Presumption of Equality as a Requirement of Fairness Wlodek Rabinowicz

1. Presumption of Equality as a Requirement of FairnessWlodek Rabinowicz

2. Presumption of Equality (PE): In the absence of relevant information that discriminates between individuals, you should treat them equally. Here: PE as a principle of fairness. Objective: To make PE more preciseand suggest why it should be obeyed and under what conditions.

3. Immediate answer: Unequal treatments should be avoided because of their arbitrariness.[Pace Joel Feinberg according to whom presuming equality is just as arbitrary as presuming inequality. (Social Philosophy 1965, p.102)]Arbitrariness considerations are important. But, they are not decisive.

4. Example:Two scholarships, S1 and a less attractive S2. Two candidates, i and j. Both candidates known to be qualified, but our limited information does not discriminate between their merits (which most probably are unequal). To give S1 to i and S2 to j, or to do the opposite, would be to treat them unequally. Unequal treatment will have to be arbitrary. But equal treatment – withholding scholarships from both – would be even more unjustified.  Avoidance of arbitrariness is not all that matters. We need to know under what conditionsit is appropriate to opt for equal treatment.

5. Obvious objection: Why not draw lots? Isn’t a 50-50 lottery a form of equal treatment? It avoids arbitrariness and still guarantees that each of the candidates gets a scholarship. (Remember: These are deserving candidates.) Drawing lots may well be the right thing to do in the scholarship example: Avoidance of arbitrariness is important. But here: An outcome-oriented (rather than a procedure-oriented) approach to fairness. From the outcome-oriented point of view: equal lottery on unequal treatments ≠ equal treatment. While the procedure is equal, its outcome is not.

6. PE justified - main idea In the absence of discriminating information, equal treatment is likely to be (more or less) unfair. But equal treatment is privileged because (and insofar as) it minimizes expected unfairness. [The scholarship case is ‘an exception that proves the rule.’]

7. Spy Story A spy is sent to an enemy territory to reach the partisans, who are somewhere in the area. Their position will be communicated to him by radio, after landing. Question: Where should he wait for the message?Answer: Ceteris paribus (in the absence of discriminating information), he should position himself in the middle of the area, to minimize the expected distance to the target.

8. Analogy: Unfairness - a kind of distance. A treatment’s degree of unfairness = its distance from the (perfectly) fair treatment. This distance measure imposes a metric on the space of possible treatments. Suggestion: Equal treatment is situated in the centre of that space. Why? In the absence of discriminating information, no spatial direction is privileged.  The treatment in the centre minimizes expected unfairness.

9. Edna Ullmann-Margalit (“On Presumptions”, J Phil 1983):(i) The core reason for presumptions lies in differential costs of potential errors. Example: Presumption of innocence.Louis Katzner (“Presumptions of Reason and Presumptions of Justice”, J Phil 1973) applied this idea to the choice between presumption of equality and presumption of inequality:Treating equals unequally is worse than treating unequals equally.My defence of PE does not depend on such a value asymmetry.The former need not be more unfair than the latter.

10. Ullmann-Margalit, cont’d(ii) Another reason for presumptions: differential probabilities of errors. We presume what is more probable. Example: Fatherhood presumption. My defence of PE does not depend on such probabilistic asymmetry either. It need not be more probable that equal treatment is fair.What I appeal to instead are the differences in expected costs of errors:With equal treatment, the expected cost of error (= the expected degree of unfairness) is minimal.

11. Principles of fairness : Constraints on procedures or constraints on outcomes. Presumption norms are constraints on procedures. But: A procedural constraint can be justified in different ways. One way: Justification in terms of the expected outcome of the procedure that obeys the constraint. This is the approach chosen here.

12. The model Individuals: I = {i1, ..., in} (finitely many) Treatments: T = {a, b, c, ...} Atreatment, a, is an n-tuple (a1, …, an), where ak specifies how individual ik is treated in a. Another notation for ak: a(ik). a is equal iff a1 = … = an.

13. Some possible interpretations of TCake-DivisionsA homogeneous object (‘cake’) is to be divided among the individuals in I. In a cake-division a = (a1, ..., an), each ak is a real number: the share of the cake that in a goes to ik. Each share  0 and the sum of shares = 1.Equal treatment: (1/n, …., 1/n)

14. Interpretations of T, cont’d Rankings Treatments: rankings of individuals in I. i≽aj - in a, i is treated at least as well as j. Appropriate interpretation when ordinal differences are all that matters. Representation of a ranking: An assignment of levels to individuals, starting from the highest one: 1 [highest], 2 [second-highest], …,. Equal treatment = (1, …, 1).

15. Interpretations of T, cont’dIndivisible goodsG - a set of indivisible objects to be distributed. Not all the objects in G need to be distributed. A treatment is an assignment of disjoint subsets of G to individuals in I.The scholarship case: G = {S1, S2}, I = {i, j}.Equal treatment = (, )

16. Conditions on Treatments A1. To every permutation p on I corresponds a permutation p on T such that for all a and i, p(a)(p(i)) = a(i). I.e., in p(a), p(i) is treated as i is treated in a. Such a simultaneous permutation on I and T: an automorphism. Notation for automorphisms: p, p’, … Example:I = {i, j, k} T = the set of cake-divisions p: ij, jk, ki. Then, for example, (2/3, 1/3, 0) p (0, 2/3, 1/3) If T instead is the set of rankings, then i jj, kpk, i _______________________________________________ A1 is a kind of completeness requirement on T.

17. Simplifying extra assumption: A2. T contains exactly one equal treatment, call it e. Note: Only the equal treatment stays invariant under all automorphisms. (I.e., for all p, p(e) = e.) Example of an interpretation that violates A2: Extended Cake-Divisions, in which part of the cake might be withheld from distribution. Then: Infinity of equal treatments. ____________________________________Def: a is structurally identical to b =df for some p, p(a) = b. Intuitively: a and b are structurally identical if we can get one from the other just by reshuffling individuals. A structure =df An equivalence class of treatments with respect to the relation of structural identity.

18. Sa – the structure of a Examples: (cake-divisions among three individuals)S(1, 0, 0) = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. S(1/2, 1/3, 1/6) = {(1/2, 1/3, 1/6), (1/2, 1/6, 1/3), (1/3, 1/2, 1/6), (1/6, 1/2, 1/3), (1/3, 1/6, 1/2), (1/6, 1/3, 1/2)}. Se= {(1/3, 1/3, 1/3)} e’s structure consists just of e. While the number of treatments in a structure may vary, this number is always finite (at most n!), even if T is infinite.

19. Measure of unfairness d(a, b) specifies how unfair treatment a would be ifb were the fair treatment. Implicit assumption: In the situations we consider, there is exactly one fair treatment according to the agent’s information (even though the agent doesn’t know which treatment it is). Standard conditions on a distance measure: d(a, b)  0 (non-negativity) d(a, b) = 0 iff a = b (minimality) d(a, b) = d(b, a) (symmetry)d(a, b) + d(b, c) d(a, c) (triangle inequality). Thus: Our model (I, T, e, d) is geometric. (T, d) is a metric space.

20. A geometric model facilitates intuitive grasp, but otherwise isn’t necessary. Symmetry and triangle inequality might be given up. Symmetry, in particular, has notable implications. Ex. d(e, a) = d(a, e) I.e., Equal treatment of unequals is just as unfair as unequal treatment of equals.An additional condition on d:Impartiality: d is invariant under automorphisms. I.e., for all automorhisms p and all a, b, d(p(a), p(b)) = d(a, b). I.e., if we move reshuffle individuals in two treatments in the same way, the distance remains the same. Example: If an individual who only deserves a small share gets all of the cake, this is equally unfair independently of who it is who gets this unfair advantage.

21. Possible interpretations of d Cake-divisions The simplest measure: (City-block) k (Minkowski measures) If k = 1, we get city-block, if k = 2, we get Euclidean distance, etcThe higher k, the more weight on larger differences.

22. Rankings Think of a ranking, a, as a set of ordered pairs (i, j) such that i≽a j d(a, b) =def the number of pairs that belong to a or to b, but not to both. [Kemeny-Snell] ________________________________________________ Indivisible Goods The scholarship example: a: i gets S1, j gets S2b: i gets S2, j gets S1e: None of the candidates gets a scholarship Distances: ab e

23. Measure of Information The agent’s information is represented by a probability distribution P on T.P(a) is the probability that a is the fair treatment. Pdoes not discriminate between individuals =def P assigns the same values to structurally identical treatments.________________________________________________ The expected unfairness of a treatment a: bTP(b) d(a, b) (For simplicity, we take P to be a finite distribution) Hypothesis 1: For every P, if P does not discriminate between individuals, e minimizes expected unfairness with respect to P.

24. What condition on the unfairness measure d is necessary and sufficient for Hypothesis 1? Structure Condition: For every structure ST, e’s average distance from S is minimal, as compared with that of the other treatments in T. Can be shown: Structure Condition  Hypothesis 1. Structure Condition holds for Rankings and for Cake-Divisions with Minkowski-distance. But it is violated in the Scholarship example. ab e Which is why Hypothesis 1 doesn’t hold in that case.

25. Illustration: Cake-Divisions, Euclidean distance, three individuals The set of treatments forms an equilateral triangle. Equal treatment in the middle. Structures consist of vertices of regular figures within the triangle (equilateral triangles or hexagons) that have e in the middle.

26. Problem:The Structure Condition isn’t transparent.Can it be derived from a set of simpler and more intuitive conditions on the unfairness measure?Conjecture: Impartiality might be one of them.

27. Minimax The maximal possible unfairness of a treatment a w.r.t. P: max{d(a, b): bT & P(b) > 0} Hypothesis 2:For all P, if P does not discriminate between individuals, e minimizes maximal possible unfairness w. r. t. P. What does it take for Hypothesis 2 to hold? Minimax: For every structure S, e’s maximal distance from S is minimal, as compared with other treatments in T. Minimax  Hypothesis 2 Structure Condition & Impartiality  Minimax

28. Extensions (1) What if T contains several equal treatments? (2) What if P does not discriminate between individuals in a limited subsetX of I? (3) What if there might be several fair treatments in T?

29. (i) What if T contains several equal treatments?Ex: Extended Cake-Divisions, where all of the cake need not be distributed.Hypothesis 3: For all P, if P does not discriminate between individuals, then for every a in T, there is some equal treatment ea such that ea’s expected unfairnessP ≤ a’s expected unfairnessP.Necessary and sufficient condition:The Generalized Structure Condition: For every a in T, there is some equal treatment ea such that for every structure S,ea’s average distance from S ≤ a’s average distance from S.This condition is satisfied by Extended Cake-Divisions, for all Minkowski measures.(ea = the vector that to every individual assigns the average of the values a assigns to different individuals.)

30. (ii) What if P does not discriminate between individuals in a limited subsetX of I? I.e., what if P is invariant under all automorphisms p such that p(X) = X? Hypothesis 4: For all P, if P does not discriminate between individuals in X, then for every a in T there is some treatment ea, X that is equal on X and is such that ea, X’s expected unfairnessP ≤ a’s expected unfairnessP. (A treatment a is equal on a set X of individuals iff for all i, j in X, a(i) = a(j).)

31. (iii) What if there might be several fair treatments in T?Then we need a modified unfairness measure: d(a, Y) – the unfairness of a on the assumption that Y is the set of all fair treatments in T. d(a, Y) = 0 iff aYWe also need a modified measure of information:P(Y) - the probability that Y is the set of fair treatments in T.P does not discriminate among individuals =df for all p, P(p(Y)) = P(Y), where [p(Y) = {b: aYp(a) = b}]The expected unfairness of a treatment a: YTP(Y) d(a, Y)What condition on the modified d will then guarantee that equal treatment minimizes expected unfairness?