the population towards the optimal strategy (or mix of strategies) and that other evolutionary factors will not deter the population from reaching this equilibrium point. Part of what is explanatory about this kind of equilibrium model is that it allows us to understand that no matter what the particular step-by-step causal trajectory of the population had been, within a certain disjunction of possible causal trajectories, the ultimate result of the population’s evolution would have been its equilibrium state. That is, the model tells us that a wide range of potential trajectories would all yield the same result, but it does not tell us which one is the actual trajectory or initial state of the target population (Sober, 1983).

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One way to understand the explanatory power of this kind of equilibrium explanation is that we know that the actual causal trajectory of the population is within the disjunction of possibilities that the model shows will lead to the same equilibrium. Michael Strevens argues that an equilibrium explanation of a ball’s arriving at the bottom of a basin is causal in this way. He says, “But it is not true that the explanation says nothing at all about the ball’s starting point and trajectory. It identifies the starting point as one of a large class, namely, all starting points at the basin’s lip, and likewise identifies the trajectory as one of an equally large class” (Strevens, 2009, p. 288). It is true that the model gives us some minimal information about the actual causal history of the evolving system by telling us it is within the disjunction of causal trajectories that would lead to the explanandum. However, citing this minimal causal information as the source of the model’s explanatory power would imply that the model is not a very good explanation compared with a model that would more precisely trace the population’s actual causal trajectory. There are, however, reasons for preferring explanations that do not cite the actual causal trajectories of particular systems (Batterman 2002, 2009; Garfinkel 1981; Pincock 2011; Weslake 2010). One reason is that citing the causal trajectory of a particular system means that the model is unable to apply to other systems subject to similar constraints and tradeoffs, but which are heterogeneous in their causes. Many philosophers have recognized the explanatory value of this kind of modal information (Garfinkel 1981; Jackson and Pettit 1992; Woodward 2003). By eliminating causal information about the particular causal trajectory of the system, the optimality model is able to capture a wider range of possible systems that are widely heterogeneous in their causal trajectories, but are subject to similar constraints and tradeoffs at the level of the population.

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Here, I do not intend to suggest that an explanation’s applying to more possible systems is always objectively better.5 Rather, I believe that the preference for generality or detail in our explanation is largely a matter of our explanatory interests (Sober, 1999). However, in many cases the generality provided by optimality explanations’ exclusion of information about the actual causal trajectory is what we want. Indeed, in many cases, the generality of our target biological explanandum dictates the need for a kind of equilibrium explanation that is independent of the initial conditions and causal trajectories of particular systems. For instance, in order to explain repeatable evolutionary patterns across causally heterogeneous populations— such as the frequency of the 1:1 sex ratio—often the causes of particular systems need to be eliminated (see Batterman, 2002 for similar kinds of pattern explanations in physics). In addition, the explanations given by optimality models are extremely enlightening precisely because they are independent of information about the actual causal trajectories of particular systems, not in spite of this feature. For one thing, it is an important piece of explanatory information about why an event occurred that the particular initial conditions and causal trajectory are not required for the target explanandum to occur. As Jackson and Pettit (1992, 177) put it, the macro-level explanation tells us that, “if the actual history described by the microcausal explanation had not obtained, the explanandum would still have occurred.”6 Although the causes in the step-by-step trajectory of the system are difference-making parts of the causal history of the explanandum, an equilibrium optimality model explains without referencing those causes. Instead, what these equilibrium explanations tell us is that the

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One problem is that measuring p-generality is notoriously difficult, but a comparison of p-generality is possible in many cases. 6 Jackson and Pettit claim that this modal information is given by a macrocausal explanation and tie the explanatoriness of the information to Lewis’ (1986) causal account. I don’t want to endorse this causal interpretation, but I do agree with Jackson and Pettit about the explanatory relevance of this model information.

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particular initial conditions and causal trajectory of the population are not important for understanding why the target explanandum occurred. By showing why the current state of the population is independent of these causal details the optimality explanation provides additional explanatory information that would not be captured by a model that represented the actual causal trajectory or initial conditions of a particular population. A key piece of explanatory information is the fact that the actual initial conditions and causal trajectory of the target system are not required for the occurrence of the explanandum because several different causal histories would have lead to the same outcome. Eliminating these causal details also allows our explanation to focus on the features of the target system that are essential for understanding the target phenomenon. Once we understand that the particular initial conditions and trajectory of the system are not important for understanding the occurrence of the explanandum, we can better appreciate the things that really matter: the structural constraints and tradeoffs that the optimality model mathematically represents and the optimization assumptions used to entail that the optimal strategy is the population’s equilibrium point. Therefore, in some instances, optimality models’ elimination of causally relevant details allows them to provide a better explanation than models that would trace the actual causal trajectory of the population. 4.2. Idealizations in Optimality Explanations. Yet even though optimality explanations do not trace the actual initial conditions or causal trajectory of a population, they may still provide a different type of causal explanation by accurately representing some causal mechanisms or causal processes that gave rise to the target explanandum. That is, the model may not tell us precisely how the selection process actually

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proceeded, but may still identify the causal processes responsible for the evolutionary outcome. However, even if optimality models can be shown to be the correct kind of causal mechanical models, they will almost always fail to meet the veridical standards required by causal approaches to explanation and modeling. This is due to the fact that optimality explanations within biology are almost always so highly idealized that they fail to accurately represent the causal mechanisms of any real-world population. The explanatory goal that motivates the introduction of these idealizations is often to capture biological explananda (e.g. patterns) that do not depend on the causes of any particular population. As a result, the best explanation of the target explanandum is often a model that does not accurately represent the causes of any realworld target system, but instead utilizes idealizations to highlight the dominant noncausal features (at the macro-level) that are responsible for the occurrence of the phenomenon across causally heterogeneous systems. To begin, optimality model’s mathematical representations of constraints and tradeoffs are often idealized representations. For example, in the foraging lapwing model the assumption is that average energy intake increases with increased distance traveled according to a perfectly asymptotic curve and that energy expended on movement is perfectly linear. But these smooth curves and constant parameters will almost always be inaccurate when compared with the causal processes in real-world populations. Moreover, in many game-theoretic optimality models, idealizations are often introduced into the payoff structure in order to use certain mathematical operations required to deduce the equilibrium state of the population; e.g. the idealization that a conflict is perfectly symmetric or that payoffs are constant across iterations of the game. These idealizations are tolerable in an optimality explanation because the mathematical representations of a biological optimality model are often only intended to capture the basic type of relationship

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between these variables at the level of the population. That is, they are intended to capture constraints, tradeoffs and fitness differences that range over aggregates of individuals—they are not meant to accurately represent the causal processes acting on individuals within the population. Second, the strategy sets of most optimality models are also idealized in that they do not accurately represent the set of strategies actually causally interacting in the population’s history. Indeed, one difficultly in constructing optimality models is that there is often no way to tell precisely which phenotypic strategies were available in the past history of the population. The assumed strategy set, however, is not intended to accurately identify precisely the set of strategies that were available, but merely aims to capture the relevant alternative strategies such that the optimal strategy can be deduced. For instance, although a sex ratio model may assume that the strategy set is the probability that each birth will be a male and can vary between 0 and 1, this is not intended to claim that in the target population there were individuals playing this range of strategies. Rather, the idealizing assumption is introduced because it is assumed that the optimal strategy will evolve regardless of which particular distribution of strategies actually occurred in the population’s history (as long as it includes the optimal strategy). That is, accurately representing the strategies available within a population is not required for the optimality model to explain the evolutionary outcome. The actual strategies available are, however, causally relevant to the causal evolutionary process that occurred. Accurately representing the causal processes (or factors) of the selection of the trait would require specifying which strategies were actually competing with one another in the population’s history. This causal information, however, is often explicitly not included in optimality explanations because these causal details are irrelevant to the optimality model’s explanation.

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In addition, in biology, a model’s optimization assumptions are almost always idealizations when compared with a population’s causal processes. For instance, in foraging models it is often assumed that natural selection will maximize average energy intake. However, this is only one thing that might influence the survival and reproduction of organisms in the population. In fact, the causal processes acting in the population often will not optimize the criterion specified by a biological optimality model. Therefore, the optimization process described by the model often will not accurately represent the actual causal selection process of the target population. The goal of a biological optimality explanation, however, is not to accurately describe the causal selection process at each step in the evolution of a trait. Rather, biological optimality models identify optimal strategies (or states) that are only outcomes or end states of an evolving system that approximates their optimization assumptions in the long run. Therefore, a model’s optimization assumptions are taken to be “adequate” if they capture the general optimizing tendency of the target system. In other words, optimality models are frequently used to explain why a system has evolved the optimal strategy regardless of whether their optimization assumptions are true of the causal processes acting in the population at any particular point along the way to that strategy. Next, the most contentious idealizations of many biological optimality models concern the way in which phenotypic strategies are inherited. This group of idealizations includes assumptions that organisms reproduce asexually and mate randomly. Both of these assumptions are often false of the target population(s) of the optimality model. They are introduced because it is assumed that changing these assumptions will have no affect on the occurrence of the phenomenon. In other words, the model’s explanation claims that the actual causal mechanisms that underlie these kinds of inheritance assumptions are not important for understanding why the

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phenomenon occurred since other inheritance mechanisms would have led to the same outcome. Therefore, we can remove these causal details, or at least specify them inaccurately, without our model losing any explanatory power. The causes underlying these inheritance assumptions are, however, important difference-makers in the explanandum’s causal history—without some causal process of inheritance nothing would have evolved. Nevertheless, an accurate representation of these causal processes is not required for the optimality model to explain the target phenomenon. Another important idealization in many optimality explanations is the assumption that the environment is constant enough that the selective pressures of the evolving system will not change significantly over time. This assumption is required in order for the optimal strategy to be an obtainable equilibrium state—i.e. the population needs time to arrive at the predicted optima before changes in environmental conditions alter the equilibrium point of the system. However, the selection pressures in a population are rarely constant in the way assumed by the optimality explanation. Accurately representing these changes in the population’s causal history, however, is not important to the explanation of the optimality model since it is assumed that, although they do make a difference to the particular trajectory of the population, the selective features represented within the optimality model will ultimately overcome their influences. Therefore, although assumptions concerning a constant environment distort the causal processes of realworld populations, they contribute to the explanation of the optimality model whenever those causal details are irrelevant to understanding the target explanandum. Finally, most optimality models assume that the population being modeled is infinite, or effectively infinite. This, however, is always an idealization; no real world population is infinite. Furthermore, population size does make a difference to every evolutionary process; i.e., drift is 23

present in every real-world population. Assuming infinite population size has the effect of setting the influence of drift to zero by utilizing various laws of large numbers. The model assumes that even if drift changes the population along the way, natural selection will have enough influence to overcome those changes in the long run. By incorporating the idealization of infinite population size, the optimal trait according to natural selection (i.e. the strategy that optimizes the model’s criterion) is what we expect to evolve. This assumption, therefore, is vital to the optimality model’s ability to explain why the current population will evolve the optimal strategy of the model. It is, however, an idealization—all populations experience drift as well as other evolutionary factors. As before, this idealization eliminates various details of the target population because they are assumed not to be important for the occurrence of the explanandum. To summarize, I have identified six kinds of idealizations frequently utilized within biological optimality explanations: (1) The mathematical curves, equations, or payoff structures of the model utilize idealizing assumptions in their representations of a system’s constraints and tradeoffs. (2) Idealized strategy sets are intended to capture the relevant alternatives rather than actual strategies causally interacting within a population. (3) The models’ optimization assumptions are often idealized such that the model only captures the general optimizing tendency of the system in the long run. (4) There are idealizations regarding how traits are inherited (e.g. asexual reproduction) and assumptions concerning random mating.

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(5) It is often assumed that environmental changes do not occur; i.e. that the selective pressures in the model are stable over the evolution of the trait. (6) Infinite population size is assumed to eliminate drift and to allow for the use of various laws of large numbers in deducing the target explanandum. These idealizations entail that biological optimality models usually provide little, if any, accurate information about the actual causes, causal processes, or causal mechanisms of any realworld population(s). Indeed, these idealizations are often introduced because it is assumed that various causes of particular target systems are not important to the explanation provided by the optimality model. In other words, these idealizations make essential contributions to the explanation provided by an optimality model because they claim that most of a system’s causal factor(s), mechanism(s), or variable(s) are not important for understanding why the explanandum occurred. Although many of these causal details are relevant to the veridical causal explanation, they are not required to explain the explanandum—other causes would have been sufficient so long as the constraints, tradeoffs, and optimization assumptions of the optimality model are satisfied. Therefore, by eliminating these causes we are able to provide an explanation that is able to apply to systems in which these causes are different. That is, by introducing various idealizations that distort or eliminate the causal details of particular systems, we are able to provide an optimality explanation that is able to apply to a wider range of possible systems (Weslake, 2010; Batterman 2002, 2009). In addition, this kind of explanation is precisely what is required in order to explain biological patterns that range over systems that are heterogeneous in most of their causal details.

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Furthermore, as before, the explanation of the optimality model now contains explanatorily enlightening information about what is not required for the occurrence of the explanandum. Part of what the idealized optimality model enables us to understand is that these causal details of particular systems are largely (or completely) irrelevant to the occurrence of the phenomena we wish to explain. Therefore, when applicable, an optimality explanation will sometimes be superior to a model that accurately represents a system’s causes because it provides us the explanatory information that most of the system’s causal details are irrelevant. What is more, these idealizations aid our grasping of the key structural relationships that are important for understanding the phenomenon. By eliminating details about the causes of particular populations, we isolate the structural relationships essential to understanding the target explanandum: those that obtain between the constraints, tradeoffs and optimality strategy with the model. If an optimality model adequately captures these dominant population-level features, the model can provide a unique kind of highly idealized equilibrium explanation despite the fact that it fails to accurately represent the causes of any real-world population. 4.3 Synchronic Mathematical Representations of Multiply Realizable Features. The mathematical representation of constraints and tradeoffs is a central component of optimality models. An optimality model, then, might be understood as a causal explanation if these constraints and tradeoffs can be understood as contributing to the explanation in virtue of representing causes of, or causal relationships that gave rise to, the target phenomenon. However, the tradeoffs represented within optimality explanations are often not causal relationships between variables. For instance, in the foraging lapwing model there is an important tradeoff between average energy intake and average energy cost of movement. However, it is strange to claim that the average energy intake of individuals playing a particular 26

strategy is causally influencing the average energy cost for individuals playing that strategy, or vice versa. What the model claims is that at the level of the population we see a tradeoff between these two averages when we aggregate over the strategies being played by individual organisms. Yet, these two population-level averages are not causally interacting. Indeed, in many cases, the tradeoffs that are central to an optimality explanation are not causal relationships. Moreover, even if two variables are both causes of some third variable, this does not entail that the tradeoff that exists between them is a cause of the target explanandum of the optimality model: the equilibrium point of the evolving system. For example, not only is it misguided to claim that average energy intake is a cause of average energy expenditure for a strategy, but it is even more puzzling to claim that the tradeoff between these two population-level averages average is a cause of anything. One reason for this noncausal analysis of these tradeoffs in optimality models is that they are simply not the sorts of things that can enter into causal relationships—they are not events, nor are they causal properties. Generally, causal explanations should provide information about the explanandum’s causal history. Yet as David Lewis explains, “a causal history is a relational structure. Its relata are events: local matters of particular fact, of the sorts that may cause or be caused” (Lewis 1986, p. 216). Other accounts of causal explanation appeal to causal properties. However, a tradeoff is not itself an event, nor is it a causal property within the population. There is a dependence relation here that is key to the ability of these features to explain the biological phenomenon; i.e. changing the tradeoffs results in a change in the predicted outcome of the evolving population. However, this dependence is not a causal one. Another reason to think that the constraints and tradeoffs of optimality explanations do not explain by accurately representing causes is that the optimality models’ mathematical

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representation of them does not reference any causal processes, or events, that unfold prior to the explanandum. The model merely identifies the optimal strategy by showing that the model’s currency is optimized by a particular strategy, given the static constraints and tradeoffs represented within the model. For instance, in Parker’s copulation model, the optimal strategy— the point at which rate of fertilization is maximized—is simply represented as the value of c at which the slope of a line that intersects the curve and passes through the origin is maximized. Nowhere does the model describe a causal process (or trajectory) that unfolds over time or any events that occur prior to the explanandum—the model merely identifies the optimal strategy by showing that rate of fertilization is maximized at the point where c = 41 minutes. Furthermore, none of the points along these mathematical curves need be instantiated by the evolving population on its way to equilibrium. These models specify endpoints of evolution by natural selection by using synchronic mathematical calculations, rather than giving a dynamical account of the causal processes or events that led to that endpoint. In most cases, an optimality model merely mathematically deduces the point (or strategy) that the model’s synchronically represented constraints and tradeoffs are expected to move the population towards without specifying the causal processes that underlie this optimization process. Indeed, an explanatory strength of many biological optimality models is that their macrolevel descriptions of these features are able to capture a range of systems that behave similarly at the population level, despite dissimilarities in their causes. The generality of these explanations is due to the multiple realizability of the macro-level properties they employ (Fodor 1974; Garfinkel 1981; Jackson and Pettit 1992; Kitcher 1984; Sober 1999). In the case of optimality models, the generality is due to the fact that the constraints and tradeoffs represented within the optimality model are multiply realizable at the level of causal mechanisms. Furthermore, this

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generality is precisely what is required in order to explain highly general biological patterns that range over systems that are widely heterogeneous in their causes.

5. Conclusion. The above analysis identifies several important features of optimality explanations in biology. Each of these features contributes to the optimality explanation by eliminating or distorting causal details that are not required in order to understand the occurrence of the explanandum. In other words, these important features move the model in precisely the opposite direction suggested by the causal approach by eliminating many of the system’s causally relevant details. What do matter in these explanations are the structural relationships between a system’s constraints, tradeoffs, and optimal strategy along with the optimization assumptions of the model. By showing how these features of a system relate to the evolution of an equilibrium state, optimality models are often able to provide an explanation that is independent of the causes of any particular real-world system. Moreover, in many cases, a model that accurately represented the system’s causes would provide a worse explanation. By eliminating the causal details of the model’s target system(s), a biological optimality model will often: (1) apply to more possible systems, (2) provide explanatory information about what is not required for the explanandum, (3) provide explanatory information about why the phenomenon would occur in other similar systems, and (4) highlight the structural relationships that are essential to understanding why the phenomenon occurred. Therefore, in many cases, it is in virtue of not accurately representing the causes of their target system(s) that optimality explanations are able to provide a special kind of explanation of biological phenomena. Consequently, our account of explanation and modeling will need to expand beyond the causal approach.

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