John Broome has developed an elegant and powerful theory of fairness. Despite these virtues, Broome's theory of fairness has not received the attention it deserves because it has not been absorbed appropriately into the current debate. In many recent discussions, descriptions of Broome's theory contain fundamental misunderstandings. Thus, it is important to lay out Broome's theory afresh. I do this in section I of the article. In section II, I show how commentators have misunderstood the basic structure of Broome's theory and I also explain why this has happened. In section III, I discuss various objections that have been raised against his theory. Once we understand the general structure of Broome's theory, we are in a better position to realize what exactly its normative implications are. My presentation will show that these implications are different from what his critics have commonly assumed.Footnote 1
I. BROOME'S THEORY OF FAIRNESS
For Broome, fairness deals with how to treat people's claims to some good. Broome's theory is best developed by focusing initially on cases in which all claims to some good are of equal strength. For such cases, his theory consists in a single principle: treat equal claims equally. (I will generalize this principle later on.)
On Broome's view, fairness judgements are relational: they participate in the relational nature of equality. Whether one of us has been treated fairly depends on how others have been treated. As with other relational judgements – for example, the claim that two people are equally rich – we cannot find out whether this comparison holds by knowing about one person's riches alone. Similarly, we cannot know whether a person has been treated fairly simply by knowing how this person has been treated. We have to compare one person's treatment with that of others. Only then can we assess whether the treatment was fair.
Suppose you can either satisfy one person's claim to some good or the claims of two other people to the same good; you cannot satisfy all three claims. Suppose all these claims are equally strong. Whatever you do, whether you satisfy the claim of the one or the claims of the two, you will have violated the requirement of fairness: you will not have treated equal claims equally. Only if a third option becomes available, namely the option to satisfy no one's claim, could you treat people fairly. You would treat them fairly by withholding the good and thus by not satisfying anyone's claim. (Equality, we know from other contexts, allows for levelling down.)
The close relationship between fairness and equality is central to Broome's theory. ‘Fairness’, Broome says, ‘is only a relative matter, a matter of how one candidate is treated relative to others’. Considering a case of equal claims, he continues, ‘so long as they all get the same amount of good, however large or small the amount may be, they have been treated with perfect fairness’.Footnote 2 Fairness requires equal treatment of equal claims. If claims ought to be satisfied, this requirement will have to stand on different grounds. Broome emphasizes this point again – equal treatment does not require satisfaction – in his later paper on this topic: ‘if all the candidates get the same quantity of the good, then fairness has been perfectly achieved, even if they each get very little, or indeed none at all’.Footnote 3
Broome's theory of fairness is not (and is not meant to be) a complete moral theory. In considering fairness, it only considers one aspect of how we ought to act. The case in which withholding the good is the only fair option shows that, sometimes, we ought to do what is unfair. Suppose the people in this case have a claim on a good that is based on their benefiting from the good. Suppose, for example, it is the kind of benefit that a doctor is able to provide. The doctor, in this case of the one versus the two, cannot satisfy all claims. However, he or she should certainly satisfy some claim. Thus, fairness, in this case, is overridden by the benefit that satisfying any of the claims provides.Footnote 4 Another illustration of why Broome's theory of fairness is not meant to be a complete moral theory is provided by cases in which we can either satisfy all claims or none at all. If the claims are equally strong, there is nothing to choose between these two options in terms of their fairness. Nevertheless, it is clear that in cases like these, we ought to satisfy all claims. Thus, fairness is not the only consideration which determines what we ought to do.
Broome's theory of fairness, I said, is an incomplete moral theory. It is also an incomplete theory of fairness. The domain of fairness has been given as how to treat people's claims. However, questions arise about the nature of claims. First, what are the sources of claims? Second, how do claims work in moral deliberation? Third, whenever claims are in play how is it settled what the claims are claims to? This kind of incompleteness might not matter. Any theory will have to leave some questions open. What is left open here does not take us into unfamiliar territory, because we understand talk of claims pre-theoretically. Children have a claim to the attention of their parents; in a society like ours, ill people have a claim to be seen by a doctor; students have a claim to being taught properly.
In all these examples the bearer of claims will benefit from the satisfaction of their claims. Benefits, and those benefits we call needs, are sometimes the source of claims. Sometimes, but not always. If, for example, I need an onion to cook my dinner (or to treat my ear infection), I might have a claim on getting one from my neighbour – as I have helped her in the past – but the number of people who owe it to me to give me an onion will be very small. So, not all needs lead to claims. Broome does not offer a complete theory of the sources of claims. He relies on our pre-theoretic understanding. ‘In different circumstances claims will have different sources. Sometimes they may arise from the candidates’ needs, from their general rights such as property rights or a right to life, from a debt of gratitude, or from something else. Sometimes claims may be the resultant of several influences.’Footnote 5
Our second question about claims was: how do claims work in moral deliberation? This question goes beyond a theory of fairness. Fairness is one part of morality, but in moral deliberation all aspects of morality will play their role. The demand of fairness is to treat equal claims equally. This is a deontological component in Broome's moral theory, in which fairness will be weighed against other goods. We have seen already that fairness can be outweighed; thus, his overall theory is a form of consequentialism in which fairness is seen as one good amongst others.Footnote 6 About claims, Broome says the following: ‘Some of these reasons [why a person should receive a good] are duties owed to the candidate herself, and others are not. I shall call the former claims that the candidate has to the good.’Footnote 7 This fits the examples I have mentioned above. Parents owe it to their children to attend to them; teachers owe it to their students to teach them well, and so on. Claims are like duties and work differently from other moral reasons, e.g. from those that concern some benefit when this benefit is not itself the source of a claim. Broome says that claims are neither weighed against other reasons nor against each other. Doing this would, as he says ‘not seem to give proper recognition to the people's separateness’.Footnote 8 Are claims like rights? Claims and rights are certainly related but they are not the same, for two reasons. First, if rights are seen as side-constraints, i.e. as determining directly what one is obliged to do (or not to do), they do not fit into Broome's general moral theory. Second, claims fall short of rights in another way as well. Claims belong to a theory of fairness and we have said that fairness is a relational concept. Thus, fairness only demands the equal treatment of claims; it does not demand their satisfaction.
We have learnt that claims do not aggregate; they cannot be weighed against each other, nor can they be weighed against other reasons. These features (as well as the examples of claims we have given above) make claims like rights. However, we have also learnt that fairness does not require that claims be satisfied. Thus, it might seem as if rights violations, on Broome's theory, were fine. How do we reconcile these two aspects of what Broome says about claims? At this point we need to make a point explicit which has been there all along. We need to separate the role of claims in a theory of fairness from other roles claims might play in an overall moral theory. When Broome emphasizes the deontological, non-aggregative aspect of claims, he is talking about the role of claims in his theory of fairness. It is important to note that the way he characterizes claims within fairness is not an extra element of his theory that would need to be justified by further argument. Their non-aggregative aspect is simply a consequence of the core principle of his theory of fairness. The requirement to treat equal claims equally entails that in conflict situations with different numbers of equally strong claims on each side, the only perfectly fair solution would be to satisfy no claims. Thus, we should not look for an explanation of why claims do not aggregate in their nature or source; claims do not aggregate for the simple reason that a denial of aggregation is part of Broome's principle of fairness. I said that it looks like his theory would, in this sense, condone rights violations. Actually, this impression is accurate, when we restrict our attention to fairness. But claims, I said, will play another role in an overall moral theory. Their obvious role is to place people under a prima facie obligation to satisfy them. Broome condones neither the violation of rights nor the neglect of claims. Claims ought to be satisfied. This requirement belongs to Broome's general moral theory; it is not part of his theory of fairness.
I have not in any way departed from Broome. He is very clear about this dual role of claims. ‘Everyone's claim to a good should, prima facie, be satisfied . . . . Call this the satisfaction requirement.’Footnote 9 This aspect of claims allows for weighing and, thus, the more claims we satisfy, the better. When it comes to fairness, however, we focus on the principle of equal treatment for equal claims, which does not itself require that claims be satisfied. ‘Claims’, Broome says, ‘therefore give rise to two separate requirements: they should be satisfied, and they should be satisfied proportionally.’Footnote 10 A theory of fairness spells out this second requirement.
After explaining Broome's principle of fairness, I said that a theory of fairness is not and is not meant to be a complete moral theory. I then said that Broome's theory is also incomplete as a theory of fairness. First, it does not provide a full account of the sources of claims. On the second question, what the role of claims in moral deliberation is, Broome has provided an answer. He distinguishes their role in fairness – equal claims demand equal treatment – from their role in a general moral theory – claims demand satisfaction. I mentioned a third area of potential incompleteness. It gets into view when we try to apply Broome's theory. Take the question whether the young should, other things being equal, get preferential treatment over the old when it comes to the distribution of scarce life-saving medical resources. The young would benefit more from these resources than the old would when we consider their effects in terms of life-years saved. Whether this difference in the size of benefits affects the strength of their claims crucially depends on what we understand their claims to be claims to. If people have a claim to a (reasonably) long life, the young will have a stronger claim. If, however, ill people simply have a claim to medical attention and treatment, differences in age will not amount to differences in claim strength. Applying Broome's theory of fairness will itself involve ethical judgements concerning what claims are claims to. This is another way in which Broome's theory of fairness is incomplete.Footnote 11
At this point I want to pause in laying out Broome's theory. We have reached a point at which we are able to look at one of Broome's critics. Brad Hooker is concerned with finding the right place for Broome's theory of fairness within a picture of morality in general and of moral reason in particular. Hooker thinks that the following tripartite map of moral reasons comes closest to Broome's view.Footnote 12 We have reasons that come from harms and benefits which lend themselves to a consequentialist treatment. We have deontological reasons that spell out side-constraints, like the reasons not to kill or torture; finally we have reasons stemming from needs, desert, agreement and, possibly, other sources, which provide Broome's theory of fairness with its distinctive domain. Hooker tries to cast doubts on Broome's theory on the basis of what he takes to be Broome's tripartite map of moral reasons. In a nutshell, he doubts whether there is a clearly defined domain for which a theory like Broome's would be needed.
If what I have said about Broome's theory so far is correct, we should not expect that Broome's view can be pressed into a picture according to which it deals with one particular set of moral reasons, like reasons stemming from people's needs. Satisfying one's need is usually a benefit to the person. So, needs are a subclass of benefits. Some benefits, whether they are needs or not, are such that some people have a claim to this benefit whilst others have not. Furthermore, claims, we have seen, give rise to two different requirements. One requirement is that they be satisfied, which for Broome falls outside fairness, another is that equal claims are treated equally. Because one and the same thing, a claim, gives rise to two different kinds of moral requirements – one that allows and one that disallows weighing – the project of determining the domain of Broome's theory by giving us a well-defined set of considerations to which the theory applies is bound to fail.
Let me return to Broome's theory of fairness. I have introduced its core principle ‘Treat equal claims equally’ and I have explained in which ways his theory is incomplete. There are two more things to be done in this section: first, we have to understand Broome's main argument for his theory of fairness and, second, I want to introduce his generalization of the core principle.
Broome says that his theory explains why it might be a good idea to distribute scarce resources by a lottery between all claimants. He writes, ‘I know of no alternative theory that explains the value of lotteries as successfully as mine.’Footnote 13 This is the main argument for his theory of fairness. It explains best when and why lotteries should be used.
Imagine a situation in which two people have the same claim on some good but there is only enough of the good available to satisfy one of these claims. It is a common view that in such a situation an equal-odds lottery should be used to distribute the good. Using a lottery in such circumstances seems fair.Footnote 14 Why, on Broome's account, is it fair?
There is a sense – we have met it before – in which holding a lottery violates the fairness principle. In holding a lottery we ensure that one person gets the good and the other does not. We don't know whose claim will be satisfied, but we know that one claim will be. Thus, we know that in holding a lottery we will not treat equal claims equally. However, there is a sense in which holding a lottery satisfies the fairness principle. In holding a lottery we treat equal claims equally in the sense that we give both parties the same chance of receiving the good. Consequently, holding a lottery is always second-best in terms of fairness. It satisfies the fairness principle in one way and violates it in another. If we withheld the good and, thereby, gave no one any chance of getting the good, we would have satisfied the fairness requirement completely. Other moral reasons, for example those captured by the Pareto Principle, which asks that we make things better for people when no one is made worse off by our doing so, speak against wasting the good. If these reasons are strong enough, we ought to distribute the good. Holding a lottery, though less fair than withholding the good, will always be fairer than giving the good to someone straight out, because in holding a lottery we satisfy the fairness requirement to an extent.
We should hold a lottery only if the reasons arising from the benefits of the thus distributed good are strong enough to outweigh the loss in fairness that a lottery brings with it. Sometimes this will be the case. The doctor of the example above should satisfy some claim. At other times, the reasons stemming from benefits will not outweigh the departure from complete fairness. We find such cases in highly competitive contexts. In these contexts no departure from perfect fairness will be tolerated. Here is an example. One's children might have incompatible preferences about what we should do as a family on a Sunday afternoon. As a parent you'd put pressure on the children to reach an agreement and use the possible withdrawal of the offer of an outing as a sanction. If no compromise is forthcoming, you might resort to offering a lottery between the options. Children, however, might reject this offer on the ground that they would rather forgo the benefit than see their sibling win the lottery. The latter, they say, would be deeply unfair. (On Broome's theory, they are making a valid point.) Similar examples can be found in other competitive contexts. Two firms might argue that some public resource that could benefit each of them but will go to only one of them should be withheld in order not to fall behind their competitor. Two states, locked in competition, might prefer that some prestigious project goes to neither of them over going to one of them with equal chances.
Let me summarize Broome's case for lotteries. Distributing goods by lotteries offers a fairness gain when compared to one party simply getting the good: there is a sense in which we treat equal claims equally when holding a lottery. Thus, lotteries are strictly preferable to the alternative of simply giving the good to one of the parties. Distributing goods by lotteries results in a fairness loss when compared to withholding the good: ensuring an unequal distribution means we have not treated equal claims equally. Whether we should use a lottery depends on whether the benefit of the good outweighs this fairness loss. In Broome's view, there is no algorithm that could relieve us of having to judge each situation in this regard on its own merits.
So far I have limited my exposition of Broome's theory to a special case. I have only considered cases in which claims are equally strong. What does fairness demand when claims are of unequal strength?
According to what is certainly the intended interpretation of the core principle – treat equal claims equally – we should make a distinction between cases when claims are equal and those when claims are unequal. Consequently, we should treat unequal claims unequally. What renders claims unequal is their strength. There are two ways to treat claims of differing strength unequally. We could favour the stronger over the weaker claim or we could favour the weaker over the stronger claim. The intended demand is, obviously, to give priority to stronger claims over weaker claims. Thus the two principles mentioned above – to treat equal claims equally and to treat unequal claims unequally – are instances of the more general principle to treat claims in direct proportion to their strength. In Broome's own words: ‘What, then, does fairness require? It requires, I suggest, that claims should be satisfied in proportion to their strength.’Footnote 15
I have generalized Broome's theory by arguing how best to extend the core idea of fairness to cases of unequal claim strength. This argument, however, is not watertight; it cannot be. Whenever we generalize, there will be more than one general principle which will capture the core idea. Here is one such alternative to Broome's generalization: treat equal claims equally and, when claims are unequal, satisfy the stronger claims. I will come back to this alternative generalization of the core principle in section III.1.
II. HOW BROOME'S THEORY HAS BEEN MISUNDERSTOOD (AND WHY)
Broome claims that his theory of fairness offers the best explanation of why we should use lotteries in some circumstances. Although I will not offer a complete survey of other attempts to justify the use of lotteries, I need to mention one such alternative account. I call it the fair-share account.
Let us start by considering cases of divisible goods. If two people have the same claim on a good because, for example, they have invested equal efforts in its production, each one should receive his or her appropriate share. In this case this amounts to getting half of the good. An equal division respects the equality of claims. Had one contributed more, this person should get a bigger share, thereby ensuring that claims are treated in proportion to their strength.Footnote 16
I have started with divisible goods because what is fair in such cases should guide our treatment of cases in which the good is indivisible (or cases in which it cannot be divided without significant loss in value). According to the fair-share view, instead of dividing the good physically, we should divide the good probabilistically, so that each claimant receives an equal chance of getting the good. If the good is divisible, each person gets the part of the good he or she is entitled to; if the good is indivisible, each gets the probabilistic part of the good he or she is entitled to. This is the fair-share view. Here is its most prominent expression.
The claim is in its essence simple: what a social lottery offers is an equal division of a good that is otherwise indivisible. A lottery constitutes a probabilistic division of the good ex ante; instead of getting one unit of the good each member of the lottery pool gets a G/P chance at one unit of the good, where G is the number of units of the good available and P is [the] number of persons in the entitlement pool. On this account a social lottery is just because it permits an equal allocation of the good consistent with the equal entitlements of the claimant pool.Footnote 17
Broome's theory of fairness and the share-fair view appeal to the same core principle: treat equal claims equally. The fair-share view claims that by using a lottery everyone gets the same good – an equal probabilistic share. In this way using a lottery satisfies the core principle.
In the quote above, there is a move from the idea of dividing the good ex ante to the claim that, thereby, the good has been allocated in equal measures to all claimants. The latter claim, however, is obviously problematic. A physical division of the good distributes the good without remainder. A probabilistic division of the good, however, does not halve the good with nothing left. If there are two claimants, probabilistic division gives each an equal chance and, in addition, one of them will get the good. A lottery is an allocation process; a probabilistic division is a division of the good in name only.
Peter Stone and Tim Henning both aim at offering comprehensive surveys of all the arguments for the use of lotteries.Footnote 18 Henning describes his project as follows.
Here is a question: are you required to give A and B an equal chance to be saved, by holding a lottery? Many authors in ethics, welfare economics, and political science answer: ‘Yes.’ For further reference, I will say that these authors endorse the Lottery Requirement, that is, the claim that we are morally obliged to let a fair lottery decide between A and B in conflict cases. In this article, I defend a skeptical view. I seriously doubt that the Lottery Requirement is true.Footnote 19
How does Henning object to Broome's theory? As a matter of fact, Henning raises no objection to Broome's view. Henning is simply not aware of Broome's theory of fairness as an independent attempt to justify lotteries. He criticizes the fair-share view and thinks that, thereby, he has criticized Broome. Peter Stone makes the same mistake.Footnote 20 Why has Broome's view met such miscomprehension? Taking the following passage in isolation, Broome seems to invite the misunderstanding of confusing his view with the fair-share view.
[When the good is not withheld] some unfairness is inevitable. But a sort of partial equality in satisfaction can be achieved. Each person can be given a sort of surrogate satisfaction. By holding a lottery, each can be given an equal chance of getting the good. This is not a perfect fairness, but it meets the requirement of fairness to some extent. It does so, of course, only if giving a person a chance of getting the good counts as a surrogate satisfaction of her claim.Footnote 21
What has misled Henning and Stone is the term ‘surrogate satisfaction’. They think Broome's idea is the following: lotteries are second-best because there isn't enough good to satisfy both claims. So claimants receive a substitute of the good, namely a chance of getting it. Stone and Henning both take it that surrogate satisfaction of a claim stands in for its real satisfaction. Understood in this way, Broome's view amounts to the same as the fair-share view. Lotteries achieve equality of distribution by thinking about the good as ‘good ex ante’ or as a ‘surrogate good’.
I said that the above passage, viewed in isolation, invites this misinterpretation because ‘surrogate satisfaction’ is used there ambiguously. In its first occurrence, the term refers to the satisfaction of the demands of fairness. Lotteries satisfy the demands of fairness in one way but not in another. Lotteries treat equal claims equally in terms of the procedure used, but violate fairness in terms of how, in the end, the good is distributed, namely unequally. The second use of ‘surrogate satisfaction’ – getting a chance of the good amounts to surrogate satisfaction of the claim to the good – means something different. It says that receiving a chance of getting the good is relevant for the satisfaction of one's claim. In other words, it is a relevant way of treating people when we assess whether people with equal claims to a good have been treated equally.
Stone says that Broome carries the logic of ‘objective equalization’ furthest.Footnote 22 By this he means that in cases where a physical division of the good is impossible, a probabilistic division provides each with a fair share of the good. Attempting to present Broome's view, Stone writes: ‘The award of an equal expected amount of the good, in lieu of an equal amount of the good, thus constitutes “a sort of surrogate satisfaction” (Broome 1990, 98).’Footnote 23 . Both Henning and Stone deny that lotteries can be justified by focusing on chances of goods; we need not forget about the good itself. Henning summarizes his criticism as follows. ‘I see no relevant sense in which chances of satisfaction are surrogate satisfaction, so that giving people equal chances would constitute a way to treat claims fairly in the distributive sense.’Footnote 24
Broome does not think that giving each claimant the same chance leads to distributional equality. Distributional equality, he rightly emphasizes, can only be achieved by withholding the good. Take the thesis that chances of a good are themselves a good.Footnote 25 Stone and Henning argue against this claim. Broome's argument is unaffected by whether this thesis is right or wrong. It does not play any role in his argument for lotteries. He says that giving everyone the same chance is a relevant way of treating people. Of course, who gets the good, and ensuring an unequal final distribution, is another relevant way of treating people. Lotteries satisfy the fairness principle in one way and not another. Whether chances of goods can stand in for the goods themselves is irrelevant to Broome's argument. There are clear statements of Broome's view that do not invite the misunderstanding we have found in Henning and Stone. Here is one such passage.
It turns out that there is a way – random selection – of treating them equally to a degree, even though they do not all equally get the good. The fairness achieved is less than perfect, not because some people do not get the good, but because the treatment is not perfectly equal.Footnote 26
It is wrong to think that chances of getting the good are relevant because they provide surrogate satisfaction (which would be an additional good of questionable status). Chances of getting the good, when equally distributed amongst people with equal claims, are relevant because such treatment is fair in a way: it satisfies the demand of fairness to an extent.
Broome's theory has fallen off the map of positions on lotteries because commentators have misunderstood his view.Footnote 27 It is time to bring Broome's theory of fairness back to the table so that it can be properly investigated.
III. OBJECTIONS TO BROOME'S THEORY
III.1. Are proportional lotteries fair?
For Broome, the core principle – treat equal claims equally – and its contrast case – treat unequal claims unequally – are instances of a more general principle, namely to treat claims in proportion to their strength. Broome explains the general principle as follows.
I do not mean ‘proportion’ to be taken too precisely. But I do mean that equal claims require equal satisfaction, that stronger claims require more satisfaction than weaker ones, and also – very importantly – that weaker claims require some satisfaction. Weaker claims must not simply be overridden by stronger ones.Footnote 28
How should the idea of proportional satisfaction of claims be implemented? Broome suggests a weighted lottery in which chances correspond to claim strength: the stronger your claim, the bigger your chance ought to be of getting the good.
This idea has met resistance. Hooker says that it strikes him as ‘highly implausible’. He continues: ‘Suppose your claims to some indivisible good are very much weightier than mine. Is there any unfairness in your getting the indivisible good rather than my getting it?’Footnote 29 Broome says that weaker claims demand some satisfaction; they should get some chance of satisfaction in a weighted lottery; Hooker says that there is no unfairness when the person with a much stronger claim receives the good. Who is right?
Hooker illustrates his case with the following example.
Suppose your claim on the medicine comes from the fact that you need it to save your life, and my claim on it comes from the fact that I need it to save my little finger. Suppose an average life is something like a thousand times more important than a little finger. So should the matter of who gets the medicine be decided by a lottery in which you have a 999/1000 chance of winning and I have a 1/1000 chance? Given that your claim is so much stronger than mine, how could it be right to take any risk that I rather than you might end up with the good? . . . Letting the stronger claim win seems completely fair.Footnote 30
Hooker's example is not ideally suited to illustrate his point. The benefits at stake in this example differ for the two parties. Thus, considerations concerning utilitarian goodness come into play. They might distort our view in favour of satisfying the stronger claim. A better example would be one in which the benefit is the same for both parties. Suppose two people have worked hard to complete some task earlier than expected. One of them needs to stay, but, with the task completed, one of them could go home and take some time off. Who should it be, if one of them has contributed much more than the other?
If the difference in contribution is big, I would feel uneasy if an appropriately weighted lottery were to send the person home who has contributed much less. It seems unfair to the person who has worked much harder. If there is to be a lottery, I'd certainly hope that the person with the stronger claim would win. This hope, it seems to me, rests on a concern for fairness.
In order to explore the normative terrain more fully, consider a variation of the above case. One person has contributed slightly more than the other to the completion of the task. Does the feeling persist that fairness requires us to send the person home who has done more, even if ever so slightly more? I, for one, would not hold this view. If their contribution was similar enough, a weighted lottery strikes me as a good idea to decide who will receive the benefit. John Broome agrees. His case is different in content but structurally the same as the one I am using. He writes: ‘When the people were equally needy, it was fair for them each to have a 50 percent chance of getting the food. Now that one person is slightly – even minutely – less needy, it is implausible that it should it fair for him to have no chance at all.’Footnote 31
When normative intuitions diverge, we can only make progress by finding some underlying rationale which we are better able to assess and which in turn explains some part of the divergence we are facing. In section I, we have seen that Hooker regards a theory of fairness as dealing with a distinct set of moral reasons. In contrast to Hooker, I explained that Broome's theory is better seen as the development of a deontological principle. We accept that equal claims need equal treatment but there is no set of reasons that exclusively falls into the domain that requires equal or, in case of claims of different strength, proportional satisfaction. The requirement of equal or proportional treatment is by itself a fundamental aspect of moral deliberation.
If, like Hooker, we see claims as a distinctive set of reasons within a theory of fairness, it is their feature of being reasons that will drive our normative thinking. First, if claims are reasons, we ought to satisfy claims. It is true on both Hooker's and Broome's conceptions that we ought to satisfy claims but, as I explained earlier, this satisfaction requirement is not and cannot be part of a relational conception of fairness.Footnote 32 A relational conception of fairness, like Broome's, will always allow levelling down. Second, cases of unequal claims become cases of unequal reasons. The principle that guides us in such cases is that we always ought to act in accordance with the strength of reasons. If the reasons are stronger on one side, this is what we must do. This principle will not be sensitive to how strongly the weight of reasons favours one side over the other. If one person has worked much harder than the other, we have a strong reason to favour him. If this person had worked only a little bit harder, we still had more reason to favour him. A weighted lottery, as demanded by Broome's theory, will never be appropriate.
I have spelled out Hooker's point of view. However, what drives his thinking – fairness deals in claims and claims are reasons – is not compatible with Broome's relational conception of fairness. Broome develops the application of a deontological principle which shares its basis, namely claims, with other aspects of his overall moral theory. Fairness as proportionality is an independent deontological principle. As such it does not violate general principles about reasons which underlie Hooker's conception. In assessing Hooker's point of view, I find that it has an implausible consequence: when claims are almost equal, a lottery at equal or almost equal odds would, according to Hooker and contrary to what I think, be unfair. On the other hand, I agreed with Hooker that in cases of very unequal claims a weighted lottery has the danger of producing substantial unfairness. As I will show in what follows, Broome's theory does not entail what Hooker and I would find objectionable. When claims are strongly unequal, Broome does not recommend holding a weighted lottery. Such a lottery would, on Broome's theory of fairness, be less fair than satisfying the stronger claim.
Let me start this explanation with what Broome says about such cases.
If the good goes to the people with the strongest claims, the others will not have been fairly treated; their claims will have been overridden. And if it goes to the other people, the unfairness will be worse. So unfairness is once again inevitable. But once again it can, if the circumstances are right, be mitigated by giving everyone a chance of getting the good. Ideally, each person's chance should be in proportion to the strength of her claim: the lottery should be unequally weighted.Footnote 33
In the case of equal claims, we distinguished two relevant ways of treating people equally. One of these ways referred to outcomes. Lotteries, we said, incur a fairness loss, when compared to withholding the good, because of the unequal outcomes they create. The other relevant way of treating people referred to the lottery as a procedure of distribution. In giving everyone the same chance, we respect fairness. The same two relevant ways of treating people need to be brought to bear in cases of unequal claims. Let us first consider the outcome-related way of treating people. If a weighted lottery rewards the person with the stronger claim, there will be some unfairness – the weaker claim has been neglected. The weaker the claim the smaller this unfairness will be. If the weighted lottery rewards the person with the weaker claim, there will again be unfairness – the stronger claim has been neglected. Neglecting the stronger claim involves weightier unfairness than neglecting the weaker claim. The bigger the difference between the stronger and the weaker claim, the bigger the unfairness will be if the weaker claim wins. This is what it looks like when we consider ways of treating people's claims that refer to the distribution that will result from our thus treating them. The second relevant aspect we need to consider is how lotteries treat people's claims in terms of being a procedure of distribution. In this sense, the weighted lottery satisfies the proportionality principle. In this sense – and this is the sense Broome stresses in the quote above – weighted lotteries are fair. He says that the fairness achieved by the weighted lottery might mitigate the distributional unfairness. The fairness achieved by the weighted lottery will, however, have to be weighed against the fairness loss a lottery brings, which will vary depending on whether the weaker or the stronger claim wins.
Let us run these ideas that involve comparisons of degrees of unfairness with an example. Let us assume we can measure claim strength precisely and consider a small-difference case in which a 51%–49% distribution of the chances to win matches the strength of the claims involved and a large-difference case in which the claim-matching chances are 90% and 10% respectively. Let me start with the small-difference case. The lottery treats claims in proportion to their strength and is, in this sense, fair. In terms of the distribution achieved, there is a risk of satisfying the weaker claim. As the weaker claim is almost as strong as the stronger claim, this additional threat to fairness is small. Should we, as Hooker's line of reasoning suggests, satisfy the stronger claim? This would create an additional fairness loss in terms of the procedure used because the person who would otherwise get a 49% chance of winning, which matches her claim strength, would get a 0% chance of winning. We need an ethical judgement that compares these two fairness losses. What is worse in terms of fairness: reducing a 49% chance to 0 or having a 49% chance of satisfying a slightly weaker claim? Broome thinks that reducing the 49% chance to 0 would be more unfair. Thus, in small-difference cases it is fair to use an appropriately weighted lottery.
The situation changes in two respects when we look at large-difference cases. First, the fairness loss when reducing a weak claim to 0 becomes ever smaller the weaker the weaker of the two claims becomes. This weakens the case for the weighted lottery. Second, the smaller chance of satisfying the weaker claim, because it gets increasingly less probabilistic weight, is offset by the increasing unfairness that would result if one satisfied an increasingly weaker claim. If the claim represented by a 10% chance actually wins, the benefit of going home would not go to the person who has worked nine times as hard. This, it seems, would be seriously unfair. In general, the weaker the claim the bigger the unfairness would be if the weaker claim were satisfied. Whether we should hold a weighted lottery or satisfy the stronger claim when the difference in claim strength is large depends again on an ethical judgement by which we compare fairness losses. What is more unfair: reducing a small chance to 0 or risking that a much weaker claim is satisfied? Whatever the answer one gives to this question, for the purpose of assessing Broome's theory, it is important to realize that this answer comes from outside Broome's theory. Broome's theory provides the question that needs to be answered. We need to answer it for ourselves. I, for one, think that reducing a very small chance to 0 is a small fairness sacrifice. The fairness loss it brings is considerably smaller than the risk of satisfying the much weaker claim, as the latter would be very unfair. Thus, Broome's theory is fully compatible with the normative view Hooker and I have endorsed: it would be unfair to use a weighted lottery when the strength of claims differs significantly. More precisely, given the right judgement about how fairness losses compare, it would be fairer, on Broome's proportionality theory, to satisfy the stronger claim than to hold a weighted lottery.
I have shown three things. First, Hooker's view of fairness – a view which sees claims as reasons so that the requirement to satisfy claims becomes part of the theory of fairness – encounters a problem: it says, implausibly in my view, that we should not use a lottery when claims differ only slightly.Footnote 34 I admit that this will be a problem only for those who share this normative intuition about what to do in small-difference cases. Second, the normative view Hooker endorses, namely not to use a lottery when claims are very unequal, is compatible with Broome's view. Broome's theory tells us that what is fair in such circumstances depends on how we answer a question about the comparison of different fairness losses. His theory needs these further ethical judgements to be applied to such cases; it does not itself answer this question.Footnote 35 It is, in my view, very plausible to answer it in a way that endorses Hooker's normative view. Third, Hooker's view is incompatible with Broome's purely relational theory of fairness. I conclude that there is no objection to Broome's theory on the grounds that it would have implausible normative implications.
In discussions of Broome's proportionality requirement, the idea that there is more than one relevant way in which, by doing something, we treat people tends to be forgotten. However, it is clear from his justification of using lotteries in equal claims cases that there are (at least) two ways in which the principle to treat equal claims equally is relevant. Broome might not have been as clear as he could have been when he considered the implementation of the proportionality principle. Proportional lotteries are fair in a way. They are unfair in another way. This further unfairness increases as the difference in claim strength increases. Thus, weighted lotteries are overall fair only when people have claims of similar strength.Footnote 36
III.2. Can't one be unfair to everyone?
It is easy to be unfair to everyone: don't treat equal claims equally. For example, satisfy one of two competing claims without satisfying the other. It might seem harder to treat everyone unfairly whilst providing the same satisfaction of all claims, because then one has treated everyone equally. On Broome's conception, it does not only seem hard to do this, it actually is impossible. Examples like the following, thus, try to challenge his conception of fairness.
Suppose I mark student essays by throwing a pair of dice. It is much quicker doing it this way. I've devised a simple system in which ‘1’ means 7, any ‘2’ means the same mark as before and the other numbers are read as what they are. My marking system produces an entirely reasonable distribution of marks. Every student has the same claim and I treat everyone in the same way. Nevertheless, I keep the system to myself because if students knew about it they might well question the fairness of my marking. On Broome's account, we know, my fairness is impeccable.Footnote 37
No doubt, students have reason to complain about my marking. For one, I do not mark according to published marking criteria. I fail my students because I don't treat their work and, thus, them with proper respect. There is no question about whether what I am doing can rightly be criticized – it can. The question is whether I am violating a requirement of fairness.
Fairness bears different relations to other moral notions. Fairness, for example, does not seem to require kindness. A superior can be fair without being kind. Fairness and justice, however, are more closely related. How can it be fair to be unjust?
Consider the following Sidney Morgenbesser story.Footnote 38 Dr Morgenbesser became something of a Columbia legend at the time of the student uprising in 1968 for being beaten when he joined a human chain against the police. When asked how he felt about this, he allegedly replied that it was unjust to beat innocent protesters but it wasn't unfair as everyone got beaten. This reply has a funny ring because of the unusual contrast between justice and fairness. Nevertheless, we understand what Morgenbesser is saying. There is a familiar moral notion, and, like Morgenbesser, I would call it fairness, for which comparisons are central.Footnote 39
Further support for a purely relational justice-related notion comes from Aristotle. In book V of the Nicomachean Ethics, Aristotle outlines his theory of justice. General justice requires lawfulness and as the law extends to all virtues general justice is complete virtue. In addition to general justice, Aristotle describes two special forms of justice: rectificatory justice and justice in distribution. In this last domain, the domain of distributive justice, Aristotle introduces a purely relational concept. ‘Hence what is just [since it requires equal shares for equal people] is in some way proportionate’ (NE 131a, 29). I mention Aristotle to put some authority behind the common-sense view that there is a need for a purely relational concept.
Different claims on some good will have different sources. These differences help us to distinguish claims that appeal to kindness from those that appeal to justice. The students’ claim to having their work properly marked is an appeal of the latter kind. Suppose I made an exception for one of the students and suppose I marked his work properly. This student, especially if the mark was poor, will have a complaint; why was he not given the chance of a result determined by the roll of the dice? He might have done much better. The lesson I draw is that however we carve up the normative domain, there will be a role for a purely relational moral concept. We can say that I treated my students unjustly but not unfairly; with equal right to represent ordinary usage we could also say that I didn't treat them fairly. If we do use an absolute conception of fairness, one that requires the satisfaction of claims, we need to be aware that we have added a notion of fairness to our moral vocabulary different from the well-established relational concept. Whatever we say, there is a need for a relational moral concept. Broome calls it fairness and has offered a theory that spells out its role in moral thought.
III.3. The calculation objection and the self-defeating argument
The final objection I want to consider comes from Kirkpatrick and Eastwood. After repeating part of Hooker's paper and endorsing his objection to Broome, which we discussed in section III.1, they object to Broome's idea of treating claims in direct proportion to their strength as follows. ‘The calculation objection argues that it is almost impossible to calculate the appropriate weights for the lottery and incorrectly calculated lotteries are unfair.’Footnote 40
I have mentioned three ways in which Broome's theory of fairness is incomplete. It does not offer a theory of the sources of claims; when applying the theory, it leaves it open what claims are claims to; fairness comparisons, which are important for deciding whether his theory endorses weighted lotteries in unequal-claims cases, are also left to considered judgement. Should we add a fourth dimension of incompleteness, namely that it does not offer a theory of how to measure claim strength?
Doing this might be premature. Often it will be easy to measure the strength of claims. If one person has worked nine hours towards completing a shared task and the other person only put in one hour, we'd know exactly which weights to set in a weighted lottery, if, that is, Broome recommended one (which he does not). Kirkpatrick and Eastwood do not consider such examples. They think of cases in which the size of benefit determines claim strength. How much more weight should be given to a person who is threatened with losing her arm whilst another person is threatened with losing her finger (and only one loss can be prevented)?
This is indeed a difficult question. Some people think that expected utility theory offers an answer. Suppose a person is able to rank all goods. If, in addition, this person's preferences satisfy certain axioms we can have numerical representations (which are unique up to positive linear transformations) of the goods in question (and of preventing their loss). Then we would need to make a case for interpersonal comparisons of utility, in order to get the weights right when different people are affected by such losses. One person who has argued that all this can be done is, coincidentally, John Broome.Footnote 41
Other people will be sceptical about introducing such precision into moral deliberations. They will say that losing an arm is certainly worse than losing a finger, but how much worse it is, they will say, is not a good question. We know from theories of measurement that we should not think of all comparisons as being guided by independently existing quantities. Sometimes, and ordinal utility theory is an obvious case in point, quantities only mirror or represent comparisons without in any way going beyond them.Footnote 42 So, according to this second view, there simply is no precise answer to the question how much worse one loss is than another.Footnote 43
How would this second view affect the implementation of Broome's theory (if, that is, Broome recommended weighted lotteries in such cases, which he does not)? We might come up with a range of numbers which is such that any number in this range would represent the idea that one loss is considerably worse than another.Footnote 44 We cannot, however, have a lottery in which ranges of probabilities determine winners. We could have a random selection of any number within this range and then have a lottery with the randomly selected number. This would be one way of solving the problem of implementation for those who think that loss comparisons will always be rough and not precise.
Kirkpatrick and Eastwood engage with neither of these alternatives. Their objection sits uncomfortably between the two views outlined. They say that it is almost impossible (at least in some cases) to measure claim strength accurately. Thus, they are committed to the existence of a precise measure of claim strength (in opposition to the second alternative), whilst denying that we can find out about this measure (in opposition to Broome's way of using expected utility theory). I see no reason that would favour this in-between position over either of these two alternatives.
If arguments in favour of their view were forthcoming, we might need to add a fourth dimension of incompleteness to our description of Broome's theory. This would not, however, undermine his theory in any way.
Having answered their calculation objection, I will be brief about their ‘self-defeating argument’. When Broome sets up his discussion of lotteries, he says that lotteries are fair and contrasts decisions by lotteries with decisions by experts. How can a decision by lottery, he asks, be better than a decision by an expert? His theory answers this question: if the expert understands fairness, he or she would also resort to a lottery. His point is to show that lotteries are not a response to epistemic limitations; they are a way of treating claims proportionally. When he sets up his discussion he mentions the possibility of biased experts. On this basis, Kirkpatrick and Eastwood write: ‘Broome argues against the use of authorities and rules in such decision procedures [which could help determine the strength of claims]. However, the use of authorities or rules seems to be required for accurately calculating claims, so Broome undermines his own theory.’Footnote 45
A theory is self-undermining if it demands something that, by its own lights, cannot be provided. Broome neither demands experts for assessing claim strength nor does he deny that there could be such people. So Kirkpatrick and Eastwood's ‘self-defeating argument’ founders in both dimensions. Think of a case in which the strength of one's claim depends on the size of the benefit one would receive. How much, for example, would your neighbour benefit from, let's say, a week's holiday in Sicily on his or her own? One would need to know a person reasonably well in order to have a reasonably good idea of how much he or she would benefit. Is Broome's comment about the possibility of biased experts incompatible with the idea that one needs to know people in order to see how much they would benefit from various things? There is no reason to think that it would be.
IV. CONCLUSION
Kirkpatrick and Eastwood characterize Broome's position in the following way. ‘The satisfaction requirement for fairness can be achieved through surrogate satisfaction in a weighted lottery.’Footnote 46 This sentence contains three mistakes. Broome has offered a relational conception of fairness.Footnote 47 I have argued that, therefore, he must exclude the satisfaction requirement from his theory of fairness (which he does). When he talks about surrogate satisfaction, he does not mean satisfaction of the claims to some benefit; he rather means that lotteries only partially meet the requirements of fairness. And, lastly, he does not think that weighted lotteries are the fairest way to treat claims of different strength proportionally. Weighted lotteries are fair in one way, but become increasingly unfair in another way as claims strengths start to diverge.
Kirkpatrick and Eastwood repeat the mistakes of others and, as described in the previous section, add some of their own. I'd like to mention two more of their mistakes. Imagine an essay the aim of which is to criticize epistemological scepticism which starts as follows. ‘Scepticism encounters a problem: do we really know that there is an external world?’ Compare this with what they say about Broome's theory. ‘The distribution of indivisible goods poses a problem for Broome's theory of fairness.’Footnote 48 The contrary is correct. How his theory deals with the distribution of indivisible goods is not a problem but the main argument Brome offers in support of his view. Their last mistake has given me the title for this article. One of their sections is called ‘Brushing Broome Aside’. When I read this, I thought ‘How unfair!’
On Broome's account, however, I can't make a judgement about whether his commentators have treated him fairly, as this would depend on how they treat other philosophers. Broome's theory of fairness has been misunderstood. Let us hope that other philosophers do not share the same fate at the hands of their commentators. If Broome is right, we are thereby committed to hoping for unfairness to happen, but John Broome, being the kind man that he is, would, I am sure, be happy with this consequence.Footnote 49
