¹ The question of what the views judge to be fairest when the kidney could be destroyed is more complicated. I return to this issue at the beginning of section II.

² Broome, John, ‘Fairness’, Proceedings of the Aristotelian Society 91 (1990), pp. 87–102CrossRefGoogle Scholar; Weighing Goods: Equality Uncertainty and Time (Oxford, 1991); ‘Fairness versus Doing the Most Good’, Hastings Center Report 24 (1994), pp. 36–9; ‘Kamm on Fairness’, Philosophy and Phenomenological Research 58 (1998), pp. 955–61; Ethics out of Economics (Cambridge, 1999).

³ Indeed, Broome's account of a claim is not independent of his account of fairness. Claims are defined as a type of moral reason to which fairness applies. That is, claims require proportionate satisfaction, are owed to individuals, and are not side constraints. Broome, ‘Fairness’, pp. 91–100.

⁴ Broome, Ethics out of Economics, p. 117.

⁵ For further illustration, suppose we have a case where we must choose between saving five persons and saving one person. Each of the persons has equally strong claims. For Broome, in this case the fairest thing to do is to toss a coin to decide whether to save five or one. Let me explain how Broome arrives at this judgement. Broome believes that lotteries are fair because (a) they provide a surrogate for fairness in outcome and (b) they can represent claims in proportion to their strength. That is, for Broome, they represent claims in proportion to their strength when, and because, they give each person a chance, proportional to the strength of his claim, of having his claim satisfied in outcome. In the above case, even though tossing the coin does not perfectly satisfy claims, it does provide a surrogate satisfaction by providing each with an equal chance. The one is given the same chance of having his claim satisfied in outcome as each of the five. Importantly, on Broome's view, when an equal division of goods is not available and a lottery is held, claims are proportionally satisfied, in a weak sense, even though there is an inequality in outcome; simply weighing claims against each other to determine which are the most weighty does not satisfy them proportionally in any sense. For further discussion of this type of problem see Broome, ‘Kamm on Fairness’. Kamm, Frances, Morality, Mortality, Volume I: Death and Whom to Save From It (Oxford, 1993)Google Scholar; Taurek, John, ‘Should the Numbers Count’, Philosophy and Public Affairs 6 (1977), pp. 293–316Google ScholarPubMed.

⁶ Broome, Ethics out of Economics, p. 120. It is not clear why Broome says that the result of a lottery will generally be that it goes to candidates who do not have the strongest claim. In two-person-weighted lotteries this will not be true. What can be said for certain is that a lottery has a lower expected fairness measured exclusively in terms of outcome.

⁷ To be clear, the alternative conception still holds 4, and so still holds that claims may achieve some form of surrogate satisfaction by being represented in a lottery, but it orders the fairness value of surrogate satisfaction and outcome satisfaction in a new way.

⁸ I return to this point in the following section.

⁹ Barry, Brian, ‘Equality of Opportunity and Moral Arbitrariness’, Equal Opportunity, ed. Bowie, N. (Boulder, 1988), pp. 23–47, at 32Google Scholar.

¹⁰ This is the view Broome ascribes to William Godwin. Broome, Ethics out of Economics, p. 115.

¹¹ Wasserman, David, ‘Let them Eat Chances: Probability and Distributive Justice’, Economics and Philosophy 12 (1996), pp. 29–49CrossRefGoogle Scholar. Wasserman's view is not the same as the alternative conception. Wasserman believes that lotteries have expressive value but no fairness value.

¹² Stone, Peter, The Luck of the Draw: The Role of Lotteries in Decision Making (Oxford, 2011)CrossRefGoogle Scholar; Wasserman, ‘Let them Eat Chances: Probability and Distributive Justice’.

¹³ Diamond, Peter, ‘Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility: Comment’, Journal of Political Economy 75 (1967), pp. 765–6CrossRefGoogle Scholar.

¹⁴ Alex Voorhoeve and Marc Fleurbaey have also made an argument for why a theory of fairness must include the distribution of chances if it is to respect the separateness of persons. Voorhoeve, Alex and Fleurbaey, Marc, ‘Egalitarianism and the Separateness of Persons’, Utilitas 24 (2012), pp. 381–98CrossRefGoogle Scholar.

¹⁵ Allow me to expand. Wasserman and Stone might argue that they agree that alternative β is superior. For example, Wasserman believes that a lottery may have ‘expressive value’, so that it can be appropriate to use lotteries to express the equal status or claims of the individuals with whom one is dealing. This response will lead back to a problem I identified earlier in section II.A. There I examined an iterated case where chances were distributed one after the other, arguing that the distribution of chances in the past should be seen as relevant for the distribution of chances in the present. The expressive view does not consider a person to have had something simply because she has had a chance that does not pay off. For that reason, it cannot explain why the fact that one person has had a chance in the past gives us a reason to give a chance to someone else in the future, ceteris paribus. By contrast, the alternative conception of fairness I have defended can explain why chances should be distributed in this way because it considers entering a claim into a lottery to it a sort of satisfaction. The alternative conception is more successful than the expressive view at explaining our intuitions about how chances should be distributed in iterated cases.

¹⁶ For comments on earlier drafts of this article I would like to thank Will Braynen, John Broome, Campbell Brown, Eamonn Callan, Chiara Cordelli, Sarah Hannan, Hyunseop Kim, R. J. Leland, David Miller, Kristi Olson, Michael Otsuka, Debra Satz, Adam Swift, Alex Voorhoeve, Juri Viehoff and the members of the Glasgow philosophy department. Special thanks are due to Mark Budolfson, who provided generous commentary at various points in its development.