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1. Introduction
   The good of those who are worse off matters more to the overall good than the good of those who are better off does. But being worse off than one’s fellows is not itself bad; nor is inequality itself bad; nor do differences in well-being matter more when well-being is lower in an absolute sense. Instead, the good of the relatively worse-off weighs more heavily in the overall good than the good of the relatively better-off does, in virtue of the fact that the former are relatively worse off. At least, that is my view. The goal of this paper is to articulate this view more precisely and offer a defence.

There are three standard answers in the literature to the question of how to measure the overall good in a society: utilitarianism, egalitarianism and prioritarianism. Utilitarianism says that overall good is total or average good, regardless of how that good is distributed across individuals. Egalitarianism says that overall good is a matter of both total or average good and also how that good is distributed, since inequality is in itself bad. Prioritarianism says that overall good is total or average moral value, where additional good accruing to an individual is less morally valuable the better things already are for her. Thus it says that equal distributions are better than unequal ones, but not because inequality is itself bad.

This paper is primarily aimed at those who are motivated by the commitments that motivate prioritarians – that equal distributions are better than unequal ones, but not because inequality is itself bad. I will show that there is another alternative to prioritarianism, which I call relative prioritarianism, and I will give reasons for preferring this alternative.

The positions in this paper each have a philosophical and formal component. However, since I am most concerned with the philosophical motivations that distinguish the positions, I will only briefly introduce the formalisms by way of examples, and leave complete formal definitions to the Appendix.

2. Set up and background
   We are here concerned with the aggregation question: once we’ve settled on what constitutes an individual’s good, how do we evaluate a distribution of individual good?

We will use as an example a society with three people. A social distribution is a list of the consequences for each individual: for example, {ANN, moderately happy life; BOB, short and difficult life; CECIL, extremely happy life} is a distribution (call this distribution D1). (Consequences are to be understood broadly, to include anything that affects individual well-being; for example, they might include an individual’s personal relationships.) We want to know how good this distribution is, and whether it is better or worse than other distributions. We will use ‘utility’ to stand in for prudential value, that is, the good of a consequence for the individual who has it, and we will assume that utility is given, whether it tracks some objective fact or is derived from preferences. So, for example, D1 might have utility values {ANN, 200; BOB, 100; CECIL, 300}. To explain the various views, let us also consider another distribution, D2 = {ANN, 200; BOB, 200; CECIL, 200}.

An aggregation rule assigns a numerical value to each distribution, such that of two distributions, the one with the greater numerical value is better. Equivalently, an aggregation rule provides a complete ranking of distributions. Whether the ranking or the aggregation rule is more ‘fundamental’ will not concern us here: we will talk in terms of aggregation rules but keep in mind that rules that give equivalent rankings are in a formal sense the same rule.

We will make some simplifying assumptions. First, we will only consider social distributions with fixed utility: we will not evaluate lotteries over social distributions, or social distributions over lotteries. While any complete theory of aggregation will have to evaluate the latter,Footnote 1 I take evaluation of the former to be more fundamental. The second assumption is that we are dealing with a fixed population of individuals. Thus, a given rule can be formulated either as evaluating total good or average good, since the two formulations will produce equivalent rankings and equivalent values up to scale. We will leave open how the correct rule should be extended to societies with different numbers of people.Footnote 2

The aggregation question presupposes that the good of a distribution can be determined from the good of its constituent parts. This rules out a number of plausible views,Footnote 3 but I simply do not have space to address them here.

To answer the aggregation question, we must do two things. We must give a formal theory of aggregation, that is, we must provide an aggregation rule. And we must give an ethical interpretation of the rule that explains what the rule means in terms of what we value. It is important to remember that formal theories don’t come interpreted, and as we will see, it is possible for a formal theory to have more than one possible ethical interpretation, or an ethical claim to have more than one possible formal correlate.

The three most prominent views are utilitarianism, egalitarianism and prioritarianism. Each can be presented in a ‘total’ or ‘average’ version.Footnote 4

Total utilitarianism says that the value of a distribution is its total utility, and average utilitarianism says that the value of a distribution is its average utility. So, the value of both D1 and D2 according to total utilitarianism is 600, and the value of both D1 and D2 according to average utilitarianism is 200. According to utilitarianism, they are equally good.

Egalitarianism holds that the fact that some people are worse off than others is bad in itself, independent of the utility values involved. As Parfit defines it, egalitarians accept the Principle of Equality:Footnote 5

Principle of Equality: It is in itself bad if some people are worse off than others.

And they typically accept this principle alongside the Principle of Utility:

Principle of Utility: It is in itself better if people are better off (on average or in total).

Thus, egalitarians hold that utility and equality each matter, though there may be some trade-off involved in doing better with respect to each of these principles.

Formally, egalitarianism takes the average or total (i.e. utilitarian) value of a distribution, and subtracts from it some value that measures the inequality in the distribution.Footnote 6 Thus, as long as D1 scores worse than D2 with respect to inequality, D2 will be better than D1. (For example, taking the average version of the view, if the inequality score of D1 is 30 and that of D2 is 0, then the value of D1 is 170 and the value of D2 is 200.) Egalitarianism says that the overall good in a society is a combination of two different sources of value: prudential value for individuals, and how that value is distributed.

Prioritarianism holds that we ought to maximize total moral value, where moral value is a concave function of utility: it is more morally valuable to increase someone’s good by a given amount the lower his absolute level of good.Footnote 7 (As one has more utility, additional bits of utility are less morally valuable. As Parfit (Reference Parfit1991: 105) says, “just as resources have diminishing marginal utility, so utility has diminishing marginal moral importance.”) The value of a distribution is then its total moral value. Although the ‘total’ formulation is normally used, the ‘average’ formulation is easily defined: the value of a distribution is its average moral value.

So, for example, we might have the following moral value function of utility, which diminishes marginally:

The overall moral value, according to the total prioritarian, of D1 and D2 will be:

And the overall moral value of D1 and D2, according to the average prioritarian, will be:

D2 is better than D1, because D2 contains more total or average moral value: when a distribution is more spread out in terms of utility, individuals with lower utility bring down the total or average moral value more than individuals with higher utility bring it up. Thus, if moral value diminishes marginally in utility, then when two distributions have the same total or average utility, the one that is less spread out in terms of utility will have a higher total or average moral value. (Equivalently: it is always better to give a fixed utility benefit to a worse-off person than to a better-off person.)

3. Prioritarian desiderata
   Prioritarians accept three claims. The first is about what utility measures, and is accepted by all three parties in the debate so far:

individual good: Utility for each person depends only on her own consequence.

For example, if the consequence I have has a particular utility for me, then its utility doesn’t go down if someone else has a higher level of utility.

individual good says that individual utility values are not relative: whatever is meant by prudential good or well-being, it is not the kind of thing that I get less of because someone else has more of it. This isn’t to say that what others have can’t make a difference to my well-being. It is that these facts are already included in the description of my consequence, the thing to which utility attaches.Footnote 8 (So, for example, if the fact that others have much more than I do means I am powerless and this makes my life less good, then the fact that I am powerless is included in the description of the consequence I get.) This may mean that consequence descriptions are somewhat complex; but once we’ve determined my utility, it doesn’t make a difference to me if other people have more or less of that. One could think of this either as a substantive point or a stipulative definition of utility.Footnote 9

The second claim that prioritarians accept is the one that egalitarians accept and utilitarians reject:

spread aversion: Of two distributions with the same total or average utility value, the one that is less spread-out in terms of individuals’ utility levels is better.Footnote 10

To avoid confusion, we should contrast the concept of utility-level spread with the concept of ‘welfare diffusion’ (Persson Reference Persson2011, Reference Persson2012). Welfare diffusion means that a given amount of utility is divided among a larger number of people (e.g. {A, 200; B, 200} is more diffuse than {A, 400}). Utility-level spread means roughly that the individuals’ utility levels are ‘farther apart’ (e.g. {A, 400, B, 0} is more spread out than A, 200; B, 200}).

The final principle that prioritarians accept is one that utilitarians accept and egalitarians reject:

no distributional good: Relational or global properties do not have intrinsic value.

By ‘relational or global’ properties I mean properties that irreducibly concern more than one person. To adhere to no distributional good is to hold that the only locations of intrinsic value are individuals, and that the only objects of intrinsic value are considerations pertaining to individuals. Properties of the distribution as a whole – like how much inequality there is – don’t have any independent value. no distributional good rules out a particular explanation for spread aversion. It rules out the egalitarian explanation that inequality is intrinsically bad (or equality is intrinsically good) – that it is in itself bad if some people are worse off than others, even if it isn’t bad for any particular person.

Call the conjunction of individual good, spread aversion, and no distributional good the prioritarian commitments.

Prioritarians explain spread aversion with reference to the fact that the worse-off should be given priority because they are the worse-off. But they also say something more specific: that those who are worse off in an absolute sense should be given priority (Parfit Reference Parfit1991: 23). We can see the primacy of absolute utility level by looking directly at the prioritarian aggregation rule P. The moral value function v transforms absolute utility level, independent of any distributional facts. How morally valuable it is to raise someone’s utility by a given amount depends on her absolute utility level, not on how much utility she has relative to someone else. Raising an individual’s utility level from, say, 100 to 200 has the same moral value whether she is the worst-off individual in a particular distribution or the best-off.

Thus, in addition to the three principles just mentioned, prioritarians accept:

absolute priority: It is more valuable to increase the utility of the absolutely worse-off than that of the absolutely better-off, regardless of their utility relative to others in the society.

For this reason, I will refer to prioritarianism as absolute prioritarianism. (Absolute prioritarianism should not be confused with the claim that the worst-off individual gets ‘absolute’ priority, in the sense that his interests trump everyone else’s. It means that priority depends on absolute utility.)

4. Rank-weighted utilitarianism
   We’ve just seen that absolute prioritarianism is an appealing way to satisfy the prioritarian commitments. It satisfies them by maintaining the ‘simple average’ or ‘simple sum’ structure of utilitarianism, but averaging or summing over moral value rather than prudential value. It thus commits itself to absolute priority.

However, there is an alternative way to satisfy the prioritarian commitments. I will show this by first introducing a formal rule and then giving it an interpretation that adheres to the three principles that make up the prioritarian concerns, as well as a fourth principle that I term relative priority. In the next section, I will argue that one can indeed hold all four principles on the basis of a single idea about value. (That the first three have been thought to rule out taking account of the relative position of the individuals in question is why, I think, this alternative has been overlooked philosophically, even though the formal rule hasn’t been.Footnote 11 )

The formal principle in its ‘total’ version is known as rank-weighted utilitarianism or the Gini family: transform the weight that each utility value gets in the total, so that the well-being of relatively worse-off individuals gets more weight and the well-being of relatively better-off individuals gets less weight. Specifically: rank the individuals from worst-off to best-off, where individuals who are tied can be put in any order. For example, in D1, Bob is 1, Ann is 2, Cecil is 3; in D2, we can put them in any order. Each rank gets a weight λ, where the weights are larger for lower ranks. Multiply each utility by the rank-weight and sum the results. For example:

In the ‘average’ version, assign weights to rank-ordered proportions (e.g. to the top 1/3 of individuals) rather than individuals, so that the weights sum to 1, and the weights of better-off groups of a given size are lower than the weights of worse-off groups of that size.

There are two equivalent ways to do the calculation for the ‘average’ version. The first: fix an importance function I(p) from [0, 1] to [0, 1], measuring the importance of the interests of the best-off group of each size p. Let I be convex: the best-off group is less and less important as its size gets smaller and smaller. (For example, the group of everyone gets importance 1; the best-off 2/3 of individuals get importance 4/9; and the best-off 1/3 of individuals get importance 1/9.) Then let the weight w of each rank-ordered group be importance of those at least as well off as that group minus the importance of those better off. This will mean that groups at the top get less weight than their size, and groups at the bottom get more weight than their size. Finally, multiply each utility by the weight of the group that obtains it. For example:

Thus, the utility values of relatively worse-off individuals get higher weight, and the utility values of relatively better-off individuals get lower weight.

The second, equivalent way to do the calculation is this. The top p-portion of individuals gets a particular weight – the importance value I(p) of this group, as above – and these weights attach to utility increments – utility that those in the top p-portion get but that those below don’t. For example, utility that everyone receives gets weight 1; utility that the top 2/3 receive over and above what everyone receives gets weight 4/9; and utility that the top 1/3 receive over and above what the top 2/3 receive gets weight 1/9:

I invite the reader to keep in mind whichever equation is more intuitive. The first equation concerns how much weight to put on the fully described consequences belonging to each individual: e.g. ‘those in the top 10% but not the top 5% have lives that look like this – how much does that fact contribute to the overall good?’. The second equation concerns how much weight to give to advantages that those in one group have but those in another do not: e.g. ‘the top 10% of people enjoy at least 10 extra years of life in addition to those things enjoyed by those less fortunate – how much does that fact contribute to the overall good?’. The differences between them won’t matter to the discussion here.

On this picture, D2 is better than D1, because D2 has a higher rank-weighted average or rank-weighted total utility, where those with relatively lower utility values are given more weight. When a distribution is more spread out in terms of utility and we put more weight on relatively worse-off individuals, individuals with lower utility bring down the rank-weighted average utility more than individuals with higher utility bring it up. Thus, if relatively worse-off individuals get more weight, then when two distributions have the same average utility, the one that is less spread out in terms of utility will have a higher rank-weighted average.Footnote 12

This aggregation rule gives priority to those who are worse-off in a relative sense: the weight given to those at the bottom of a particular distribution is higher than the weight given to those at the top of that distribution. The welfare of those who are relatively better off makes less of a difference to the overall good than the welfare of those who are relatively worse off.

W (rank-weighted utilitarianism) is a sort of dual to P (absolute prioritarianism): instead of transforming the utility values (with v) and keeping group sizes as weights, we transform the weights (with I or w) and keep utility values. It is unfortunate that v has been sometimes called an (absolute) prioritarian ‘weighting’ function instead of a ‘value’ function, since it transforms utility values, not the weight that each utility value gets – and the key formal difference between absolute and relative prioritarianism is which function gets transformed. To maintain precision, I will refer to the absolute prioritarian v as a value function and the relative prioritarian I or w as a weighting function – more on the philosophical distinction between value and weight in subsequent sections.

W and P do not give rise to the same orderings of social distributions: there are orderings that can be captured by W but not P, and vice versa.Footnote 13 So they are not merely notational variants of each other with different philosophical interpretations; instead, they say different things about which distributions are better than which others, and in what way the interests of the worse-off should matter more.

When weighted-rank utilitarianism is discussed in the philosophical literature, it is interpreted as an egalitarian principle, for reasons I will discuss shortly. But it needn’t be. I will argue that a particular ethical interpretation of W is a plausible way to satisfy the prioritarian concerns: individual good, spread aversion, and (perhaps surprisingly!) no distributional good. This interpretation accepts these claims and, instead of absolute priority, accepts:

relative priority: It is more valuable to increase the utility of the relatively worse-off than that of the relatively better-off, regardless of their absolute utility.

It is more valuable to increase the utility of the relatively worse-off, the view says, because the utility of the relatively worse-off counts more in determining the total good than the utility of the relatively better-off – it is weighed more heavily. The overall good in a society comes from one source – how things go for each individual – but some instantiations of this (some individuals) contribute to or reflect the overall good more than others do. How good things are for worse-off individuals in a distribution makes more of a per capita contribution to how good things are for the group, because their good more strongly determines how good that distribution is than the good of better-off individuals does .Footnote 14 This is the crux of the view.

Call this view the view I’ve just described relative prioritarianism. Relative prioritarianism is characterized by the formal principle W in conjunction with the philosophical view that how good things are for worse-off individuals in a group more strongly determines how good things are for the group than how things are for better-off individuals does.

It is easy to see that relative prioritarianism accords with spread aversion – the claim that of two distributions with the same average utility, the one that is less spread out in terms of utility is better – and relative priority – the claim that it is more valuable to give a bit of utility to a group of a particular size than to give that same bit of utility to a better-off group of the same size.

A bigger puzzle is how it can make sense to take a relative position into account while still holding that relative position does not affect individual utility, nor are relational properties a separate source of value.

5. Assessment and weight in aggregation problems
   Before showing that relative prioritarianism satisfies the remaining two claims, it will be helpful to make a distinction that arises in aggregation problems – problems of evaluating the whole from its constituent parts – more generally. The distinction is between which attributes of the whole matter to the evaluation, the assessment of each instance of an attribute, and the weight that each instance gets in determining the assessment of the whole.

Consider the problem faced by an engineer who wants to determine the strength of a chain composed of many links. Let us assume that we have two hypotheses for how the strength of a chain relates to the strength of its links: its strength might either be an average of the strength of its links, or it might be the strength of its weakest link. Both aggregate the same attribute (strength), and, furthermore, they agree about how to assess this attribute (how to measure individual chain strengths). But they differ in that they assign different weights to bearers of the attribute.

Or consider the problem faced by an Olympic committee that wants to determine how to calculate a gymnast’s overall score, based on her scores for four events. Let’s say that the committee members score a gymnast’s performance on each event according to the gymnastic ability she displayed, and they agree that the overall gymnastic ability displayed by an athlete should be the average of her four event scores, but they disagree about how to measure gymnastic ability: some members of the committee think it should be determined by execution, while others think it should be determined by execution and difficulty. Here, the members agree about the attribute to be aggregated (gymnastic ability) and about the weight of each bearer of the attribute (i.e. each event); but they disagree about how to assess the attribute.

Finally, consider two deans evaluating philosophy departments. They agree about how to measure individual professors’ research output (we should count publications), and about how to weight them in determining a department’s overall research output (we should sum individual research outputs), but they disagree about how overall department strength relates to research output: one dean thinks department strength just is research output, whereas the other thinks it is research output combined with faculty diversity. Thus, they disagree about which attributes matter to overall department strength. Here, they agree about both assessment and weight of research output in determining overall department research output; but they disagree about whether this attribute is the only thing that matters in their evaluation of department strength.

Two important things to note. First, the assessment of an attribute remains the same whether it is part of an aggregation problem or not, and it is stable across different aggregation problems: the strength of an individual chain link doesn’t change when paired with other links; the execution score of a routine doesn’t change when paired with other routines. By contrast, the weight of an attribute is essentially a feature of aggregation itself: there isn’t an answer to ‘how much does this attribute contribute’ apart from knowing ‘contribute to what whole?’.

Second, which attributes are to be aggregated naturally corresponds to what intrinsically matters in the evaluation. The dean that thinks diversity matters to a department thinks that it matters intrinsically: for him, it’s not that diversity contributes to what ‘really’ matters, a department’s research output; rather, it is a separate part of a department’s strength. By contrast, the members of the Olympic committee and the engineer’s two hypotheses agree about what intrinsically matters to their overall evaluation – gymnastic ability on individual events and individual link strength, respectively.