I.i. The role of chances of benefiting

Before outlining and objecting to Broome's view, I should illustrate the role of ex ante chances of benefiting for fairness, for claim satisfaction and for permissibility. Consider the following case:

The Chancy Doctor: A doctor can conduct successful life-saving surgery on two of her patients in two separate, easy operations. Both patients are equally badly off, and the doctor has no other obligations. Instead of just conducting the surgery on both patients, she decides to roll two genuinely random six-sided dice. If and only if the first die rolls a one, she will choose to conduct surgery on the first patient. If and only if the second die rolls between one and five, she will choose to conduct surgery on the second patient. This is the only difference in the doctor's treatment of the two patients. The first die rolls one and the second four, so she actually conducts successful surgeries on both patients. She does not tell the patients about her lottery, she never runs such a lottery again nor does anyone ever find out about it.

Even though the Chancy Doctor gives the same life-saving benefit she owes to both patients, her action is patently unfair and wrong because of the differences in the ex ante chances of benefiting she gives to the two patients. It would be even less fair if she gave one patient a 100 per cent chance to be treated but gave no chance to the other. It is clear that people's chances of benefiting are relevant to what is fair and what is wrong. While I will not assume any contentious theory about which chances of benefiting are fair in all cases, for the purposes of this article I assume that some distributions of chances are fair.

The Chancy Doctor does not appropriately satisfy the potential beneficiaries' claims. A potential beneficiary X has a claim that Y provide a benefit B to X if and only if Y has a duty to X to provide a benefit B to X. Claims generate patient-relative reasons, where a patient is a person or thing that undergoes an action of another. This meaning of ‘patient’ contrasts with the meaning of ‘agent’. Because the doctor can easily save both of their lives, fairness intuitively requires her to give both patients an equal 100 per cent chance of benefiting. To do so is the appropriate way to satisfy the patients' claims.

I.ii. Broome's synchronic view of claim satisfaction by lottery

How do chances to benefit satisfy potential beneficiaries' claims on scarce, indivisible benefits? According to John Broome, fair lotteries provide surrogate satisfaction of claims by giving all claimants an appropriate chance of benefiting as a surrogate for actually receiving the benefit. His view is synchronic. Those who are not selected to benefit by the lottery receive surrogate satisfaction of their claim and thus no longer have any claim on the benefit.Footnote ⁴ Those who are selected by the lottery (i.e. those who win) have their complete claim intact and thus we ought to give the benefit to the potential beneficiaries who still have claims on the benefit.

Broome motivates this view by arguing that it explains the absurdity of ignoring the first lottery and running a second lottery amongst precisely the same potential beneficiaries. On Broome's synchronic view, after the first lottery the number of available benefits is equal to the number of claimants and so all claimants can be benefited. If there were no changes to potential beneficiaries’ claims after the first lottery (and no other morally relevant changes), Broome contends that there would be equally good reason to run a second lottery as to run the first. Furthermore, if there were no morally relevant changes between lotteries then any number of iterative lotteries could be just as justified as the first lottery. Call the implausibility of these iterative lotteries, iterative absurdity.

According to Broome, it is absurd to tamper with a previously run lottery that has already provided surrogate satisfaction of some potential beneficiaries’ claims by running a subsequent lottery between the exact same potential beneficiaries. This is because, Broome argues, doing so gives potential beneficiaries with no claims on receiving the benefit a chance of receiving the benefit. While I agree that running subsequent lotteries in such cases is absurd, I argue that iterative absurdity can be avoided without adopting Broome's synchronic view and furthermore that Broome's view has implausible implications.

I.iii. Objections to Broome's synchronic view

While initially plausible, Broome's synchronic view commits him to take morally irrelevant features as morally relevant and undermines his own view about the fairness of giving claimants an equal chance of benefiting. On Broome's view, lottery winners retain their complete claim on the benefit, whereas lottery losers have no claim on the benefit. This leads to particularly implausible implications in temporally extended cases such as the following.

Stormy Seas: On a ship, two sailors (A & B) fall overboard during stormy weather and will drown if not saved. Each sailor has an equally strong claim to be saved by the ship's captain. Unfortunately those overboard can only be saved through the use of life buoys. There is only one such buoy and it can save just one person. Call each set of people (sometimes a singleton) that can be benefited together an outcome group. In this case, A is an outcome group and B is another outcome group. There are no morally relevant differences between either outcome group. Having read and agreed with the moral prescriptions of Broome and other philosophers, the captain opts to run a genuinely random lottery that gives each claimant an equal one-half (50 per cent) chance of winning the lottery and selects one winner (B).Footnote ⁵ Given his long experience with games of chance, he runs the lottery quickly without endangering the lives of those he is about to save. However, after he has run the lottery but before he throws out the buoy, another wave hits the boat and two more sailors (C & D) fall overboard. What should the captain do now?

On Broome's synchronic view, because A has lost the lottery, A has no claim on receiving the benefit. However, B, C & D all have equally strong claims on the benefit, but not all of them can be benefited, so according to Broome's synchronic theory, their claims give the captain reason to run a second lottery between them which gives them all equal one-third chances of winning. According to Broome, when we have to distribute a scarce, indivisible benefit amongst potential beneficiaries who all have an equally strong claim on the benefit, their claims give us reason to run a lottery that gives each claimant an equal chance of winning a lottery.

While doing so gives B, C & D equal chances of winning the second lottery, it does not give all potential beneficiaries (A to D) equal chances of benefiting (i.e. being saved). Those who fall overboard later (in this case, C & D) have a greater chance of being saved as they participate in fewer lotteries and thus have less chance of losing their claim. In this case, C & D have to win just one lottery, which gives them a one third chance each in order to be saved. However those who fell in the beginning (A & B) have to win two lotteries, one with a one half chance of winning, one with a one third chance of winning, in order to be saved. A & B each have a significantly lower chance of being saved (one sixth) than those who fell overboard later. This is implausible as it gives the time the people fell overboard excessive moral importance.

This is more implausible the more subsequent groups fall overboard. Call potential beneficiaries who have their claims on the benefit created after a lottery has been run subsequent claimants. If a third group of subsequent claimants fall overboard after the second lottery has been run, then there is more discrepancy between potential beneficiaries’ chances of benefiting. On Broome's synchronic view, the later one or more potential beneficiaries develop their claims on the benefit (and thereby develop their claims after one or more lotteries have been run), the greater chance they have to benefit. This implausibly gives an intuitively morally irrelevant feature of this case great moral importance. As it is a structural feature of the case, not the particular details, that is necessary for the objection, the objection can be generalized. Furthermore, there are structurally similar actual world cases. For example, if lotteries run in accordance with Broome's synchronic view were used to determine who should receive donor kidneys, those who suffered from renal failure more recently would have a greater chance of receiving a kidney.

As it is the time that people develop their claims relative to the time the lotteries are conducted that affects their chances of benefiting, Broome's synchronic view also has implausible implications in other temporally extended cases. Imagine Stormy Seas except that because the captain was slightly slower in running his lottery, C, but not D, falls overboard before he runs the lottery. This changes the case so that D has a one half chance of being saved but A, B and C only each have a one sixth chance of being saved. Again, Broome's view implies that a feature of the case that is morally irrelevant (the slight difference in speed in which one conducts a lottery) has been elevated to something of life-and-death importance.

Another type of temporally extended case also presents problems for Broome's account of claim satisfaction. Imagine Stormy Seas, except that instead of C & D falling in after the first lottery has been run, the captain finds a second equally functional life-saving buoy. Call this case Extra Buoy. It is absurd in Extra Buoy to claim that while B has a claim to be saved as he won the first lottery, A has no such claim.

Broome has suggested in conversation that this apparent absurdity can be explained away by considering specifically what A & B have a claim on. If, prior to the lottery, they have a claim not just on the general benefit of having their life saved, but on the specific benefit of having their life saved by the first buoy, then this explains why the lottery loser, A, has no claim on the first buoy. Call this the specific benefits view.

The specific benefits view is implausible. Reconsider Extra Buoy. According to the specific benefits view, B could justifiably complain if they are given the second buoy instead of the first as it would not satisfy their specific claim. On this view, there are reasons to give the first buoy to B and the second buoy to A as opposed to the other way around. Intuitively, it does not matter who gets which buoy, as long as both of their lives are saved. Broome's view once again raises something that is intuitively morally irrelevant to something of significant moral importance.

Another problem for Broome's synchronic view emerges if we try to extend it in plausible ways to cases where the lotteries run are not perfectly fair. Consider a case where we can just save one of two lives, as in the start of the Stormy Seas case. Now stipulate that in order to decide whom to benefit, a slightly unfair lottery is run – one that gives A a 49 per cent chance of benefiting and B a 51 per cent chance of benefiting. It is unfair because claimants that are not morally different in any way are treated unequally.

Broome does not consider this case, so I have to consider how his theory could be extended in a consistent way to account for it. As the lottery is not completely fair, as it would be if they were each given a 50 per cent chance of benefiting, it is implausible that the lottery would result in either claimant having their claim completely satisfied or having no claim on the good after the lottery. Instead, the most plausible theory extension would be to treat the claimants as if they have had their claims partially satisfied. That is, the claimants each have less of a claim on the good after the lottery than before, but they still have some weakened claim on the good. The partially fair lottery is, after all, more fair than running a lottery that just gives A 100 per cent chance of benefiting and B no chance of benefiting. How much are those claims partially satisfied?

Here, a Broomean theory faces a dilemma. On one horn is the view that claimants would have their claims equally partially satisfied, as they are given chances of benefiting equally distant (1 per cent) from the fair chance of benefiting (50 per cent). That would result in each claimant having a claim on the good that is less strong than they had previously but equally as strong as each other's claim. There are two problems with this horn. First, if both beneficiaries still have claims that are equally strong, then the objectionable iterative absurdity arises. In fact, this variant of iterative absurdity is even more implausible than the original presentation of iterative absurdity. This is because as each time an imperfectly fair lottery is run, their claims are weakened, until they have nearly no claims at all. Second, that they each have their claim equally partially satisfied is implausible. Consider the following analogy. If you owe two people each one dollar and give one 51 cents and the other 49 cents, you have not equally satisfied their claims. Given this, it seems, in this imperfectly fair case, that their claims are unequally partially satisfied, which is the other horn of the dilemma.

On this second horn, in line with Broome's view that lottery losers in fair lotteries completely lose their claim, the loser of the somewhat unfair lottery has a less strong claim on the benefit than before. In order to avoid iterative absurdity, this claim has to be weakened enough so that it is no longer fair to run a lottery between previous lottery losers and winners. However, this view gives rise to the same implausible implications we saw before in temporally extended cases like Stormy Seas. It claims that previous lottery losers have insufficiently strong claims to be included in subsequent lotteries, so subsequent claimants will have greater chances of benefiting than claimants that developed their claims prior to a lottery being run. As Broome's theory already has problems with subsequent claimants and is motivated by its avoidance of iterative absurdity, it would be best for a Broomean to embrace that horn of the dilemma.

These cases show that someone acting in accordance with Broome's synchronic view does not directly provides claimants with equal chances to receive the benefit, but instead provides claimants with equal chances to lose their claims in each lottery. Those who win the lottery are not absolutely any better off than they were before the lottery as they have the same claim both before and after the lottery. Broome's lotteries only indirectly determine potential beneficiaries’ chances of benefiting because those who receive the benefit are those who still have their claims after the last lottery that results in the number of claimants equalling the number of available benefits. This not only causes Broome implausibly to elevate the importance of these morally irrelevant features, it undermines one of his other views.

Broome contends that fairness in cases such as these requires all claimants receive an equal chance of benefiting. Broome's synchronic view in temporally extended cases such as Stormy Seas gives each claimant an equal chance of winning each lottery but different chances of actually benefiting. On Broome's account, it is equal chances of benefiting, not equal chances of winning each lottery, which are the fair surrogate for actually receiving the benefit. This part of Broome's view explains the unfairness of the second synchronic lottery in Stormy Seas well – it gives A & B each a one sixth chance of survival compared to C & D's one third chance of survival when fairness requires each claimant to have an equal one quarter chance of survival. As a result, Broome's synchronic view of claim satisfaction by lottery undermines his view of fairness in temporally extended cases such as Stormy Seas. This is because it results in distributive lotteries that are unfair by Broome's own theory and the reason for running lotteries in the first place was to ensure a fair claimant selection procedure. I consider and reject a possible response to this undermining objection in the next section.

While Broome's synchronic theory is able to explain iterative absurdity, it does not give a plausible account of temporally extended cases where either subsequent claimants appear (e.g. Stormy Seas) or additional benefits become available (e.g. Extra Buoy), or cases where only partially fair lotteries are run. We need a view that plausibly accounts for these four features. I evaluate alternatives to Broome's view in section II.

II.i. Diachronic alternatives

Two diachronic views of claim satisfaction by lottery are more plausible than Broome's in two ways. On one such view, those who lose the lottery have their claim qualitatively weakened but not negated. Call this view diachronic weakening. On the other view, those who win the lottery have their claim qualitatively strengthened. Call this view diachronic strengthening. Both of these views can explain iterative absurdity. This is because it would be absurd to run a lottery between those with sufficiently stronger claims and those with weaker claims.Footnote ⁶ Both views would also explain why both A & B have claims to be saved in Extra Buoy (though they claim, somewhat oddly, that A has a less strong reason than B to be saved). Furthermore, these two views have an intuitively plausible feature, namely that they maintain that lotteries that give claimants a less than 100 per cent chance of benefiting do not fully satisfy their claims on the benefit. After all, intuitively, full satisfaction of a claim on a benefit is only achieved by actually receiving the benefit. This is why it would be fairest to give all claimants the benefit they have claims on, which is possible in cases like Extra Buoy and the Chancy Doctor but is impossible in cases of scarcity like Stormy Seas. In these two ways, both diachronic views are more plausible than Broome's.

The differences between these three views can be illustrated by the following table. The numbers indicate the strength of that individual's claim on the benefit. The starting strength of the qualitatively equal claims is arbitrary (fifty). However, zero indicates that the beneficiary has no claim on the benefit. Furthermore, stipulate that when one claim's strength is ten or more units greater than another claim's strength, it is not fair to run a lottery between those claimants to decide whom to benefit as one claim is significantly stronger than the other. This avoids iterative absurdity. These numbers are provided only for illustration. It is not essential for my argument that the strength of claims can be precisely measured quantitatively.

While more plausible than Broome's synchronic view, these diachronic views should also be rejected. This is because both views also lead to the implausible conclusions of Broome's account in temporally extended cases like Stormy Seas. Diachronic weakening would claim that those who had lost previous lotteries would not have sufficiently strong claims to be included in subsequent lotteries such as Stormy Seas. After all, this is how diachronic weakening avoids iterative absurdity. In this way, diachronic weakening is similar to Broome's synchronic view that in temporally extended cases like Stormy Seas fairness requires us to run lotteries between previous winners and subsequent claimants while excluding previous lottery losers.

Diachronic strengthening has an even more implausible implication, namely that those who win the first lottery have significantly stronger claims than lottery losers and those who fall overboard after the first lottery. On such a view, both lottery losers and subsequent claimants have qualitatively equal claims that are insufficient to justify running subsequent lotteries that give them any chance of surviving. For this reason diachronic strengthening is less plausible than diachronic weakening.

There is one problem with all of the aforementioned accounts that gives us some guidance to a better theory. In order to avoid iterative absurdity, the theories maintain that potential beneficiaries’ claims on the benefit change depending on whether or not they win the first lottery. However, the differing strengths in claims on the benefit that were adopted to avoid iterative absurdity lead to implausible implications for lotteries involving subsequent claimants. This problem can be solved when one recognizes that potential beneficiaries have two types of claims that are relevant to benefit distribution cases like these.

II.ii. The dual structure view

By stipulation, in this article's cases all potential beneficiaries have qualitatively equally strong claims on the benefit. These are the claims that I have focused on so far. For brevity, call these their benefit claims. Benefit claims are satisfied, ex post, when the benefits are actually distributed. In addition to these benefit claims, I posit that potential beneficiaries also each have a claim not to be treated inappropriately. This is a matter of procedural ex ante fairness. Call these their procedural claims. It is these procedural claims that are satisfied by a lottery that gives claimants appropriate, fair chances of benefiting. I call my view that posits both benefit claims and procedural claims in these cases the dual structure view.

The dual structure of the claims parallels the dual structure exhibited in the Chancy Doctor case. Even though both patients were successfully given the life-saving treatment they had a claim on and thus had their benefit claims fully satisfied, the doctor's act was wrong and the patients were treated unfairly. This is because while their benefit claims were satisfied, their procedural claims were not.

Importantly, my dual structure view also gives an explanation of iterative absurdity, vindicating the intuition that there is no reason to run a second (or a third, or a fourth. . .) lottery with the exact same claimants. On my dual structure view, the procedural claim has already been met by the first fair lottery and running a second lottery would invalidate a previously fair procedure for no reason.

Another advantage of my view over Broome's is that on my view benefit claims cannot be lost due to the results of a procedurally fair lottery. This explains why people who lose the lottery still have benefit claims and therefore offers the best account of temporally extended cases where the number of benefits available for distribution increases, like Extra Buoy. In Extra Buoy, both A and B have equally strong benefit claims after the lottery. The dual structure view does not maintain, like the synchronic and diachronic views discussed above, that A has less of a claim to be saved. In such a case, according to my view, the earlier lottery can be ignored as it is no longer a fair procedure. Instead of giving A or B each a one half chance of benefiting, in Extra Buoy, both claimants should receive a 100 per cent chance of benefiting, as in the Chancy Doctor case.

My dual structure view also explains why it is fair to run a new lottery that gives all potential beneficiaries an appropriate chance of benefiting in temporally extended cases such as Stormy Seas. This is because when subsequent claimants appear previously conducted lottery procedures are no longer fair. They are no longer fair as on all views of the fairness of lotteries in benefit distribution cases (weighted and unweighted), the fair chances of benefiting depend on the number of outcome groups. On unweighted lotteries, all claimants are always given an equal chance of benefiting. On weighted lotteries, claimants are sometimes given an unequal chance of benefiting, often weighting chances based on outcome group size. However, in this case where all outcome groups are of equal size and there are no other morally relevant differences between outcome groups, they both prescribe giving all claimants equal chances of benefiting. By adding subsequent claimants in cases like Stormy Seas, the fair chances of benefiting change and therefore previously conducted lotteries are no longer fair. On both weighted and unweighted lotteries, the fair chance of benefiting is 25 per cent. As a result, the original lottery that only included A & B is not fair when claimants C & D are added. This explains why procedural fairness requires us to run a new lottery procedure between all potential beneficiaries in Stormy Seas and ignore the now unfair results of previous lotteries. This is similar to the dual structure view's prescription that the first lottery in Extra Buoy should be ignored once the second buoy becomes available.

In a similar way, the dual structure view can explain lotteries that are not completely fair. When imperfectly fair lotteries are run, like the 49/51 per cent lottery discussed in section I.iii, potential beneficiaries' procedural claims are not fully satisfied. They are merely partly satisfied. It would more fully satisfy those procedural claims (and be fairer) to ignore the results of a previously run imperfectly fair lottery and run a new, perfectly fair lottery.

The dual structure view provides a good explanation of all of the four desiderata set out in section I. It explains iterative absurdity, temporally extended cases where either subsequent claimants (e.g. Stormy Seas) or additional benefits appear (e.g. Extra Buoy), as well as cases where only partially fair lotteries are run. Furthermore, the dual structure it posits parallels the structure evident not just in the above-mentioned cases of benefit scarcity (i.e. when not all potential beneficiaries can be benefited) but also cases like the Chancy Doctor where all claimants can be benefited.

II.iii. Defending the dual structure view

Broome has objected that lottery winners of earlier, previously fair lotteries have a justified complaint that their earlier winnings are ignored. This complaint is particularly salient if earlier losers won a subsequent lottery. Broome has suggested in conversation that this may be avoided by running subsequent lotteries that take into account earlier results. They do so by only running subsequent lotteries between previous winners and subsequent claimants, while adjusting the probabilities of subsequent lotteries to ensure all claimants have an appropriate chance of benefiting.

To see how this would work, consider Stormy Seas again. Instead of running a subsequent lottery that gives either B, C & D or A, B, C & D an equal chance of winning, one may run a lottery that gives B a 50 per cent chance of winning, and C or D each a 25 per cent chance of winning. This would ensure that A, a previous loser, cannot be saved by the results of a subsequent lottery, while ensuring all claimants A, B, C and D have an equal 25 per cent chance of being saved. For A or B to be saved, they each have to win two lotteries each of which they have a 50 per cent chance of winning, so they each have a 25 per cent chance of benefiting. For C or D to be saved, they each have to win only one lottery, of which they have a 25 per cent chance of winning. Note that on Broome's synchronic view, this account will result in giving claimants with claims of equal strength (subsequent claimants and former lottery winners) different chances of winning lotteries.

However, I object to Broome's claim that lottery winners have a justified complaint that their earlier winnings are ignored. To see that they have no such justified complaint, consider a variation on the Extra Buoy.Footnote ⁷ Call it the Defective Extra Buoy. The difference is that the first buoy is faulty so while it can be used to save someone's life, being saved using it is an anxiety-inducing affair. This is because the defective buoy is not a stable flotation device and results in the person being unpleasantly dunked under the water multiple times before being saved. Now imagine that after the first lottery is run, a normal, non-defective buoy becomes available. In this case, it seems absurd that the first lottery winner would have a complaint if their earlier win on the lottery for the defective buoy was ignored and they were saved by the non-defective buoy.

Defective Extra Buoy also illustrates a further objection to the specific benefits view. It is absurd to think that the first claimant who wins (and thus has his claim intact) has a claim on the defective buoy and should be given that particular defective buoy, and the other potential beneficiary who lost the lottery should be given the non-defective buoy. On such a view, the first lottery winner would have a justified complaint if they were given the non-defective buoy instead of the defective buoy which they purportedly have a claim to be given. This is highly implausible. However if the first lottery winner has no justified complaint in Defective Extra Buoy that their earlier winnings are ignored, why should they have a parallel complaint in the case where the potential benefits are equal, as in Stormy Seas? I see no principled, non-ad hoc distinction that can be plausibly drawn between these two types of cases and therefore reject the claim that lottery winners of earlier lotteries would have a justified complaint that their earlier winnings are ignored.