Additivity and the Wimsatt Lloyd Approach

Wimsatt (1980) and Lloyd (1988) both argue that the concept of additivity holds the key to the levels-of-selection question. Additivity is another word for linearity. If two factors combine additively to produce a given effect, this means that the effect is a linear function of each factor's contribution. Less than perfect additivity means that the difference made by one factor depends on the other factor's contribution, so the factors interact. Non-additivity is one way that emergence might be made into a mathematically respectable notion.2

Additivity plays an important role in population genetics, where it is often important to know whether genes combine additively in the production of phenotypes, or whether they interact. Usually, geneticists talk not about additivity per se but additivity of variance.3 Perfectly additive variance means that differences between entities, with respect to a given variable, are fully explained by differences in the independent contributions of some factors. So for example, if genotype fitness is the variable and the factors are two alleles A and B, the variance in genotype fitness is perfectly additive if and only if (waa — wab) = (wab- wbb), that is, genotype fitness is a linear function of allelic 'dosage'. This means that fitness differences between genotypes are fully explained by differences in the number of A alleles they contain. The concept of additive variance is easily generalized beyond population-genetic contexts.

The idea that additivity might be relevant to the levels of selection is prima facie quite plausible. In a multi-level scenario, if all the variance in collective fitness is additive, then fitness differences between collectives are fully explained by differences in their particle composition; from which one might infer that particle-level selection is doing all the causal work. (This seems to be the intuition driving Wimsatt and Lloyd.) Interestingly, this suggestion ties in with Sewall Wright's (1980) approach to the levels question. For Wright, the distinction between 'genic' and 'organismic' selection, in the context of his 'shifting balance' theory of evolution, hinged precisely on additivity. Dominance and epistasis, which generate non-additive variance in genotype fitness, shift selection from the genic to the organismic level, according to Wright.4

2 This suggestion has surfaced in the literature on occasion. Thus for example, Vrba (1984) defines an emergent character as one which is 'related by a nonadditive composition function to characters at lower levels' (p. 19).

3 Strictly speaking, 'additivity' refers to the pattern of functional dependence itself, 'additive variance' to the pattern of statistical variation that the dependence gives rise to, so the concepts should not be equated. An effect could depend additively on two factors and yet show no additive variance, if the factors happen not to vary in the population. This is the basis of the distinction between 'statistical' and 'physiological' epistasis drawn by Wolf, Bradie, and Wade (eds.) (2000). Epistasis means non-additivity, in effect.

4 However, Wright is using both 'genic selection' and 'organismic selection' in a nonstandard sense. By the former, he means directional selection within a large panmictic

Figure 4.1. Particles of two types nested within collectives

Figure 4.1. Particles of two types nested within collectives

More recently, Michod (1999) has argued that non-linear interactions play a key role in transitions between units of selection.

Despite its prima facie plausibility, the Wimsatt/Lloyd idea has been heavily criticized in the literature (Lewontin 1991; Godfrey-Smith 1992; Sarkar 1994; Sober and Wilson 1994). My strategy will be to use the foregoing analysis of cross-level by-products to try to adjudicate the debate.

Consider a multi-level scenario with strict nesting and a fixed number of particles per collective, as before. Particles are of two types, A and B, found in differing proportions in different collectives (Figure 4.1). Consider a collective character Z which varies in the population, that is, Var (Z) = 0. It is possible that a collective's Z-value depends on the proportion of A-particles that it contains. If the dependence is perfectly linear, as in Figure 4.2A, then all the variance in Z is additive. This means that differences in collectives' Z-values are fully explained by the differing proportions of A-types that they contain. A non-linear pattern of dependence is shown in Figure 4.2B. This means that at least some of the variance in Z is non-additive. Note that we can take Z to be fitness itself—either fitness! or fitness2—permitting us to ask whether all the variance in collective fitness is additive.

The simplest version of the Wimsatt/Lloyd proposal is that if all the variance in fitness at a level is additive, then selection is not acting at that level—all the action is at a lower level.5 More precisely, I will take the additivity proposal to be the conjunction of two claims:

(I) if there is collective-level selection, there must be non-additive variance in collective fitness, and population of the sort associated with Fisher (1930). By the latter, he means diffusion of co-adapted genotypes between partially isolated demes of a species. Note also that Wright's original presentation of the shifting-balance theory does not use the terminology of genic and organismic selection; see for example Wright (1931).

5 This formulation is closer to Wimsatt (1980) than to Lloyd (1988).

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