If in Doubt, Treat’Em Equally
A case study in the application of formal methods to ethics
Wlodek Rabinowicz
Department of Philosophy, Lund University
Abstract:The so-called ‘presumption of equality’ requires that individuals be treated equally in the absence of relevant information that discriminates between them. Our objective is to make this principle more precise, viewed as a principle of fairness, and to determine why and under what conditionsit should be obeyed.
Presumption norms are procedural constraints, but their justification can be sought in the possible or expected outcomes of the procedures they regulate. This is the avenue pursued here. The suggestion is that in the absence of information that would discriminate between the individuals, equal treatment minimizes the expected unfairness of the outcome. Another suggestion is that equal treatment under these circumstances also minimizes the maximal possible unfairness of the outcome. Whether these suggestionsare correct or not depends on the properties of the underlying unfairness measure.
1. Introduction
This paper examinesthe so-called Presumption of Equality, which enjoins us to treat different individuals equally, if we can’t discriminate between them, in relevant respects, on the basis of the available information. I will view this principleas a requirement of fairness – more precisely, as a procedural principle whose goal is topromote fair outcomes. I would like to make Presumption of Equality more precise and determine why and under what conditionsit should be obeyed. The paper can be viewed as a case study the purpose of which is to show how problems in ethics can be amenable to formal methods.
Why,then, should Presumption of Equality be obeyed? A rather obvious answer is that, in the absence of relevant discriminating information,unequal treatment is bound tobearbitraryand thus could invite a suspicion of partiality. I don’t think, though, that this is all there is to it. While arbitrariness considerations are important,they are not decisive. In some cases in which the discriminating information is missing, equal treatment might be wrong, and unequal treatment might be right,despite thearbitrariness of the latter. To give an example, suppose individualsi and jcompete for two scholarships,the more attractive scholarship Aand the less attractiveB.Your task is to make the decision. While it is given that both applicants are qualified, your information doesn’t discriminate between their merits: While you might know that one of the candidates is more deserving, you have no cluewhich candidate it isand there is no time to gather further information.Since the scholarships aren’t equally attractive, to give one to i and the other to j would be to treat the applicantsunequally and to that extent arbitrarily.At the same time, you could treat them equally by withholding the scholarships from both. However,such ‘levelling down’ would be even more unsatisfactory given thatthey bothdeserve a scholarship. Avoidance of arbitrariness is not everything that counts. To justify equal treatment, in those cases in which such treatment can be justified, we need to rely on other considerations.
At this point, I expect theobjection: Why not simply decide the scholarship assignment bya toss of a coin? This would give each of the applicants afifty-fiftychance of getting the more attractive scholarship, but even in case of a loss the applicant will not come away empty-handed: She will receive the less attractive scholarship instead. Such a lottery, it seems,is itself a form of equal treatment. By tossing a coin, we would avoid arbitrariness but at the same time see to it that both applicants do get scholarships, which they deserve.
It is true that, in the case at hand, drawing lots or tossing a coin is the obvious thing to do. Avoidance of arbitrariness is important. I would deny, however, that an equal lottery on unequal treatments itself amounts to equal treatment. On an outcome-oriented approach to fairness, this is not so. While an equal lottery on unequal outcomes gives different individuals equal chances, the outcome of the lotterywill still be unequal. And inequality in outcome may well matter from the point of view of fairness.
Here is the suggestionI would insteadwant to examine: Principles of fairness can be constraints on procedures or constraints on outcomes. Presumption norms constrain procedures, but procedural constraints could be justified in terms of the expected outcomes of the procedures they regulate. This avenue is pursued in my paper.In particular, equal treatment should be chosenbecause and to the extent that it minimizes the expected unfairness.When the available information does not discriminate between the individuals concerned, it will typically be the case, I would like to suggest, that expected unfairness will be at its lowest if individuals are treated equally.Not always, though. As will be seen, the scholarship example provides an exception to this rule.
It has been suggested that all presumption normsare grounded, to some extent at least,in ‘the differential acceptability of the relevant sorts of expected errors.’[1] Thus, for example, presumption of innocence in criminal lawisjustified by the asymmetry between potential moral costs of punishing an innocent andthose of letting a guilty person go free. My approach to Presumption of Equality is different. As will be seen, for this principle to hold, the moral cost of treating equals unequally need not be greater than that of the equal treatment of unequals.
Some presumptions might have more to do with the differential probability of errors than with their differential costs. In such cases, what is being presumed is deemed to be soprobable as to be used as a default assumption. But my argument does not assume that equality in deserts is probabilistically privileged in this way.Indeed, it may be very improbable that the individuals in the case at hand are equally deserving.
While the justification I offerdoes not appeal to the differentialcosts of errors or to the differences in error probabilities, it does appeal tothe differences in the expected costs of errors. In the absence of discriminating information, the suggestion is that we should treat individuals as if they were equal because such treatment minimizes the expected moral cost of error: the expected unfairnessin treatment. To get the main idea, consider an analogy. Suppose that, in wartime, you are being sent to make a rendezvous with a group of partisans behind the enemy lines. You will be parachuted somewhere in the area where the partisans operate. Only afterwardsyou will be notified by radio of their exact location. Where in the area should you make your landing in order to minimize the expected distance to the rendezvous point? Barring special considerations (road conditions, an irregular shape of the area,etc.), the answer is obvious: You should position yourself rightin the center. Ifthe probability of partisan locationisuniformly distributed over the area, or at least if it is distributed symmetrically with respect to the different geographical directions, the expected distance to the target will be minimal in the center.
Now, unfairness mightalso be seen as a kind of distance between the way individuals are treated and the way they deserve to be treated. A treatment is more unfair,the farther awayit is from the (perfectly) fair treatment.Its expected unfairnesscan therefore be seen as its expected distance from the fair treatment. Now, the conjecture is that treating people equally is like positioning yourself in the center of a designated area. Just as choosing the center-point minimizes expected distance to the target in the absence of information that discriminates between the geographical directions, equal treatment is conjectured to minimizeexpected distance from the fair treatment, i.e. to minimimize expected unfairness,in the absence of information that discriminates between the individuals.
I want to examine under what conditions equal treatment will in fact have this feature. What conditions on the unfairness measure do we need to impose to guarantee this result? I will also inquire what happens if we instead choose a‘minimax’ approach, i.e., if we opt for a treatment which is least unfair if ‘worst comes to worst’. In other words, I willalso examine under what conditions equal treatment minimizes maximal possible unfairness.
2. Individuals and Treatments
In my discussion, Iwill make use of an abstract model that allows of different interpretations.Clearly, there are dangers in abstraction: Unless we are cautious, we might inadvertently leave out important considerations. But, on the positive side, our conclusions gain in generalityin this way and we avoid unnecessary complications.
Two maincomponents of the model area non-empty finite set I = {i1, ..., in} of individuals who are to be subjected to treatment and a non-empty set T = {a, b, c, ...} of possible alternative treatments.(Other components will be introduced later.) Every treatment a in T is a vector (a1, …, an), where ak(1 kn) is the way individual ik is treated in a. a is equal iff for all individuals ik and im, ak = am. We shall sometimes use the notation a(ik) for ak.
The model allows of different interpretations. The interpretations I will suggest below are themselves relatively abstract. Each may in turn be exemplified in many different ways.
Interpretation 1: Cake-divisions
A‘cake’is a homogeneous object that is to be divided among the individuals in I. A treatment ais a particular division of the cake and thus may be seen as a vector of real numbers, (a1, ..., an), witheach akbeing the share of the cake assigned bya to ik.Each such shareis non-negative and together they sum up to one. Tis the set of all possible vectors of this kind. The equal treatment divides the cake equally among the members of I: (1/n, ..., 1/n).
Representing cake-divisions in this schematic waymeans that we viewthem as types rather than tokens. Thus, iftwo cake pieces are of the same size, this interpretation does not distinguish between givingone piece to i and the other to jor assigning these pieces the other way round. Since the cake is homogenous, the question who gets which equal-sized piece is irrelevant from the point of view of fairness. There is therefore no reason to make this distinction in the model. This is a general feature of ourapproach. Treatments are interpreted as types that incorporate the relevant characteristics of the treatment tokens. As a result, any two treatments in the model are supposed to be relevantly different from each other.
Interpretation 2: Rankings
On this interpretation, a treatment is a ranking of the individuals in I:T is the set of all such possible rankings.That a treatment a ranks i at least as highly as it ranks j means that i is treated in aat least as well as j. Such an interpretation is appropriate when the ordinal differences between the individuals are all that matters from the point of view of fairness, i.e., whenfairness only requires that the more deserving individuals should be better treated and that the equally deserving individuals should be treated equally well.
Formally, a ranking may be represented as an assignment of ordinal numbers, 1, 2, 3 …, to individuals, with1being the highest level in the ranking, 2 the second highest level, and so on. The assignment of levels starts from the highest one and continues downwards.Thus, the equal treatment is a ranking in which every individual is assigned the same highest level: (1, …, 1). (A different, but equivalent representation of rankings will be made use of below, in section 3.)
Interpretation 3: Indivisible goods
Suppose that G is a set of indivisible objects that are to be distributed, with or without remainder, among the individuals in I. Each treatmentain Tis an assignment of disjoint subsets of G to individuals in I. Thus, a(i) is the subset of G that treatment a assigns to an individual i. For some i, a(i) may be empty, and for distinct i and j, a(i) a(j) = . But it is not required that a(i)iI = G: Some objects in G may be withheld from the distribution.The scholarship case provides an example. There, G consists of the two scholarships, A and B,and the possible treatments amount to different possible partial or total distributions of G among the two individuals involved.
What would the equal treatment consist in, according to this interpretation? If the objects in G are relevantly different from each other, as in the scholarship case, then the equal treatment must be(, …, ), i.e. the distribution in which everyi is assigned the empty subset of G. But if G contains sufficiently many relevantly similar objects, say, A1, ..., An, the number of equal treatments would rapidly rise: One such treatment, , would assignA1 to i1, A2 to i2, ..., and An to in; another, ,would assign A2 to i1, A3 to i2, ..., and A1 to in; etc. These treatments, however, would be relevantly similar to each other, thereby violating the ‘relevance’ restriction we have imposed on the model. To satisfy this restriction, wewould need to re-interpret the notion of a treatment.Thus, we couldpartition G into equivalence classes with respect to the relevant similarity relation and then let a treatment be a function thatspecifies, for every individual i,how many objects i is to receive from each equivalence class. Given this re-interpretation, we would not have to make irrelevant distinctions. Thus, for example, if the equivalence class C contains the objects A1...,An, then the two equal treatments and , described above, would not be distinguished. Both would be tokens of the same treatment (type) that consists in each individual being assigned one object in class C.
However, such an interpretation, while satisfying the relevance restriction, would still violate another constraint I want to impose: For simplicity, I exclude decision problems in which there is more than one equal treatment available.And the case described above would be an example of such a problem: The individuals could be here equally treated either by not being given anythingat all or, say, by being given exactly one C-object each.
The same problem would arise, by the way, in Cake-Division, if we allowed divisions in which part of the cake is withheld from the distribution. The number of equal treatments would then increase from one to infinity: The set of real-numbered vectors (a1, ..., an) such that a1 = ... = an 0 and a1 + ... + an 1 is non-denumerable.
Here, I want to exclude such interpretations. To keep the model relatively simple, I will be assuming that there is a unique equal treatment in T. In addition, I will assume that T is closed under permutations on individuals. Thus, we impose two conditions on the set of treatments:
A1. For every permutation fon I and every a in T, T contains some b such that for every i in I, b(f(i)) = a(i).[2]
A2. There is a unique element of T, call ite, such that e is an equal treatment.
We have seen that A2 is a controversial claim. So is A1, which is a kind of completeness requirement on T. If T is the set of actually available treatments, this set might in some cases be too small for A1 to be satisfied. Thus, suppose the agent can give the whole cake to i1, but, for some reason, he cannot give it to i2. Then the set of available cake-divisions contains (1, 0, 0) but not (0, 1, 0). Under these circumstances, T wouldn’t be closed under permutations on individuals that assign i2 to i1: There is no available treatment that gives to i2what (1, 0, 0) gives to i1. Here, however, I want to ignore this difficulty. I shall assume that T is sufficiently ‘roomy’ for A1 to be satisfied.[3]
If A1holds, every permutation f on I induces a permutation fT on T such that for every treatment a and every individual i, fT(a) treatsf(i) in the same way as atreatsi: fT(a)(f(i)) = a(i).A1 implies the existence of such a permutation on T. And it is easily seen that this permutation must be unique. Had there been two of them, fTand fT’, then there would exist somea in Tsuch that fT(a) ≠ fT’(a). This would require the existence of some i in I for whichfT(a)(f(i)) ≠ fT’(a)(f(i)). Which contradicts the assumption that, for every i,both fT and fT’treatf(i) in the same way as atreats i, i.e, that they both assign a(i) to f(i).
Now, suppose that f is a permutation on I and let fT be the permutation on Tinduced byf.I will refer to the union of f and fT as an automorphismand use symbolsp, p’, etc, to stand for different automorphisms on IT.Intuitively, an automorphism is a simultaneous permutation of individuals into individuals and of treatments into treatments, in which the former permutation induces the latter:
Automorphism: An automorphism, p, is a simultaneous permutation of I and of T such that for all i in I and all a in T, p(a)(p(i)) = a(i).
Observation: Every permutation on I is included in exactly one automorphism.[4]
This notion of an automorphism will come handy below. Here follow some examples. Suppose that I= {i1, i2,i3} and let T be the set of cake-divisions among the members of I. One automorphism would then permutei1into i2, i2into i3 and i3into i1. This wouldeffect a corresponding permutation on cake-divisions. For example, (0, 2/3, 1/3) would be transformed into (1/3, 0, 2/3). Analogously, if Tis the set of rankings of I = {i1, i2, i3}, the automorphism that permutes i1 into i2, i2 into i3 and i3 into i1 effects a corresponding permutation on rankings.For example, it transforms the ranking with i1on top, followed by i2 and i3,in that order, into a ranking with i2 on top, followed byi3 and i1. In general, it is easy to see that only the equal treatment estays invariant under all automorphisms:For allautomorphisms p, p(e) = e, and for all aT, if ae, then for some some automorphism p,p(a) a.
We now define an equivalence relation that’s going to play an important role in the discussion below - the relation of structural identity between treatments.
Structural Identity:A treatment a is structurally identical to a treatment b iff there exists some automorphism p such that p(a) = b.
Intuitively, this relation obtains between two treatments if we can get one from the other just by moving individuals around, so to speak. In the case of Cake-Divisions, this means thatthe cake in both treatments is cut in the same way and the only difference consists in to whom the shares are distributed. In the case of Rankings, we have structural identity iftwo rankings would look exactly the same if one in each of themwere to replace individuals by individual variables. Finally, in the case of Indivisible Goods, two assignments of goods are structurally identical if they both involve the same collection of individual ‘baskets’ (subsets of G). The difference between structurally identical goods-assignments only appears when it comes to the question which basket goes to whom.
Structural identity is an equivalence relation, i.e., it is reflexive, symmetric, and transitive.[5] Therefore,we can partition T into structures, which are equivalence classes of treatments with respect to the relation of structural identity. I shall refer to different structures as S, S', S", etc. As an example,suppose that T is the set of cake-divisions among the individuals in I= {i1, i2, i3}. Consider a cake-division a = (1, 0, 0). Its structure consists of three treatments:
(1, 0, 0), (0, 1, 0) and (0, 0, 1).
On the other hand, the structure ofb = (1/2, 1/3, 1/6) consists of six treatments.In b, each individual gets a different share and there are six ways in which we can assign three different shares to three different individuals. As for the equal division,e = (l/3, 1/3, 1/3),its structure contains nothing but e itself.