1.1. The practical importance of individual fairness

Understanding the fairness and unfairness of lottery distribution procedures is especially important due to the significant historical and contemporary use of such procedures. Lottery distribution procedures have been used for thousands of years, with early recorded uses including the distribution of land by lot in the Old Testament and political positions in Roman and Greek city states.Footnote ² Some contemporary uses include the US permanent residency (‘green card’) lottery and lottery-allocated Wimbledon tickets. Lotteries have also been advocated (and in some cases, conducted) as a fair method of distributing scarce goods such as donor organs, health insurance and vaccines.Footnote ³

Practically, there are at least two reasons why it is important to be able to ascertain how unfair differences from the most fair distribution of chances are. Sometimes the most fair distribution of chances is unfeasible. For example, say it is most fair for a particular potential beneficiary to receive a 50% chance of benefiting. However the only feasible chancy procedure available is a single coin flip that has a 60% chance of landing heads and 40% chance of landing tails. Is it equally unfair to provide the good only if the coin lands heads as it is to provide the good only if the coin lands tails? Even though both are 10% different from the most fair chance of benefiting, I argue for a negative answer to this question in the next section. In cases where the most fair chance distribution is unfeasible, a measure of the unfairness of feasible alternatives can provide a prescription about which of the feasible options is most fair. This is important for anyone who claims that fairness is a value or that there are reasons to act fairly. Second, for any pluralist about the good, who recognizes both the value of fairness and one or more other values, there is a potential for axiological trade-offs. This is evident in cases where lotteries are conducted even though there are other distribution procedures that could predictably lead to outcomes that are better in terms of some other value, such as utility. Arguably, this is the case in Oregon's previous decision to determine Medicaid enrolment eligibility by lottery rather than distributing healthcare solely on the basis of cost-effective utility maximization. It is therefore important to know how much worse, in terms of fairness, some chance is in comparison to another. This enables one to ascertain how much value is foregone when a less than ideally fair chance distribution is chosen, and this is important in working out whether trade-offs with other values are worthwhile.

1.2. Specifying the problem

This paper defends a theory of how individually (un)fair it is that a potential beneficiary B gets a certain chance P of receiving a good G that they have an individual claim on receiving from a benefit distributor D. By goods G, I refer to states of affairs the obtaining of which would be good for a beneficiary.Footnote ⁴ Here I take a capacious view of what would be good for a beneficiary to include avoiding some burden or harm. This type of good is evident, for example, in the claims certain Oregon residents had on their government to provide them Medicaid so they could avoid the burden of being sick without health insurance. By chances P, I mean the type of chance that fair lotteries are supposed to distribute.Footnote ⁵

B has a claim on D to provide B a good G if and only if D has a defeasible or non-defeasible duty to B to provide good G to B.Footnote ⁶ This claim generates a particular kind of reason for D to provide good G to B. For example, if Andrew promises to give Ben a doughnut, then Ben has a claim on the provision of that doughnut from Andrew. Ben's claim is an individual claim because it is not determined by comparisons with any other claimant's claims or how others are treated. This claim grounds a defeasible duty that requires Andrew to give Ben a doughnut. Furthermore, it is unfair if Ben does not get the doughnut from Andrew. Some claims are claims on receiving a chance of a good and this type of claim is the focus of this paper. This type of claim is evident in the Sole Lottery Ticket case. Tara's purchase gives her a claim on a particular chance of receiving a sporting good. A similar claim on a chance of a good is also generated when someone makes a roulette bet, where the chance they have a claim on depends on what number(s) or colour they bet on, and the respective number of winning to losing pockets. For example, if someone bets on only the number 10 on an American wheel with 38 pockets, they have an individual claim on a 1/38 chance of winning. Chancy distributions in such cases can be fair even if the good is not scarce.

Furthermore, in general when a claimant has a claim on the provision of a good and the claimed chances of that good are not explicitly specified, the claimant has a claim on a 100% chance of receiving that good. This is evident in the doughnut case.Footnote ⁷ Andrew's promise gives Ben a claim on a 100% chance of receiving a doughnut from Andrew. We can see this by considering a case where Andrew will bake Ben a doughnut if and only if Andrew rolls six on one roll of a six-sided die. In this case, even if Andrew rolls a six and Ben gets a doughnut from Andrew, Ben has a justified complaint against Andrew. Whenever a claim to a good is violated by a benefit distributor and it is thus individually unfair for the claimant, the claimant has a justified complaint against the benefit distributor because the distributor has not fulfilled a duty they owe to the claimant. Therefore our judgments about the existence of justified complaints of this kind provide evidence about unfairness. Andrew has treated Ben unfairly by not fully meeting Ben's claim.

Now, some terminology about chances. First, call a potential beneficiary's most fair chance of benefiting their most fair chance. Unless otherwise specified, I assume that each claimant has a claim on a particular chance of benefiting and, relatedly, has a particular most fair chance. I relax this assumption at the end of Section 2.1. For brevity, whenever I refer to chances hereafter, I am referring to chances of benefiting. Call differences above the most fair chance positive deviations. Someone whose most fair chance is 50% but receives a 60% chance is subject to a positive deviation. Conversely, call differences below the most fair chance negative deviations. Someone who has a most fair chance of 50% but receives a 30% chance is subject to a negative deviation. The size of a deviation is the difference in percentage points between it and the most fair chance. The size of the positive deviation example in this paragraph is 10. The size of the negative deviation example in this paragraph is −20. The absolute size is the non-negative value of the size.

1.3. On individual fairness and unfairness

This paper focuses on individual fairness and unfairness. In order to ascertain the individual unfairness of chances for a particular claimant, one only needs to consider the individual claimant's most fair chance and deviations from it and not how that individual's chances relate to the chances others get. It is important to distinguish the question this paper answers from an important, related, interpersonal question that Broome and others ask and is prevalent in the literature on fair lotteries – namely, how unfair is the overall distribution of different chances to different claimants? For example, if the most fair chance for each and every claimant in a group is 30%, and all claimants receive this 30% chance except a few who receive no chance at all, how unfair is the overall distribution to all claimants that the few do not get the most fair chance that the others get? This is a type of overall interpersonal fairness comparison that I will not focus on in this paper because it does not concern individual fairness.Footnote ⁸ Because individual fairness is only one type of fairness, the individual unfairness of deviations does not exhaust all potential sources of unfairness.

The distinctness of individual fairness from Broome's interpersonal comparative fairness can be further underscored by modifying Sole Lottery Ticket to include ninety-nine other ticket purchasers each with a claim equally strong as Tara on a 1% chance of winningFootnote ⁹ . According to Broome, if all of the ticket purchasers receive no chance of winning and thus are all treated equally, it is (comparatively) fair to all. However, I contend that Tara's 0% chance of winning alone makes the chance distribution individually unfair to Tara.

While this type of unfairness has been overlooked in much of the literature on fair lotteries, it has not been overlooked by the philosophical literature more broadly. Feinberg (Reference Feinberg1974: 300) writes of a similar non-lottery treatment: ‘If we treat everybody unfairly, but equally and impartially so, we have done each an injustice’. In such a case, both Feinberg and I agree that each and every individual can be treated unfairly, even if all individuals have individual claims that are equal in all morally relevant respects and they are all treated equally. Feinberg (Reference Feinberg1974: 300) recognizes the importance of each individual claim when he writes: ‘what is ‘due’ the other person is not a share or portion of some divisible benefit or burden; hence it is not necessary for us to know what is due others in order to know what is due the person with whom we are dealing. His rights-or-deserts alone determine what is due him; and once we have come to a judgment of his due, that judgment cannot be logically affected by subsequent knowledge of the condition of other parties’.

With this qualification in mind, some may deny that what I have called individual fairness is really a type of fairness. They may claim that fairness only concerns interpersonal comparisons of the kind I have just ruled out and my paper is about some other topic. While I agree the English word ‘fairness’ is sometimes used in this interpersonal way, it is not used in this interpersonal way exclusively. As Wallace (Reference Wallace1994) and Hooker (Reference Hooker2005) have separately argued, the word ‘fair’ is used in distinct ways both in everyday language and in moral philosophy. Philosophers such as Feinberg (Reference Feinberg1974) and lawyers such as Levine and Pannier (Reference Levine and Pannier2005) use the word ‘unfairness’ in the same way as I do.Footnote ¹⁰ In addition, others have endorsed views about unfairness that do not involve any of the aforementioned interpersonal comparisons. For example, Wallace (Reference Wallace1994) has defended the influential view that it is unfair to blame an individual agent for doing an act Φ if that individual does not have the general capacity to appreciate relevant moral reasons and to regulate her behaviour in light of those reasons.Footnote ¹¹ This general capacity for appreciation and regulation fundamentally depends only on facts about the individual agent and not on interpersonal comparisons. As Wallace (Reference Wallace1994) and others show, there are perfectly respectable types of ‘fairness’ both in ordinary language and in the philosophical literature that are distinct from the interpersonal type that is typically the focus of the literature on fair lotteries.

Furthermore the intuitive judgments about unfairness in the Sole Lottery Ticket case support the view that there is an important and natural use of the English word ‘fairness’ that corresponds with what I have named individual fairness. If Tara receives no chance of winning the lottery, it is intuitively unfair to Tara, even if there are no others to make interpersonal comparisons with. Similarly, no matter how many doughnuts Andrew gives to people other than Ben, there remains a question about whether or not Ben's individual claim is appropriately satisfied. This is what makes the level of Ben's claim satisfaction involve comparisons about different states of affairs possibly affecting Ben rather than interpersonal comparisons between Ben and other claimants. The question of whether or not Ben's individual claim is satisfied (and thus whether his situation is individually fair) is independent of others' claims and whether or not they are satisfied. I am here being explicit about the specific type of fairness this paper focuses on. Individual fairness concerns the satisfaction (and lack of satisfaction) of each individual claimant's individual claims.

In focusing on individual fairness, I do not reject the existence or importance of other types of fairness, such as Broome's interpersonally comparative fairness. Like Hooker (Reference Hooker2005) and Wallace (Reference Wallace1994), I contend that a philosophical theory of fairness that comprehensively vindicates all of the varying uses of the word ‘fairness’ as well as the intuitive judgements about how fairness works in both cases and principles must posit multiple types of fairness. Individual fairness is just one type of fairness and in a pluralist account of fairness is complementary to other types of non-individual fairness.

While I have argued that my topic is a particular type of fairness, the central view in this paper does not fundamentally depend on this particular use of this variously used term. For the purposes of this paper, I need to claim that some deviations are worse with respect to some value. Applied to the Sole Lottery Ticket case, it is worse in one respect if Tara receives less than the 1% chance of winning she has a claim on. This is the type of value I measure in this paper. I have given intuitive reasons that the particular type of value in question is a type of fairness. However I leave it open to readers to substitute some other name for the value in question if they prefer. For example, you could substitute ‘most appropriate’ instead of ‘most fair’; or ‘individual justice’ instead of ‘individual fairness’.Footnote ¹² With the exception of section 2.6 where I discuss an implication of my view for the Scarce Drug case and for Broome's theory of fairness, whenever I refer to fairness or unfairness hereafter, I will be referring to what I call individual fairness or individual unfairness.

Now that I have defended the account of fairness this paper focuses on, I provide an account of the relation between unfairness and disvalue. It is bad when someone receives an unfair chance. To see this, keeping fixed that Tara does not win the prize, it is better if Tara receives her most fair chance (1%) than if she unfairly receives no chance. This paper will focus on how non-instrumentally unfair it is that a claimant does not receive their most fair chance. For brevity, from here onwards, unless otherwise specified, when I claim something is better or worse, I will be referring to whether it is non-instrumentally better or worse, regarding this type of fairness. Where necessary, I distinguish an increase in valuable well-being (i.e. something that is better for some person) from an increase in valuable fairness by referring to the former as being ‘personally better’. In this paper I will not take a stand on whether or not it is personally better for B to be treated fairly than for B to not be treated fairly.

2.1. Positive and negative symmetry

The Simple View depends on only the absolute size of the deviation and therefore entails what I call:

Stipulate that Alex Fifty’s most fair chance is 50%. According to Positive-Negative Symmetry, it is equally unfair that Alex Fifty receives a 60% chance or a 40% chance. The negation of Positive-Negative Symmetry is:

Against the Simple View, I argue for this asymmetry.Footnote ¹³ For reasons specified in Section 1.2, judgements about the existence of a claimant's justified complaint against the benefit distributor provide evidence about unfairness. If Alex Fifty receives less than her most fair chance, as happens if she receives a 40% chance, intuitively she has a justified complaint of this kind. However if Alex Fifty receives a 60% chance, she does not. This is because if a claimant receives more than what they have a claim on, it is not unfair and the claimant has no justified complaint that their claim has been violated.Footnote ¹⁴ To see this, return to the case from Section 1: if Ben receives a 100% chance of receiving two doughnuts from Andrew, he has had his claim fully satisfied. This is because in this situation, Ben receives the one doughnut he has a claim on and something above and beyond what he has a claim on (i.e. the second doughnut). If Ben has no other unsatisfied claims by stipulation, the way Ben has been treated in this situation is fair.Footnote ¹⁵ Analogously, if Alex Fifty receives a 60% chance, she gets the 50% chance she has a claim on and something above and beyond what she has a claim on. This is evident in the intuitive sense in which her 60% chance is partially composed of a 50% chance (analogously, Ben's two doughnuts is partially composed of a single doughnut).

The lack of claimants’ justified complaints when they receive more than their most fair chance supports a particular version of Positive-Negative Asymmetry, namely:

These intuitive judgments suggest that any intuitively plausible measure of the unfairness of deviations should vindicate a particular version of Positive-Negative Asymmetry, namely Ideally Fair Positive Deviations.

When we relax the assumption that there is always a particular most fair chance that the individual has a claim on, and consider cases where an individual has a claim on a range of chances above a particular chance and has a range of most fair chances, an intuitive implication of Ideally Fair Positive Deviations is evident. To see this, consider a variant of the Alex Fifty case where the only difference is that Alex-At-Least-Fifty has a claim on receiving at least a 50% chance and any chance of 50% or above is their most fair chance. According to Ideally Fair Positive Deviations, there is no difference in unfairness for chances above 50% for Alex-At-Least-Fifty and Alex Fifty. For example, it is just as fair for Alex-At-Least-Fifty to receive a 60% chance as it is for Alex Fifty to receive that same 60% chance. This is an intuitively welcome implication. There is no intuitive difference in the amount of fairness of any particular deviation that both Alex Fifty and Alex At-Least-Fifty are subject to.Footnote ¹⁶ In cases where there is a range of claimed, most fair chances (e.g. 20–40%), we can treat it as if their most fair chance is the lowest point in the claimed range when ascertaining the unfairness of deviations.

2.2. Linear changes in fairness

Keeping the most fair chance fixed, I argue that the disvalue of negative deviations from the most fair chance can be represented by a linear function of the size of the deviation. We can see this is intuitively supported by considering subsequent equally sized changes from the most fair chance. For Alex Fifty, is the change from a 50% chance to a 49% chance equally bad regarding unfairness as the change from a 49% chance to a 48% chance or a 25% chance to a 24% chance? While I can see some intuitive reasons why these changes may not be of equal importance to Alex Fifty's preferences, I argue that these intuitive reasons do not justify the view that these changes are of unequal importance to fairness. If Alex Fifty has risk averse preferences (i.e. Alex Fifty attaches greater preferential weight to increases in her chances of a good when they are low than when they are high), the change from 25% to 24% will be worse regarding her preferences than the change from 49% to 48%.Footnote ¹⁷ However it is important to distinguish preferences from fairness, as otherwise our intuitions regarding one subject can adversely affect intuitions about the other. In general, a claimant's preferences are not an accurate guide to how fairly they have been treated. To see this, note that a claimant will prefer having a greater chance of benefiting, even if that greater chance of benefiting is not fairer, as is the case when a claimant receives more than the most fair chance. While these changes may not be of equal preferential importance, these changes are intuitively equally bad regarding fairness. These intuitive judgements about fairness support the following view:

Equally Weighty Deviations is implied by the Simple View, and is one intuitive merit of that view. The intuitive judgements about the shifts in Alex Fifty's chances in this subsection show that a plausible measure of the unfairness of deviations should vindicate Equally Weighty Deviations.

2.3 The proportional view

In this section I argue that the unfairness of a negative deviation depends on what proportion of the claimant's most fair chance that deviation constitutes. Furthermore, I argue that all other things being equal, this allows us to compare the unfairness of negative deviations between individuals who have different most fair chances.

Consider two claimants that only differ in their most fair chance – Alex Fifty and Alex Two. The difference between the two claimants is just that Alex Two's most fair chance is 2% instead. We can imagine Alex Fifty and Alex Two are each subject to a 1% negative deviation. With the change, Alex Fifty now has a 49% chance of benefiting and Alex Two has a 1% chance of benefiting. Intuitively, even though the size of the deviation is the same, the effect on individual fairness is different for Alex Fifty than Alex Two. It is worse, regarding fairness, that Alex Two is subject to this deviation.

Consider now Alex Twenty, whose most fair chance is 20%. A 1% negative deviation makes a larger difference to unfairness if Alex Twenty is subject to it than if Alex Fifty is subject to it. Similarly, a 1% negative deviation is more unfair when it applies to Alex Two than when it applies to Alex Twenty. These intuitions support the view that the impact on unfairness of absolutely equally sized deviations is dependent on the beneficiary's most fair chance. It remains a question precisely how it depends.

One defensible view is that how unfair a deviation is does not fundamentally depend on the absolute size of the deviation but instead what proportion of the beneficiary's most fair chance the deviation constitutes. This is supported by our earlier intuitive judgements about Alex Fifty, Alex Two and Alex Twenty. A 1% deviation is a much larger proportion of Alex Two's most fair chance than Alex Fifty's. For Alex Two, it constitutes a change of half of her most fair chance. For Alex Fifty, it is only 1/50th of her most fair chance. Our intuitions in cases like these are captured by the following view:

Proportional Deviations and the abovementioned intuitive judgements about this trio of cases are compatible with Equally Weighty Deviations. This is because even though the size of all of the deviations is the same in the three cases (−1), the three cases involve different most fair chances. Equally Weighty Deviations only claims that for any given most fair chance, the disvalue of negative deviations from the most fair chance can be represented by a linear function of the size of the deviation. Equally Weighty Deviations implies that the change for Alex Fifty from a 50% chance to a 49% chance is no worse, regarding unfairness, than the change from a 49% chance to a 48% chance. It is compatible with the claim that the change for Alex Two from a 2% chance to a 1% chance for Alex Two is worse and more unfair than the change from a 2% chance to a 1% chance for Alex Fifty.

Furthermore, judgements about Alex Fifty, Alex Twenty and Alex Two show that our intuitions allow us to make inter-individual comparisons of individual fairness when the only relevant difference between individuals is the most fair chance.Footnote ¹⁸ A negative deviation of 25% for Alex Fifty is proportionally equivalent to a negative deviation of 1% for Alex Two. Both deviations constitute a change of half of her most fair chance. It is intuitively equally bad regarding fairness for Alex Fifty to be subject to a 25% negative deviation as it is for Alex Two to be subject to a 1% negative deviation. While Proportional Deviations is a view of how unfair a particular chance of benefiting is to each individual, the proportion can be used to compare the unfairness of different individuals’ deviations. All other things being equal (e.g. the strength of different individuals’ claims), a 25% deviation for Alex Fifty is equally bad regarding fairness as a 10% deviation for Alex Twenty or a 1% deviation for Alex Two.

The intuitive judgements in this subsection support a measure of the unfairness of deviations that vindicates Proportional Deviations.

2.4. A supplementary argument for Positive-Negative Asymmetry

In this section, I provide a supplementary argument in support of Positive-Negative Asymmetry. I argue that if one accepts Proportional Deviations, Positive-Negative Symmetry is not tenable and thus we have reason to accept its negation. This reason is independent of the argument I previously gave for Positive-Negative Asymmetry. Therefore, even if you do not accept my earlier argument for Positive-Negative Asymmetry, there is another reason to accept it.

Consider the view that captures both Proportional Deviations and Positive-Negative Symmetry. Call this view the Symmetrical Proportional View. This view can be represented by the following equation:

$$\begin{equation*} {\rm{For\; all\; values\; of\; }}{f_{{\rm{(B)\;}}}}{\rm{and \;}}{{{p}}_{{\rm{(B)}}}}{\rm{,\; then\; SPU = }}\left( {{\rm{| }}{f_{{\rm{(B)\,}}}}{\rm{ -\, }}{{{p}}_{{\rm{(B) }}}}{\rm{| }}} \right){\rm{ / }}{f_{{\rm{(B)}}}} \end{equation*}$$

Like the Simple View, this equation's output represents how unfair a deviation is – the larger the output, the more unfair the deviation.

In order to see the incompatibility of Proportional Deviations with Positive-Negative Symmetry, consider the unfairness of deviations when an individual's most fair chance is 0%. Call such cases Zero Cases. When Proportional Deviations is combined with Positive-Negative Symmetry, the resulting view is not able to provide an answer about how unfair deviations are in Zero Cases. This is because any deviation at all will not be a proportion of the most fair chance. That is because it is impossible to divide by zero. This is a significant problem for any view that vindicates Positive-Negative Symmetry. I can see no theoretically defensible way of combining Positive-Negative Symmetry and Proportional Deviations that gives a non-ad hoc account of Zero Cases. If we accept Proportional Deviations, the fact that the Proportional Deviations is incompatible with Positive-Negative Symmetry gives us further reason to accept Positive-Negative Asymmetry.

2.5. Meeting the three intuitive criteria

While the Simple View is initially plausible, like many simple philosophical views it generates counterintuitive results. In particular, I have argued that it generates implausible results by entailing Positive-Negative Symmetry, and by denying Proportional Deviations. I have argued that we should adopt a measure that vindicates the following four criteria: Positive-Negative Asymmetry (and in particular, Ideally Fair Positive Deviations); Equally Weighty Deviations; and Proportional Deviations. As Ideally Fair Positive Deviations entails Positive-Negative Asymmetry, we can proceed by considering only the latter three criteria as necessary conditions on an intuitively plausible unfairness measure. I now present a piecewise function that captures these three conditions. Call the measure of deviations’ unfairness according to this piecewise function the Asymmetrical Proportional View (APU). This piecewise function's outputs represent the unfairness of deviations – the larger the output, the more unfair the chance is.

We can represent this view about the unfairness to an individual B who receives a negative deviation (i.e. a smaller chance than their most fair chance, f _(B), the chance they have a claim of a particular strength on) using the following equation:

$$\begin{equation*} {\rm{For\; all\; }}{f_{{\rm{(B)}}}}{\rm{\, -\, }}{{{p}}_{{\rm{(B)\,}}}}{\rm{ > 0,\; then\; APU = }}\left( {{f_{{\rm{(B)}}}}{\rm{\, - \,}}{{{p}}_{{\rm{(B)}}}}} \right){\rm{ / }}{f_{{\rm{(B)}}}} \end{equation*}$$

This part of the view respects Equally Weighty Deviations. To see this, keeping the most fair chance (f _(B)) fixed, note that doubling the size of the negative deviation will always lead to a doubling of the unfairness of that deviation. Furthermore, note that each subsequent equally sized negative change in chances of benefiting (e.g. the change from a 50% chance to a 49% chance and the change from a 49% chance to a 48% chance for Alex Fifty) will constitute a change of the same proportion of the most fair chance (e.g. each 1% change constitutes a 1/50th proportion of Alex Fifty's most fair chance).

According to Ideally Fair Positive Deviations, all positive deviations are as fair as the most fair chance. This can be represented with the following equation:

$$\begin{equation*} {\rm{For\; all\, }}{f_{{\rm{(B)}}}}\, - {{{p}}_{{\rm{(B)}}}} \le {\rm{0; APU = 0}} \end{equation*}$$

This part of the view has two other merits. Like the Simple View, it claims that when the actual chance of benefiting is the most fair chance, there is no unfairness. Second, it claims that because all deviations in Zero Cases are positive deviations, all of those deviations are ideally fair. This feature of my view avoids the problems that the Symmetrical Proportional View has with Zero Cases.

Not only does the Asymmetrical Proportional View capture the three intuitively supported conditions, it also has three practically important merits. First, it can be used to generate a complete, transitive ordering from most fair to least fair chances of benefiting according to the ‘at least as unfair as’ relation. The higher the equation's output, the more unfair the deviation. For a particular individual with a claim of a particular strength on receiving a chance of a good, some chance p ₁ is at least as unfair as chance p ₂ if and only if the functional output when p ₁ is a functional input is greater than or equal to the functional output when p ₂ is a functional input. The ideally fair chances are the most fair chance, and all positive deviations from it. The larger the negative deviation, the more unfair it is. Furthermore, because unfairness is bad, when all other things are equal, the ‘at least as bad regarding unfairness’ relation is co-extensive with the aforementioned ‘at least as unfair as’ relation. Assuming we know what the feasible options are, the Asymmetrical Proportional View can be used to ascertain what the most fair feasible chance(s) of benefiting are. Second, the Asymmetrical Proportional View is a cardinal measure of how unfair any deviation is. For example, a negative deviation that constitutes one half of a claimant's most fair chance is twice as unfair as a negative deviation that constitutes one quarter of that same claimant's most fair chance. As noted earlier, this is important for trade-offs with other values. Finally, the Asymmetrical Proportional View can be used to make certain inter-individual comparisons of unfairness. It explains, for example, that when all other things (such as claim strength) are equal, a 1% negative deviation for Alex Two is twenty-five times worse, regarding unfairness, than an 1% negative deviation for Alex Fifty. I will now outline an important implication of this third feature before discussing more general implications of my paper's arguments for broader debates about fairness and fair lotteries.

2.6. Implications of the Asymmetrical Proportional View

The Asymmetrical Proportional View entails that, when all other things are equal, it is significantly worse, regarding fairness, that those with a small most fair chance are subject to an absolutely small (but proportionally large) negative deviation than those with a large most fair chance subject to that same absolutely small negative deviation. One implication of this is that we should be more concerned about small negative deviations to those who have absolutely low most fair chances than to those who have absolutely high most fair chances. For example, consider a rigged roulette wheel with 100 pocketsFootnote ¹⁹ numbered 0 through 99. This wheel is rigged against the number ‘1’ so there is a 0% chance the ball will land on ‘1’. All other numbers have a 1% chance of being selected except for ‘0’ which has a 2% chance of being selected. Gamblers are unaware of the rigging of the wheel and the bets are being sold by the casino as if each number has an equal 1% chance of being selected. According to my Asymmetrical Proportional View, all other things being equal, using this rigged roulette wheel is more unfair to someone who bets once to win only when the number 1 is selected (giving them a claim on a 1% chance of winning and a 1% most fair chance) than someone who bets once to win when any one of the fifty odd numbers is selected (giving them a claim on a 50% chance of winning and a 50% most fair chance). This is because even though both bets are subject to an equally sized negative deviation (−1) because there is no chance that ‘1’ will be selected, the deviation constitutes a larger proportion of the ‘1’ bet (1/1) than the ‘odd numbers’ bet (1/50). A casino that uses such a rigged wheel owes greater compensation to those that were treated more unfairly by spins of this wheel than those who were treated less unfairly.

Finally, I highlight some implications of this paper for more familiar lottery allocation cases like Scarce Drug and for Broome's influential theory (Broome Reference Broome1990, Reference Broome1998). In order to distinguish individual fairness from all-things-considered fairness I will no longer abbreviate ‘individual fairness’ to ‘fairness’. On my view, there are distinct types of fairness (including individual fairness), that can contribute to whether or not something is all-things-considered fair. In this paper, I have not argued for the existence of types of fairness other than individual fairness, and will assume that the type of interpersonally comparative fairness that Broome defends is an extant type of fairness. However I disagree with Broome that his type of fairness is the only extant type of fairness.

This disagreement can be illustrated by considering what I call the Any Equal Chances View. The Any Equal Chances view is implied by Broome's more general view that in cases like Scarce Drug, fairness requires the proportional satisfaction of claims (Broome Reference Broome1990).Footnote ²⁰ The Any Equal Chances View is that in cases like Scarce Drug, all other things being equal, if all claimants each receive an equal positive chance of receiving the same benefit they each have a claim on, it is all-things-considered equally fair, no matter what the precise percentage is. For example, in such cases, all other things being equal, Broome contends that there is no difference in all-things-considered fairness between giving all potential beneficiaries the same 50% chance of benefiting and giving all potential beneficiaries the same 5% chance of benefiting. This is because Broome contends that there is only one type of fairness, comparative fairness, and thus all-things-considered fairness is solely determined by comparative fairness. Comparative fairness depends only on interpersonal comparisons between claimants – if all claimants with equal claims receive equal treatment, it is fair. Broome rejects other types of fairness, such as individual fairness. The example above illustrates that the Any Equal Chances View has a counterintuitive implication.

My view about individual fairness explains why this implication of Broome's Any Equal Chances View is incorrect by explaining why there is a difference in all-things-considered fairness in the example above. While there is no difference in Broome's comparative type of fairness between both claimants in Scarce Drug getting an equal 50% chance of cure and both getting an equal 5% chance of cure, there is a difference in individual fairness. If the arguments in my paper are sound, each Scarce Drug patient has a claim on a 100% chance of a cure and it is less individually fair to each patient if they each get 5% chance of cure than if they each get a 50% chance of cure. According to the Asymmetrical Proportional View, it is significantly worse (almost twice as bad) regarding individual fairness that each Scarce Drug patient receives a 5% chance of cure than if they receive a 50% chance of cure. In Scarce Drug, the smaller the chance of cure to each individual, the more individually unfair the chance allocation is. In summary, my view about individual unfairness explains why it is not equally all-things-considered fair to give all Scarce Drug patients an equal 50% chance of cure as it is to give them all a 5% chance of cure; and relatedly my arguments provide a pro-tanto reason to reject two of Broome's views: Broome's Any Equal Chances View and Broome's view that there is only one extant type of fairness, comparative fairness.