Egalitarianism under Severe Uncertainty

Decision-makers face severe uncertainty when they are not in a position to assign precise probabilities to all of the relevant possible outcomes of their actions. Such situations are common — novel medical treatments and policies addressing climate change are two examples. Many decision-makers respond to such uncertainty in a cautious manner and are willing to incur a cost to avoid it. There are good reasons for taking such an uncertainty-averse attitude to be permissible. However, little work has been done to incorporate it into an egalitarian theory of distributive justice. We aim to remedy this lack. We put forward a novel, uncertainty-averse egalitarian view. We analyse when the aims of reducing inequality and limiting the burdens of severe uncertainty are congruent and when they conflict, and highlight practical implications of the proposed view. We also demonstrate that if uncertainty aversion is permissible, then utilitarians must relinquish a favourite argument against egalitarianism.

"By 'uncertain' knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty; (...). Even the weather is only moderately uncertain. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper (...). About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know." 2 Following seminal work by Daniel Ellsberg, this form of uncertainty is also commonly referred to in the economic literature as "ambiguity." 3 An uncertain (or ambiguous) situation so defined contrasts with a merely risky situation in which the decision-maker is in a position to assign precise probabilities to all relevant potential outcomes of the alternatives they have to choose between. We emphasize that in the senses in which we employ the term, both "risk" and "uncertainty" are subjective notions-they pertain to the beliefs about the chances of all relevant outcomes of their decisions that a rational decisionmaker is in a position to form on the basis of their prior beliefs and the evidence available them.
Uncertain situations in this sense are common. For example, with regard to many policy-relevant possibilities, the International Panel on Climate Change (IPCC) reports only as our point of departure a recent, pluralist egalitarian theory of distributive justice under risk. We propose and defend a novel extension of this view for uncertain situations and trace some key implications for policy decisions.
We proceed as follows. In Section II, we summarize the egalitarian view for decisionmaking under risk that we take as our point of departure. In Section III, we introduce a cautious, or "uncertainty-averse" decision criterion that we will appeal to throughout. On this criterion, uncertainty represents a burden in the sense that it reduces the value of a prospect. In subsequent sections, we explore novel implications generated by the interplay of the twin aims of reducing the burden of uncertainty and limiting inequality. In Section IV, we discuss cases where these aims are congruent. We show that our view provides novel reasons to direct resources towards those who have worse prospects or outcomes than others. In Section V, we consider cases where uncertainty aversion and inequality aversion are in tension. We show that our view weakens the egalitarian impulse to ensure that everyone sinks or swims together, since it gives great weight to eliminating the possibility of Equality for Possible People?" Ethics 126 (2016): 929-5. By contrast, the philosophical literature on egalitarianism under uncertainty focuses primarily on the question whether egalitarian principles can be derived from John Rawls's veil of ignorance, which creates an uncertain situation by denying people knowledge of the probability of ending up in any particular position in society. See Rawls, A Theory of Justice, 6 collective misfortune. In Section VI, we provide a new perspective on the debate between utilitarians and egalitarians. We demonstrate that if aversion to uncertainty is permissible, then utilitarians cannot wield a favourite argument against egalitarians. We summarize our principal findings and their relevance for a range of policy decisions in Section VII, where we also return to our opening Swine Flu case.
Before proceeding, we emphasize that our aim is merely to propose an egalitarian view which incorporates a set of rationally and morally permissible (rather than required), differential attitudes towards risk and uncertainty. In the service of this aim, we assume orthodox decision theory under risk, because leading alternatives to the orthodoxy under risk see conformity with the orthodoxy as rationally permissible. 7 But we pair it with an unorthodox (though well-known) decision principle under uncertainty, which yields the orthodoxy in the special case in which the decision-maker assigns precise probabilities to each outcome. The challenges to the orthodoxy posed by uncertainty are unique, since they involve a decision-maker lacking the ability to form reasoned beliefs of the kind that play a central role in making the orthodoxy plausible. It is therefore coherent, and indeed common, to endorse the orthodoxy in risky cases but not in cases of uncertainty. 8 This approach also allows us to focus squarely on unexplored issues involved in confronting uncertainty.

II. EGALITARIANISM UNDER RISK
In this section, we describe the egalitarian view for decisions under risk which we aim to extend to uncertain cases. In order to proceed at pace to novel ideas in subsequent sections, we do not offer a full defence of this view, which is provided elsewhere. 9 We shall refer to it as follows: Pluralistic egalitarianism: We should aim to improve people's prospects for wellbeing, raise total well-being, and reduce inequality in both people's prospects and in their final well-being (how well their lives end up going).
With respect to each individual's fate, we will assume that we are concerned with the distribution of a cardinal, interpersonally comparable measure of lifetime well-being derived from idealized preferences satisfying the von Neumann-Morgenstern axioms under risk. On this measure, a prospect has higher expected well-being for a person just in case it would be preferred after rational and calm deliberation with all pertinent information while attending to that person's self-interest only. 10 One prospect has the same expected well-8 being as another for a person just in case such deliberation would yield indifference between the two prospects.
To illustrate our egalitarian view under risk, imagine the following situation of a resource allocation manager in the National Health Service. Two ten year-old children, Ann and Bea, have just been diagnosed with an illness which, if untreated, will leave them completely blind and with a lifetime well-being of 50 (a moderately good quality of life); if fully cured, each would have a lifetime well-being of 80 (a very good quality of life). 11 Both are strangers to the decision-maker and to each other. Unfortunately, the resources at the decision-maker's disposal do not suffice to fully cure both Ann and Bea for sure. Below, we will describe the alternatives open to them. In order to link up with Ellsberg's paradigmatic presentations of risky and uncertain alternatives, risk will be represented by a random draw from an urn which is known to contain only 50 red balls and 50 black balls. 12 (While it may seem odd to speak of providing treatments that are effective conditional on the draw of a ball of a particular colour from an urn, this is merely a device to depict treatments for which the decision-maker rationally assigns precise probabilities to each possible outcome.) Table   satisfaction and that the specified idealized preferences fully track the size of this other thing. See Otsuka and Voorhoeve, "Why It Matters that Some Are Worse Off than Others: An Argument against the Priority View," 37 (2009): 171-99, 172-3, n. 3. 11 The numbers given for life-time well-being correspond roughly to those yielded by the well-known Health Utilities Index, Mark III (which is a von Neumann-Morgenstern measure of health-related well-being), for, respectively: living ten years in full health followed by seventy years being completely unable to see; and living 80 years in full health. See The Health Utilities Index, Mark III. http://www.healthutilities.com/hui3.htm [Accessed July 25, 2017]. On this scale, zero well-being is a life of equivalent value to the person to never having existed. 9 1 lists the final well-being for Ann and Bea given each possible draw from this urn for each of the alternatives we will presently consider. 13 We use redr and blackr to represent the possible draws from this risky urn, and pred and pblack for the probability of these draws.

Philosophy & Public Affairs
Throughout, for simplicity, we will consider only Ann's and Bea's well-being; we will not consider how their level of well-being relates to that of further people. there is, on this view, less unfairness overall when each is given an equal shot at a cure than when one child is given a cure outright and the other has no chance at receiving it. 14 Next, suppose that the decision-maker has a further alternative available: Equality under Risk: this treatment will either cure both children, or be wholly ineffective for both, with each result being equally likely.
On our egalitarian view, this alternative is superior to the preceding two. For by ensuring that all are in the same boat, it eliminates all unfair inequality without loss of expected total well-being.
Finally, suppose that the following alternative also becomes available: Equality under Certainty: this treatment will improve both Ann's and Bea's condition to that of a merely partial, but still substantial, visual impairment. We will consider both cases in which the level of well-being associated with this partial impairment is precisely halfway between the well-being associated with complete blindness and a full cure and cases in which this level falls short of this halfway point. The shortfall is given by a cost c, with 0 ≤ c < 15. For some, sufficiently small positive cost (c > 0), Equality under Certainty will still be chosen over the first two alternatives, because it eliminates all inequality with only a small reduction in expected total well-being. However, our egalitarian view will also regard it as inferior to Equality under Risk, because the latter offers each individual better prospects while ensuring equality.

III. A CAUTIOUS CRITERION FOR DECISIONS UNDER UNCERTAINTY
We shall now explain how we propose to extend our pluralist egalitarian view to cases of uncertainty. Let us start with a simple, one-person case. Suppose that Ann will go wholly blind unless she is treated. You must either provide Ann with an established, risky treatment, which, given the extensive evidence available, you confidently believe has a 0.5 chance of curing her and a 0.5 chance of having no effect on her, or instead provide her with a novel, maximally uncertain treatment, which will either lead to a full cure, or else be entirely ineffective. There is no further information available on the probabilities associated with these possible outcomes of the experimental treatment; nor do you possess precise prior beliefs about the probability of its effectiveness. Which treatment(s) is it morally permissible for you to provide? And: which would you choose?
There is evidence that many decision-makers' answer to the latter question would be: the merely risky treatment. Part of this evidence is that in a wide range of experiments involving self-interested choices, a large share of decision-makers (typically: a majority) strictly prefer a prospect in which they gain on the toss of a fair coin to the same gain on an event about which they know only that its probability may be anything in a range from 0 to 1. 15 They thereby display what is known as "uncertainty (or ambiguity) aversion" on their own behalf. (Those who are indifferent between this risky and uncertain prospect are commonly described as "uncertainty [ambiguity] neutral;" those who strictly prefer the uncertain prospect are known as "uncertainty [ambiguity] seeking".) And though there is less data on choices which concern others' interests only, uncertainty aversion appears to It is easy to see the appeal of such an uncertainty-averse attitude. In this situation, by hypothesis, the combination of your evidence and prior beliefs do not offer a compelling basis for a unique and precise assignment of probabilities to particular outcomes to the novel treatment. We submit that, in your decision-making, it is neither rationally nor morally required to arbitrarily adopt a single precise assignment of probabilities to each possible outcome of the novel treatment. Instead, it is permissible to take account of the full range of relevant probability assignments. In other words, you may consider everything from the worst probability distribution over the outcomes "wholly ineffective," and "full cure" that is consistent with your evidence and prior beliefs (viz., that the novel treatment provides Ann with no chance of a cure) through to the best probability distribution consistent with this information and these beliefs (viz., that it is sure to cure her), without reducing them to a single probability distribution. Moreover, while you should assign some decision weight to both the worse and better possible probability distributions over outcomes, quite how much decision weight to assign to each is, within a considerable range of sensible weights, up to you. And, we submit, cautiously assigning somewhat greater decision weight to the worse possible probability distributions than to the better ones is in this sensible range. 17 Such caution in the face of an inability to arrive at precise probabilistic assignments is the central motivation for uncertainty aversion.
Our claim is not that uncertainty aversion is the only reasonable attitude. It is merely that at least a moderate degree of such aversion is perfectly sensible. Caution of the kind outlined, could, we believe, be offered as a good reason for a choice of the risky treatment over the uncertain to anyone concerned with Ann's welfare. Interestingly, research suggests that people share our view in this regard-in an experiment in which participants had to try to persuade others of the rationality of their choices between risky and uncertain alternatives, participants typically accepted explanations for uncertainty-averse choices. 18 Despite its appeal, our claim is controversial among decision theorists, some of whom argue that a rational decision-maker must always somehow manage to decide in a manner that is compatible with assigning precise probabilities to every possible outcome. 19 Here, we will not re-litigate this issue. Instead, we will explore what would follow if, as we believe, a degree of uncertainty aversion were both rationally and morally acceptable. This question is worth exploring because, for the reasons just given, uncertainty aversion strikes us, many everyday decision-makers, and a considerable number of experts as a reasonable attitude, and it gives rise to underexplored issues of distributive justice.
Many uncertainty-averse decision criteria have been proposed. For illustrative purposes, we will here use a simple but popular criterion first put forward by Leonard decision-maker's information and prior beliefs. 20 Our conclusions hold for all other leading criteria, including those that give some weight to all probability distributions that the decision-maker regards as consistent with their evidence and beliefs. 21 On what is known as the α-Hurwicz or α-maxmin criterion, one values each person's prospect at α × its expected value given the least favourable probability distribution consistent with one's information and prior beliefs, plus (1 -α) × its expected value given the most favourable probability distribution that is so consistent, where 0 ≤ α ≤ 1 is the Hurwicz pessimism-optimism index. Uncertainty aversion involves giving more decision weight to the least favourable possible probability distribution than to the most favourable one; in other words, it involves taking α > 0.5. (The criterion reduces to orthodox decision theory when a decision-maker employs a single probability distribution.) In what follows, we will assume a decision-maker who has a fixed, permissible degree of uncertainty aversion both when they evaluate a prospect for the sake of a single individual and when they evaluate a multi-person prospect. This implies a constant α > 0.5 for all decisions.
By way of illustration, consider the experimental treatment with which we opened this section and which we represented by a case in which Ann is cured if and only if a red ball is draw from a wholly uncertain urn. An uncertainty-averse decision-maker who 20 Leonard Hurwicz, "Optimality Criteria for Decision Making under Ignorance," Cowles Commission Discussion employs the α-maxmin criterion will consider both the most pessimistic assessment of the information available-according to which there are no red balls in this urn-and the most optimistic assessment-according to which it contains only red balls. Moreover, they will give at least somewhat greater weight to the former than to the latter. Because of this cautious form of evaluation, they will regard the uncertain treatment as less good for Ann, in prospect, than giving her a risky treatment which would carry a 0.5 probability of a full cure and a 0.5 probability of being wholly ineffective. For example, a moderately uncertainty-averse decision-maker for whom α = 0.6 will regard Ann's wholly uncertain prospect as equivalent to a treatment with an expected value of 62 units of well-being, or 3 units of expected well-being less than this risky treatment. (Despite the fact that the criterion permits us to assign such equivalents to uncertain prospects, the value of an uncertain prospect when applying this criterion is not an expected value, because the decision weights applied to different possible probability distributions are not probabilities. When we are discussing uncertain and/or risky prospects, we therefore use the more general term "prospective value.") While in this simple case of Ann's experimental treatment, this criterion gives weight to both the worst and best possible outcome, this is only because, in this example, the most pessimistic assignment of probabilities is a certainty of failure and the most optimistic assignment of probabilities is a certainty of a cure. Whenever the decision-maker can rule out such extreme probability distributions, α-maxmin gives weight not to the worst and best outcomes, but to the lowest and highest expected values that the decision-maker assigns to the prospect. By way of illustration, suppose that the decision-maker gained more information about this experimental treatment, so that the uncertainty involved was reduced as follows: they now rationally conclude the treatment has between a 0.25 and 0.75 chance of curing Ann. The α-maxmin criterion evaluates this revised prospect as follows: α times the most pessimistic assessment of its expected value (viz., that there is a 0.25 chance of curing Ann), plus (1 -α) times the most optimistic assessment of its expected value (that there is a 0.75 chance of curing Ann). Or, filling in the numbers: A moderately uncertainty-averse decision-maker for whom α = 0.6 will therefore regard this partly uncertain treatment as equivalent to a treatment with an expected value of 63.5 units of well-being, or precisely in between the value of the aforementioned wholly uncertain treatment and the value of the aforementioned merely risky treatment, which has a 0.5 chance of effecting a cure. This illustrates that, on this criterion, reducing the range of uncertainty also, naturally, reduces the depressing effect it has on the value of a prospect.

IV. WHEN REDUCING UNCERTAINTY AND LIMITING INEQUALITY ARE CONGRUENT
We will now review ways in which adding uncertainty aversion to our egalitarian view generates novel implications. We first focus on cases in which the aim of reducing uncertainty does not conflict with the aim of reducing inequality. (We deal with conflicts between these aims in the next Section.) Suppose that a decision-maker must choose between the aforementioned Equal Risk, Unequal Final Well-Being and the following: Equal Uncertainty, Unequal Final Well-being: This treatment will either cure Ann and leave Bea wholly blind, or, instead, cure Bea and leave Ann wholly blind, with no information available about the probability of either outcome.
These two alternatives are displayed in the top two rows of Table 2. We use redu (blacku) to signify the event of a red (black) ball being drawn from an uncertain urn.
Our pluralistic view requires that we take account of both the distribution of being is certain: one person will be fully cured, another will go wholly blind. One can therefore say while one of these alternatives contains individual-level uncertainty, neither contains any population-level uncertainty. All things considered, Equal Risk, Unequal Wellbeing is therefore more choiceworthy, but only because it avoids the depressing effect of uncertainty on the value of individuals' prospects. Now imagine a choice between the aforementioned Equality under Risk and the following: Equality under Uncertainty: This treatment will either cure both children, or leave them both to go wholly blind, with no information available about the probability of either outcome.
These alternatives are depicted in the bottom rows of Table 2. In this case, both a concern for individuals' prospects and a concern for the prospective value of the possible anonymized distributions of final well-being point in the same direction. Equality under Uncertainty offers each individual a less valuable prospect. Moreover, it generates population-level uncertainty, because the decision-maker is uncertain about the anonymized distribution of final well-being. This lowers the value of Equality under Uncertainty, because the worst possible probability distribution (i.e., that the probability that both individuals are cured is 0) receives greater weight than the best possible probability distribution (i.e., that the probability that both are cured is 1). An uncertaintyaverse view will therefore have two reasons for judging that it is better to opt for Equality under Risk.
So far, we have analysed cases in which, while keeping inequality constant, a decision-maker can ensure less uncertainty. Now we will consider a case in which, keeping total uncertainty constant, a decision-maker can equalize its burden.
Suppose that a decision-maker must choose between the following: Unequal Uncertainty: Ann is given a novel treatment which will either cure her or instead leave her wholly blind, with no information about the probability of either outcome. Bea is given a distinct treatment which will either, with probability 0.5, cure her, or instead, with probability 0.5, leave her wholly blind.
Equal Moderate Uncertainty: Ann and Bea are each given different distinct, moderately uncertain treatments, each of which will either offer a full cure or instead leave its recipient wholly blind. For each of their treatments, the probability of a cure ranges from 0.25 to 0.75.
These alternatives are represented in Table 3.
The choice between Unequal Uncertainty and Equal Moderate Uncertainty can be thought of as follows. Ann and Bea each face a draw from a separate urn. Each receives a cure if a red ball is drawn from their urn; if a black ball is drawn, their treatment is ineffective. The decision-maker must fill each urn with 100 balls. They have four bags of 50 balls each: two risky bags containing an equal mix of red and black balls, and two wholly uncertain bags about which the decision-maker has no information except that they can be any proportion of red and black. If they empty both uncertain bags into Ann's urn and both risky bags into Bea's urn, then they generate Unequal Uncertainty. By contrast, if they fill each urn with one uncertain and one risky bag, then they generate Equal Moderate Uncertainty. The former places all the burden of uncertainty on Ann's prospects. By contrast, the latter equalizes the burden of uncertainty. Moreover, it is natural to suppose that the total burden created by the uncertain balls is not increased when they are divided equally. 22 From the perspective of the distribution of the value of individuals' prospects, therefore, Equal Moderate Uncertainty is clearly superior. 22 Indeed, according to the α-maxmin criterion, under Unequal Uncertainty, the value of Ann's prospects is 50α + 80(1 -α) and the value of Bea's prospects is 65. Under Equal Moderate Uncertainty, the value of Ann's prospects is α(0.25 × 80 + 0.75 × 50) + (1 -α)(0.75 × 80 + 0.25 × 50); the same is true of Bea's prospects. In both cases, the total value is therefore 145 -30α. An egalitarian view which incorporates α-maxmin will therefore hold Equal Moderate Uncertainty is superior because it distributes this total more equally. We must also consider the prospective value of the possible anonymized distributions of final well-being associated with each of these alternatives. Using the αmaxmin criterion, it is sufficient to consider only the worst and best among the possible probability distributions, which are listed in Table 3 are in their favour. This makes it better from the perspective of the prospective value of the anonymized distribution of final well-being. 25 We can conclude that our uncertainty-averse egalitarian view yields the plausible verdict that one should distribute the burden of uncertainty equally.
Our view does not merely posit a novel object of egalitarian concern (the disvalue of uncertainty); it also lends additional force to the egalitarian aim of directing aid towards those who end up less well off than others. By way of illustration, suppose that our decisionmaker must choose between Equal Uncertainty, Unequal Final Well-being and Equality under Certainty. For convenience, both are represented in Table 4. Recall that c is the cost of achieving both equality and certainty, with 0 ≤ c < 15. 25 More precisely, and writing v{80, 50} for the value of a distribution in which one person gets 80 and another 50, etc., the prospective value that the α-maxmin criterion assigns to the distribution of final well-being under Equal Moderate Uncertainty minus the prospective value that it assigns to the distribution of final well-being under Unequal Uncertainty is: This expression will be positive just in case "levelling up" from (80, 50) to (80, 80) generates more moral value than "levelling down" from (80, 50) to (50, 50) destroys. And this will be true on our pluralistic egalitarian view. This is because the added 30 units of well-being in the levelling up scenario are valuable both because they are good for a person and because they reduce inequality. By contrast, the disvalue of the lost 30 units of wellbeing in the levelling down scenario is tempered by the fact that their loss reduces inequality.  prospects, an uncertainty-averse decision-maker should therefore be willing to incur a cost (c > 0) to eliminate this uncertainty.
Turning to the prospective value of the possible distributions of well-being, a drawback of Equal Uncertainty, Unequal Final Well-being is, naturally, the certainty of outcome inequality. An inequality-averse decision-maker will therefore be willing to pay a price (c > 0) to eliminate this inequality.
In sum, both uncertainty aversion and inequality aversion will prompt us to incur a cost to remove inequality. Moreover, jointly, they will justify paying a higher price to achieve equality than either alone would. To see why, suppose for the moment that our decision-maker remained inequality averse, but became indifferent to uncertainty (that is, their α = 0.5). They would then evaluate Equal Uncertainty, Unequal Final Well-being as equivalent to Equal Risk, Unequal Final Well-being (the latter is described in Table 1).
Suppose that to eliminate the inequality of final well-being in these alternatives, it is right to incur up to, but no more than, a cost to each person of c* units of expected well-being. We can then say that, for an uncertainty-neutral, but inequality-averse decision-maker, both for a cost larger than c*. It follows that an uncertainty-averse egalitarian view justifies incurring a larger cost in order to achieve both equality and certainty than an uncertaintyneutral egalitarian view would countenance.
Let us summarize the distinctive implications of our view uncovered in this section.
First, and straightforwardly, it will favour situations in which a better basis is available for assigning probabilities to outcomes. This is illustrated by the stylized choices in Table 2. In real-world cases, the view will therefore display a tendency to favour policies with an extensive evidence base over ones with a minimal evidence base, keeping other things equal. Under these circumstances, it will also favour familiar over unfamiliar treatments for patients, and make the provision of the latter harder to justify. 26 Second, the proposed view posits an additional object of egalitarian concern, namely, the burden of uncertainty. In general, it implies that there is unfairness in situations in which some face a greater burden of uncertainty than others, either because there is less information about the likelihood of the possible threats they face, or because there is much more at stake for them. The case outlined in Table 3 provides a stylized example. A realistic case in which such inequality is of concern is climate policy. For it is likely that people in poorer nations who inhabit marginal lands and who are dependent on the weather for their livelihood face larger burdens of uncertainty than urbanites in wealthy countries.
Third, in uncertain situations in which we know that one person's good fortune will be the counterpart of another's misfortune, uncertainty aversion and inequality aversion point us in the same direction. For, in such situations, steering benefits to whoever turns out to be less fortunate reduces the stakes for each person and thereby lessens the burden of uncertainty; of course, it also reduces inequality in final well-being. Table 4 provides an example where this process of improving the lot of the worst off can be pursued to the point of equality, but the impulse towards equality will be present even when perfect equality cannot be reached. A policy issue to which this may be relevant is the taxation of returns from complex financial instruments held by pension funds on behalf of ordinary workers. These instruments are uncertain prospects that will yield both winners and losers.
Measures that dampen the variability of returns (e.g., taxing gains and allowing a tax deduction for losses) will be valuable both because they reduce uncertainty and because they decrease inequality in final well-being. 27

V. WHEN REDUCING UNCERTAINTY AND LIMITING INEQUALITY ARE AT ODDS
We now turn to cases in which uncertainty aversion and inequality aversion pull in opposite directions. By way of illustration, imagine a choice between the aforementioned alternatives Equal Uncertainty, Unequal Final Well-being and Equality under Uncertainty depicted in Table 5. 27 While some investors in such instruments may be adequately informed and wholly responsible for their choice of investments (so that a responsibility-sensitive egalitarian may have attenuated reason for being concerned with inequalities that result), many of the returns also come to participants in funds who should not be held fully responsible for the relevant investment decisions (see, for example, Michael Lewis, The Big Short (New York: Norton, 2010), ch. 6, recounting how, among others, Japanese farmers' unions and European pension funds were exposed to swings in the value of collateralized debt obligations (CDOs). Inequalities in final well-being between such investors should concern egalitarians. Electronic copy available at: https://ssrn.com/abstract=3881962 In this case, in terms of the prospects they grant individuals, both alternatives are equivalent. However, in terms of the prospective value of the possible distributions of final well-being they generate, matters are less straightforward. If the decision-maker chooses Equal Uncertainty, Unequal Final Well-being, they can be certain that one person will be cured and one will go wholly blind. But this absence of population-level uncertainty about the anonymized distribution of final well-being comes at the cost of guaranteed inequality.
If, instead, they choose Equality under Uncertainty, there will be guaranteed equality, since Ann and Bea will either both go wholly blind or both be cured. But the fact that they will either sink or swim together generates substantial uncertainty regarding the value of the distribution of final well-being.
The view we are developing is silent on the choice between these alternatives. It assumes only that some degree of inequality aversion is required and that some degree of uncertainty aversion is both permissible and adopted by the decision-maker. Consistently with these commitments, decision-makers may decide this case differently. A decisionmaker who is strongly inequality averse but only slightly uncertainty averse will choose Equality under Uncertainty; a decision-maker who is only slightly inequality averse but very averse to uncertainty will favour Equal Uncertainty, Unequal Final Well-being. Here, we will not argue that one of these ways of trading off the two concerns is uniquely right; the key conclusion is simply that our view incorporates some resistance to the egalitarian impulse to bind people's fates together.
One further important lesson about how uncertainty aversion changes egalitarianism is this. Under risk, our egalitarian view embodies a tendency to allocate benefits away from the fortunate and towards the unfortunate when and only when the former are (or would be) better off than others and the latter are (or would be) worse off than others. This is illustrated by the fact that in choosing between Equality under Risk and Equality under Certainty in Table 1, our view is unwilling to pay any cost in order to redistribute from the fortunate potential futures of Ann and Bea to their less fortunate potential futures. By contrast, under uncertainty, our view embodies a tendency to direct benefits from the fortunate towards the unfortunate even when these are merely two potential futures of the same person and there is no inequality. To see this, compare Equality under Uncertainty with Equality under Certainty, which are both depicted in Table 6. Due to uncertainty aversion, for some sufficiently small, positive c, Equality under Certainty is superior to Equality under Uncertainty; this is indicative of the cost our view is willing to pay to reduce the burden of uncertainty by directing benefits away from Ann and Bea's better potential futures towards their worse potential futures. To be sure, on our view, the strength of this tendency to allocate benefits away from the fortunate and towards the unfortunate will be strongest when it both reduces inequality between people and reduces the disvalue of uncertainty for each person. Nonetheless, the predilection in uncertain situations to direct resources towards all victims of misfortune is striking. It follows from this view that we have special reason to make provision for possible adversity to which we are not in a position to assign a precise probability, even if this adversity would, if it occurred, equally affect everyone in the collectivity with which we are concerned. A realistic case may be the purchase of a financial hedge for uncertain and volatile revenues from a country's collectively-owned natural resources, to protect against a downturn that would depress all citizens' livelihoods. 28

VI. UNCERTAINTY, PARETO, AND ANTI-EGALITARIANISM
We shall now demonstrate that the permissibility of uncertainty aversion should lead us to reassess a common argument against the kind of egalitarianism we have been developing.
Consider the following principle: Pareto for Prospects: If, for every person, a first alternative provides a more valuable prospect than a second, then the first alternative should be chosen over the second.
The motivation for this principle is that a social decision-maker should choose an alternative that could be chosen for the sake of each person, if such an alternative exists.
The principle holds that an alternative is more choiceworthy for a person's sake just in case it has greater prospective value for that person.
We draw two conclusions from these findings. Electronic copy available at: https://ssrn.com/abstract=3881962 sake of improving total final well-being, why would one not also be permitted to do so for the sake of reducing inequality?
Our second conclusion is substantive. By focusing only on each individual's prospects, considered separately, Pareto for Prospects is insensitive to how combinations of these prospects give rise to possible patterns of final well-being. But these patterns matter, both because they determine the fairness of the eventual outcome and because they determine how much uncertainty about the value of the distribution of final well-being a social decision-maker faces. After all, a decision-maker is in a very different situation when some will definitely end up better off than others than when no such inequality can arise.
They are also in a different situation when they can accurately predict the impact of their policies at a population level than when they must contend with a wide range of possible population-level outcomes and are not in a position to assign precise probabilities to these outcomes. That Pareto for Prospects does not permit decision-makers to take account of these differences is, we submit, a reason to reject it.

VII. CONCLUSION
Uncertain situations are ubiquitous. A common and, we have argued, permissible attitude in such situations is for a decision-maker to keep in mind the full range of distributions of probabilities over outcomes consistent with their limited evidence and prior beliefs and to respond cautiously to this range by giving some extra decision weight to the less favourable probability distributions within it. We have here explored what would follow if such an uncertainty-averse attitude were incorporated into a plausible egalitarian theory.
We highlight four implications of this view. First, and most straightforwardly, it favours risky alternatives over alternatives for which the decision-maker is not in a position to assign precise probabilities to possible outcomes. It therefore regards as more difficult to justify courses of action that avoidably involve uncertain possibilities of success and failure.
For example, it will make it more difficult to justify enrolling people in an experimental treatment when an alternative is available with well-established risks. 31 Second, our view identifies the disvalue of uncertainty as a new object of egalitarian concern. It can thereby justify special efforts to improve the prospects of those who face a larger burden of uncertainty, by improving information-gathering and making provision to improve their lot should the current, imprecisely estimated odds prove to be against them.
A pertinent case is the possible effects of climate change on people whose lives and livelihoods would be threatened by large temperature rises or by changes in precipitation.
Third, in uncertain situations, uncertainty aversion serves as a counterweight to the egalitarian impulse to bind people's fates together. For, in such situations, ensuring that either everyone or no one is benefitted creates an uncertain possibility of collective misfortune, which the view will regard as especially problematic.
Fourth, in uncertain situations, the proposed view justifies directing benefits away from the fortunate and towards the unfortunate both when they are different people and when they are merely different possible futures of the same person. It therefore adds force to the egalitarian impulse to resolve interpersonal trade-offs to the benefit of the less welloff. It also introduces a novel reason, namely, the reduction of the depressing effect of uncertainty on the value of individuals' prospects, to resolve intrapersonal trade-offs in a manner that makes special provision for the least fortunate potential future of the person concerned. This implication is relevant to justifications for a social safety net. As egalitarians have commonly understood this institution, it serves both to reduce inequalities and improve people's prospects by mitigating risks pertaining to income and health. 32 But where such a safety net must operate not merely under conditions of risk but also under uncertainty, our view provides the following additional reasons for its maintenance and expansion. By improving the fate of the worst off, a social safety net reduces both the depressing effect that individual-level uncertainty has on the value of individuals' prospects and population-level uncertainty about the value of the distribution of final well-being.
In closing, let us revisit our initial case involving the UK's policy in the face of Swine Flu. Analyses of the actual decision-making process reveal that key decision-makers in the civil service and UK Government evaluated all courses of action exclusively in terms of their impact in the event of a devastating epidemic that would, if the more extensive of the contemplated policies were not implemented, claim between 65,000 and 750,000 lives. 33 Consequently, the government chose to invest the maximal amount in response to the threat. As it turned out, the virus was not particularly dangerous and the death toll was 457.
Of course, this outcome alone does not show that the decision-making process at the time 32 See, e.g., Nicholas Barr, The Economics of the Welfare State, 5 th edition. (Oxford: Oxford University Press, 2012); Michael Otsuka, "How to Guard against the Risk of Living too Long: The Case for Collective Pensions," in was flawed. However, some commentators have derided the UK Government's exclusive focus on the worst range of outcomes as excessively cautious and wasteful. They have argued that, instead, it should have made an "informed guess" assignment of precise probabilities to each possible outcome and then conducted a standard expected value calculation for each policy alternative. 34 This orthodox approach, however, has its own problems, since it demands that public decision-makers come up with a precise probability for every relevant outcome when even experts are unable to do so. Using such arbitrary probabilities would seem hard to justify publicly. And in the absence of such a unique probability distribution, the proposed expected value analysis cannot be carried out.
The approach we have outlined here, is, we submit, an attractive alternative to both the Government's exclusive focus on the worst case scenario and to orthodox expected value analysis. There are two points at which our proposed approach differs from "assuming the worst." First, it focuses special attention on the least favourable probability distribution over outcomes consistent with available data and reasonable prior beliefs. That is, it focuses on the worst expected value rather than on the worst outcome. It is only in a case of complete ignorance that the worst expected value and the worst outcome are identical.
And, as commentators have pointed out, the historical record of flu pandemics, alongside incoming information about the nature of the H1N1 virus, ensured the situation was not one of complete ignorance. This, they claim, made it unreasonable to reason as if the worst case was certain to occur. 35 If these observers are correct, then among all the probability distributions over outcomes in a sensible range, even the most negative would assign some probability to scenarios other than the worst.
Second, uncertainty aversion does not require giving decision weight only to the most negatively assessed expected value; it merely involves cautiously giving such gloomy estimates of expected value somewhat greater decision weight than more favourable assessments. While this leaves a degree of freedom about these decision weights, a choice of weights for high-stakes public decision-making under uncertainty strikes us as an unavoidable value judgment that is best openly debated.
Our critique of the UK Government's exclusive focus on the worst case does not, of course, show that substantial investment in pandemic preparedness was unwarranted.
Indeed, our uncertainty-averse, egalitarian view identifies several reasons to value such investment at the moment of decision. First, by improving individuals' outcomes in the event that the pandemic turned out to be severe, it mitigates the depressing effect of uncertainty on the value of individuals' prospects. Second, by limiting the number of deaths in this unfortunate eventuality, it alleviates population-level uncertainty. Finally, given that the impact on people's lives would be highly disparate (with some, including many young people, dying and many more surviving), an investment in saving lives if the pandemic had proven to be severe lowers the prospective degree of inequality in final well-being. In sum, besides demonstrating the value of an approach which treats decisions under uncertainty differently from merely risky decisions, this case highlights a central conclusion of our analysis, which is that uncertainty aversion both reinforces egalitarian reasons and provides new reasons to improve the lot of the unfortunate.