Gerard Vong: “Weighing up Weighted Lotteries”. Précis by Nate Sharadin

Welcome to what we expect will be a very interesting and productive discussion of Gerard Vong‘s “Weighing up Weighted Lotteries: Scarcity, Overlap Cases, and Fair Inequalities of Chance.” The paper is published in the most recent issue of Ethics, and is available here. Nate Sharadin kindly agreed to contribute a critical précis, and it appears immediately below. Please join in the discussion!

Gerard Vong’s “Weighing Up Weighted Lotteries: Scarcity, Overlap Cases, and Fair Inequalities of Chance” has two main aims. The first is to convince you that the fair procedure for distributing benefits in equal conflict cases, i.e., cases where all claimants have equally strong claims to the benefits, there aren’t any other morally relevant differences between them, and where the benefit cannot (say, because of indivisibility) be distributed equally to all, must use a weighted lottery. The second is to convince you that, among weighted lotteries, the fairest weighted lottery procedure is his, what he calls the exclusive lottery procedure. I learned a lot by reading and thinking about Vong’s paper, and I’m pleased to have the chance to introduce it here. I’ve divided this too long precis into two sections. The first section lays out Vong’s arguments and some of his technical machinery. The second raises a worry with his approach. If you’ve already read Vong’s paper, you can skip right to the second section. If you haven’t, or would like a refresher, read both.

1. Unweighted, Weighted, and Composition-Sensitive Lotteries

The argument against unweighted lottery procedures is straightforward. Take an equal conflict case. An unweighted lottery theory tells us that fairness requires we give each and every claimant an equal chance of benefiting. So far, so good. But now imagine that there is some overlap between claimants in different outcome groups, i.e., groups of claimants that can be benefited. Then, an unweighted lottery procedure entails that there is no fair distribution of chance of benefit. Here is why: Suppose we can benefit A&B, A&C, D&E, or D&F, and that we cannot benefit no one. Then, as Vong points out, while we can give each outcome group equal chances, we cannot give each claimant equal chances. Hence the unweighted lottery procedure entails that no chance distribution is fair. But that is implausible: pretty clearly, there is a fair way to distribute the chance of benefiting in such cases, viz. to divide it equally across the outcome groups.

(The natural revision of such unweighted lottery theories — an equal-as-possible unweighted lottery — is, Vong argues, also unacceptable: it implies that, in situations where we can give each and every benefit an equal zero chance of benefiting, doing so is more fair than any alternative non-equal distribution of the chances. But, again, that is implausible.)

Vong diagnoses the problem with unweighted lottery theories as driven by their failure to appreciate the importance of two features of overlap cases: our considered judgments about absolute fairness and outcome group composition. Take the first of these first. Comparative fairness is a matter of how one claimant’s claims are treated as compared to other relevantly similar claimant’s claims. If all equally strong claims are treated equally (e.g., by contributing equal chances to the unweighted lottery), then comparative fairness is fully realized. If all we’re concerned about when we’re concerned about fairness is comparative fairness, then, in equal zero benefit cases what it’s most fair to do is give every claimant an equal zero chance of benefiting. Again, that’s implausible on its face. Hence, absolute fairness: absolute fairness is what is promoted when a claimant’s chances of receiving a benefit to which they have a claim are increased without regard to others’ (potentially equally strong) claims. If all we’re concerned about when we’re concerned about fairness is absolute fairness, then, in equal zero benefit cases what it’s most fair to do is give the outcome group with the most claimants a 100% of benefiting. But, that too is implausible on its face. What we need to do then, Vong argues, is pay attention to both notions of fairness: we should be pluralists about fairness and care instead about ‘all-things-considered’ fairness, comprising both notions. I’m going to return to the idea of all-things-considered fairness and the way in which it’s composed of both comparative and absolute fairness below, in raising a worry for Vong’s account.

What about outcome group composition? Vong’s idea is that the fair lottery procedure must be composition-sensitive, i.e., sensitive to the the composition of the outcome groups (again, the groups that can be benefited that in turn contain the claimants). But there are many ways of being sensitive to the composition of outcome groups. How can we constrain out options? Vong’s suggestion: by reflecting on our considered judgments about the relative importance of comparative and absolute fairness in contributing to all-things-considered fairness. Importantly, as we just saw, comparative and absolute fairness can push in different directions when it comes to our selection of an outcome group. In equal conflict cases comparative fairness requires that each claimant contribute equally to the selection of the outcome group. But absolute fairness, since it requires in general that all claimants have a 100% of receiving a benefit they are due, will in equal conflict cases simply pick the largest outcome group and assign that outcome group a maximal chance of benefiting (since that is how we can maximize absolute fairness).

Vong’s judgment (and here he agrees with his opponents) is that, in equal conflict cases, is is the most comparatively fair procedure that is most all-things-considered fair; in other words, whatever composition-sensitive procedure we select, it must be such that each equally worthy claimant contributes equally positively to the outcome group selection. Again, I’ll return to this idea, below, in raising a worry with Vong’s approach.

This idea, that comparative fairness is what matters most, narrows the field: no simple ‘assign-the-largest-outcome-group-100%’ composition-sensitive procedure will satisfy the requirement to be comparatively fair. But it still leaves a wide range of possible procedures, among which these three: the equal composition-sensitive lottery, the exclusivecomposition-sensitive lottery, and the iterated individualist composition-sensitive lottery.

Equal composition-sensitive lotteries work like this: the chance contribution of a claimant, A, to each group of which A is a member is a proportion both of the number of claimants, c, and the outcome groups, Oa, of which A is a member, i.e., each claimant contributes a chance equal to: (1/c) / Oa to every group of which A is a member. Intuitively: if there are 5 claimants divided into outcome groups like so: A&B, A&E, C&D, then B, C, D, and E each contribute their (1/c/Ob-e=1/5/1=) 20% to their respective groups, while A contributes (1/c/Oa=1/5/2=) 10% to both A&B and A&E. The advantages, in terms of comparative fairness, are clear: each claimant makes an equal contribution (hence, equalcomposition-sensitive) to the chances. But, like its unweighted brethren, it falls to our judgments about overlap cases. Consider Vong’s example. Suppose we have 1,000 people numbered sequentially, and three outcome groups: G1, comprising claimants 1-500, G2 comprising claimants 501-1,000, and G3, comprising claimants 2-999. (The existence of G3 is what makes this an overlap case.)

The equal composition-sensitive lottery delivers the ‘startingly implausible result’ that it is most fair to give a greater than 50% chance of saving either G1 or G2. As Vong points out, this is an ‘affront to absolute fairness’: it simply cannot be that all-things-considered fairness requires giving 998 claimants a worse chance of benefiting than 500.

What about exclusive composition-sensitive lotteries? Claimant A is (by stipulation) exclusive relative to claimant B just in case A is in at least one but not all outcome groups of which B is a member. Thus, if a claimant is not exclusive relative to another claimant, then the first claimant is in either all or none of the outcome groups of the second claimant. Then, exclusive composition-sensitive lotteries work like this: the chance contribution of a claimant, A, to each group, W, of which A is a member, is a proportion both of the the number of claimants in a group that are exclusive relative to A (uA,W) and the sum of the claimants that are exclusive relative to A in every group to which A belongs (eA), i.e., each claimant contributes a chance equal to: uA,W /eA.

Intuitively, the idea is this: in deciding the positive contribution a claimant, A, should make to the chance some outcome group of which she is a member is selected, we should ignore other claimants who will either definitely be benefited if A is benefited, or who will definitely not be benefited if A is benefited. As Vong explains, this disjunctive condition captures the intuitive ideas that claimants who aren’t in any of the relevant claimant’s outcome groups are irrelevant (why should they play a role in determining A’s contribution?) and claimants who are in all of the relevant claimant’s outcome groups are also irrelevant (why should it matter, since they’ll receive the same chance in any case!). In other words, only differences in composition make differences.

You can see the appeal of such an approach. It handles overlap cases such as the one causing trouble for equal composition-sensitive lotteries with ease: doing the math, the result is that G1 and G2 each have a ~.2% chance of benefiting, whereas G3 has a ~99.6% of benefiting. That is not an affront to absolute fairness. Moreover, it is comparatively fair: all equally worthy claims had an equal positive impact  on the outcome group selection.

This brings us to iterated individualist lotteries. These work exactly as they say on the tin: the procedure for selecting an outcome group is to first run an equal-chance individual lottery, i.e., a lottery where each claimant with an equal claim is given an equal chance at winning. Then, we iterate that equal chance lottery across claimants who are members of outcome groups of which the winner of the first lottery is also a member until we arrive at a unique outcome group. For instance, if the outcome groups are (again): A&B, A&E, C&D, then we first run an equal chance lottery over with 20% chances over A-E. Assume A wins. Then, we run an equal chance lottery with 50% chances over B and E. Assume B wins. Then, the outcome group selected is A&B.

Iterated individualist lotteries have the exactly the same plausible results as exclusive lottery procedures in overlap cases. I’ll leave the math to Vong’s paper, but the result is that in our G1-G3 scenario from before, G1 and G2 each receive (on the iterated lottery view) a ~.2% chance and G3 receives a ~99.6% chance.

So: How shall we decide between iterated individualist lotteries and Vong’s preferred account, the exclusive composition-sensitive lottery? Here’s the separating case, according to Vong: overlap cases involving multiple subset-groups. An overlap case has this feature when one or more outcome groups are proper subsets of more than one maximal group (an outcome group that is not a proper subset of another outcome group). For instance, consider a case where you can benefit A, A&B, A&B&C, or D. In such cases, the intuitive judgment is that it’s absolutely fairer to benefit A&B — a subset group — than it is to benefit (just) A — also a subset group — and similarly that it’s absolutely fairer to benefit A&B&C — the maximal group — than (just) A&B or (just) A. Because subset groups (such as A or A&B) can in principle be selected by either the iterated or the exclusive lottery procedures, and because such selections are intuitively unfair, we need a way to proceed. Vong’s suggestion is straightforward: in both cases, we (re)iteratelotteries.

For the individualist lottery, the procedure is simple: just run lotteries until all winners are members of only one maximal group. From our example: If A is selected in the first run, run a lottery on B and C. If C is selected, you’re done. If instead B is selected, run a lottery on C (it’ll win!). Hence, if A (a subset group) is selected in the first run, whatever happens in this case you’ll end up with a maximal group of which A is a member, viz. A&B&C.

For the exclusive lottery, things are somewhat, though not much, more complex. The basic idea is exactly the same: we simply iterate — this time exclusive — lotteries until all the winners are members of just one maximal group. In our example from before, we arrive at the same result.

Here, then, is how to use such cases to decide between the (re)iterated individualist lottery procedure and Vong’s preferred (iterated) exclusive lottery procedure. Take a case with multiple subset groups with the following structure: as before, we have 1,000 claimants. The outcome groups that can benefit are, however, more diverse. They are: (as before) 1-500, 501-1000, and (newly this time) each individual and each possible pair contained in 1-1000 (e.g., 1&2, 1&3, … 1&1000, 2&3, 2&4… 2&1000). So: many more possible outcome groups, and many subset groups.

The results in this case (again, I’ll leave the math to the paper) is that the iterated individualist lottery procedure gives the following chances to the outcome groups:

1-500: ~24.975%

501-1000: ~24.975%

Neither group of 500 (a maximal group that is a pair, e.g., 2&999): ~50.050%

Whereas the iterated exclusive lottery procedure gives the following chances to the same outcome groups:

1-500: ~33.3%

501-1000: ~33.3%

Neither group of 500 (a maximal group that is a pair, e.g., 2&999): ~33.3%

Vong’s judgment is that the results delivered by the iterated individualist lottery are “deeply implausible”. This is because, he says, when everyone is a member of a group of 500 it would be “clearly unfair” to make it more likely that two people benefit than that some group of 500 does so. And that is exactly what the iterated individualist procedure does. Not so, happily, with his own view: on that view, the chance of one of the groups of 500 benefiting is significantly greater (double) the chance of just two people benefiting.

Hence, while both theories are indistinguishable in terms of their comparative fairness — they both treat equal claimants’ equally worthy claims equally — they can be distinguished on grounds of absolute fairness. The exclusivelottery procedure is more absolutely fair. Hence, since both are equally comparatively fair, the exclusive lottery procedure is more all-things-considered-fair.

1. All-Things-Considered Fairness and Outcome Groups

The remainder of Vong’s paper addresses what’s come to be called the ‘awkward conclusion’ in non-guaranteed benefit cases, and his remarks there are worth considering. But in what follows I’ll just focus on his proposal for driving a wedge between the iterated individualist lottery and his preferred exclusive lottery procedure by using cases involving multiple subset-groups.

I’ll put my point two different ways. The first is rhetorical, but it’s meant to warm you up to the more constructive suggestion that follows. Rhetoric, then, first: If we think there’s something perverse, from the point of view of fairness, about it being more likely that some pair of individuals benefits than that one of two groups of 500 benefits (of which every member of each pair is also a member of at least one of the groups), then it seems to me this is because there’s something perverse, from the point of view of fairness, about some pair of individuals benefiting having any chance at all of benefiting in the face of the possibility of benefiting one of two groups of 500 of which each pair is a member of at least one. In other words: we certainly ought not to accept a procedure that gives some (any) pair that is a maximal group higher chances of being selected than the (much) larger groups of 500, but that in turn is because we ought not to accept a procedure that gives pairs that are maximal groups any chances at all when the groups of 500 (again, of which each member of a pair is a member of at least one) are possible beneficiaries.

Vong, I imagine, would reply that we’re constrained in our choice of procedure first by comparative fairness. And any attempt to zero out the chances of the pairs that are maximal groups will fail to respect the requirement to allow each equally strong claimant to have an equal positive effect on the outcome group selection (the requirement of comparative fairness defended earlier in the paper). But I am not so sure this is right, for two reasons.

First, Vong’s quite sensible suggestion, early on in his paper, is that there’s a “reasonable compromise between comparative and absolute fairness” (332). What’s odd about this is that he then endorses the view that, in equal conflict cases (such as the one we’re now grappling with), “comparative fairness is the most important type of fairness” (ibid.). But saying there’s a reasonable compromise between comparative and absolute fairness, and then judging the fairness of procedures first by what’s comparatively fair, and only then ranking them in terms of absolute fairness doesn’t, to me, seem like a reasonable compromise between comparative and absolute fairness. Instead, it seems to me like comparative fairness shoving absolute fairness out of line, gobbling up its meal, and then telling absolute fairness to eat its fill (of the scraps). So, it’s not clear to me that Vong’s judgments about cases — which I share — show that we ought to be constrained in our choice of fair procedures, in the way Vong thinks we are constrained, by comparative fairness.

Second, I think we can reach a reasonable compromise between comparative and absolute fairness that delivers the judgment that, above, I invited you to join me in making, viz. that the groups of 500 should each receive a 50% chance of benefiting, and that the pairs that are maximal groups should have zero chance of benefiting. Here is how: we simply disallow (i.e., ignore) outcome groups when the members of those groups can be given the same, or higher chance of benefiting by disallowing them. I lack the space to defend this procedural requirement in detail here, and to explain fully why I don’t think it’s ad hoc, but I’ll say a couple things.

First, let’s see how the suggestion works in the present case. (I’ll use the exclusive lottery for illustration, but it works for the iterated individualist lottery too. That, in part, is why it’s a challenge to Vong’s proposal. It removes the wedge he drives between the two procedures.) The chance of (say) claimant 2 being benefited if we allow the pairs that are maximal groups to stand as outcome groups is equal to the chance of the group 1-500 being selected (33.3%) plus the chance of a pair that’s a maximal group containing 2 (e.g., 2&999) being selected (vanishingly small), which sum to less than 50%. The chance of claimant 2 being benefited if we disallow the pairs that are maximal groups to stand as outcome groups is clearly greater than this, since it’s exactly 50%. Similar remarks go for claimant 3. And for claimant 4. In each case, we can improve a claimant’s odds of receiving the benefit to which they have a claim if we disallow the outcome groups that are the maximal pairs containing them from standing as potential beneficiaries in the lottery. My suggestion is that if we do so, we are promoting absolute fairness (by increasing the odds that those who have a claim on a benefit will receive that benefit — the odds of the groups of 500 go to 50%) and, moreover, that we are not treating anyone’s claim on the benefit unequally, since each claimant is subject to the same procedural vetting via their outcome group membership.

You might think that in following this suggestion we’d somehow be treating the pairs that are maximal groups unfairly, since we’re not allowing them to stand as potential beneficiaries in the lottery. But this, I think, misunderstands the nature of outcome groups. To admit that some group is an outcome group that we can benefit (and so enter it into the lottery) is already to make a choice from the point of view of the fair distribution of benefits. The complex convergent case Vong presents that has led us where we are is a case in point. If we allow the pairs that are maximal groups to be considered potential beneficiaries in our lottery, we thereby sacrifice absolute fairness on the altar of (a particular conception of) comparative fairness. But we are not required, by a reasonable conception of all-things-considered fairness, to do so.

One last way of driving home the point. Consider lifeboat cases containing no overlap. Absolute fairness, given its requirement to maximize the chances that each individual receives the benefit they are due, requires us to give 100% chance to the largest outcome group. Vong argues, and I agree, that this is unacceptable. Spreading the chances among the potential outcome groups, assuming they are not proper subsets of maximal groups, is the obeisance absolute fairness must make to comparative fairness. What I’m urging here is that, in cases where absolute fairness can be promoted without worsening any (and in our present case improving all) claimants’ chances of receiving the benefit they are due by dissolving (i.e., refusing to consider distributing the benefit to) certain outcome groups, this is what fairness requires. I do not think it is a violation of comparative fairness, since each claimant’s equal claim is being treated equally in the vetting, and I do not think outcome groups should be understood per se as valid claimants on benefits (it is instead their members who are claimants). But if you think it is an affront to comparative fairness, then think of it as the corresponding tithe comparative fairness makes to absolute fairness in their mutual compromise.

13 Replies to “Gerard Vong: “Weighing up Weighted Lotteries”. Précis by Nate Sharadin”

1. I want to sincerely thank Nate Sharadin for his excellent critical précis on my paper. Not only did Sharadin provide a clear summary and a thoughtful objection to my paper, he defended a new positive view about fairness that is worthy of careful consideration. In my response I want to evaluate that positive view and its defense. This evaluation is important both because the positive view is important in its own right and because Sharadin uses it to support his objection.

Sharadin argues for a new requirement of fairness. He argues that fairness requires that we allocate groups of potential beneficiaries (i.e. outcome groups) zero chance of benefiting when doing so instead of running my exclusive weighted lottery (or the iterated individualist lottery) results in an alternative distribution such that the individual members of those outcome groups can be given the same or higher chance of benefiting (“we simply disallow (i.e., ignore) outcome groups when the members of those groups can be given the same, or higher chance of benefiting by disallowing them”). Call this the new requirement.

Sharadin demonstrates how this new requirement works by applying it to my case where you can benefit one of the following groups: 1-500, 501-1000, each individual, and each possible pair contained in 1-1000 (e.g., 1&2, 1&3, … 1&1000, 2&3, 2&4… 2&1000 …). In this case, because individuals can be given the same or higher chance of benefiting by excluding the maximal outcome pairs (i.e. those that are not proper subsets of larger outcome groups) from the exclusive lottery procedure (i.e. only distributing positive chances of benefiting to the two outcome groups of 500 individuals), Sharadin contends that it is fair to exclude those maximal outcome pairs.

This new requirement has the following merit that I do not dispute: because it increases the chances that claimants receive the benefits that they have claims on relative to my exclusive lottery, it better promotes absolute fairness than my exclusive lottery. It also offers a unified explanation of why non-maximal outcome groups and some maximal outcome groups should not be allocated any chance of benefiting. Furthermore, Sharadin argues that the new requirement is not a violation of comparative fairness because of the following conjunction: each claimant’s equal claim is not being treated unequally according to this requirement and outcome groups should not be understood per se as valid claimants on benefits.

My response to this new requirement has three parts: (I) I will object to Sharadin’s argument that this new requirement of fairness is not a violation of comparative fairness; (II) I will demonstrate that increases in absolute fairness in cases that are relevantly similar to the cases that the new requirement applies to can be unfair; (III) I will suggest that Sharadin’s chance allocation is more plausible as a requirement of morally permissible action in certain circumstances rather than a requirement of fairness.

(I) Let’s consider Sharadin’s claim that his new requirement of fairness is not a violation of comparative fairness because of the aforementioned conjunction. Assume, for the sake of argument, that outcome groups should not be understood per se as valid claimants on benefits. If we grant this, then Sharadin’s conjunction gives a sufficient condition for a requirement being comparatively fair: a requirement is comparatively fair if that requirement does not treat claimants’ equal claims unequally. However, this sufficient condition is vulnerable to a counterexample. For consider the implausible requirement that in equal conflict cases, all claimants and all outcome groups receive an equal 0% chance of benefiting. Because this requirement does not treat each claimant’s equal claim unequally, by Sharadin’s lights, this requirement is comparatively unfair. Yet, for reasons I give in my paper, this requirement seems comparatively unfair. My paper defends an explanation for why this requirement is comparatively unfair: a necessary condition for a requirement’s being fair is that in equal conflict cases it gives all claimants with equal claims an equal positive impact on the outcome group selection procedure. If this is correct, then Sharadin’s conjunction does not provide a compelling defense of his new fairness requirement.

Let’s now evaluate the second part of Sharadin’s conjunction, namely that outcome groups should not be understood per se as valid claimants on benefits. There are some reasons that support the view that outcome groups per se can be valid claimants on benefits. First, consider how claims to be benefited are often generated between individuals: through agreements, promises and contracts. Now note that groups (e.g. organizations) can gain valid (and legally protected) claims to be benefited in just the same ways: through agreements, promises and contracts. Group agents, like individual agents, can engage in these activities that ground claims to benefit. Relatedly, I am convinced by List & Pettit’s arguments in their book, Group Agency, that group agents exist and are irreducible to individual agents (and their beliefs & preferences).

That said, in the equal conflict cases that are the focus of the paper and the précis, outcome groups do not have claims on benefiting by stipulation. So let me here concede that in equal conflict cases, the outcome groups should not be understood per se as valid claimants on benefits: the outcome groups are equally claimless.

Nevertheless, there may still be reasons from comparative fairness to not treat equally claimless outcome groups unequally. These reasons are not based in the violation or infringement of the groups’ claims. After all, the groups have no claims. Instead, these reasons are based in the fact that comparative fairness requires one to treat like things alike.

The new requirement fails to treat equally claimless outcome groups equally because it treats some equally claimless maximal outcome groups differently from other equally claimless maximal outcome groups. That is, it does not treat like outcome groups alike. In the case where you can benefit one of the following groups: 1-500, 501-1000, each individual, and each possible pair contained in 1-1000 (e.g., 1&2, 1&3, … 1&1000, 2&3, 2&4… 2&1000), Sharadin suggests that some claimless maximal outcome groups (e.g. 2&999) should be treated differently than other claimless maximal outcome groups (e.g. 1-500) because you can increase the chances that individuals (e.g. 2) have to benefit by giving the maximal outcome group pairs no chance of benefiting and instead distributing all of the chances of benefiting between the maximal outcome groups 1-500 and 501-1000 (50% each).

In addition to the theoretical explanation given above, there is some evidence from experimental economics that there are fairness-based reasons to not treat equally claimless groups unequally. Karni, Salmon and Sopher ran an experiment where paid participants were knowingly arbitrarily divided into three equally sized, non-overlapping groups they labelled A, B and C. This arbitrary division ensured that each group is equally claimless. In excess of the study participation payment that individual participants had a claim to, there was an additional (unpromised) \$15 lottery prize for one of the three groups. Those in group A chose the allocation of chances to be used in deciding who in groups A, B and C won this \$15 prize. Their study showed that a substantial proportion of subjects in group A were willing to sacrifice their own chance of winning in order to effect a more equal, more fair chance allocation procedure between the three groups. Those in group A also tended to treat those in group B and C equally. Furthermore, while those in group C did not affect the chances that groups had to win the monetary lottery, they were asked to select the chances of benefiting that they believed were fair. Karni et al. wrote that “the distribution of A and C choices appear to be remarkably similar”. Thus there is experimental evidence that people are willing to sacrifice their own chances of benefiting in order to make the chances of benefiting more equal amongst equally claimless groups and furthermore that people think that doing so is fair. On many plausible views regarding the methodological significance of competent people’s judgments, this experiment’s results support the view that it is fair to treat equally claimless groups equally. This suggests that it is unfair to treat equally claimless outcome groups unequally. As previously demonstrated, the new requirement does precisely this: it treats some claimless maximal outcome groups differently from other claimless maximal outcome groups. This further contributes to the reasons to reject the new requirement.

In summary, the conjunction that Sharadin offers in support of new requirement’s comparative fairness does not entail that requirement’s comparative fairness because some requirements that satisfy that conjunction are comparatively unfair. Furthermore, even if the conjunction does entail the requirement’s comparative fairness, I have presented reasons to doubt the second part of the conjunction. For these reasons, I do not endorse the new requirement. That said, I believe that something like the new requirement may be defensible, and after presenting a case-based response to the new requirement in (II), I will argue in (III) that a deontic variant of the new requirement is more plausible than the new requirement.

(II) The new requirement ‘ignores’ outcome groups in cases where one or more claimants can increase their chance to receive the benefit that they have a claim on (thereby promoting absolute fairness) without reducing the chances any individual has to benefit. Sharadin contends that this ignoring is fair: it is absolutely fairer than not ignoring those groups while also not being comparatively unfair. To assess the plausibility of this, consider a different equal conflict case in which one or more claimants can have their chance to receive the benefit that they have a claim on increased without reducing the chances any individual has to benefit. Suppose, for example, there are only two potential beneficiaries, A and B, who each have an equal claim to receive the same benefit. My exclusive lottery (like most other lottery procedures) implies that it is fair to give each of A and B an equal 50% chance of benefiting. However, imagine now that it is possible to increase the chance that B benefits to 70% while maintaining a 50% chance of benefiting A. This type of case could arise when potential beneficiaries’ chances to benefit are independent of one another. For an example of this type of case, imagine a case where A and B each have a claim on some particular indivisible prize, there are two ticketed lotteries that each independently award that particular indivisible prize, but the tickets are structured such that one lottery has a 50% chance of awarding the prize and the other has a 50% or 70% chance of awarding the prize. In this case, it seems intuitively unfair (because comparatively unfair) to increase B’s chance of benefiting to 70% even when that increase does not reduce the chances any individual has to benefit. While this case is not the same as the kind of case that the new requirement applies to, it does share a relevantly similar feature (namely that one or more claimants can increase their chance to receive the benefit that they have a claim on without reducing the chances any individual has to benefit). If this 50%/70% distribution is unfair in this case, Sharadin owes us an explanation for the asymmetry between this case and the kinds of cases that the new requirement applies to: why in some cases an increase in one or more claimants chance to receive the benefit they have a claim on (without reducing the chances any individual has to benefit) is unfair in this case but not the cases that the new requirement applies to. I am uncertain whether this explanatory burden can be met.

(III) Despite what I have said in response to the new requirement, there is still something appealing about it. After all, why shouldn’t you give one or more individuals a greater chance of benefiting when it does not come at the expense of any other individuals’ chances of benefiting? I think this appeal can be captured by a nearby deontic variant of the new requirement. Here’s the deontic variant: in equal conflict cases where the value of the benefits distributed justify a chance allocation that is not all-things-considered as fair as possible (e.g. in cases where that value is more important than conducting the most fair possible procedure), moral permissibility requires that we allocate outcome groups (even maximal outcome groups) zero chance of benefiting when doing so instead of running the most fair possible procedure (which, I contend, is the exclusive lottery) results in an alternative distribution such that the individual members of those outcome groups can be given the same or higher chance of benefiting. This deontic variant is consistent with everything I wrote in my paper, because the paper focused on what is procedurally fair rather than what is all-things-considered morally permissible to do, while arguing that fairness-based reasons are a subset of all of the reasons there are.

This deontic variant of the requirement is relatively plausible because it increases the chances that more valuable benefits are distributed while also being more absolutely fair than the exclusive lottery on its own. The value of the benefits combined with the additional absolute fairness can, in certain cases, make a procedure that is comparatively unfair permissible by justifying the loss of comparative fairness. For example, this is justified when the benefits are very valuable (e.g. saving the lives of young healthy individuals who will go on to lead a long and happy lives) and it is a one-off distribution case where comparative fairness is not very important (as opposed to certain persisting contexts, such as legal courts, where comparative fairness is very important).

Finally, consider a variant of the case I presented in (II) that has these two features regarding the high value of benefits and the low importance of comparative fairness: in such a case it is intuitively permissible to give A 50% chance of benefiting and B a 70% chance of benefiting even though doing so is unfair. That unfair allocation not only maximizes the chances that A and B will live a long and happy life, doing so is the most absolutely fair chance allocation possible. There is no asymmetry between that case and the cases that the deontic variant of the new requirement applies to, so the deontic variant does not suffer from the same explanatory burden that the new requirement does.

To summarize: I have argued that the deontic variant of the new requirement is more plausible than the new requirement because it is not subject to the criticisms of the new requirement above and it can be theoretically motivated by the distinction between fairness-based reasons and reasons not based in fairness. That said, I have not provided decisive reasons for the deontic variant of the new requirement, and for this reason I am unsure if it is true. It is, however, a view I think is worthy of further evaluation.

In conclusion, I want to sincerely thank Daniel Star and PEA Soup for hosting and organizing this discussion. I am also grateful to Aidan Penn, who provided numerous insightful comments on my response to Sharadin’s précis. I would also like to thank Nate Sharadin again for his excellent comments. He elegantly presented my view and offered both a criticism and a new view about fairness that are worthy of continuing consideration. I hope that you, the reader, will join us in doing so in the discussion below.

References:

List, C., & Pettit, P. (2011). Group agency: The possibility, design, and status of corporate agents. Oxford University Press.

Karni, E., Salmon, T., & Sopher, B. (2008). Individual sense of fairness: an experimental study. Experimental Economics, 11(2), 174-189.

2. Marie Kerguelen Feldblyum Le Blevennec says:

First, I want to say that I am very excited to be taking part in this discussion of Gerard Vong’s thought-provoking article, and that I am grateful to Nate Sharadin as well for his helpful critical precis.

I think Vong’s article very clearly lays out the issues that unweighted lotteries face when it comes to equal conflict cases, and makes a strong case for preferring exclusive composition-sensitive lotteries in equal conflict cases. With that said, I am sympathetic to Sharadin’s worry that we need a genuine compromise between absolute and comparative fairness.

In fact, I would like to expand upon Sharadin’s thought by presenting an example of an equal conflict case in which I think absolute fairness is at least as, if not more important than, comparative fairness, in order to press a bit on Vong’s claim that comparative fairness is the most important type of fairness, as well as on Vong’s claims about the Awkward Conclusion.

Vong claims in his article that what is morally wrong about the Awkward Conclusion, and what makes it seem unintuitive, has nothing to do with fairness. However, in the Andrew vs. Ben case he discusses, what seems wrong (in my view) is the reduction of Andrew’s chances for the sake of comparative fairness. In other words, the problem with the awkward conclusion is that it disrespects Andrew’s claim by reducing his chances, which conflicts with what Andrew is owed in virtue of having a claim. And as Sharadin points out, absolute fairness requires us to maximize the chances that each individual receives the benefit they are due in virtue of having a claim. So, the awkward conclusion is overall morally wrong and unintuitive (which I think we all agree on) precisely because it is absolutely unfair to Andrew, despite its being comparatively fair. This suggests that Andrew vs. Ben is a case in which absolute fairness is at least as important as comparative fairness.

So, I am wondering what Vong (or anyone else reading this) would have to say in response to this example. In particular, I am wondering whether this line of argument shows that Sharadin is right to suggest that a genuine compromise between comparative and absolute fairness could involve greater concessions when it comes to comparative fairness (in at least some cases) than Vong suggests in his article.

3. Thank you for your comment, Marie Kerguelen Feldblyum Le Blevennec. The relative priority of absolute fairness and comparative fairness (the two components of all-things-considered fairness in equal conflict cases) is an important topic. I welcome the opportunity to think about this topic further through the use of the Andrew v Ben equal conflict case from Section V of my paper. In my response, I want to clarify my paper’s assessment of what is fair and what is right in that case.

As a reminder to readers of the paper, in that case, attempts to save Andrew’s life have a 100% chance of success, whereas attempts to save Ben’s life have a 50% chance of success, even though their claims to be saved are equally strong and indistinguishable. In the paper, I argue that it is not implausible that a lottery that equalizes their chance of being successfully saved, by allocating a 2/3 chance of attempting to save Ben and a 1/3 chance of attempting to save Andrew, is all-things-considered fair. However I also argue that such a chance equalization is morally wrong (even though it is fair) because it is significantly worse than an unfair procedure that just saves Andrew (giving him a 100% chance of benefiting).

Le Blevennec is correct to point out that such a chance equalization is less absolutely fair to Andrew than some other ways of acting because it allocates Andrew a lower chance of receiving the benefit he has a claim to. So absolute fairness counts against such a chance equalization (in favor of just saving Andrew). I did not intend to imply otherwise in the paper (see footnote 43 on page 346 of the paper).

Where Le Blevennec and I disagree is that Le Blevennec seems to think that absolute fairness is the only reason to prefer just saving Andrew over such a chance equalization. Le Blevennec writes that the problem with such a chance equalization is “that it disrespects Andrew’s claim by reducing his chances, which conflicts with what Andrew is owed in virtue of having a claim …(and) is overall morally wrong and unintuitive (which I think we all agree on) precisely because it is absolutely unfair to Andrew, despite its being comparatively fair. This suggests that Andrew vs. Ben is a case in which absolute fairness is at least as important as comparative fairness.” I believe that both absolute fairness, *and* the value of saving Andrew’s life, count in favor of just saving Andrew. Combined, the two are able to outweigh the reasons from all-things-considered fairness in favor of chance equalization. However, because in the paper I defend the relative priority of comparative fairness over absolute fairness in equal conflict cases, I contend the reasons based in comparative fairness in favor of chance equalization outweigh the reasons based in absolute fairness in favor of just saving Andrew. Thus it is all-things-considered fair to equalize their chances of benefiting even though it is not optimal with respects to absolute fairness. This is how I resist the conclusion that absolute fairness is at least as important at comparative fairness in the Andrew v Ben case even though it is morally wrong to equalize Andrew and Ben’s chances of being saved.

Notwithstanding my defense of the relative priority of absolute and comparative fairness in Section III of my paper, there may be other reasons to reject the priority of comparative fairness over absolute fairness in equal conflict cases. However, as I have argued above, I am able to explain the wrongness of equalizing Andrew and Ben’s chances of benefiting without threatening that relative priority.

4. I don’t quite follow how the counterexample you discuss in (ii) is supposed to work. I must be missing something. I don’t understand how to imagine simultaneously running two ticketed lotteries, 1 and 2, that independently award a particular indivisible prize, P, where A is the only entrant into 1, and B is the only entrant into 2, such that I can increase B’s chances of obtaining P (as a result of ‘winning’ 2) without thereby decreasing A’s chances of obtaining P (as a result of ‘winning’ 1). I mean: I see how I can increase B’s chances of winning 2 independently of A’s chances of winning (or losing) 1; that’s just because they’re ‘independent’ lotteries. But since they’re both distributing P, and there’s just one P, they’re not *really* independent when it comes to the chances that A or B receives P. That’s because B’s win excludes A’s obtaining P and vice versa…. Right? What am I missing?

5. Thank you for your comment, Nate. I should have explained the two independent lotteries better to avoid misunderstanding. In particular, I shouldn’t have written ‘same benefit’ without clarifying what I meant. I apologize for that. I meant that the benefit was of the same type for both A and B, rather than it being precisely the same benefit (analogous to lifesaving conflict cases – each beneficiary receives the same type of lifesaving benefit, but Anna does not actually receive Ben’s life when Anna is saved). So potential beneficiaries need not be competing over precisely the same prize. With that in mind, let’s give a more filled out example of the type of case I had in mind for illustration.

Sam has promised his two tennis-loving children that they will sit in the stadium to see their favorite players at the next Wimbledon. One loves Roger Federer, the other Rafael Nadal, and both players may retire soon! Unfortunately, between the time Sam made the promise, and the time that the tickets became available, Sam has come under unforeseeable financial hardship beyond his control. As a result, he cannot afford to buy the tickets for the seats. However, fortunately for Sam, there happen to be lotteries for Wimbledon seats (as is common with some popular sporting events). For simplicity’s sake, let’s stipulate that each raffle ticket for lottery 1 gives its owner a 1% chance of winning a seat at Federer’s only Wimbledon game, and that each raffle ticket for lottery 2 gives its owner a 1% chance of winning a seat at Nadal’s only Wimbledon game. There are fifty Federer and fifty Nadal tickets left from the official Wimbledon website, so Sam buys them. Trying to avoid disappointing his children, he scours the internet for more raffle tickets for days, and finds a legitimate seller for 20 more Nadal tickets on Craigslist, and none anywhere else. Is it fair for Sam to buy them? If the objection I raise in (II) works in the way I intended it to, the answer is that it is not fair. It is unfair because it is comparatively unfair (despite the increase in absolute fairness).

On a separate note, I should mention that some lotteries have no prize winner after a drawing. Think, for example, of many jackpot lotteries run by the government. For this reason, it is sometimes possible to increase the chances that one potential beneficiary has of receiving a prize without reducing the chances of someone else receiving that prize in the same lottery draw. To return to the jackpot example, imagine Alice has a ticket with the numbers 1, 2, 3, 4, 5 on it, and the lottery requires all five numbers to be identical to the winning draw to win the only prize, namely the jackpot. Betty subsequently buys a ticket with the numbers, 10, 20, 30, 40, 50 on it. Even if Alice and Betty are the only ticket holders for the lottery draw, Betty has increased her chances of winning by buying a ticket without decreasing Alice’s chance of winning (because winning the prize is dependent on the drawn numbers and Alice and Betty have different winning numbers).

Thank you to both Marie and Nate for their comments. Please let me know if my responses were not helpful in clarifying of my position, or if you have any follow-up questions.

6. Johann Frick says:

Many thanks to Gerard, for this excellent (and intricate!) paper, and to Nate for your extremely lucid critical précis.

I’m delighted to have the opportunity to put a couple of questions to you, Gerard. I’ll break them up into separate comments.

My first question concerns the basic motivation for the weighted lotteries approach. You believe (and so does Nate, judging by his précis) that in a choice between saving the life of one stranger, A, and the lives of five other strangers, it would be comparatively unfair to A (though perhaps still the right thing to do all-things-considered) to simply rescue the greater number. Rather, you hold, comparative fairness requires us to organize a weighted lottery that would give A a 1/6 chance of being rescued.

In support of this claim about comparative fairness you write on pp. 331-2: “Comparative fairness in conflict cases (in general) requires that potential beneficiaries’ claims to benefit that are similar (or dissimilar) in some relevant respect are treated relevantly similarly (or dissimilarly) in the outcome group selection procedure. (…) this requirement expresses equal and sufficient concern and respect to each equally worthy claimant. In equal conflict cases, because each and every individual potential beneficiary has an equally strong claim to the provision of an equal benefit, the general comparative fairness requirement implies that comparative fairness requires that each equally worthy claimant have an equal positive impact on the outcome group selection procedure.”

But consider this rejoinder by a proponent of the “save the greater number”-proposal: “My procedure, too, treats people’s equal claims equally, by counting each person’s claim as a pro tanto reason of equal strength in favor of the option that will save their respective life. I thereby allow each claimant to have an equal positive impact on the outcome group selection procedure, and show them equal and sufficient concern. In the words of Derek Parfit: ‘Each counts for one. That is why more count for more.’”

Presumably you reject this way of interpreting the thought that we should treat people’s claims equally and show them equal and sufficient concern, since you think it supports your weighted lottery proposal in particular. Could you say a bit more about why?

7. Johann Frick says:

My second question concerns cases with overlapping outcome groups. However, as a warm-up, consider first the following

Non-Overlapping Case: I can either save (A and B) or (C and D) or (E and F).

Here, fairness plausibly requires me to give an equal 1/3 chance of being rescued to all three outcome groups.

Now let us modify this case to make it one with overlapping outcome groups:

Overlapping Case: I can save either (A and B) or (A and D) or (E and F).

In this case, your preferred exclusive composition-sensitive lottery procedure would have us give a 30% chance of rescuing the first outcome group, a 30% chance of rescuing the second group, and a 40% of rescuing the third group. (The equal-composition sensitive lottery, incidentally, yields the same result in this case). This is because B, D, E, and F each contribute their baseline chance of 20% to their respective groups, whereas A, being a member of two outcome groups, contributes only 10% to (A and B) and (A and D) respectively.

Comparing the Overlapping and the Non-Overlapping cases, B and D are disadvantaged, relative to E and F, by the fact that they share an outcome group with someone, namely A, who also features in a further outcome group. Indeed, despite the fact that there are fewer people with a claim to be rescued in Overlapping Case than in Non-Overlapping Case, B’s and D’s chances of being rescued *decrease* compared to the former case. But that hardly seems fair, one might think. “Why should B and D’s chances of rescue go down,” the thought goes, “just because of who else happens to be in their outcome group?”

The composition-sensitive lottery procedure, we might say, has ‘negative externalities’ for those who happen to share an outcome group with other claimants who feature in multiple outcome groups. However, should a complete theory of fairness in life-saving cases not take account of such externalities (as your view seems not to), and don’t we have pro tanto reasons of comparative fairness against creating them?

(Of course, one possible alternative, namely giving an equal chance of being rescued to each outcome group in Overlapping Case, also seems imperfect from the point of view of comparative fairness. It would give A a chance of survival that is even more disproportionate than under your preferred procedure (66% instead of 60%). But it does seem, at least, that there are multiple competing desiderata of comparative fairness here, in a way that your view doesn’t fully capture).

8. Johann Frick says:

My third point concerns your discussion of the so-called “awkward conclusion” in Section V, which purports to challenge a weighted lottery proposal like yours. To recap, this challenge arises in “nonguaranteed cases”, such as the following:

Andrew vs. Ben: If I do nothing, Andrew and Ben will both drown. I can either rescue Andrew, with a 100% chance of success, or attempt to rescue Ben, with a 50% chance of success.

Katharina Berndt Rasmussen argues that in this case, a view like yours implausibly implies that, to decide whom to save, we should hold a weighted lottery which gives Ben a *greater* chance of being chosen than Andrew. For, in that way, Andrew’s and Ben’s chances of survival are more equal. (Perhaps, for instance, we should hold a lottery that gives Ben a 2/3 and Andrew a 1/3 chance of winning. That way, both have a 1/3 chance of survival). But that seems quite counterintuitive. Surely, we shouldn’t make it probable that we will attempt to rescue the person whom we have *less* of a chance of saving!

Your response to this challenge, in essence, concedes that holding such a weighted lottery may be what fairness requires in this case. However, you point out that the right action *all-things-considered* isn’t always that which is fairest. In the case of Andrew vs Ben, the right thing to do all-things-considered may well be to save Andrew, since this produces a considerably better expected outcome.

However, I want to suggest that this response is much too concessive. There is a better way of responding to Berndt Rasmussen’s challenge, namely to deny that this is the kind of case where fairness requires us to hold a lottery of any kind.

The central claim of your paper is that (weighted) lotteries are appropriate in equal conflict cases, i.e. cases where all claimants have equally strong claims to your help, but you can’t help them all. By contrast, as you yourself note on p. 347, “if claims are not equal or roughly equal, nothing in this article entails that lottery procedures are fair.” Thus, suppose I could save Chris from losing his life or Dora from losing her leg, but not both. Clearly, holding a lottery (even a *weighted* lottery) to decide whom to help would be inappropriate here – and not just because reasons of fairness are decisively outweighed by reasons of other kinds. Rather, since Chris’s claim to be helped is *much* stronger than Dora’s (because Chris has a much greater stake in being helped than does Dora), it is in no way unfair to Dora that I simply help Chris.

We should give a parallel analysis of the Andrew vs Ben case. A person who could be given a 100% chance of survival, instead of dying for sure, has considerably more at stake than someone who could only be given a 50% chance of survival. So, contrary to what you say on p. 346, Andrew’s and Ben’s claim to your help are not equally (or nearly equally) strong. Rather, Andrew’s claim is considerably stronger than Ben’s. (If you are not convinced, imagine that you could give Ben only a 1% of survival). So this is *not*, in fact, an equal conflict case, and therefore your view does not commit you to the awkward conclusion that fairness requires us to give a Ben an equal or greater chance of being rescued than Andrew. For all you have argued in your paper, you are free to assert that we should just save Andrew, and that doing so would be in no way unfair to Ben.

9. Thank you for your questions, Johann. First question first: there are many things that can be said in response to the claim that saving the greater number in equal conflict cases is fair (because it expresses equal and sufficient concern to each individual). For example, I find much of what Broome says about the unfairness of benefiting the greater number in his paper ‘Fairness’ to be compelling (though I argue fervently against him about many other matters regarding fairness). Sections II and IV of that paper are particularly relevant here. However, in this response, let me focus in on what I can say that is distinctive given the views I defend in my paper.

To bring out the difference in the way that a ‘save the greater number’ advocate and I respond to each individual’s claim, consider the following lifesaving equal conflict case with two differently sized outcome groups. A shortsighted lifesaver looks at two sinking boats knowing that he can save the occupants of only one. Even though the lifesaver is shortsighted, he knows that one boat clearly has more occupants than the other, but he doesn’t know precisely how many more. If saving the greater number is fair and he knows it, then he knows he’s acting fairly if he saves the occupants of the boat that contains the greater number. However he knows that without knowing the number of occupants in the less occupied boat (or for that matter, either boat). On this view, there seems to be no fairness-based reason for the lifesaver to get his glasses and count the numbers of individuals on each boat (assume that doing so would not endanger the wellbeing of any of the occupants). If that’s right, then what really matters for the ‘save the greater number’ advocate in cases like these is that one group is largest, not counting each individual’s claim. Given that, it doesn’t seem to me that the ‘saving the greater number’ advocate is truly counting each person’s claim as a pro tanto reason of equal strength and thus demonstrating each individual equal and sufficient concern. What this case shows is that you don’t actually have to count each person’s claim to save the greater number. This, I contend, is a reason to reject the view that saving the greater number is fair because it expresses equal and sufficient concern to each individual by appropriately counting each person’s claim.

The same cannot be said of my exclusive lottery. If the view about the fair exclusive lottery in my paper is true, the lifesaver has a fairness-based reason to get his glasses to ascertain how many people are on each boat and run a lottery that actually takes each individual’s claim into account in the chance distribution. You can also imagine overlapping variants of this case which further highlight what really matters to the ‘save the greater number’ advocate – all that they need to know is that they’re saving the largest group, and don’t need to know which individuals are in which groups. On such a view, the individuals’ claims don’t truly matter in themselves, they only matter derivatively insofar as they contribute to the largest group. However on my view, individual’s claims really matter as evidenced by the fact one needs to know the individual composition of all outcome groups to run the exclusive lottery. For example, on my view, but not the saving the greater number view, it makes a difference whether or not one is in multiple outcome groups. Put another way, the ‘save the greater number’ advocate can ascertain who it is fair to benefit by counting individuals, but does not have to. In order for me to ascertain whom it is fair to benefit, I need to know not only how many individuals there are, but how they compose different outcome groups.

I don’t want to take us too far afield, but there’s a somewhat analogous debate between total utilitarians and egalitarians in population ethics. Total utilitarians say that they are counting each person for one (by taking into account each individual’s contributions to total utility) and thereby treat people equally, but egalitarians deny that type of counting is all that matters for equality. What matters instead of (or in addition to) total utility is how individuals’ wellbeing compares to other individuals’ wellbeing (e.g. as measured by the Gini Coefficient). Analogously, I deny that the type of counting that the ‘save the greater number’ advocate engages in is all that matters for fairness.

What else matters? While you focus on comparative fairness in your comment, I actually think that the type of fairness that the ‘save the greater number’ advocate may be latching onto is absolute fairness rather than comparative fairness. After all, it is the correct response to absolute fairness to count each person’s equally strong claim as grounding a pro tanto reason in favor of benefiting that individual. The satisfaction of more claims results in more absolute fairness. For this reason, just saving the greater number is the absolutely fairest possible procedure. If one is a fairness monist (absolute fairness only), I can see how one would think that saving the greater number is all-things-considered fair. What I contend, though, is that this view fails to appropriately take into account another part of all-things-considered fairness, namely comparative fairness.

Thanks again for your questions, Johann! I look forward to answering the next one.

Reference:

Broome, J. (1990, January). Fairness. In Proceedings of the Aristotelian Society. Vol. 91, pp. 87-101.

10. Gerard: That’s helpful, thanks. I now see what the kinds of cases you had in mind are. And I think I just disagree with you about whether it’s unfair; in the case you give, I think it wouldn’t be comparatively unfair for Sam to buy the extra 20 tickets. (Although: it would be, if he were unwilling to buy more tickets if they became available for the other child…. but that’s not the case as described.)

I have the intuition that your judgment in this case makes your view liable to a kind of leveling-down objection; after all, no one’s chances are made any worse off if Sam buys the additional tickets, and Sam has made everyone’s chances the highest he possibly can if he does buy them (and not if he doesn’t). Another way to put the same intuition: I just don’t see how either child could have a reasonable fairness-related complaint if he buys the tickets, but I see pretty clearly how one of them could have a complaint if he fails to do so! I’m not sure, though, and I’ll have to think more about it.

11. Johann’s second question next: I think that there is a satisfactory answer to “Why should B and D’s chances of rescue go down… just because of who else happens to be in their outcome group?” I argued for the importance of outcome group composition in Sections I through IV of my Ethics paper, but let me say more here by addressing your particular overlapping case. In that case, there is a relevant difference between each of B and D and each of E and F. To bring this difference out, I will adopt your practice of considering a sequential pair of cases.

Start with the following non-overlapping equal conflict case of singletons: can save one of A (Alice) or B (Betty) or D (Diana) or E (Edna) or F (Frances).

Clearly this case, its fair to give each claimant an equal 1/5 (20%) chance of benefiting.

Now let’s modify the singleton case to your overlapping case: can save either (A and B) or (A and D) or (E and F). As you rightly note, my exclusive lottery gives 30% chance of benefiting to each of (A and B) and (A and D) and 40% chance of benefiting to (E and F).

Relative to the case of singletons, this is an increase in everyone’s chance of benefiting. However B’s and D’s chances of benefiting respectively do not increase as much (+10% each) as E’s or F’s chances respectively (+20% each). Can this difference be justified? Yes. To see how, let’s consider the perspective of each of the individual claimants, starting with A.

A can look at their change in outcome group membership to the second overlapping case and note that the two outcome groups of which they are now a member are relevantly similar (the shift from A to (A and B) and (A and D) are relevantly symmetrical and so should have the same change in chance of benefiting. This is evidenced in the fact that Betty could have been represented by either the letter B or D in this pair of cases – it doesn’t matter which (mutatis mutandis for Diana). In this case, any fairness-based reason to allocate chances of benefiting to (A and B) apply equally to (A and D). Indeed, this is precisely what the exclusive lottery prescribes (30% chance of benefiting to each of these pairs).

However E and F’s shift (joining an outcome group with each other) is not the same as A’s shift. It is relevantly different, because E and F, unlike A, have only one outcome group that they can possibly contribute their baseline chance to. However E and F’s shifts are relevantly symmetrical – any fairness-based reason to increase E’s chance of benefiting also applies to F. This is evidenced in the fact that Edna could have been represented by either the letter E or F – it doesn’t matter which (mutatis mutandis for Frances). We should expect their chances of benefiting to increase by the same amount, and indeed that is what my exclusive lottery prescribes (+20%).

Similarly B can look at their change in outcome group membership to the second overlapping case and note that D’s shift is precisely the same as theirs (i.e. a singleton joining an outcome group with A in it). In this case, any fairness-based reason to increase B’s chance of benefiting also applies to D. Whatever change applies to B should apply to D and my exclusive lottery correctly captures this by giving them each a 10% increase in their chances of benefiting.

Now, let’s ignore the transition and just look at your overlapping case. Its clear that (A and B) and (A and D) are relevantly symmetrical to each other while being relevantly different from (E and F). Any fairness-based reason to allocate (A and B) any chance of benefiting also applies equally to (A and D), but not necessarily to (E and F). Indeed, these differences are respected by the exclusive lottery. I endorse the ‘negative externalities’ you point out as a feature, not a bug. I do think (A and B) and (A and D) are relevantly different from (E and F), even if preceded by your non-overlapping case.

A separate question you ask is do we have pro-tanto reasons from comparative fairness against creating outcome groups where some individuals suffer from these ‘negative externalities’? (in virtue of the fact that they share one or more outcome groups with claimants who are in multiple outcome groups). Given what I have argued for above and in the paper, I don’t see why we would have reasons *from comparative fairness* to avoid creating such outcome groups. I do, however, think that there are reasons from absolute fairness to not decrease claimants’ chances of receiving the benefit that they have a claim on, and in so far as the creation of some ‘negative externalities’ does so (as, for example, in the shift between your comment’s two cases), there are reasons from absolute fairness to avoid doing so. But this is not special to ‘negative externalities’, but also applies to any cases where claimants’ chances to receive what they have a claim on are reduced.

I hope this response is helpful and clarifies how my account is supposed to be responsive to different fairness considerations in your pair of cases. Please don’t hesitate to follow up if need be.

12. Now I will respond to Johann’s third comment about how Section V of my paper responds to Katharina Berndt Rasmussen’s ‘Awkward Conclusion’ objection. Instead of the view I defend there, Johann suggests an alternative. Namely, we should not consider cases in which attempts to benefit claimants do not have equal chances of success as involving equally strong claims, because claimants with lower chances of that kind should be treated as having weaker claims than claimants with higher chances of that kind. In the Andrew vs Ben case, Johann suggests that Andrew’s claim is considerably stronger than Ben’s. So strong, Johann contends, that it is not unfair just to save Andrew by giving him a 100% chance of benefiting.

In an earlier version of my paper, there was a part of Section V where I endorsed a view similar to Johann’s suggestion. It was similar in that it suggested that some cases in which attempts to benefit claimants do not have equal chances of success do not involve equally strong claims and thus are not equal conflict cases. It was not precisely the same view as Johann’s because I didn’t endorse the view that I ascribe to Johann in the last sentence of my last paragraph: I did not claim it was not unfair to just save Andrew. Instead, I wrote that because such cases were not equal conflict cases that those cases were beyond the scope of my paper. I think there’s important work to be done about how lotteries may or may not be fair in cases where the strengths of potential beneficiaries’ claims differ greatly, and did not want to take a side in that debate there.

However a reviewer suggested removing that part of Section V because Katharina Berndt Rasmussen restricts her objection to equal conflict cases. I agreed with that removal, partly because while I think that Rasmussen’s objection is somewhat weakened by that part, it does not address it completely. This is because one can conceive of cases in which attempts to benefit claimants do not have equal chances of success yet they do involve equally strong claims. This is because the strength of claims to be benefited depend not only on the magnitude of the benefit or the chance that an attempt of benefiting will be successful, but on other facts about how the claim was generated. For example, a claim that is generated by a mutually agreed upon legal contract between the potential beneficiary and benefit distributor is significantly stronger than a claim that was generated by an implicit but reasonable expectation between the two parties. This is why when I define equal conflict cases in Section IA of the paper, I take care to specify both that potential beneficiaries’ claims are equal in strength, *and* that those claims are to an equal benefit. So I think one can conceive of equal conflict cases that are non-guaranteed cases and Johann’s suggestion is not an effective response to those cases.

To summarize, while I am sympathetic to part of Johann’s suggestion, I do not think it completely addresses all of the cases that Rasmussen’s objection applies to. However the response in Section V of my paper does address all of those cases.

13. Nate*, it is interesting that our intuitive judgments differ in this parental case. I have been considering more complex variants of your new requirement, including variants that are less susceptible to the objections I raised in my initial response above. One way I was varying your requirement was changing the conditions under which outcome groups were to be ‘disallowed/ignored’. For one, I wasn’t sure why you stipulated that individuals in those groups had to have the same OR higher chance of benefiting. I understand why you want them to be higher, but why disallow/ignore the group if it makes no difference at all to any of their chances of benefiting (i.e. disallowing leads to all of the individuals having the same chance as they would without disallowing)? Can you say more about why your condition would include that? Would it be better to have the following nearby sufficient condition for disallowing/ignoring a group: there is a pareto improvement in the chances of benefiting of individuals in the disallowed group (i.e. disallowing that group increases the chances of benefiting at least one individual, and no one has their chances of benefiting reduced)? I have also considered other variants of your new requirement, but am yet to generate a good taxonomy of them so don’t think what I would say would contribute meaningfully to the discussion here. However different variants of your new requirement may have different implications for the parental case, which is why I find our disagreement here particularly interesting.

On a separate note, you are right that my view is subject to a kind of leveling down objection (that is somewhat analogous to the leveling down objection to egalitarianism in population ethics). My response to such an objection is somewhat analogous to the response that some egalitarians make in population ethics, namely that equality is not all that matters. On my view, comparative fairness is not all that matters for fairness or for what is all-things-considered morally justified. Absolute fairness and the value of the benefits distributed also matter. For this reason, it may be all-things-considered wrong to level down in some cases (e.g. the parental case), but leveling down does improve fairness. I think that’s a defensible view, so I’m not worried about the leveling down.

In any case, I am glad that you seem to agree with my fairness pluralism, even if you disagree about the relative priority of comparative and absolute fairness.

*I hope that it is OK that I have been referring to you as Nate rather than Nathaniel, though please let me know if it is not. I was just following the original post, but noticed that you have been posting as Nathaniel.