¹ The aim of this article is to lay out Broome's theory and to defend it against criticisms and misunderstandings. I don't mean to convey that no doubts about the correctness of Broome's theory remain. Philosophers, me included, will, in time, present their objections. We need to understand a theory fully before we can discuss it. This article aims to prepare for such a discussion.

² Broome, J., ‘Selecting People Randomly’, Ethics 95 (1984), pp. 38–55 CrossRefGoogle Scholar, at 43.

³ Broome, J., ‘Fairness’, Proceedings of the Aristotelian Society 91 (1990), pp. 87–101 CrossRefGoogle Scholar, reprinted in his Ethics out of Economics (Cambridge, 1999), pp. 111–22, at 116.

⁴ For ease of exposition, I will simply follow Broome in contrasting fairness, which, in some cases, requires withholding the good, with what he calls ‘general or utilitarian goodness’ or simply ‘goodness’, which requires distributing the good. Benefits are the typical example of what belongs to this latter category.

⁵ Broome, ‘Selecting’, p. 44.

⁶ In this article, I focus on fairness and will avoid unnecessary engagement with Broome's general moral theory, which he develops in Broome, J., Weighing Goods: Equality, Uncertainty and Time (Oxford, 1991)Google Scholar and in Broome, J., Weighing Lives (Oxford, 2006)Google Scholar. By bringing fairness, the content of which is determined by a deontological principle, into a weighing process, he ‘consequentializes’ fairness.

⁷ Broome, ‘Fairness’, p. 115.

⁸ Broome, ‘Fairness’, p. 116.

⁹ Broome, ‘Fairness’, p. 118.

¹⁰ Broome, ‘Fairness’, p. 118.

¹¹ At the beginning I have called Broome's theory of fairness a powerful theory. What I mean by this is that it helps us to structure moral debates. Take the case of preferential medical treatment for the young over the old. Broome's theory allows us to distinguish three different levels of engagement with this question. The first level, which I have described above, concerns the nature of claims. Do people have a claim to live reasonably long and happy lives or not? Once we have established which claims are in play – and there might be several – we need to establish their strength. We need to consider, for example, whether the size of a benefit, in the cases under discussion, affects the strength of one's claim to the benefit. (We don't think that all differences in the size of benefit count. We do not think that a generally happy and contented person has a stronger claim than a less happy and less contented person.) Debates on these two levels will interact in various ways before we reach a considered view. Broome's theory then tells us what fairness requires. It does not tell us what we ought to do. Sometimes it may be allowed or even required to be unfair. This is a third level of debate. Broome's theory helps us to structure a moral debate in its complexity.

¹² See Hooker, B., ‘Fairness’, Ethical Theory and Moral Practice 8 (2005), pp. 329–52CrossRefGoogle Scholar, at 334.

¹³ Broome, ‘Fairness’, p. 121.

¹⁴ Here is some relevant background to Broome's discussion. The development of utility theory in the wake of Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior (Princeton, 1944)Google Scholar has inspired attempts to set utilitarianism on a solid axiomatic basis. A leading proponent of this approach was John Harsanyi (see Harsanyi, ‘Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility’, Journal of Political Economy, 63 (1955), pp. 309–21). Harsanyi derives utilitarianism from the following three axioms. (1) Individual decision-making satisfies the axioms for expected utility maximization. (2) Social welfare can be written as an increasing function of individual expected utilities. (3) Social choice satisfies the axioms for expected utility maximization. Peter Diamond (see Diamond, ‘Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility: Comment’, Journal of Political Economy 75 (1967), pp. 765-6) has raised the following objection to the third axiom. It is a general feature of utility theory that if two alternatives have the same utility value any probabilistic mixture between what are indifferent options will share the same value with these options. In a situation of scarcity we can provide one person with the good of another. If both options have the same utility value, a lottery which gives both of the potential recipients an equal chance of getting the good, as well as any other lottery, will have the same value as either of the two options. This consequence of utility theory, Diamond claims, is incompatible with what people feel about justice and fairness. In a social context, where the good of more than one person is at stake, a lottery is preferable because it is fair. Broome's theory of fairness offers an explanation of Diamond's claim.

¹⁵ Broome, ‘Fairness’, p. 117.

¹⁶ If we'd generalize the idea that equal claims should be treated equally in a different way, the person with the stronger claim should get all of the good. If this seems unfair – it does to me – Broome's proportionality principle looks more plausible than the alternative generalization.

¹⁷ Kornhauser, L. A. and Sager, L. G., ‘Just Lotteries’, Social Science Information 27 (1988), pp. 483–516 CrossRefGoogle Scholar, at 495f.

¹⁸ See Stone, P., The Luck of the Draw: The Role of Lotteries in Decision Making (Oxford, 2011)CrossRefGoogle Scholar and Henning, T., ‘From Choice to Chance? Saving People, Fairness, and Lotteries’, Philosophical Review 124 (2015), pp. 169–206 CrossRefGoogle Scholar.

¹⁹ Henning, ‘From Choice’, p. 170. Note that, at the outset, Henning seems to neglect Broome's view, which does not endorse the Lottery Requirement as stated. On Broome's view, we are only obliged to use a lottery if the fairness loss of lotteries is compensated by the benefit of distributing the good. I have argued that this condition will not always be fulfilled.

²⁰ See Stone, P., ‘Why Lotteries Are Just’, Journal of Political Philosophy 15 (2007), pp. 276–95CrossRefGoogle Scholar; Stone, P. (ed.), Lotteries in Public Life: A Reader (Exeter, 2011)Google Scholar; Stone, Luck of the Draw.

²¹ Broome, ‘Fairness’, p. 119.

²² See Stone, Lotteries, p. 9. The term ‘objective equalization’ comes from Eckhoff, T., ‘Lotteries in Allocative Situations’, Social Science Information 28 (1989), pp. 5–22 CrossRefGoogle Scholar, reprinted in Stone, Lotteries, pp. 161–76, at 163.

²³ Stone, Luck of the Draw, p. 61.

²⁴ Henning, ‘From Choice’, p. 177.

²⁵ I discuss this thesis with reference to Broome's overall moral theory in Piller, C., ‘Valuing Knowledge: A Deontological Approach’, Ethical Theory and Moral Practice 12 (2009), pp. 413–28CrossRefGoogle Scholar.

²⁶ Broome, ‘Selecting’, p. 46.

²⁷ I have concentrated on commentators who aim to describe the whole domain of how to justify lotteries. I have chosen them because their failure to understand Broome has a negative impact on discussions about fairness. Henning and Stone, however, are not alone in mischaracterizing Broome's theory. Here is another attempt to capture Broome's view: ‘The lottery acts as a surrogate which partially satisfies every candidate's claim’ ( Kirkpatrick, J. and Eastwood, N., ‘Broome's Theory of Fairness and the Problem of Quantifying the Strengths of Claims’, Utilitas 27 (2015), pp. 82–91 CrossRefGoogle Scholar, at 85). It should be clear by now that using a lottery partially satisfies fairness; it does not partially satisfy a person's claim to the good. The same comment applies to the following description. ‘According to John Broome, fair lotteries provide surrogate satisfaction of claims by giving all claimants an appropriate chance of benefiting as a surrogate for actually receiving the benefit’ ( Vong, G., ‘Fairness, Benefiting by Lottery and the Chancy Satisfaction of Moral Claims’, Utilitas 27 (2015), pp. 470–86CrossRefGoogle Scholar, at 472).With the obvious exception of Broome himself, there is hardly anyone who is able to describe his theory adequately.

²⁸ Broome, ‘Fairness’, p. 117.

²⁹ Hooker, B., ‘Fairness’, Ethical Theory and Moral Practice 8 (2005), pp. 329–52CrossRefGoogle Scholar, at 349.

³⁰ Hooker, ‘Fairness’, p. 349. See Stone, ‘Why Lotteries’, pp. 285f., for a similar view. Stone argues that claim strength should determine who receives the good. Whenever someone has a stronger claim than someone else, he or she should get the food. Stone calls this a demand of impartiality.

³¹ Broome, ‘Selecting’, p. 48.

³² Hooker regards it as an improvement of Broome's theory if the satisfaction requirement becomes part of a theory of fairness. He writes: ‘If the requirement of proportionality is central to fairness, fairness requires the greatest possible proportionate satisfaction of claims’ (Hooker, ‘Fairness’, p. 341). Broome's relational account of fairness needs to keep the fairness principle separate from any satisfaction requirement. This separation is entailed by the relational nature of fairness.

³³ Broome, ‘Fairness’, p. 120.

³⁴ Broome (see ‘Selecting’, p. 40) provides the following further argument in support of the intuition that we should use a lottery when claims are almost equal. Lotteries bring some fairness to an unfair situation. (The situation is unfair because a lottery will result in an unfair distribution and we might be morally required to distribute the good.) If lotteries were only used in situations in which claims are exactly equal, any change in claim strength would be more important than the fairness that lotteries bring. Fairness, however, is an important good that cannot be outweighed that easily. For me, this is a good argument. In our dialectic, however, I cannot put much weight on this argument as a defender of Hooker's conception would regard it as question-begging. (According to such a defender of Hooker, we cannot appeal to the value of fairness in cases, like our small-difference cases, in which, according to Hooker, it would not be fair to use a lottery.)

³⁵ This is a third dimension in which Broome's theory of fairness is incomplete.

³⁶ Has Broome really been less clear than he, perhaps, should have been? Considering a case where several people have roughly but not precisely equal claims, he says that it might be fairer to have an equal odds lottery than to give the good to the person with the strongest claim. Whether this is so, ‘depends on a complicated judgement. . . . The likelihood of this less fair result [namely not having satisfied the strongest claims] will have to be weighed against the contribution to fairness of the lottery itself’ (Broome, ‘Fairness’, p. 120). Whenever he endorses weighted lotteries, he only considers cases in which the differences in claim strength are very small. Summarizing his view he writes: ‘A lottery should be held when, first, it is important to be fair and, secondly, the candidates’ claims are equal or roughly equal’ (Broome, ‘Fairness’, p. 120). Let me add that this presentation of Broome's theory will also have answered the worries raised in Lazenby, H., ‘Broome on Fairness and Lotteries’, Utilitas 26 (2014), pp. 331–45CrossRefGoogle Scholar, which, for reasons of space, I cannot discuss in any detail.

³⁷ Hooker (‘Fairness’, p. 340) makes the same point. If I owe two people some money but I don't repay my debt and burn my money instead (or, more plausibly, simply keep it to myself), have I treated them fairly?

³⁸ I owe my knowledge of this story to Michael Rosen.

³⁹ Sometimes the story is told the other way round. Beating innocent protesters is marked as unfair whilst justice was done by beating everyone. I find this way of telling the story less appealing, but what alone matters is that an aspect of justice or of something related to justice is captured by a purely relational notion.

⁴⁰ Kirkpatrick and Eastwood, ‘Broome's Theory’, p. 86.

⁴¹ See Broome, Weighing Goods and Broome, Weighing Lives.

⁴² See Ellis, B., Basic Concepts of Measurements (Cambridge 1966)Google Scholar for a thorough philosophical discussion of these issues.

⁴³ Even on standard expected utility theory (without interpersonal comparability) the question whether something is twice as bad as something else cannot be answered. Quantities measured on an interval scale allow for comparisons between differences but not for the kind of comparison we are considering. A temperature scale is the most familiar example of an interval scale. It does not make sense to say that one thing is twice as hot as another.

⁴⁴ If I'd be pressed to specify this range for the case at hand, I'd say that the range in which the stronger claim falls goes from approximately 75% to about 98%. By this I mean that any number in this range would represent the idea that the loss of an arm is much worse than the loss of a finger.

⁴⁵ Kirkpatrick and Eastwood, ‘Broome's Theory’, p. 90.

⁴⁶ Kirkpatrick and Eastwood, ‘Broome's Theory’, p. 87.

⁴⁷ Kirkpatrick and Eastwood (‘Broome's Theory’, n. 7) say that they have been made aware that, according to Broome, withholding the good is fair. We have seen that this is the essential feature of Broome's relational theory of fairness. Kirkpatrick and Eastwood, however, regard it as a ‘subtlety’ which, they say, they will ignore.

⁴⁸ Kirkpatrick and Eastwood, ‘Broome's Theory’, p. 85.

⁴⁹ I have presented previous versions of this article at a conference on Contractarianism in Rennes in 2012 and at a conference on Social Contract Theories in Lisbon in 2014. On both occasions I have received helpful comments. I am very grateful to Brad Hooker for his detailed comments. Responding to them has substantially improved this article. I thank John Broome for his comments and, more broadly, for his inspiring work. I thank those of my students who found some enjoyment when they understood Broome's theory of fairness.