Brock’s example

Brock asks us to consider the following distribution (table 1).1

I suggest, although Brock does not say so, that we should assume we are dealing with the distribution of life years. Moreover, we could assume that the initial distribution is, say, that all three persons will die at age 50 if not provided with either policy 1 or 2.

Policy 1 will provide 10 additional life years to person A, 20 life years to person B and 20 to C. Policy 2 will provide 11, 15 and 25 life years to persons A, B and C, respectively. As noted by Brock, distribution 1 is more equal than 2, while the position of the worst off, A, is better in 2 than in 1. This is a case where equality of outcomes will not support improving the position of the worst off. Notice that Brock does not claim that egalitarianism will never support the position of the worst off, but that it will not always do so. Notice also that Brock appeals to an “intuitive” judgement of equality, namely that at face value, distribution 2 “looks” more unequal than distribution 1. The differences in outcomes between A, B and C are smaller under 1 than 2 (B−A = 10; C−B = 0 under 1 and B−A = 4; C−B = 10 under 2). Obviously, it would be good to use more formal measures of inequality of distributions and I return to this below.

The importance of introducing this example, however, is that it enables us to explore the distinction between egalitarianism and prioritarianism in the context of the distribution of health. If it can be shown that prioritarianism and egalitarianism are distinct ethical principles, with distinct implications for judging distributions, they may not be open to the same objections. For example, Parfit’s levelling down objection may create problems for some versions of equality of outcome, while not for prioritarianism.i

Prioritarianism and egalitarianism

Brock draws on Parfit’s distinction between prioritarianism and egalitarianism. Parfit has argued that egalitarians should not be concerned about strict equality, but rather should be concerned with giving priority to the worse off.3 The “leveling down objection” is introduced to suggest that our concern for what we call equality actually is a concern for the worse off. As Parfit notes on what he calls telic egalitarianism:

On the widest version of this view, any inequality is bad. It is bad, for example, that some people are sighted and others are blind. We would therefore have a reason, if we could, to take single eyes from some of the sighted and give them to the blind. That may seem a horrific conclusion.3

Brock, in assessing Parfit’s argument, also concludes that leveling down provides a strong objection to strict telic egalitarianism:

[I] believe “leveling down” objections do not just show that we are moral pluralists, but rather call into question a noninstrumental commitment to equality in outcomes in general, and whether such a commitment explains our special concern for the worse off in particular.1

If equality of health were valuable for its own sake, then there would be something better about the situation if the sighted were to become blind compared with the state where some are sighted and some are blind (ie, inequality). That is to say, while the situation might be worse on balance, at least there would be one better thing about it—namely, there would be more equality of health. Parfit argues that a situation cannot be better if it is not better for anyone, and it is not better for anyone if the healthy become sick while everyone else remains the same (although this view has been challenged by others4 5 6).

In other words, concern for equality, according to Parfit, should more correctly be understood as the view that the worse off a person is, the more important it is, morally speaking, to help that person. This view, which Parfit originally called non-relational egalitarianism, is now usually referred to as “prioritarianism”. (Quoting Nagel, Parfit observes that prioritarianism has an indirect or “built-in bias” towards equality.3) The prioritarian justification for favouring the worst off rejects the claim that equality of health is valuable in itself, and instead establishes priority to the worse off as a more plausible interpretation of the ideal of equality.

There are many ways to deal with this objection and I shall not reject a non-instrumental commitment to equality by defending an instrumental commitment to equality, even if I think such a defence is possible. I shall rather pursue another avenue, the one indicated but not further developed by Brock—as Brock says: “we are moral pluralists”. I argue that we can best deal with the levelling down objection by acknowledging a pluralist approach that combines fairness and goodness in the distribution of life years.7 To do so, I need first to discuss some recently introduced univariate measures of health inequality—equality and priority—and examine their properties.

Judging the inequality of distributions: the extended Gini

Adam Wagstaff has suggested that the extended Gini index—applied to pure health outcomes instead of income—captures and makes explicit both aversion to inequality of health and a special concern for the worst-off individual (or group).8 9

Consider first the standard Gini. It can be written as:8 10

where n is the population size, h_i is the health for person i, μ(h) is the average level of health in the population, and R_i is the relative rank of the ith person (rank 1 is the rank of the best off). When the Gini coefficient is written in this form, it is easier to see that the rank-order imposes an implicit priority weight. The weights are given by the element (2R_i−1), so that for three groups ranked as number 3, 2, and 1 the weights are 5, 3 and 1. So, the standard Gini, has “a built in bias towards prioritarianism”. In the terminology of the economist, it satisfies the condition of diminishing transfers, that is, a given transfer from those with much health to those with less health decreases inequality more when it is made at the lower tail of the distribution than when it is made at the upper tail.

The extended Gini makes these weights explicit. Bleichrodt and van Doorslaer write the extended Gini in the following form:8 10

where the notation is the same as above. The parameter v, often called inequality aversion, reflects the relative weights assigned to the health of different persons (or groups with the same health). When v = 1, there is no inequity. When v = 2, the equation equals the standard Gini coefficient. For v>1, a larger priority weight is assigned to the lowest-ranked group as compared with the higher-ranked groups. The interpretation of the Gini measure is as follows: If the extended Gini  = 0, the distribution is perfectly equal; if the extended Gini  = 1, the distribution is perfectly unequal. The Gini compares and measures the difference between all persons.

Brock’s example reconsidered

With Wagstaff’s extended Gini at hand, we can now revisit Brock’s example. I have calculated the extended Gini for different values of v (table 2).

We see from table 2 that when v = 2, as in the standard Gini, distribution 1 is judged as more equal than 2. If v = 4.9, then the two distributions are judged as the same (equally unequal). When v = 6 (or >4.9), distribution 2 is judged to be more equal than 1 (0.331<0.347). This is so because the higher the v, the more weight is given to the lowest-ranked position as compared with the higher-ranked positions. When Brock’s example is reconsidered in this way, by the use of the extended (priority-weighted) Gini, we see that there exists one measure of inequality that—assigning a high weight to the worst position compared with the better-off positions—will judge distribution 1 as more unequal than 2.

So this is a case where the Gini measure, typically seen as a measure of inequality, in Brock’s carefully designed example, will also support improving the position of the worst off. This is so because the extended Gini incorporates inequality among other things—including priority weights.

The extended Gini compared with priority-weighted aggregation

Another measure for the evaluation of the two distributions, avoiding any appeal to equality, would be simply to construct a measure of priority-weighted aggregation. This is a function that is additively separable and is often written in the following form:

g  =  w(g₁(h₁)) + w(g₂(h₂)) + …. w(g_n(h_n)),

where g denotes overall goodness (welfare), h₁ is the health of person 1, g₁ is the function that assigns value to h₁ and w is a function that assigns a value to g₁(h₁), etc. Each person’s good is determined by that person’s health only. The simplified well-known formula is:

A simple numeric example can help us understand priority-weighted aggregation. Consider the distributions in table 1 again. Let us assign a high weight (31) to the person with worst health, a somewhat lower weight (30) to the next person with somewhat higher health, and so on, and the weight 1 to the person with best health. The weights are arbitrarily chosen and assign a high weight to the position for the worst off with lowest health. This gives the following values for overall goodness:

Distribution 1: g  =  (31×10) + (2×20) + (2×20)  = 390

Distribution 2: g  =  (30×11) + (3×15) + (1×25)  = 400

Distribution 2 has the highest-weighted aggregate and is, given the priority weights defined, therefore best.

Priority-weighted aggregation assigns weights according to who are worst off, but does not, as does the extended Gini, also measure the difference in health between all persons. The extended Gini actually combines weights according to who are to the worst off with a measure of the absolute difference between all persons in the distribution. Another difference between priority-weighted aggregation and the extended Gini is that the former is not sensitive to levelling down (that goes without saying), while the latter is.

To see this, consider again Brock’s example. We now change the health variable of the best-off person in distribution 2; that is, we change the number of life years for person C from 25 to 20. We get the distribution shown in table 3, and the corresponding extended Gini.

We see that distribution 2* is judged more equal than distribution 2 for all values of v chosen here.

This is so simply because the difference between A, B, and C is smaller. Being susceptible to levelling down is an undesirable property, at least for an overall measure for assessing distribution. The fact that the extended Gini is sensitive to levelling down may be seen as problematic. Should we therefore reject the ideal of equality, and also reject the extended Gini as one possible measure of equal health?

Broome’s argument

Before I argue that the extended Gini has a relevant role in the evaluation of distributions, I will introduce an argument showing that we can best deal with the levelling down objection by acknowledging a pluralist approach to the distribution of well-being—and life years. John Broome has presented such an argument for the distribution of well-being, and I shall try to present it here. Broome shows that one can construct a simple measure of fairness F (incorporating the value of equality) which is, when combined with a measure of goodness G, not open to the levelling down objection:

It is easy to construct measures G and F, and form a combined measure from them, in such a way that levelling down can never be accounted a good thing.11

Broome’s simple formula is G + aF. If F has value, the formula assures that the overall value of these combined measures will always increase if F increases. Broome holds that fairness is always good for someone (for example when equality is increased); so fairness has value. If people have equal claims on a good, say life years, fairness is achieved when people have their claims satisfied equally, or in proportion to the strength of their claims.12 In a pluralist notion of distribution, fairness and goodness will need to be weighed against each other. Broome shows that the formula G + aF implies that levelling down is always a bad thing. I only refer to Broome’s argument here; I shall not go through his proof here. The main point I want to emphasise is that Broome demonstrates that a pluralist notion of distribution—which combines fairness with goodness—is not open to the levelling down objection (for a related argument, see Fleurbaey 7). Indeed, Broome argues that his formula implies that “levelling down is always a bad thing”.

Measuring the combined value of fairness and goodness in health: the achievement index

Instead of going through Broome’s argument relying on distributions of welfare, I shall provide my own argument concerning the distribution of life years—a natural unit of distribution of health outcomes. There exists one measure that combines goodness and fairness, or, to be more precise, priority-weighted health gain combined with one notion of fairness (more equality in health outcomes). This is the achievement index.

Wagstaff has developed the achievement index that is based on the extended Gini measure. This index measures the combined value of promoting equality in health outcomes and maximising average health. The achievement index, A(v), can be written as follows:8

A(ν)  =  μ(h)(1−EG(ν)),

where μ(h) is average health in the population, and EG(v) is the extended Gini. Wagstaff’s achievement index captures exactly those concerns that a pluralist framework for just distribution has identified as normatively important, such as (a) aversion to inequality of health, (b) a special concern for the worst off individuals or groups and (c) explicit judgements about the appropriate trade-off between equality concerns (a) + (b) and average health.ii

With the achievement index at hand, let us return to Brock’s example again (table 4).

First, we compare distribution 1 and 2. We see that the achievement index is higher for distribution 1 compared with 2 when v = 2 (moderate priority weight). This means that distribution 1 is judged better than 2 because it is more equal. When v = 4.9 and above, distribution 2 is judged better than 1 because more weight is assigned to the worst off. This is how it should be.

What then about the achievement index? Is it sensitive to levelling down? In distribution 2*, person C’s health outcome is reduced from 25 to 20 life years without giving any more life years to others. If we compare distribution 2 and 2*, we see that distribution 2 is judged better than 2* for all values of v chosen. The achievement index for distribution 2* approaches the value for distribution 2 for higher values of v but will never be higher.

This is so for a wide range of reasonable values of v (v = 1 → v = 10), as can be seen from figure 1. The achievement index is never sensitive to levelling down.

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Figure 1

Achievement index (AI) for distribution 2 and 2* for values of v ranging from 1 (no priority weight) to 10 (high priority weight).

We see from fig 1 that the achievement index for distribution 2* will never be higher than for distribution 2 (shown here for reasonable values of v up to 10). This implies that levelling down is never a good thing—as measured by the achievement index.

If I am right, the achievement index is a measure of the distribution of health that combines concerns for fairness (equality) and goodness (average priority-weighted health outcomes). And this measure is not sensitive to levelling down. In other words, levelling down is never a good thing for such a combined pluralist measure of just distribution.

Broome’s formal argument against the priority view

Before I conclude, I need to clarify one important objection against the priority view, originally set out against priority-weighted aggregation, but apparently also important for the use of the achievement index. Broome argues that the priority view makes a distinction between the quantity of a change in well-being, and the value of the change, and this is methodologically and theoretically problematic.13

Broome holds, correctly in my view, that the quantity of well-being gets its meaning through the valuation of uncertain prospects, and if so, that “the quantity of wellbeing will turn out to be exactly the same as the value of wellbeing”. This means that if the value (the goodness) of a gain in well-being is correctly established, it is not possible to assign another value (based on priority according to the initial level of well-being for the worst off) to that quantity:

A prioritarian still needs to separate the value of wellbeing from the quantity of wellbeing. Perhaps she can find a way of doing so. But until she does, her theory is shaky.13

Is this objection also valid against priority weights as used in the achievement index? I believe not. Broome’s objection is not problematic for the achievement index as defined above, because it measures changes in advantage in terms of a natural unit—life years— gdand assigns a value to that change from lower to higher levels of health outcomes. So this measure incorporates goodness by assigning values to life years according to priority (where the achievements of few life years are considered worse than more life years). The quantity of life years is not the same as the value of health outcomes measured as life years.

Conclusions

In this paper I have discussed and accepted Brock’s example showing that prioritarianism and egalitarianism may judge distributions of health outcomes differently. I have also examined whether prioritarianism and egalitarianism—properly understood and properly combined with concerns for average health—is open to the levelling down objection. I have accepted the argument that levelling down is never a good thing—all things considered. But I hold that equality often is. Fleurbaey has made a related argument—that the distinction between egalitarianism (in its pluralist version) and prioritarianism mostly has to do with the reasons for, rather than the content of, judgements about distributions.7

When equality, prioritarianism and aggregation are combined, as fairness and goodness are in the achievement index, egalitarians can still promote the value of equality. I have also suggested—although a more thorough investigation of the properties of the achievement index is needed—that this measure is (a) always sensitive to inequality in health, (b) is always sensitive to average health, (c) can assign priority to those with lowest health outcomes and (d) is never sensitive to levelling down.

The contribution of this argument is, I believe, that a theoretically attractive normative view that combines fairness with goodness can be operationalised into a quantitative measure of advantage that also has robust mathematical and logical properties that have a clear foundation in well-established measures of health inequality.10 Another important conclusion is that the argument presented here combines the best of two plausible normative views—namely, that equality matters, as does priority to the worst off.

The use of the achievement index is grounded in a pluralist moral view. I therefore add this note to Brock’s important contribution to the bioethics literature on the distribution of health outcomes: levelling down is never a good thing for egalitarians if they adopt a pluralist theory that integrates fairness with goodness. Equality is not the only value egalitarians promote. But equality is so important that we should not reject it.