On Fair Lotteries
1. Introduction: When James Watson and Francis Crick submitted to Nature their groundbreaking paper relating DNA structure to protein synthesis, they faced a choice. In what order were their names to be listed? Would it be “Watson and Crick,” or “Crick and Watson?” They resolved the matter by tossing a coin (Crick, 1988, p. 66).1
Watson and Crick—I shall use the same ordering of names that they did—faced a problem of justice in placing their names on their seminal paper. There was something of value to be had—top billing on a groundbreaking paper. Either Watson or Crick, but not both, could enjoy this good. Each scientist could give good reasons for listing his name first; both had, after all, worked hard to produce the results contained in the paper. Thus, each had a claim to the good. If the claim of one had been clearly stronger than that of the other, then justice would have required that the superior claimant receive the good. Had, for example, Watson done more work on the project, a case could have been made for “Watson and Crick” on the basis of merit. Alternatively, if Watson had been more well-established, and therefore less in need of a boost in reputation, then grounds of need might have recommended the ordering “Crick and Watson.”2 In short, had a uniquely
strong claim existed, the use of a coin toss would have made no sense.
In this case, however, the reasons that could be given on behalf of “Watson and Crick” were just as good as the reasons for “Crick and Watson;” neither man’s claim to top billing was stronger than the other’s. As a result, claim strength proved indeterminate in deciding who ought to receive the good. In effect, there was a tie. Some sort of tiebreaking procedure was needed to resolve this indeterminacy. Hence the coin toss.
1 Francis Crick, What Mad Pursuit: A Personal View of Scientific Discovery (New York: Basic Books,
1988), p. 66.
2Merit and need are two of the most natural sources of claims. On their respective roles when justice is at
The case of Watson and Crick is not an aberrant example. There are many allocative decisions in which considerations of justice prove indeterminate.3 People need
organ transplants, and society is reluctant to decree one life to be more valuable than another. Would-be emigrants from the Third World wish to enter the United States in numbers far greater than that country would realistically consider admitting; most of them have no better claim to admission than any of their rivals. Troops are needed in times of war. Young men (and possibly women) must be called to service, but not all are needed; exemptions are available, and there’s often no particular reason to give these exemptions to some rather than others. Ties must be broken in all of these cases, and so tiebreaking is anything but a marginal phenomenon. Moreover, while non-academics may dismiss the Watson and Crick case as trivial, nobody can treat lightly such topics as organ transplantation or military conscription. Sometimes, the resolution of indeterminacy is literally a matter of life and death.
But although ties must be broken when considerations of justice prove indeterminate, it remains true that not all tiebreakers are equally just. True, tossing a coin was not the only option available to Watson and Crick. They could have rolled a die, with an even number leading to “Watson and Crick” and an odd number leading to “Crick and Watson.” Alternatively, they could have drawn from a deck of cards, with the high card (or low card) granting the right to be listed first. But suppose Crick had proposed an alphabetical listing. Or imagine that Watson had proposed a listing by age, from youngest to oldest. (Watson was born in 1928; Crick was born in 1916.) Few would defend such
3 Indeed, even within the world of scientific discovery, Watson and Crick were not the first to employ the
tiebreakers as just. In short, the demands of justice do not end once the candidates with the strongest claims to a good have been identified. These demands must also be satisfied when one of these candidates is chosen as the ultimate recipient of the good.
This raises the question of how to distinguish just from unjust tiebreaking procedures. Philosophers concerned with justice have devoted surprisingly little attention to this question. Rather, they have assumed that some paradigmatic tiebreaking procedures—coin tosses, balls drawn from an urn, drawn straws—are just. They have attempted neither to define the class of just tiebreaking procedures more precisely nor to explain why all and only all the procedures in this class should be counted as just.4
There have been few efforts to break this rather surprising philosophical silence. The most prominent of these efforts is George Sher’s classic paper “What Makes a Lottery Fair?”5 Sher uses the term lottery to describe any tiebreaking device that could be
used to resolve indeterminacies with respect to the allocation of goods. He then sets out to distinguish between fair lotteries, which can be used to resolve indeterminacies justly, and unfair lotteries, which cannot. Sher defends a broad definition of a fair lottery, one that would encompass coin tosses, such as that used by Watson and Crick, but also many other procedures. But Sher not only offers a definition of a fair lottery; he provides an argument as to why it is just to use a fair lottery to break ties in this manner.
Since the publication of Sher’s seminal paper, there have been few sustained efforts to examine the relationship between lotteries and justice.6 While this literature has
4 For example, in his study of the concept of equity, H. Peyton Young notes that ties between applicants for
a good with equally good claims “could be resolved by a chance device, such as tossing a fair coin.” But Young neither delineates what counts as a “chance device,” nor indicates why chance devices are superior to other tiebreakers, such as selecting the applicant with the nicest smile. See H. Peyton Young, Equity: In Theory and Practice (Princeton, NJ: Princeton University Press, 1994), p. 33.
5Noûs 14 (1980): 203-216.
6 See, e.g., Lewis A. Kornhauser and Lawrence G. Sager, “Just Lotteries,” Social Science Information 27
produced valuable results, it has failed to confront systematically the complete problem posed by Sher. Some works attempt to explain why justice might require resort to a lottery, while relying upon an intuitive understanding of what a lottery is. Others grapple with the problem of defining a fair lottery, but take for granted the relationship between lotteries and justice. Few connect together the tasks of definition and defense, and none confront Sher’s argument systematically. Has Sher provided an adequate definition of a fair lottery? And has he successfully linked fair lotteries to justice? This paper offers answers to these two questions.
Section 2 lays out Sher’s definition of a fair lottery and explains why he believes justice requires using such a lottery as a tiebreaker. In section 3, I argue that Sher’s definition suffers from three critical failings. It fails to guarantee that a fair lottery will even exist; it fails to include all potential tiebreaking devices which intuition suggests should count as fair lotteries; and it allows the extent of the class of fair lotteries to depend upon arbitrary factors. In section 4, I point out where Sher’s definition goes wrong and offer two corrections. Section 5 concludes by suggesting why justice might demand that ties be broken using a fair lottery satisfying the definition I endorse. It also draws lessons from my disagreement with Sher regarding contractarianism and justice.
Before proceeding, a small terminological problem is worth settling. A tiebreaking procedure—a method for resolving indeterminacy—is needed whenever there is a set of people with maximally strong claims that is larger than the quantity of goods to be distributed. Each of the persons in this set has a claim that is as strong as the claim of every other person in the set; moreover, each person in the set has a stronger claim than
anyone outside of the set.7 Some of the people outside of this set may have weaker
claims; others may not have claims at all. (In effect, their claims are of “zero” strength.) A healthy person has no claim to an organ transplant, whereas a moderately sick person may have a weak claim. It is cumbersome to reiterate all of this each time I must refer to an individual in this set, and so I shall refer to each such individual as a strongest claimant. This terminology is a little misleading. One normally takes the adjective “strongest” to denote uniqueness; if there is a strongest claimant, then that seems to imply that this person is also the strongest claimant.8 I do not intend this implication. It is the
fact that there are multiple strongest claimants—more than there are goods to go around —that prompts the need for tiebreaking in the first place. This is not the neatest terminological solution, but it will have to do.
2. Sher on Fair Lotteries: “It is generally agreed,” writes Sher, “that when two or more people have equal claims to a good that cannot be divided among them, the morally preferable way of allocating that good is through a tie-breaking device, or lottery, which is fair.” Sher aims to discover “(a) exactly what conditions are necessary and sufficient
7 I assume that the relationship “has at least as strong a claim as” constitutes an ordering, in that it is
complete, reflexive, and transitive over the set of all possible claimants. This assumption would be violated if, for example, some claims were incommensurable—if one were unable to say, for example, whether a sickly young child or a healthy older person had a stronger claim to a vital organ transplant. For more on incommensurability, see Ruth Chang, ed., Incommensurability, Incomparability, and Practical Reason
(Cambridge, MA: Harvard University Press, 1997). While there is much to be said regarding incommensurability, it introduces a complicating factor that I shall avoid here.
8 A good example of the confusion that can arise from such terminological problems occurs in Plato’s
for a lottery to be fair, and (b) why it should be morally preferable to allocate indivisible contested goods through lotteries which satisfy these conditions.”9
It is worth emphasizing the expansive nature of the term “lottery” as used by Sher. On his account, a lottery is any procedure that can be employed to decide how to resolve an indeterminacy that arises in the distribution of a good.10 This definition
obviously includes all the standard devices one would regard as lotteries, such as tossing a coin or drawing balls from an urn. But it also includes other tiebreaking rules that do not involve any of the randomness traditionally associated with lotteries. It includes, for example, a procedure that awards the good to the tallest person, or the wealthiest, or the best friend of the person implementing the tiebreaker. The inclusiveness of the set is in some respects good; Sher is concerned, after all, to distinguish the good (fair) ways of resolving indeterminacy from the bad (unfair) ways, and if he can make this distinction hold for every possible device for resolving indeterminacy, so much the better. But in other respects it poses problems; many people, I suspect, might equate what Sher calls “fair lotteries” with lotteries in general. It is therefore important to keep Sher’s broad definition in mind in considering the definition he ultimately offers of a fair lottery.
Sher begins his argument with the following tentative definition of a fair lottery: (P1) A lottery L held at t is fair if and only if the hypotheses that different claimants
will win are all equally well confirmed by the totality of evidence available at t.
9 Sher, “What Makes a Lottery Fair?” p. 203. Sher also makes plain that the avoidance of bads can be
treated in the same way as goods. Avoiding the military draft matters in much the same way as receiving an organ transplant, and so what counts as a fair lottery for distributing the latter will do likewise when the former is concerned (p. 214).
10 Other authors have held similarly expansive accounts as to what should count as a lottery. Lewis
Kornhauser and Lawrence Sager, for example, write that “A lottery allocates a benefit (sometimes called a ‘prize’) among a designated group of potential beneficiaries (‘candidates’ who comprise a ‘pool’)
This definition has several clear advantages going for it. It certainly accords with our everyday intuitions regarding what a fair lottery looks like. It also clearly includes procedures like drawing straws, or coin tosses—indeed, all of the paradigmatic examples one might choose to offer of fair lotteries. It can also explain the intuition that a lottery’s fairness “depends on its affording all claimants an equal chance” in a way that avoids philosophical difficulties. If the idea of an “equal chance” were interpreted ontologically, for example, then the question of whether any lottery could ever be fair would depend on whether or not the universe was deterministic. The tentative definition with which Sher begins his analysis eludes this problem by focusing on what people know, or might reasonably be expected to know given the best available evidence.11 A coin toss is thus
fair on this definition, no matter what the physics of a nickel navigating its way through the air may ultimately turn out to involve.12
Despite these advantages, Sher rejects this definition as too restrictive. (The reasons for this rejection will be discussed later.) After an extended argument, and two more tentative definitions (P2 and P3, which I shall not discuss here), Sher offers a fourth, more elaborate definition.
(P4) A lottery L is fair if and only if there is no person q such that
(a) q desires that L’s contested good be awarded to a particular claimant or type of claimant, and
(b) q’s desire is not shared by all the claimants to L’s contested good, and
11 Sher, “What Makes a Lottery Fair? p. 204. The idea of equal chances cannot rest on a frequentist
conception of probability because, as Sher notes, such a conception cannot assign chances to single events, which is what is needed here.
12 At least as traditionally understood. Recent mathematical models suggest that coin tosses may be slightly
(c) q knows that his performing an action of type A will increase the probability of his desires being satisfied, and
(d) q performs an action of type A on the basis of this desire and the belief-component of this item of knowledge.13
Sher summarizes this definition as stating that “any lottery initiated to aid a preferred claimant or type of claimant is unfair unless its initiator’s preference is shared by all the claimants involved.”14 It is this move to a more complex definition, as well as the
reasoning behind the move, that I wish to contest here.
Sher regards his more complex definition of a fair lottery as adequate because he believes it captures the logic of what it means for individuals to have equally strong claims to a good. “It is no part,” he writes,
of our concept of strongest claims to goods that a person with such a claim is entitled to delegate the relevant good as he prefers. Many claims to goods, such as claims to jobs or admission to competitive institutions, are nontransferable. It is part of our concept of strongest claims to goods, however, that when someone has such a claim, no one else is entitled to enjoy or dispose of the relevant good as he alone prefers. If n has the strongest claim to G, then any other person who either arrogates G to himself or delegates it to another on the basis of preference different from n’s is ipso facto infringing on n’s rightful claim to it.15
If someone wants to distribute the good in a particular manner, and n doesn’t like it, then n has the right to say no. To have a strongest claim to a good, Sher argues, is at the very least to have the right to veto a procedure for distributing the good.
13 Sher, “What Makes a Lottery Fair?” p. 212. 14 Sher’s emphasis. Ibid., p. 211.
When there are multiple strongest claims to a good, this implies a veto held by each of the strongest claimants. Sher’s description of the implications of this position regarding lotteries is worth quoting at length:
More specifically, since it is part of the concept of claims to goods that no one without the strongest claim to G may delegate G as he pleases, the mere fact that n, o, and p share strongest claims to G must itself entail that no person different from them can legitimately take any step aimed at awarding G to a person or type of person who he, but not all the claimants, favors. Any non-claimant who took such a step—including the step of advancing a lottery with this aim—would ipso facto violate at least some claimant’s strongest equal claim to G. Furthermore,
given the equality of their claims, it also follows that no member of n, o, and p may himself take any step towards directing G to a person or type of person whom he, but not all the others, favors. Any claimant who took such a step—including again the step of advancing a lottery with this aim—would necessarily violate the others’ equal claims by attempting to impose his preferences on them. Taken together, these considerations entail that no person at all may legitimately advance any lottery with the purpose of awarding the contested good to a person or type of person whom he, but not all the claimants, favors. This conclusion, however, is identical to the principle whose grounding we have been trying to understand.16
In other words, no tiebreaking device can be employed without the agreement of all strongest claimants. Each of these individuals with a strongest claim must have a veto.
Sher is reluctant to specify too much about the nature of claims to a good. This reluctance makes him go so far as to say that “a satisfactory explanation of the binding
force of fair lotteries must await the construction of a full theory of the grounding of obligations.”17 He therefore restricts himself to saying that, whatever else claims do, they
provide the sort of veto over tiebreaking procedures that he believes is specified by P4. This suggests that a veto of this kind is a necessary condition for a fair lottery. Further, by suggesting that there remains much to be said about the nature of claims, Sher implies that there may be more conditions required in order for a lottery to be truly fair besides surviving the veto power enjoyed by each of the strongest claimants. It turns out, however, that he both sees P4 as necessary and sufficient for a lottery to qualify as fair (as evident from the definition of P4), and views P4 as equivalent to the requirement that each individual with a strongest claim have a veto over the tiebreaking procedure chosen. This must mean that any tiebreaking procedure that remains unvetoed by any individual with a strongest claim must ipso facto be fair. In effect, the strongest claimants jointly possess the power to decide how the good should be distributed.18
Before moving on, I should note that technically, Sher’s definition speaks of what the strongest claimants desire, not of what receives their consent. According to Sher, it does not matter if any claimant was explicitly asked for permission to use a particular lottery in order to resolve the indeterminacy at hand. Indeed, it does not even matter if the agent in charge of making the allocation knows that the tiebreaking procedure he has adopted accords with the desires of the entire set of strongest claimants; the mere fact that
17 Ibid., p. 216, n. 7.
18 In effect, Sher would have the strongest claimants constitute a collegium. In social choice parlance, a
collegium consists of a rule for rank ordering alternatives that designates a specific group as having special powers over the ordering. When any member of that group ranks one option x over another option y, then regardless of the rankings held by non-group members, the rule cannot rank y over x. Furthermore, when all group members rank x ahead of y, then the rule must rank x ahead of y, again regardless of the rankings held by non-group members. On the nature of collegiums, see David Austen-Smith and Jeffrey S. Banks,
it does so accord is sufficient.19 But this is only a small complication. It is clear from
Sher’s defense of the definition he proposes that what matters is whether the strongest claimants would grant permission if asked, regardless of whether or not permission is in fact asked. I shall return to this point in my conclusion to this paper.20
3. Objections to Sher’s Definition: It is immediately apparent that Sher’s definition is only plausible given certain tacit restrictions. These restrictions would prevent, for example, the set of strongest claimants from jointly doing whatever they like with the good—throwing it away, for example, or giving it to some non-claimant. While there may be scenarios in which justice permits the strongest claimants to do what they will, this clearly does not hold for the general case. Perhaps the strongest claimants to a candy bar can make whatever use of it suits their fancy, but the strongest claimants to a kidney transplant can, at most, agree to a method of allocating the transplant to one of their own.
Even with such restrictions, however, Sher’s definition of a fair lottery remains subject to three serious objections. First, there is no reason to expect that any tiebreaking procedure will satisfy the definition. Second, even if some tiebreaking procedures do satisfy the definition, the paradigmatic cases of fair lotteries may not be among them. Third, the class of lotteries that count as fair will vary with a variety of arbitrary factors.
How might P4 leave the class of lotteries empty? Consider the following example, taken with some modifications from Sher.21 Suppose that n and o comprise the set of
strongest claimants to a good. Claimant n would like the good, but he considers it far
19 Sher, “What Makes a Lottery Fair?” p. 211.
20 In their exploration of lotteries, Kornhauser and Sager concern themselves “with social allocations that
do not involve express agreements of the competing claimants to the pertinent social good.” They “refer to instances of such social imposition of allocations by lot as ‘social lotteries.’” See Kornhauser and Sager, “Just Lotteries,” p. 484. Kornhauser and Sager thus approach the problem of defining lotteries in a manner diametrically opposite to that of Sher.
more important to deprive o of the good. Similarly, o would like the good, but is even more anxious to make sure n doesn’t get it.22 If n and o were each strongly committed to
denying the other the prize, then n would veto lotteries which gave o a positive probability of receiving the good, and o would do the same for lotteries that might award n the good. Thus, no lottery would survive the screening process.
In such a case, either no solution would ever be reached, or else the only solution upon which both n and o might agree would be to destroy the good. While the latter option has some precedent within the literature regarding fairness and indivisible goods,23
it does not really qualify as a tiebreaking device. It is exactly the sort of option that must be excluded from consideration if the idea of the strongest claimants jointly agreeing upon the allocation of the good is to prove at all plausible. But if a proper solution must award the good in question to somebody in order to qualify as a “tiebreaker,” then it would appear that no tiebreaker is fair in the case of n and o.
The example involving super-rivalrous claimants n and o has other disturbing implications. On Sher’s account, a coin toss would not constitute a fair lottery in this example. To qualify as a fair lottery, the good could not be awarded using a procedure that either n or o might veto. A coin toss, however, would give o a chance of winning the good that n would find unacceptably high, and vice versa. The same logic applies to
22 This could happen, for example, if the good in question were a positional good. The value of possessing a
positional good depends in part upon the positional goods possessed by others. A person might value a sports car, not only because it is flashy, but because it is the flashiest car in his neighborhood. If his neighbor obtains a flashier sports car, part of the value of his own car is thereby lost. The good n and o both desire might function like this, granting so much status to the ultimate recipient (and thereby lowering the status of the other party dramatically) that both will do anything necessary to avoid losing out. On positional goods, and the pernicious impact they can have, see Fred Hirsch, Social Limits to Growth
(Cambridge, MA: Harvard University Press, 1976) and Robert Frank, Choosing the Right Pond: Human Behavior and the Quest for Status (New York: Oxford University Press, 1987).
23 In the Jewish tradition of commentary upon the Talmud, there is a case discussed in which two people are
drawing straws, rolling dice, picking a name out of a hat, etc. Sher’s definition thus excludes even paradigmatic cases of fair lotteries under certain circumstances.24 As will
be discussed in the next section, Sher intends his refined definition to be an expansive one, a definition that will include the obvious cases of fair lotteries and some others as well. Unfortunately, the definition he proposes does not work this way in practice.
Finally, note that there will still be many situations in which a coin toss (or drawing straws, etc.) would count as a fair lottery. Not everyone is possessed by the same kind of envy that motivates n and o in the example given. However, the fact remains that the envy of n and o is an important factor in determining whether a lottery is fair in the situation at hand. Other factors will also matter—for example, the bargaining power of the various claimants. A claimant whose need or desire for the good is less immediate than that of her rival could probably hold out for a lottery more favorable to her than a mere coin toss.25 This means that a coin toss will count as fair in situations where all
claimants have relatively equal bargaining power but not in situations where this power is unequally distributed. And this is an odd result; intuitively, such factors as envy and bargaining power should not effect the fairness of the tiebreaking mechanism.
All three of these objections to Sher’s definition have a common root. Sher makes what counts as a fair lottery in a given tiebreaking scenario dependent upon features of that scenario that have no business playing a role. This is inevitable given Sher’s definition in terms of actual agreement or acceptability to the entire class of strongest
24 Sher himself recognizes that, if n were anxious enough to prevent the good from going to o, it would be
irrational of n to consent to a coin toss to resolve the indeterminacy. Unfortunately, he interprets this to mean that n would not consent to a fair lottery. See Sher, “What Makes a Lottery Fair? pp. 215-216, n. 7. He does not seem to notice that his definition requires n to consent to a lottery before it could be described as fair. He repeats this oversight elsewhere, where he acknowledges that a claimant might object to a lottery that he can foresee would not favor him, but denies that this renders the lottery unfair (p. 215, n. 6).
25 On the importance of the discount rate in determining what bargains are struck, see Jack Knight,
claimants. It may be the case that no lottery can meet this definition in a given context. It may also be the case that none of the canonical examples of a fair lottery will meet this definition in a given context. But regardless, whether or not a lottery meets this definition depends crucially upon the nature of the preferences of the set of strongest claimants, as well as the distribution of bargaining power within this set. And whether or not a tiebreaking procedure counts as fair should not depend upon such factors at all.26
4. An Alternative Definition: In order to construct a more defensible definition of a fair lottery, I shall revisit P1—the first definition of a fair lottery that Sher considers, a definition he ultimately rejects. This definition reads as follows:
(P1) A lottery L held at t is fair if and only if the hypotheses that different claimants will win are all equally well confirmed by the totality of evidence available at t. Sher offers two reasons for rejecting this definition. First, P1 “does not specify which persons must know at t that the entrants’ probabilities of victory are equal.” Sher is willing to assume tentatively that it is the controlling parties of a lottery who must have this knowledge, but later on he modifies this assumption, and he generally considers the problem posed by this question to be a serious one for P1. Second, P1 excludes numerous lotteries that he believes intuitively ought to count as fair. In particular, it excludes any lottery “whose controlling parties know nothing about the entrants’ respective probabilities of victory.”27 Sher would count some, but not all, such lotteries as fair. 26 If someone tosses a coin just to see which side lands face up, does this count as a fair lottery? On Sher’s
account, it is impossible to see how the question could even be answered. Sher defines a lottery, after all, as a tiebreaking device; this implies that a coin tossed in order to break a tie might count as a fair lottery, whereas the exact same coin tossed for some other purpose does not. This result is counterintuitive. Admittedly, lotteries might reasonably be defined in terms of their use, just as with other tools (hammers, etc.). But the definition of a tool should not depend on its actual use for that purpose. A hammer is something useful for pounding nails into wood, but it remains a hammer even when it is not being used for this purpose. Similarly, a lottery (fair or otherwise) is a procedure that can be used to break ties, but it remains a lottery even when it is not being so used. By tying the definition of a lottery so closely to actual use, Sher introduces another irrelevant factor into the process of distinguishing fair lotteries.
It is these two objections that lead Sher to reconsider P1, a reconsideration that leads him ultimately to P4. These two objections, however, are not as telling as Sher seems to think. Consider the first objection—that the definition fails to specify which persons must know that the outcomes of the lottery are equally probable. Sher does not distinguish between probability and confirmation in terms of evidence. He therefore takes for granted a definition of probability in terms of warranted belief. When a process is capable of generating several possible outcomes—like a coin toss, capable of yielding heads or tails—each outcome has high or low probability depending on whether the evidence for predicting the outcome will occur is strong or weak, relative to the evidence for predicting the other outcomes will occur. This is what I refer to as an intersubjective conception of probability, in that it grounds probability in evidence that any individual may, in principle, access. This is in contrast to an objective conception, which makes probability out to be a physical property of the process itself; or a subjective conception, which defines probability in terms of how certain a particular individual is about the outcome of the process. Whereas the objective conception defines probability in terms of what is, regardless of what anyone knows about it; and the subjective conception defines it in terms of what someone believes about the world; the intersubjective conception defines it in terms of what one is justified in believing about the world.
the beliefs of different individuals will be effectively identical,28 but this does not result in
a truly intersubjective understanding of probability. Probability is not a two-way relationship between a belief and a body of evidence, as in the intersubjective conception, but a three-way relationship between belief, evidence, and an individual possessing both. Nevertheless, the subjective conception is also not universally accepted. I shall not attempt to defend the intersubjective conception further here.29 Instead, I shall assume it
is valid, as Sher does, and examine the implications of this assumption.
Assuming the intersubjective conception of probability is valid, as Sher does, renders the first of Sher’s concerns regarding P1 a non-issue. Sher contends that if one is to speak of the probabilities attached to a process, as in a fair lottery, one must also identify an individual to whom the probabilities are attached. This is untrue. If a statement of probability specifies a relationship between evidence and a proposition, then that relationship holds independently of the beliefs of any particular individual. If a body of evidence exists, and if that evidence warrants a prediction that an event will occur with probability p, then one is justified in speaking of that probability without referring to the beliefs of any specific individual. Probabilities are, on this conception, like scientific theories or poems. One is justified in speaking of Special Relativity, or “The Wasteland,” existing in contemporary society without specifying any particular individual who knows either of them. One can do the same with a claim that a body of evidence warrants a certain level of support for predicting a certain event.30 Granted, the relationship between
28 If two individuals each access the same large body of information about a process, their probability
assessments regarding the outcomes of that process will grow closer together. At the limit they will converge, regardless of how far apart they were before the information is obtained.
29 For a more thorough comparison of the three conceptions, see Peter Stone, “What Is a Fair Lottery?”
Unpublished Manuscript (Stanford University, 2007). The most important statement of the intersubjective conception occurs in John Maynard Keynes, A Treatise on Probability (New York: Dover, 2004).
30 Probability statements, like scientific theories and poems, become what Karl Popper describes as “world
a society and a probability assessment remains a bit vague on this story, just like the circumstances under which it is proper to say that some scientific theory is “known” to a society. But this does not change the fact that both types of claims are perfectly intelligible. And this fact obviates the need for P1 to incorporate any reference to specific actors. If the evidence available to a society suggests that each outcome of a lottery will happen with equal probability, as specified in P1, then that lottery is fair as far as that society is concerned. Sher’s first objection to P1 thus fails to hold.
Sher’s second objection to P1 arises from his wish to count as fair lotteries about which nothing is known. If probabilities are assigned to outcomes based on available evidence, then how does one assign probabilities when no evidence is available? One possibility is to assume that, in the absence of evidence, all possible outcomes are equally likely—in other words, to treat it as just another fair lottery. There is some merit in this position. After all, if nothing is known regarding which outcome a lottery will produce, then trivially the evidence provides equal support for each possible predicted outcome. But this argument—which relies upon the principle of insufficient reason—depends critically upon how the outcomes of the lottery are specified.31 Suppose that a die were to
be rolled, and nothing was known about its properties. If the possible outcomes of the die roll were enumerated as “1”, “2”, “3”, “4”, “5”, and “6”, then the argument would assign a probability of 1/6 to each outcome. If, however, the outcomes were enumerated as “1” and “2 through 6”, the argument dictates that each be assigned a probability of ½. These two conclusions contradict each other, but resolving the contradiction requires some
Press, 1972).
“natural” way to specify the set of outcomes. And few probability theorists today believe this is possible.
With all this in mind, Sher regards lotteries about which nothing is known as qualitatively different from lotteries about which something is known. One can make use here of the distinction, first drawn by Frank Knight, between risk and uncertainty.32
Knight distinguished between processes whose outcomes could be assigned probabilities in a reasoned manner, which he described as involving risk, and those for which no non-arbitrary probability assignment was possible, which he described as involving uncertainty. Accordingly, we can distinguish between lotteries with risk and lotteries with uncertainty. This makes possible a further distinction between lotteries with risk for
which all outcomes are equiprobable, and lotteries with risk for which different outcomes have different probabilities, both of which are distinct from lotteries with uncertainty.
Thus, there are equiprobable lotteries with risk, non-equiprobable lotteries with risk, and lotteries with uncertainty. P1, according to Sher, counts only the first type of lottery as fair, whereas Sher would like to add at least some elements of the third as well.
As justification for such a change, Sher offers the following example:
let us imagine that n, o, and p [all of whom have equal claims to some good G] are all about to undergo surgery, and that m [who must decide how to break the tie], knowing nothing about their medical histories, decrees that G will be awarded to whichever claimant has some physical characteristic (say, the smallest liver) which will be discovered during the surgery. Although this lottery is somewhat bizarre, it does not intuitively seem unfair.33
The “liver lottery” is a lottery with uncertainty, and it should count as fair; therefore, P1 is in need of revision. Or so Sher argues.
Setting aside Sher’s concern with what the “controlling parties” know in a lottery (in this case, what m knows about the livers of n, o, and p), I fear that in this case the bizarre nature of the example may obscure a serious objection to the proposed lottery.34
Claimants n, o, or p—or, for that matter, any impartial bystander observing the situation —would be inclined to wonder about m’s choice of the liver lottery. Why did m select this lottery, and not one of the more standard fair lotteries, such as drawing straws? It is of course possible that m selected the lottery for no reason in particular, on a “whim” (which, barring considerations of convenience, is the only way one could choose one fair lottery over another).35 But it is also very possible that m knows something about the liver
lottery that others do not. This would not be difficult; after all, by assumption nothing whatsoever is generally known about the liver lottery. Perhaps m has gleaned some fact about livers that would justify predicting one of the three claimants as the winner, either with certainty or with some high probability. Either way, m could avoid such reasonable suspicions only by employing a lottery about which something is known—in other words, a lottery with risk, not with uncertainty.
More generally, it is very difficult to establish the limits of knowledge. It is easy to demonstrate that a given piece of knowledge exists; one simply has to produce it. But it is hard to imagine proving that this knowledge does not exist. One can always claim not to have it, but such behavior is also compatible with dishonestly withholding the
34 On the difficulties posed by philosophers’ use of bizarre thought experiments, see Kathleen V. Wilkes,
Real People: Personal Identity without Thought Experiments (Oxford: Clarendon Press, 1988), ch. 1; and Daniel Dennett, “Cow-Sharks, Magnets, and Swampman,” Mind and Language 11 (1996): 76-77.
35 In effect, the selection between equally defensible fair lotteries is itself made by a sort of lottery with
knowledge. This poses a serious problem for lotteries. A lottery is only fair, after all, if all the information known about it suggests that all of its outcomes are equiprobable. This may in fact be the case. It may even be the case that it is known to be the case; that is, it is known that this information is known and that it suggests all of the lottery’s outcomes are equiprobable. But it is difficult if not impossible for it to be known that this truly is all of the information known about the lottery. And where doubt exists on this question, the fairness of the lottery is in question.
As a general rule, this poses a bigger problem for lotteries with uncertainty than for lotteries with risk. Intuitively, the more that is known about a process, the harder it is to generate new information about the process that is radically at odds with the old information. Coin tosses, for example, have been observed for centuries, and after countless coin tosses being carried out in that time period by countless agents, the evidence strongly suggests that this lottery produces each of its possible outcomes with approximately equal probability.36 In light of this fact, it would be unreasonable for
anyone to worry about someone devising a way to predict the outcome of coin tosses. It might happen, but the odds are overwhelmingly against it. It is much more reasonable to fear that someone may have devised a way to predict the outcomes of a completely unknown process. After all, the discovery of even a little information would presumably give its possessor an edge in terms of predicting outcomes, unless that evidence suggested that all outcomes were equally probable.
For this reason, the use of lotteries with uncertainty, where fair lotteries with risk exist, is indefensible. The former generates a legitimate concern that the latter does not, at
36 The probabilities might not be exactly equal. See n. 12. Also, note that this presupposes an ordinary coin
least not to the same extent. It might be reasonable to call a lottery with uncertainty a fair lottery, but only if one is prepared to call a (equiprobable) lottery with risk a fairer lottery. This suggests that “fairness” is not a dichotomous property; it is not the case that a lottery is fair or not, but that some are fairer than others. This introduces an added level of complexity to the problem of defining a fair lottery, but this complexity can be overcome provided that one is willing to define the class of fair lotteries as equivalent to the class of maximally fair lotteries. To allocate a good using a less-than-maximally fair lottery, when a maximally fair one is available, would thus be unjust.37
This argument is sufficient to defeat Sher’s contention that, intuitively, some lotteries with uncertainty must be included within the definition of a fair lottery. Thus, his second objection to P1 also falls short of defeating it.
5. Conclusion: In this paper, I have defended a relatively straightforward definition of a fair lottery, a definition that squares closely with what one might call paradigmatic cases of attractive tiebreaking processes (tossing a coin, drawing straws, etc.). According to this definition, a fair lottery is one whose outcomes are equiprobable, where the probability of an outcome is equal to the level of warrant for predicting that the outcome will occur, relative to the level of warrant for predicting alternative outcomes. Sher objects to this definition because it makes no reference to the agent with whom the probability must be associated, and because it ignores lotteries involving uncertainty. But as I have shown, probability, at least when properly perceived in this context, need not make any reference to the beliefs of any particular agent. Moreover, under reasonable assumptions regarding uncertainty, lotteries with uncertainty are inferior to equiprobable lotteries with risk. And
37 Barring extraordinary situations. If the only available maximally fair lottery required much greater
so it would be unjust to use one of the former when one of the latter is readily available. Sher’s objections therefore turn out not to be fatal to his earlier effort to define a fair lottery, and given the difficulties generated by the final definition he endorses, the second should give way to the first in our thinking about lotteries and justice.
Sher does, however, consider one further advantage that his preferred definition enjoys. This approved definition carries on its face the reason for using it. In other words, if one understands Sher’s definition of a fair lottery, then the reasons for using a fair lottery to break ties involving justice are ready at hand. Therefore, if one rejects Sher’s definition in favor of the one I defend, one must still confront the question—why a fair lottery? By way of a conclusion, I shall offer a brief answer to this question.38
Sher’s explanation of the justice of fair lotteries (as he defines them) is essentially a contractarian one. The individuals who must make use of such a lottery would agree to its use, and for this reason it is just. The contractarianism in Sher’s account is quite literal; he would require that all of the strongest claimants find the lottery acceptable before it can count as fair. This definition, as noted before, renders the appropriateness of a given lottery dependent upon the bargaining power of the participants involved. This seems counterintuitive; the entire purpose of a theory of justice, after all, is to specify what people should accept, not what they will accept. For this reason, it makes more sense from a contractarian perspective to ask, not what the strongest claimants would accept, but what they could reasonably accept.39 If no strongest claimant could reasonably
38 This answer is developed at greater length in Peter Stone, “Why Lotteries Are Just,” Journal of Political
Philosophy 15 (2007): 276-295.
39 The following relies heavily on the contractarianism of Thomas Scanlon. Scanlon’s account, with its
reliance on reasonableness, is, I believe, the one best suited to explaining why fair lotteries are just. See T.M. Scanlon, “Contractualism and Utilitarianism,” in Amartya Sen and Bernard Williams (eds.),
object to the use of a lottery, then regardless of that claimant’s actual willingness to consent, the lottery ought to be regarded as just.
The definition of a fair lottery that I defend encompasses the entire class of tiebreaking procedures that would receive reasonable consent. Consider, for example, a tie between claimants n, o, and p to a good G. Four tiebreaking procedures are available:
(1) A procedure that would award G to n with certainty.
(2) A procedure that would award G to n with probability qn, o with probability
qo, and p with probability qp, with qn > qo and qn > qp.
(3) A procedure that would award G to each strongest claimant with unknown probability.
(4) A procedure that would award G to each claimant with probability 1/3.
These options exhaust the set of possible tiebreaking procedures available in this situation. (I exclude such options as destroying the good, producing more of the good, awarding the good to a weaker claimant, etc.) (1) barely looks like a lottery at all; it singles out a claimant for reasons unrelated to justice. (2) might be described as a weighted lottery with risk; it “weights” the process towards one claimant. (3) is a lottery with uncertainty, and (4) is an equiprobable lottery with risk.40
For obvious reasons, both (1) and (2) would reasonably be rejected by claimants o and p. Their claims to G, after all, are just as good as that of n; why should n expect to receive G with a higher probability (whether it be 1 or qn) than they do? In the previous
40 If n, o, and p were super-rivalrous claimants—each more anxious to deny G to the others than to receive
section, I argued that all three claimants would reasonably reject (3) where (4) is available; the danger that some unknown factor will allow some claimants to be favored over others is greater for lotteries with uncertainty than for lotteries with risk. And so (4) alone could not be reasonably rejected by any strongest claimant (or anyone else, for that matter). The contractarian case for the fair lottery (as I define it) is therefore established.
Whether one prefers my definition of a fair lottery—and the contractarian defense that can be made for it—to that defended by Sher ultimately depends on one’s view of contractarianism. For a fundamentally different attitude towards contractarianism underlies the two approaches. The difference is well-captured by Thomas Scanlon in a well-known review of Robert Nozick’s Anarchy, State, and Utopia. Scanlon contrasts Nozick’s approach with that of John Rawls in the following way:
The contrast between Nozick’s and Rawls’ views on political obligation illustrates the important difference between two types of consent theory. In theories of the first type, actual consent has a fundamental role as the source of legitimacy of social institutions. Theories of the second type start from the assumption that the institutions with which political philosophy is concerned are fundamentally non-voluntary. These institutions are held to be legitimate if they satisfy appropriate conditions, and the idea of hypothetical consent enters as a metaphorical device used in the formulation and defense of these conditions. Questions of actual consent arise only as internal questions of liberty, that is, as questions about what options acceptable institutions must leave open to those living under them.41
41 Thomas Scanlon, “Nozick on Rights, Liberty, and Property.” Philosophy and Public Affairs 6 (1976):
Sher’s approach to contractarianism parallels that of Nozick, as mine parallels the approach of Rawls. Answering all skeptical concerns about my definition and defense of fair lotteries would therefore require a defense of one strand of social contract theory against another. And such a defense is beyond the scope of this paper.42
Peter Stone
Department of Political Science Stanford University peter.stone@stanford.edu
42 In general, I accept Scanlon’s argument that Nozick’s brand of contractarianism winds up endorsing
consequences that
