¹ This is true at least when (i) we are dealing with a one-person case or with many-person cases in which everyone involved in a gamble will end up equally well-off; and (ii) we consider these individuals’ well-being in isolation. When these conditions do not hold, it may be inappropriate to do what is in the expected interest of each. See Fleurbaey, and Voorhoeve, , ‘Decide as You Would with Full Information! An Argument Against Ex Ante Pareto’, Health Inequality: Ethics and Measurement, ed. Eyal, Nir, Hurst, Samia, Norheim, Ole, and Wikler, Dan (Oxford, forthcoming)Google Scholar; Fleurbaey, , ‘Assessing Risky Social Situations’, Journal of Political Economy 118 (2010), pp. 649–80CrossRefGoogle Scholar; Otsuka, Michael and Voorhoeve, , ‘Reply to Crisp’, Utilitas 23 (2011), pp. 109–14CrossRefGoogle Scholar; and Otsuka, ‘Prioritarianism and the Separateness of Persons’ (this issue).

² See Gauthier, David, Practical Reasoning (Oxford, 1963), pp. 123–7Google Scholar; Rawls, John, ‘Constitutional Liberty and the Concept of Justice’, Nomos VI: Justice (New York, 1963), pp. 98–125Google Scholar and A Theory of Justice, 2nd, rev. edn. (Oxford, 1999), sects. 5 and 39; Nagel, Thomas, The Possibility of Altruism (Princeton, 1970), pp. 134–42Google Scholar, and ‘Equality’, in Mortal Questions (Cambridge, 1979), pp. 106–27; and Nozick, Robert, Anarchy, State and Utopia (Oxford, 1974), pp. 32–3Google Scholar.

³ Otsuka, Michael and Voorhoeve, Alex, ‘Why It Matters that Some Are Worse Off than Others’, Philosophy and Public Affairs 37 (2009), pp. 171–99CrossRefGoogle Scholar. See also Otsuka, , ‘Prioritarianism’. For the Priority View, see Derek Parfit, ‘Equality or Priority?’, The Ideal of Equality, ed. Clayton, Matthew and Williams, Andrew (Basingstoke, 2002), pp. 81–125Google Scholar. All references to ‘the Priority View’ and ‘Prioritarianism’ are to the ‘pure’ version of this view which Parfit defines on p. 103 of this article.

⁴ Practical Reasoning, p. 126.

⁵ ‘Why It Matters’, pp. 182–3.

⁶ Martin O'Neill, ‘Priority, Preference and Value’ (this issue).

⁷ McKerlie, Dennis, ‘Egalitarianism and the Difference between Intrapersonal and Interpersonal Judgments’, in Egalitarianism: New Essays on the Nature and Value of Equality, ed. Holtug, Nils and Lippert-Rasmussen, Kasper (Oxford, 2006), pp. 157–73, at p. 158, n. 2Google Scholar.

⁸ See Arneson, Richard, ‘Postscript to “Equality and Equal Opportunity for Welfare”’, Equality: Selected Readings, ed. Pojman, Louis and Westmoreland, Robert (Oxford, 1997), pp. 238–41Google Scholar and Temkin, Larry, ‘Inequality: A Complex, Individualistic, and Comparative Notion’, Philosophical Issues 11 (2001), pp. 327–53CrossRefGoogle Scholar.

⁹ We assume Albert is a child in order to eliminate the consideration that a morally motivated stranger has reason to do what will maximize Albert's expected utility out of respect for his autonomy. (One could argue that a respect for autonomy would require such expected utility-maximization if Albert was an adult and one assumed a measure of utility of the kind elaborated in the following footnote. See Otsuka and Voorhoeve, ‘Why It Matters’, pp. 183–6.)

¹⁰ Besides assuming that utility is measured on a cardinal scale, we do not here assume any particular measure of utility. Neither do we assume that the right way to make decisions under risk is given by orthodox decision theory. We do, however, assume that the interests that are taken into account in this measure do not include an interest in the fairness of the distributive process or distributive outcomes. Individual utilities therefore do not incorporate information on the degree to which the latter interest is satisfied. (In this respect, we depart from Fleurbaey, ‘Assessing’, on pp. 653–4.) This assumption makes it easier to discuss the fairness of distributions of well-being.

One can make more concrete claims about the size of the loss for which non-intervention is justifiable to Albert, and therefore permissible in this one-person case, if one assumes a measure of utility that is derived from idealized preferences satisfying the Von Neumann–Morgenstern axioms. On this measure, an alternative has higher expected utility just in case it would be preferred for the individual's sake after rational and calm deliberation with all pertinent information while attending to this individual's self-interest only. An alternative has the same expected utility as another alternative just in case such deliberation would yield indifference between the two alternatives. If the individual is a normally capable adult with well-formed preferences, the person expressing this preference can be taken to be an ideally rational version of this adult. If the individual is incapable of such deliberation (as we suppose the ten-year-old Albert in our example is), the person expressing the preference can be taken to be an ideally rational guardian who has deliberated with exclusive focus on his charge's self-interest. On this measure, there is good reason to regard non-intervention as permissible whenever it has an expected utility greater than or equal to intervention (that is, for any l in the specified range). After all, on this measure, it is true by stipulation that a risky alternative will have an expected utility greater than or equal to the utility of a secure alternative just in case, from the perspective of Albert's self-interest, the value of the chance of the benefit offered by the risky alternative outweighs or perfectly balances the value of the chance of the loss associated with this alternative. However, the arguments in the main text do not depend on the truth of this claim.

¹¹ We assume that Albert and Bob have no contact with, influence on, or power over each other, and are and will remain ignorant of each other's existence. The badness, if any, of the inequality that would result from non-intervention is therefore entirely the product of the mere fact that one of the two is worse off than the other, and not of the quality of their relationships with each other.

¹² This is a plausible assumption. If there were no risk of loss, then intervention would rob Albert of a chance of a benefit that comes without a risk of harm to anyone. Intervening when there is no risk of loss would therefore be akin to levelling down for the sake of maintaining equality, which egalitarians generally judge to be wrong, all things considered. And once is it granted that Albert could be left exposed to a costless (for him) chance of a gain when this could generate inequality, why should he not be granted this chance when it is accompanied by a risk of a relatively small loss to him? Notwithstanding its plausibility, we consider in section III what would follow if this assumption were false.

¹³ If the resulting inequality is bad, then the largest loss for which non-intervention is permissible in the Two-Person Intrapersonal Case will be smaller than the largest loss for which non-intervention is permissible in our one-person case, because it requires a greater net expected benefit to Albert in this two-person case to balance or outweigh the expected badness of the inequality.

¹⁴ Here is another way of putting our central claim. The largest loss for which non-intervention is permissible in the Two-Person Intrapersonal Case is larger than the largest loss (if any) for which non-intervention is permissible in the contrasting interpersonal case.

¹⁵ ‘Equality or Priority?’, p. 105.

¹⁶ ‘Equality or Priority?’, p. 104 (emphases in original).

¹⁷ Rabinowicz, Wlodek, ‘Prioritarianism for Prospects’, Utilitas 14 (2002), pp. 2–21CrossRefGoogle Scholar, argues that this ‘ex post’ approach is the version of the Priority View as applied to risky prospects that comes closest to respecting prioritarian intuitions.

¹⁸ Note that we need not assume that an adherent of the Priority View makes decisions under uncertainty by maximizing expected moral value. All that is required for our argument is that if, in each state of the world, one alternative yields the same outcome as another alternative, then a prioritarian is indifferent between these alternatives.

¹⁹ There is a different prioritarian view, which holds that we should apply prioritarian weighting to individuals’ expected utility, rather than to the utility that they end up with, when these two differ. On this ‘ex ante’ Priority View, it is more important to provide an equally large improvement in expected utility to someone with low expected utility than to someone with high expected utility. (See Epstein, Larry G. and Segal, Uzi, ‘Quadratic Social Welfare Functions’, Journal of Political Economy 100 (1992), pp. 691–712CrossRefGoogle Scholar.) For criticism of ex ante prioritarianism, see Otsuka and Voorhoeve, ‘Why It Matters’, pp. 197–8; Otsuka, ‘Prioritarianism’; and Fleurbaey and Voorhoeve, ‘Decide as You Would’. Yet another version of the Priority View gives weight to both this ‘ex ante’ view and the ‘ex post’ view discussed in the main text, by giving more weight to (i) a given improvement in a person's utility the lower his utility outcome and to (ii) a given improvement in expected utility the lower his expected utility. Parfit defends such a view in ‘Another Defence of the Priority View’ (this issue).

²⁰ ‘Equality or Priority?’, p. 85.

²¹ ‘Equality and Priority’, Utilitas 17 (2005), pp. 299–309.

²² Note that we do not question (c) or similar decision-theoretic principles like the sure-thing principle. (Here, we differ from Diamond, Peter, ‘Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility: Comment’, Journal of Political Economy 75 (1967), pp. 765–6CrossRefGoogle Scholar.) Instead, we believe that a morally relevant part of an outcome in a given state of the world may be what individuals would have achieved in a different state of the world. In this, we follow Broome, John, Weighing Goods (Oxford, 1991), pp. 111–15Google Scholar.

²³ Recall that 0 < l ≤ 0.2.

²⁴ We are grateful to Michael Otsuka for stressing this point in correspondence.

²⁵ See Cohen, G. A., ‘On the Currency of Egalitarian Justice’, Ethics 99 (1989), pp. 906–44CrossRefGoogle Scholar, esp. 908; Arneson, , ‘Postscript’, and ‘Equality of Opportunity for Welfare Defended and Recanted’, The Journal of Political Philosophy 7 (1999), pp. 488–97CrossRefGoogle Scholar; and Temkin, , ‘Inequality’, and ‘Equality, Priority, and the Levelling Down Objection’, The Ideal of Equality, ed. Clayton, and Williams, (Basingstoke, 2002), pp. 126–61Google Scholar.

²⁶ ‘Inequality’, p. 334.

²⁷ ‘Inequality’, pp. 327–8 and 337–8, emphasis in the original.

²⁸ ‘Inequality’, pp. 338–9.

²⁹ ‘Postscript’, p. 240.

³⁰ See Sher, George, ‘What Makes a Lottery Fair?’, Noûs 14 (1980), pp. 203–16CrossRefGoogle Scholar, and Wasserman, David, ‘Let Them Eat Chances: Probability and Distributive Justice’, Economics and Philosophy 12 (1996), pp. 29–49CrossRefGoogle Scholar.

³¹ Those who believe that the Risky Treatment 2 would be fairer in this sense need not, of course, think it preferable all things considered, since they may care about the higher levels of well-being enjoyed under Non-Risky Treatment 2.

³² See, for example, Diamond, ‘Cardinal Welfare’; Broome, John, ‘Fairness’, Proceedings of the Aristotelian Society, ns 91 (1990), pp. 87–101CrossRefGoogle Scholar; and Otsuka, ‘The Fairness of Equal Chances’ (MS).

³³ See Otsuka, ‘The Fairness’.

³⁴ In our cases, it is assumed that individuals’ chances of receiving a good are non-transferable, so that they have no opportunity to exchange their as-yet-unrealized chance for some other good that will contribute to their well-being. We can also assume that the individuals concerned do not know their chances of receiving a benefit, so that they derive no benefit from the mere anticipation of the possible gain. The chance of receiving a benefit therefore makes no contribution the quality of life of a person who does not end up receiving it.

³⁵ ‘Let Them Eat Chances’.

³⁶ Alternatively, egalitarians could claim that if, as Wasserman claims, inequalities in chances (as opposed to outcomes) do not affect fairness, then, notwithstanding the initial appearances, non-intervention is no less justifiable in the interpersonal case than it is in the intrapersonal case. In support of this claim, one could argue that the initial perception of the unfairness of the inequality of expected utilities in the intrapersonal case was a false perception which explains, but does not justify, the initial, and erroneous, judgement that non-intervention is less justifiable in the interpersonal case. However, even if we were to accept Wasserman's view, we would not accept this debunking explanation of this difference in the justifiability of non-intervention between our central pair of cases. The difference between, on the one hand, Albert running a risk of harm for his sake and, on the other, Albert running that risk for the sake of another who will be better off, is so clear that we believe one should look for a non-debunking explanation of its importance. Indeed, the Competing Claims View described in section V provides such an explanation which is not dependent on accepting the disputed claim about the fairness of equal chances.

³⁷ See Otsuka and Voorhoeve, ‘Why It Matters’, pp. 181–2.

³⁸ See Parfit, , Reasons and Persons (Oxford, 1984), pt. IVGoogle Scholar, and Otsuka, ‘Prioritarianism’.

³⁹ Versions of this article were presented at the Brocher Foundation, CMU, the LSE, Manchester University, the NIH, Turku University, UCL, the University of Bayreuth, and the University of Exeter. We thank those present for their comments. We also thank Michael Otsuka for extensive comments on and discussion of the claims we advance in this article, and Gustaf Arrhenius, Richard Bradley, John Broome, Franz Dietrich, Nir Eyal, Chiara Felli, Johann Frick, Joe Mazor, Thomas Porter, Wlodek Rabinowicz, and Matthew Rendall for written comments or helpful discussion.