The Levelling-Down Objection and the additive measure of the badness of inequality

Abstract The Levelling-Down Objection is a standard objection to monistic egalitarian theories where equality is the only thing that has intrinsic value. Most egalitarians, however, are value pluralists; they hold that, in addition to equality being intrinsically valuable, the egalitarian currency in which we are equal or unequal is also intrinsically valuable. In this paper, I argue that the Levelling-Down Objection still minimizes the weight that the intrinsic badness of inequality could have in the overall intrinsic evaluation of outcomes, given a certain way of measuring the badness of inequality, namely, the Additive Individual-Complaints Measure.


Introduction
The Levelling-Down Objection is a standard objection to monistic egalitarian theories where equality is the only thing that has intrinsic value. For an illustration of the objection, consider the following distributions: In distribution A, person P 1 is better off than person P 2 , whereas, in distribution B, they are equally well off. Suppose that P 2 is worse off than P 1 in A through no fault of their own. Then, clearly, B is more equal than A. But, since P 1 is better off in A than in B and everyone else is at least as well off in A as in B, it seems that A should be overall intrinsically better than B. Hence equality can't be the only thing that is intrinsically valuable, because, if it were, B would be overall intrinsically better than A (Parfit 1995: 26-28).
Nevertheless, the Levelling-Down Objection doesn't work against pluralistic egalitarianism, where equality is not the only thing of intrinsic value. Larry S. Temkin proclaims: 1 I, for one, believe that inequality is bad. But do I really think that there is some respect in which a world where only some are blind is worse than one where all are? Yes. Does this mean I think it would be better if we blinded everybody? No. Equality is not all that matters. (Temkin 1993: 282) Likewise, Joseph Raz (2009: 32n7) argues that it's implausible that there would be any intrinsic value in an equal distribution of something intrinsically neither good nor bad. Hence, to be plausible, egalitarianism must be pluralistic; because, in addition to equality, the egalitarian currency in which we are equal or unequal must also be of intrinsic value. And then the Levelling-Down Objection doesn't work any more, because the loss for P 1 in B of the additional unit of well-being might be intrinsically worse than the inequality in A. (We assume here that the currency of egalitarian equality is well-being, but nothing crucial for our discussion will depend on this assumption.) In this paper, I shall argue that a version of the Levelling-Down Objection still minimizes the weight that the intrinsic badness of inequality could have in the overall evaluation of outcomes given an additive way of measuring the intrinsic badness of inequality.

The Additive Individual-Complaints Measure
Temkin favours an additive measure that adds up all undeserved pairwise differences in well-being: the ultimate intuition underlying egalitarianism is that it is bad : : : for some to be worse off than others through no fault of their own. : : : . [T]he additive principle reflects the view that if it really is bad for one person to be worse off than another through no fault of his own, it should be even worse for two people to be in such a position. Similarly, the relative to all those better off view of complaints reflects the view that if it is bad to be worse off than one person through no fault of your own, it should be even worse to be worse off than two. : : : . After all, to paraphrase the basic insight of the utilitarians, more of the bad is worse than less of the bad. (Temkin 1993: 200-201) Here, Temkin suggests The Additive Individual-Complaints Measure The intrinsic badness of the inequality in distribution X equals the sum of the absolute value of all well-being differences (not due to the fault of the worse off) for all distinct pairs of individuals in the population. 2 1 See also Scanlon (1976: 9-10). 2 I am following, more or less, the interpretation in Arrhenius (2013: 79). Temkin (2001: 346-348) also allows that inequality matters less at higher levels, but this complication won't matter for our discussion.
Given the Additive Individual-Complaints Measure, we have that the intrinsic badness of the inequality in A is 1 unit of badness, since there is only one distinct pair of individuals and the absolute difference between their well-being levels is 1. And the intrinsic badness of the inequality in B is 0 units of badness, since the members of the only distinct pair of individuals have the same amount of well-being. To avoid the Levelling-Down Objection, the loss of one unit of badness of the inequality in A when we level down to B must count less for overall intrinsic value than P 1 's loss of one unit of well-being. Hence the Levelling-Down Objection puts an upper limit to how much inequality can matter compared with well-being in the balance of overall intrinsic value.
This might not seem very worrying for pluralistic egalitarianism. We can, however, construct a more severely limiting version of the Levelling-Down Objection which puts this upper limit at an arbitrarily low level. This basically rules out that inequality has any influence on the overall intrinsic value of an outcome, compared with losses and gains in personal wellbeing. Consider the following distributions, where n (the size of the population) can be arbitrarily large: The Additive Individual-Complaints Measure entails that when n gets larger the intrinsic badness of the inequality of C gets worse, with no upper bound on how bad it could get. The intrinsic badness of the inequality of C will be equal to n 1, since each additional person at well-being level 1 creates exactly one new pair of people with a well-being difference between them: the pair of that new person and P 1 . So we get that the inequality in C is arbitrarily bad given a large enough n. The intrinsic badness of the inequality in D, however, remains at 0 no matter how large n gets, since D never gets any people who differ in well-being. So, given that n is large enough, we have that D is much better than C with respect to inequality. On the other hand, there seems to be only one respect in which C is better than D, namely, P 1 has slightly higher well-being in C than in D.
We can sum up the advantages of C and D relative to the other as follows: Advantages of C over D: P 1 has one more unit of well-being in C than in D.
Advantages of D over C: D has no inequality, whereas the inequality in C is intrinsically bad to an arbitrarily great extent.
To avoid the Levelling-Down Objection, we must still claim that C is overall intrinsically better than D. And we must claim this regardless of how small the unit of well-being is. Thus we get that any amount of badness of inequality counts less than a minimal gain in the well-being of one person. The relevance of the badness of inequality must therefore be negligible compared with the relevance of the value of well-being for comparisons of overall intrinsic value. Hence this limiting version of the Levelling-Down Objection shows that the relative importance of inequality compared with well-being for overall intrinsic value must be zero (or arbitrarily close to zero), given the Additive Individual-Complaints Measure. Note that this objection only applies to measures where the intrinsic badness of the inequality in C grows without an upper bound as n gets larger. 3 Measures based on the average rather than total of the differences between individuals avoid the problem. 4 Likewise, the Gini-coefficient measure also avoids this problem. The Gini coefficient can be calculated by multiplying the sum of all pairwise well-being difference with 1=2n 2 µ, where n is the number of individuals and µ is the average level of well-being. 5 These more complicated measures, however, give up the straightforward additive relationship between the badness of undeserved individual differences in well-being and the overall badness of inequality, which Temkin found compelling in the above quotation.

Positive egalitarianism
Temkin proposes an objection of his own to the Additive Individual-Complaints Measure. His objection concerns comparisons of inequality between distributions for populations of different sizes. The Additive Individual-Complaints Measure entails the following counter-intuitive conclusion: The Repellant Conclusion For any world F, let F 's population be as large (though finite) as one likes, and let the gaps between F 's better-and worse-off be as extreme as one likes, there will be some unequal world, G, whose population is "sufficiently" large such that no matter how small G's gaps between the better-and worse-off might be G's inequality will be worse than F 's (even if everyone in G is better off than everyone in F). (Temkin 1993: 218) 3 Temkin (2012: 328-329) suggests a capped model for ideals, which would put an upper bound on how much the badness of inequality can count against other ideals. This wouldn't solve the problem, however, because a capped model would still have to imply that the badness of inequality has almost no weight in large populations. 4 See, for example, Rabinowicz (2003: 62). 5 See Sen (1973: 31) and Donaldson and Weymark (1980: 70).

The Repellant Conclusion can be visualized as follows:
Here, worlds F and G are each represented by two combined boxes, corresponding to the better-off and worst-off halves of their populations. The width of the boxes corresponds to the number of people, and the height of the boxes corresponds to their well-being level. The dashed parts of the top of the two G-boxes reflect that these boxes could be much wider. The Repellant Conclusion says, roughly, that, for any world like F, there is a world like G which has worse inequality than F.
In response to variable populations cases of this kind, Gustaf Arrhenius (2013: 85) proposes a positive form of egalitarianism where, in addition to pairwise relations of inequality being intrinsically bad, pairwise relations of equality are intrinsically good. This form of egalitarianism could avoid the Repellant Conclusion, since there are many more pairwise relations of equality in G than in F; so, if the positive value of these pairwise relations of equality were given sufficient weight compared with the negative value of pairwise relations of inequality, G would be better than F with respect to the combination of equality and inequality (Arrhenius 2013: 87).
Could this positive egalitarianism also save the additive approach from the limiting version of the Levelling-Down Objection? It cannot. It makes the problem worse: In addition to D having fewer pairwise relations of inequality than C, we also have that D has more pairwise relations of equality than C. There are n 1 more pairwise relations of inequality in C than in D. Accordingly, there are n 1 more pairwise relations of equality in D than in C. So, in addition to D being arbitrarily better with respect to inequality than C, it is also better with respect to equality, since D has all the equality relations C has plus some further ones. Hence this positive egalitarianism just introduces a further respect in which D is better than C. And then, given an additive measure of the intrinsic goodness of equality, the Levelling-Down Objection yields that the relative importance of equality for overall intrinsic value compared with well-being must be zero (or arbitrarily close to zero). 6 F G 6 Note that this also holds given Arrhenius's (2013: 89) proposal that only large differences in well-being have negative value; small differences in well-being have positive value rather than negative, because they are roughly equal and thus close to perfect equality. As long as relations of perfect equality are better than relations of rough equality (which seems plausible), D will still be better than C with respect to the combination of equality and inequality.