Equal Chances versus Equal Outcomes: When Are Lotteries Fair and Justified?

I. WINNER TAKES ALL ECLIPSES EQUAL PORTIONS AS WELL AS EQUAL CHANCES

Where a fully divisible good has always decreasing marginal utility among people who are equally well off and equally able to convert resources into welfare, its division into equal portions will give rise to more welfare in the aggregate than giving the whole good to one person. That fact would provide utilitarian reason, above and beyond the fairness of an equal distribution among those with equally strong claims, to divide such a good into equal portions rather than giving all an equal chance of receiving the whole good.⁵ At the other extreme, there are cases in which a division of the good itself renders it worthless: e.g., the division of a heart among those who need a lifesaving transplant of an entire heart to survive. The same can be said of a Solomonic division of a baby in two to settle a custody dispute. A division of time-shares of the baby rather than the baby itself into equal parts would provide a superior resolution of such a dispute. In the case of the lifesaving drug with which this article opened, there is, however, no division into either equal spatial or temporal portions which renders the drug other than worthless. The best we can do is provide equal chances of the entire drug. Here we are driven to the proposal to distribute this good by lot.

In some cases, even if a good can be divided into equal spatial or temporal portions that retain significant use value, there remains a case for distributing chances of the undivided good rather than dividing the good into portions. Consider the following two examples which will figure prominently in the remainder of this article:

Life extension: Unless they receive a life-extending good, 100 people now aged 20 will all die at the age of 81 after having previously enjoyed excellent health. Suppose that each person could either have (i) a 1 in 100 chance of 100 more months—which is 8.3 more years—in excellent health, to just past the age of 89, or (ii) a certainty of a 1/100^th portion of 100 months, which would yield a life extension of one more month in excellent health: from 81 years to 81 years plus a month. Here each might rationally prefer a 1/100^th chance of 8.3 more years to the certainty of an extra month. This in spite of the fact that an extra month of life is of significant positive value.

Money: Suppose that each of 500 Americans of somewhat above average income and wealth could either have (i) a 1 in 500 chance of $50,000 or (ii) a certainty of $100, which is a 1/500^th portion of $50,000. Everyone might rationally prefer the former lottery option. This need not be because they regard an extra $100 as worthless. They might all regard such a relatively modest sum as worthwhile rather than wholly trivial. Let us stipulate that they would spend this money to purchase something useful or enjoyable. Let us also stipulate that their preference for a lottery is not on account of any desperate need for $50,000. It’s not the case, for example, that they need to raise this amount for a lifesaving operation. Rather, let us suppose that it would be useful for a down payment on a house even though their current rental arrangement is perfectly adequate, or for a nice home improvement, even though their house is not in any disrepair.⁶

These examples illustrate that an equal portion of a good might be worth less to each person than an equal chance of receiving the whole good even when that equal portion is of significant, or at least non-trivial, value.

One can draw on this observation to provide an initial response to Hare’s suggestion that what we value is living rather than a lottery chance of living. Recall that, in the course of defending this view, Hare maintains that ‘If I live then I am not significantly worse off for having had a low chance of living.’ I grant that, when one’s lottery chance of living is relatively low, the value of that chance will pale into insignificance in comparison with the benefit of living that is bestowed on the winner of the lottery. The winner of the aforementioned life extension lottery, for example, would value the extra 100 months of life he receives far more highly than the mere 1 in 100 lottery chance of receiving these hundred more months. One cannot, however, infer from this that what we value is living rather than chances of living. This is because similar claims of relative insignificance can be made regarding the certainty, rather than a chance, of living for a greater rather than a lesser number of months. A 1 in 100 chance of living 100 more months pales in significance in comparison with actually living 100 more months; but so does living only one extra month in comparison with living 100 more months. Moreover, as the above example involving life extension illustrates, it might be rational for everyone to prefer the 1 in 100 chance of living 100 more months to the certainty of one more month. Hence one cannot infer, from the fact that one greatly prefers the whole prize of many more years of life, that what one values is living rather than the chance of living.

Further insights can, however, be unearthed from the passage from Hare that I quoted in the introduction to this article, which might be thought to support the claim that living and the tangible goods of life, rather than a chance of those goods, is what we should care about. I shall respond to these further challenges to the significance of lottery chances in the next section.

II. THE INSTRUMENTAL VALUE OF LOTTERY CHANCES

III. WHEN ARE LOTTERIES PERFECTLY FAIR?

In this section, I shall provide an account of some conditions which are necessary and sufficient to ensure that a distribution by lot perfectly realizes the value of fairness. These conditions part company with Broome’s influential account of perfect fairness.

When one is indifferent between a given chance p (0 < p < 1) of an entire good and the certainty of a specified portion of that good, the latter is the certainty equivalent of the former. Let us stipulate that, in the case of a particular good, everyone is indifferent between a 1/n chance of the entire good and a 1/n portion of that good for certain. Here the 1/n portion of the good is everyone’s certainty equivalent of a 1/n chance of the good. Suppose that a distribution of equal portions 1/n of this good would perfectly achieve fairness among equal claimants, as Broome claims it would. Why wouldn’t a lottery with 1/n chances for all also perfectly achieve fairness, given that each values this chance the same as the certain outcome which Broome maintains would perfectly realize fairness?

Here are grounds to challenge the claim that such gambles involving lottery chances, which are valued the same as equal portions, perfectly realize fairness: the equal portion and the equal chance are equivalent merely ex ante; yet we know in advance that the outcome of the lottery will involve inequality. Suppose that there are two gambles of the same expected prudential value for each, involving equal-sized risks to each, but in one of which the risks are perfectly correlated and the other in which they’re inversely correlated.¹⁶ Even though these two gambles are ex ante equivalent in their expected prudential value, the former remains more fair than the latter, since it guarantees an equal outcome, whereas the latter guarantees an unequal outcome, and the two gambles do not differ in any other morally relevant respect.¹⁷ For similar reasons, equal portions might be more fair than an ex ante equivalent gamble whose outcome is guaranteed to be unequal.

The mere fact that an outcome will give rise to inequality does not, however, provide sufficient grounds to condemn it as unfair. That is one of the main lessons of luck egalitarianism. If the unequal outcome arises through no choice or fault of anyone, then it is unfair. But an unequal outcome might be perfectly fair if it arises through the choices of individuals. If the ant and the grasshopper are morally responsible agents whose different choices regarding work, leisure, and consumption provide the full explanation of the ant’s abundance and the grasshopper’s destitution, it follows that this inequality of outcome is fair. More to the point of our current discussion, we can also illustrate the luck egalitarian fairness of unequal outcomes with cases involving risk. The inequalities arising from gambles might be fair instances of what Dworkin has labelled ‘option luck’, which is roughly luck to which one has exposed oneself as the result of one’s voluntary choices, in contrast to ‘brute luck’, which is unchosen and unavoidable.¹⁸

Let us consider cases in which an unequal outcome is the result of the choice of each to gamble. Suppose, in variants of the above two lottery examples involving life extension and money, that each individual has the option of choosing for herself whether to receive a 1/n portion of the good for certain or else a 1/n chance of the entire good of 8.3 years of extra life in the one case or $50,000 in the other case.¹⁹ Suppose, moreover, that each person regards a lottery as more valuable to equal portions. If each rationally chooses the lottery option, these are cases in which the resulting inequalities of outcome would constitute perfectly fair instances of option luck. It is crucial, here, that the lottery chances that different people are provided involve equal odds. The mere fact that each would prefer the lottery chance they’re offered to an equal portion would be insufficient to render unequal lottery chances perfectly fair. Such a lottery would remain unfair because it fails to pass the envy test to which I referred in the introduction to this article, thereby failing to provide equally valuable chances to all with equally strong claims.

Now suppose that it is not possible to provide equal portions for some and equal lottery chances for others. Rather, the good must either be divided into equal portions for all or equal lottery chances for all. If, as before, everyone prefers the lottery, and they express this preference, then the good ought to be distributed by lot rather than divided into equal portions. The resulting inequalities of outcome would remain perfectly fair instances of option luck. We should reach the same conclusion even if nobody has the opportunity to express their preference for a lottery or an equal portion, yet we know that they each prefers the lottery: the good ought to be distributed by lot rather than divided into equal portions. The resulting inequalities would constitute perfectly fair option luck on the grounds, among others, that we know that each would have chosen such a lottery involving equal chances if provided with the opportunity.

Now suppose that a division of the good into equal 1/n portions is not possible. We cannot divide the 8.3 extra life years into such equal portions of one month for each of the 100 people. These years must all go to a single person if they go to anyone. Similarly, we cannot divide the $50,000 into such equal portions of $100 for each of the 500 people. The money must all go to a single person. But let us suppose that the preferences of the individuals are as we have been assuming above. In this case, the following counterfactual is true: even if it were possible to divide the good into such equal portions, everyone would still prefer a lottery involving equal chances. Even if the lottery is imposed for lack of an actual alternative involving equal portions, it might remain perfectly fair on grounds, among others, that it satisfies the envy test and everyone would have agreed to it, rather than to equal portions, had there been a choice between these two options. Here we have hypothetical choice, but not from under an imposed veil which deprives everyone of knowledge of their circumstances and their conceptions of the good and the preferences that arise from it. In this case, and the previous ones, involving hypothetical choice, it’s the fact that this option is preferred, rather than any appeal to consent, which is of normative significance.²⁰ I would maintain that a lottery involving equal 1/n chances needn’t be regarded here as a second best in the dimension of fairness to an impossible equal division of the good into valuable 1/n portions. Rather, such a lottery would perfectly realize fairness.

This claim marks a departure from Broome’s account of the fairness of lotteries. On his view, a lottery which distributes a scarce, indivisible good among people with equal claims is inevitably unfair at least to some degree, ‘because some candidates will get the good and others will not’. On account of this fact, Broome maintains that the distribution of none of this good to anyone is the only feasible course of action that will ‘perfectly achieve’ fairness. Broome also maintains that one needs to balance and trade off the ‘demands of fairness’ against the competing ‘demands of overall satisfaction’. He writes that ‘In some circumstances, no doubt, it will be very important to be fair, and in others fairness may be outweighed by expediency.’²¹ In the latter case, on Broome’s account, an indivisible good should be allocated to a single person by lot rather than to nobody, in sacrifice of perfect fairness to the satisfaction of claims.

If, however, a hypothetical (because unfeasible) distribution of the indivisible good into equal 1/n portions constitutes a benchmark which is dispreferred by all to a lottery which presents each with highest equal odds, why isn’t the lottery perfectly fair? In order to account for the perfect fairness of such a lottery, I need to draw attention to something special about the two cases under discussion involving life extension and money: namely, the benchmark of a sufficiently good, risk-free point of reference involving equal outcomes with which to compare the gamble of a lottery.

To illustrate what such a benchmark involves, I shall first point to a case in which it is clearly absent. Recall the case of a single indivisible lifesaving drug with which I opened this article, in which each of a number n>1 of young adults will die soon unless they receive this drug. Here there is no sufficiently good, risk free benchmark involving equal outcomes which is such that people would find it rational to choose a lottery for a single drug over it. The only actually available risk-free alternative involving equal outcomes is one in which all are certain to die because untreated. This alternative is risk-free even though not harm-free, since risk involves a non-trivial 0<p<1 chance of harm. Here the harm is certain. This is clearly not a sufficiently good risk-free alternative. Moreover, a hypothetical risk-free equal benchmark involving the certainty equivalent of a lottery chance of the lifesaving drug of 1/n would not be sufficiently good either, as it would involve a significantly degraded certainty of life: for example, the certainty of life for a small number of further years, which falls well short of the number of years one could expect if fully cured of this disease. In such cases where one is compelled to gamble by virtue of the lack of a sufficiently good, risk-free alternative, the rationality of such a gamble does not fully transform unfair brute bad luck into fair option luck.

By contrast, the risk-free benchmarks in the two lottery examples I introduced in Section I are sufficiently good in absolute terms. One more month of life for certain beyond one’s 81^st birthday is the certainty of ‘fair innings’. Similarly, one isn’t forced, by necessity, to opt for a gamble for a chance of a $50,000 gain in preference to the certainty of an extra $100. In both cases, choosing the lottery does not involve resignation to unfairness on account of the unsatisfactory nature of an alternative involving equal portions. Here, giving everyone an equal chance of the entire good isn’t a second best, when it comes to fairness, as compared with giving everyone an equal portion of that good if only it could be divided into goods of sufficiently high positive value. Rather, one’s choice of a lottery over a sufficiently good benchmark involving a division into equal portions constitutes an unproblematic case of option luck.

As an illustration to motivate this view, I once described a case in which each of two individuals has the same choice between an option whose outcome is certain and a high-risk option: (i) planting crops on a plateau, which is sure to yield a harvest of food which will allow him to live at a modest level comfortably above subsistence, or (ii) planting crops in the hills, which is 50% likely to yield just enough for bare, Spartan subsistence and 50% likely to yield a rich abundance of food which will allow him to feast in splendid luxury. If planting in the hills yields abundance, the person who chose the plateau has no grounds for complaint of unfairness that he’s poorer than the person who chose the hills, since he freely chose not to pay the cost (of exposure to the same risk in this case) which was a necessary condition of abundance. If the hills yield nothing more than bare subsistence, the person who chose them has no grounds for complaint of unfairness that he is poorer than the person who chose the plateau, since he had the same sufficiently good alternative whose outcome was certain; nevertheless, he found it worth his while to expose himself to this risk for the prospect of great gain. Similar things can be said about a case in which both individuals choose to gamble by planting in the hills and the one person’s crops yield an abundant harvest and the other’s bare subsistence: the inequality is fair, since they each had the same sufficiently good alternative whose outcome was certain; nevertheless, they each chose an option which carried with it the same known risk of loss.²²

In the article from which this example is drawn, I claimed that a sufficiently good, risk-free alternative in absolute terms was a necessary, but not a sufficient, condition of the fairness of such a choice to gamble.²³ I maintained that it was not a sufficient condition on the following grounds. It would be unreasonable to opt for such a risk-free alternative in favour of a gamble in cases where the downside risk of a gamble is no worse than the alternative involving certainty, or it is slightly worse, but the expected value of the lottery far exceeds the option involving certainty. Whereas I still think it would be reasonable to reject this alternative in favour of such gambles, I am now inclined to endorse the view that such a sufficiently good alternative is not merely necessary, but also sufficient to ground the full fairness of such a gamble. I now favour this view for the following reason. When the risk-free benchmark is sufficiently good in absolute terms, one is not compelled, by the undesirability of the risk-free benchmark, to take the gamble. Rather than being repelled by the intrinsic badness of the risk-free benchmark, one is merely attracted to a gamble which departs from this benchmark by the lure of its greater expected value. One might think that the fact that the latter gamble is ‘too good to pass up’ renders it coerced, and for that reason unfair. But this would be a mistake. In any sense of ‘coercion’ which is inconsistent with fairness, ‘too-good-to-pass-up’ gambles are not coercive.

For reasons I have offered in my above discussion of a varying sequence of cases, I would also now extend my account of the perfect fairness of lotteries to hypothetical as well as actual choice: i.e., to cases in which a lottery involving equal chances would have been chosen over an alternative involving equal portions which are sufficiently good in absolute terms. Such lotteries are perfectly fair even in the absence of actual choice, where such hypothetical choice reflects people’s actual preferences for lottery chances of a good. Perhaps the value of equality is realized to a greater degree when there is a division of a tangible good into equal portions rather than the equal chance of an unequal outcome that a lottery confers. Nevertheless, the equiprobable lottery is a perfectly fair procedure of allocation just in case it provides a distribution of equal chances to equal claimants which all rationally prefer to a division into equal portions that each constitute a sufficiently good risk-free alternative to that lottery.

I shall conclude with some observations regarding the light my account sheds on a memorable hypothetical case of James Fishkin’s involving a random reassignment of newborns to different parents at birth so that each has the same chance of being raised by wealthy parents as any other. Fishkin maintains that ‘equal life chances, in a quite precise sense, would result from the arrangement’, since ‘[a]ny newborn’s chance of reaching any highly valued position would be precisely equal to that of any other newborn infant’.²⁴ Few, however, would maintain that such a birth lottery would render the outcomes fair no matter how great these inequalities. As Brian Barry has observed:

But surely there is something very weird about a notion of [fair] equal opportunity such that a system of unequal opportunity (e.g., massive differences in life chances depending on one’s parentage) could be magically transformed into one of [fair] equal opportunity simply by switching babies in their cradles and leaving everything else the same. Suppose that this has in fact been carried out secretly over many years in some country marked by great inequalities of opportunity in the ordinary sense (a rigid caste system if you like).²⁵

My account can explain why such a birth lottery would fail to render things perfectly fair. The hypothetical benchmark of a risk-free alternative involving equal outcomes would need to involve a fairly dire state of affairs for it to be rational to reject it in favour of a gamble involving an equal chance of being born into different places in a rigid caste system or other highly unequal arrangements. Hence, such a lottery would not be chosen against the benchmark of a sufficiently good risk-free alternative. For this reason, the choice would be less than completely free. Rather it would, to some extent, be forced by necessity. Insofar as the choice to engage in a lottery was forced, the complaint of unfairness of the inequality of its outcome is not completely silenced by the fact that the lottery was chosen. It is not completely silenced even when this lottery involved equal odds among equal claimants. If, by contrast, entry into such a lottery were fully free, by virtue of the fact that it was chosen against the benchmark of a sufficiently good risk-free alternative, that fact would eliminate any complaint of unfairness regarding the inequality of the outcome. The fact that this lottery was freely chosen against a background of equal opportunity, rather than forced by necessity, renders it perfectly fair.

Competing Interests

The author declares that he has no competing interests.