Extrapolated Social Preferences

Transcription

1 Extrapolated Social Preferences Maya Eden World Bank July 17, 2017 Abstract This paper proposes a simpler interpretation of Harsanyi s impartial observer idea. It postulates an impartiality axiom with respect to the distributions of a certain benchmark good. The axiom requires society to be indifferent between each distribution of the good and an equiprobable lottery among all distributions that differ from it only in the matching between individuals and allocations (and not in the size distribution of allocations). Combined with a Pareto condition, this impartiality requirement characterizes a complete social ranking, which is distinct from Harsanyi s. Applying this impartiality requirement to all possible benchmark goods generates a partial social ranking, which nonetheless may be broader than the Pareto ranking. JEL Classification: D63 Keywords: Utilitarianism, egalitarianism, certainty equivalent Contact address: Maya Eden, Development Research Group, The World Bank, 1818 H Street, NW, Washington D.C ; meden@worldbank.org. I am grateful for helpful conversations with Paolo Piacquadio, Uzi Segal and John Weymark. This paper reflects my own views and not necessarily those of the World Bank, its Executive Directors or the countries they represent. 1

2 1 Introduction The construction of Harsanyi s social welfare function builds on the idea of an impartial observer (Harsanyi [1953], Harsanyi [1955] and Grant et al. [2010]). Harsanyi proposes that the ethical desirability of different distributions of goods should reflect the preferences of an impartial observer over identity lotteries that assign equal probabilities to being each individual in society. The ethical appeal of this construction is that, under the veil of ignorance, there is no reason to prefer one individual over another, so the welfare of each individual should be valued equally. Yet, Harsanyi s construction has been criticized for demanding too much from the impartial observer (see Pattanaik [1968]). In Harsanyi s construction, identity lotteries are interpreted not only as lotteries over allocations of goods, but also as lotteries over preferences. 1 In particular, if risk preferences are heterogeneous, then the risk preferences of the impartial observer will be an outcome of the identity lottery. This creates several difficulties. For example, which risk preferences should be used for evaluating the identity lotteries? What does it mean to have a lottery over possible risk preferences? This paper pursues a simpler interpretation of the impartial observer idea. Rather than requiring the impartial observer to evaluate lotteries over the joint distribution of allocations and preferences, the impartial observer is asked only to evaluate lotteries over the distribution of a subset of allocations. In particular, it is assumed that there exists some subset of allocations called an impartial base, such that society is indifferent between any distribution of elements of that subset and a random permutation of that distribution. For example, if society consists of two individuals, 1 and 2, and B is an impartial base, then, for any b, b B, social preferences are assumed indifferent 1 Karni and Weymark [1998] illustrate that Harsanyi s impartial observer theorem can be derived under the assumption that the impartial observer s preferences are defined only over lotteries in which there is an equal chance of being each individual in society, rather than an arbitrary probability distribution over the set of identities. 2

3 between a distribution in which individual 1 gets b and individual 2 gets b, a distribution in which individual 1 gets b and individual 2 gets b, and a lottery in which each of these distributions occurs with equal probabilities. I show that a standard Pareto condition can be used for extrapolating a complete social ranking from this impartiality requirement. I refer to this social ranking as the extrapolated social ranking, and show that it can be represented by a social welfare function, which I refer to as the extrapolated social welfare function. When risk preferences are identical across individuals and satisfy the expected utility axioms, extrapolated social preferences can be represented by a utilitarian social welfare function. 2 Whenever there is at least one individual with maximin risk preferences, the extrapolated social ranking can be represented by the Rawlsian egalitarian social welfare function, which ranks distributions by their least desirable allocations (Rawls [1974]). The functional form of the extrapolated social welfare function therefore depends crucially on the distribution of individual risk preferences. In addition, the extrapolated social ranking depends on the choice of impartial base. I establish that, unless individual preferences are identical, different impartial bases imply different extrapolated social rankings. At the same time, I illustrate that the partial ranking consistent with all possible extrapolated social welfare functions is broader than the Pareto ranking. This paper is related to a rich literature on social choice that builds on the idea of an impartial observer. The construction of a social welfare function based on an impartiality requirement shares with Segal [2000] and Piacquadio [2017] (among others). In Segal [2000], impartiality requires indifference between any two dictatorial allocation rules. In Piacquadio [2017], there is a symmetry requirement with respect to the distribution of individual choice 2 More precisely, if individual preferences are represented by the expected utility function u, then social preferences can be represented by the social welfare function I i=1 u(a i), where a 1,..., a I are individual allocations. See Weymark [1991] and references therein for a discussion on the relationship between this social welfare function and the philosophical notion of utilitarianism. 3

4 sets (or opportunity sets). Here, a symmetry requirement is applied to the distribution of base-equivalents ; for example, if the impartial base is the set of deterministic allocations, then the social welfare function is symmetric with respect to the distribution of certainty equivalents. The concept of base equivalents is a generalization of the concept of certainty equivalents for impartial bases consisting of non-deterministic allocations. 2 Setup There are I individuals, indexed i = 1,..., I, and J goods, indexed j = 1,.., J. Individual preferences, i, are defined over the set of all finite-state lotteries, A: A = {{(c(s), p(s))} S s=1 S N; c(s) R J +; p(s) (0, 1]; S p(s) = 1} The elements of A, which I refer to as allocations, are finite-state lotteries of the the form {(c(s), p(s))} S s=1, where S N is a natural number representing the number of positive-probability states, c(s) R J + is the vector of goods realized in state s, and p(s) is the probability of state s. Note that A consists of lotteries over any finite number of states. 3 The set of distributions, D, consists of lotteries over the profile of allocations: D = {{(a 1 (s),..., a I (s), p(s))} S s=1 S N; a i (s) A; p(s) (0, 1]; s=1 S p(s) = 1} Elements of D are finite-state lotteries of the form {(a 1 (1),..., a I (s), p(s))} S s=1, 3 This setup is more restrictive than in Harsanyi [1955], in which individual preferences are defined on the set of lotteries over a common social state. Here, individuals are assumed to care only about their own consumption and not about the consumption of others. s=1 4

5 where a i (s) A is the lottery allocated to individual i in state s and p(s) is the probability of state s. To economize on notation, I make use of several homomorphisms between R J +, A, A I and D. For c R J +, the object c A is identified with the degenerate lottery that delivers c with certainty (the allocation {(c, 1)} A). Similarly, for (a 1,..., a I ) A I, the object (a 1,..., a I ) D is identified with the distribution {(a 1,..., a I, 1)}. Given an allocation a A, the distribution a D is identified with the symmetric distribution {(a,..., a, 1)}. Conversely, for a distribution d D, the allocation [d] i A is identified with the lottery d projected on individual i. 4 If d D is a symmetric distribution (meaning that [d] i = [d] k for all i, k), then the element d A is identified with the allocation [d] i A (which is independent of i). Using this notation, it is useful to define individual preference rankings on the set of distributions D as d i d if and only if [d ] i i [d] i. Note that for a A, the element a D is identified with the symmetric distribution (a,..., a), and thus d i a is the same as [d] i i a. I use the standard notation i and i to denote the indifference relation and the strict preference relation. I assume that, for each i, the individual preferences i can be represented by a strictly increasing utility function, u i : A R. 5 I restrict attention to environments in which u i either satisfies the von Neumann and Morgenstern expected utility axioms (u i ({(c(s), p(s))} S s=1) = S s=1 p(s)u i(c(s))), or takes the maximin form (u i ({(c(s), p(s))} S s=1) = min s w i (c(s)) for some continuous w i : R J + R). Individuals are assumed to have (correct) common beliefs about objective probabilities. 4 Explicitly, if d = {(a 1 (s),..., a I (s), p(s))} S s=1 D and a i (s) = {(c i (s, s), p i (s, s))} S s =1 A, then [d] i = {{c i (s, s), p(s)p i (s, s)} S s =1 }S s=1 A. 5 The utility function u i is strictly increasing if, for every pair c 1, c 2 R J +, where c 1 = (c 1 1,..., c 1 J ) and c2 = (c 2 1,..., c 2 J ), if c1 j c2 j for all j, then u i(c 1 ) u i (c 2 ), and, if the inequality is strict for some j, then u i (c 1 ) < u i (c 2 ). 5

6 3 Extrapolated social welfare functions This section presents a construction of a set of extrapolated social welfare functions. The following section contains a representation theorem, stating that social preferences can be represented by an extrapolated social welfare function if and only if they are consistent with certain axioms. Definition (Consensus ranking). 1. For a given preference profile { i } I i=1, the consensus ranking is a partial ordering on A defined as a a if and only if a i a for every i, and a < a if and only if a i a for every i. 2. A subset of allocations B A is consensus-ranked if each pair of its elements b, b B is consensus-ranked. A subset of allocations B A is consensus-ranked if all of its elements are identically ranked by all individuals. Definition (Base). A subset of allocations, B A, is a base for the preference profile { i } I i=1 if (a) B is consensus-ranked for the preference profile { i } I i=1 and (b) for each i there is a base function, ˆb i : D B, which satisfies ˆb i (d) i d and, if d i d, then ˆb i (d) = ˆb i (d ). A base is a consensus-ranked subset of allocations that is accompanied by individual mappings between the universe of allocations and their base equivalents. For example, if there is only one good, then the set of deterministic allocations C = R + is a base. Note that C is consensus-ranked because all individuals agree that more consumption is better. The base functions, ĉ i, are the mappings between distributions and their individual certainty equivalents. These functions satisfy the definition of base functions because, by definition, individual i is indifferent between any distribution d and its individual certainty equivalent ĉ i (d) = ĉ i ([d] i ), and, if d i d, then i s certainty equivalents of d and d are the same. 6

7 There are many other bases. To illustrate, for a = {(c(s), p(s))} S s=1 A and λ R +, define the allocation λ a A as the lottery in which c(s) is multiplied by λ in each state (λ a = {(λc(s), p(s))} S s=1). Note that, for each lottery a A, the set {λ a λ R + } is a base because (a) if λ < λ, then λ a first-order stochastically dominates λ a, and hence the set is consensusranked; and (b) the restrictions on individual preferences guarantee that, for each distribution d D, there exists a unique λ such that d i λ a. The concept of a base equivalent is a generalization of the concept of a certainty equivalent. The base functions allow for any distribution d to be uniquely mapped to a distribution of base elements (ˆb 1 (d),..., ˆb I (d)), such that all individuals are indifferent between the two distributions. Every base provides a possible benchmark for interpersonal comparability. For a given distribution d, the statement ˆb k (d) ˆb i (d) implies that all individuals agree that i s base equivalent of d is better than k s base equivalent of d. If individual welfare is measured in base equivalents, then this can be interpreted as d being more favorable to i than to k. This construction shares with Piacquadio [2017]. In Piacquadio [2017], the set of allocations is a union of nested opportunity sets, which represent different feasibility constraints. Opportunity sets are consensusranked because, for each individual, a larger opportunity set is associated with higher welfare. Thus, a sequence of nested opportunity sets implies an interpersonally-comparable measure of welfare. Similarly, the construction here creates interpersonal comparability through a consensus-ranking of allocations. Let S I denote the set of all possible permutations of {1,..., I} (note that there are I! such permutations). Define Π : D D as the mapping between distributions and their random permutations, where the random permutation of a distribution is given by: Π({a 1 (s),..., a I (s), p(s)} S s=1) = {{(a π(1) (s),..., a π(i) (s), p(s) )} S I! s=1} π SI (1) 7

8 The random permutation Π(d) assigns equal probabilities to all possible permutations of the allocation profile. Note that Π(d) is a symmetric distribution, in which all individuals face the same probability of obtaining each allocation in the distribution. In this sense, the random permutation is similar to Harsanyi s identity lottery. However, the concepts are distinct: while Harsanyi s identity lottery involves the simultaneous determination of allocations and preferences, the random permutation is a lottery over allocations only. The following theorem associates each base, B, with a social welfare function, e B : D B. Theorem 1. Let there be a base, B, with ˆb i its associated base functions. For every n N, define a set of functions {e n i,b : D B} n N,i=1,...,I as follows: e 0 i,b(d) = ˆb i (d) and, for any n 0, e n+1 i,b (d) = ˆb i (Π(e n 1,B(d),..., e n I,B(d))) There exists a unique function e B : D B such that, for every d and every i, lim n u i (e n i,b (d)) = u i(e B (d)). The proof, together with other omitted proofs, is provided in the appendix. Note that, as bases are consensus-ranked, the functions e B : D B can be interpreted as a social welfare functions, in which each pair of distributions d, d D is ranked according to the consensus-ranking of e B (d ) and e B (d). I refer to e B as the social welfare function extrapolated from B. Figure 1 illustrates the convergence process underlying the construction of e B in a simple example in which there are only two individuals, 1 and 2, and B is the certainty base. Initially, e n 2,B (d) < en 1,B (d). The (n + 1)-th iteration is computed by asking each individual to evaluate the random permutation Π(e n 1,B (d), en 2,B (d)), using his own risk preferences. The allocations en+1 1,B (d) and e n+1 2,B (d) are then computed as the individual certainty equivalences of this random permutation. Note that, while it remains the case that e n+1 2,B (d) < e n+1 1,B (d), the allocations en+1 2,B (d) and en+1 1,B 8 (d) are closer than the allocations

9 Figure 1: The construction of e n+1 i,b (d) when I = 2 and B is the certainty base IC1 IC 2 IC1 n e 1,B (d) B IC 2 p=1/2 n+1 e 1,B (d) n+1 e 2,B (d) e n 2,B (d) Π(e n n 1,B(d),e 2,B(d)) p=1/2 The axes represent two equiprobable states, and the curves labelled IC 1 and IC 2 are the indifference curves of individuals 1 and 2, respectively (individual 1 is risk neutral and individual 2 is risk averse). As B is the certainty base, it can be represented by the 45 degree line. The point labelled Π(e n 1,B (d), en 2,B (d)) is a lottery that delivers en 1,B (d) with probability 1/2 and e n 2,B (d) with probability 1/2. The values of en+1 1,B (d) and en+1 2,B (d) are the individual certainty equivalents of this lottery. 9

10 e n 2,B (d) and en 1,B (d). Theorem 1 establishes that, as m, em 1,B (d) and e m 2,B (d) converge to a single point in B. Although extrapolated social welfare functions are derived as a limit of an iterative process, there are some special cases in which they can be written in a simpler form, as illustrated by the following corollaries. Corollary 1. Assume that individual preferences are identical and represented by some utility function u( ). Then, any extrapolated social welfare function is an ordinal transformation of u(π( )). In particular, if preferences are identical across individuals and satisfy the von Neumann-Morgenstern expected utility axioms, then any extrapolated social welfare function is an ordinal transformation of the utilitarian social welfare function in Vickrey [1960], I i=1 u([d] i)/i. 6 Another special case of individual preferences generates egalitarian extrapolated social welfare functions. The following corollary establishes that, if there is at least one individual with maximin preferences, then any extrapolated social welfare function evaluates allocations by their least-desirable base equivalents: Corollary 2. If there exists i such that i are maximin preferences, then, for every base B, e B (d) = min k {ˆb k (d)}. The above corollary states that, whenever there is a single individual with maximin risk preferences, the extrapolated social welfare function is egalitarian with respect to base elements: in particular, for (b 1,..., b I ) B I, e B (b 1,..., b I ) = min k {b k }. If d B I, then e B (d) is the minimal baseequivalent of d. Note that this result does not require that all individuals 6 In the case of common individual preferences, the social ranking is consistent with the Suppes-Sen ranking (Suppes [1966], Sen [1970]). When individual preferences are identical, the Suppes-Sen ranking of two distributions d, d D is equivalent to the firstorder stochastic dominance ranking of Π(d ) and Π(d). Thus, if d Suppes-Sen dominates d, then u(π(d )) u(π(d)). 10

11 have maximin preferences - only that a single individual does. 7 Taken together, Corollaries 1 and 2 illustrate that the properties of the extrapolated social welfare function depend strongly on the underlying distribution of individual preferences. For the same base B, the extrapolated social welfare function e B is an expected utility function when preferences are identical and satisfy the expected utility axioms, but an egalitarian social welfare function when there is a single individual with maximin risk preferences. 4 Axiomatic foundations This section studies the conditions under which social preferences can be represented by an extrapolated social welfare function. Social preferences are defined as a complete and transitive order on D, and are denoted with (as well as and to indicate the strict preference relation and the indifference relation). Definition (Impartial base). A set of allocations B A is an impartial base if: 1. The set B is a base for the preference profile { i } I i=1 { }; and 2. For every distribution (b 1,..., b I ) B I D, social preferences satisfy (b 1,..., b I ) Π(b 1,..., b I ). 7 Together with Theorem 2, Corollary 2 therefore provides a potential axiomatic foundation for the Rawlsian maximin approach, which does not rely on any extreme assumptions regarding the distribution of risk preferences. Rawls [1974] justifies the maximin approach using the same reasoning as Harsanyi s impartial observer theorem, combined with an extreme assumption that all individuals have maximin risk preferences with respect to identity lotteries. Here, extreme risk aversion is required for only one individual. For other justifications of the egalitarian principle that do not rely on individual risk preferences, see Gevers [1979], Donaldson and Roemer [1987] and Howe and Roemer [1981]. See also the discussion in chapter 5 of Roemer [1996]. 11

12 The first clause of the definition states that an impartial base is a base for the preference profile { i } I i=1 { } (and not just { i } I i=1). This imposes two additional restrictions on B. First, the consensus ranking on B must be consistent with social preferences: for each pair b, b B it holds that b b if and only if b i b for every i. Since social preferences are defined on the set of distributions D, and since the allocations b, b A are identified with the symmetric distributions (b,..., b ), (b,..., b) D, this condition is interpreted as the requirement that (b,..., b ) (b,..., b) if and only if b i b for every i. The second restriction is that there exists a social base function, ˆb : D B, which maps each distribution d to an element ˆb(d) B such that d ˆb(d). Note that, as a distribution, the allocation ˆb(d) A is identified with the symmetric distribution (ˆb(d),..., ˆb(d)) D. The existence of a social base function therefore requires that, for each d D, there exists ˆb(d) B such that (ˆb(d),..., ˆb(d)) d. The social base function is similar to the mapping between distributions and their equally-distributed equivalents as defined in Kolm [1968], Atkinson [1970] and Fleurbaey [2010]. The second clause of the definition states that, if B is an impartial base, then, for every (b 1,..., b I ) B I, social preferences satisfy (b 1,..., b I ) Π(b 1,..., b I ). While the first clause of the definition is a technical requirement which may be easily satisfied by many different bases, this second clause is the key defining feature of an impartial base. The first axiom is stated as follows: Axiom (Impartiality). There exists an impartial base. Similar to Harsanyi s impartial observer ideal, this impartiality axiom combines a symmetry requirement with some degree of consequentialism. Note that Π(b 1,..., b I ) = Π(b π(1),..., b π(i) ) for every permutation π. Thus, if B is an impartial base and (b 1,..., b I ) B I, then, for every permutation π, (b 1,..., b I ) Π(b 1,..., b I ) = Π(b π(1),..., b π(i) ) (b π(1),..., b π(i) ). This symmetry requirement implies that social preferences are indifferent with regards to the distribution of impartial-base elements across individuals. 12

13 In addition to this symmetry requirement, the existence of an impartial base demands some degree of consequentialism: social preferences must be indifferent between a deterministic distribution of impartial-base elements, (b 1,..., b I ) B I, and a lottery that assigns equal probabilities to all of its permutations. As, by symmetry, (b 1,..., b I ) (b π(1),..., b π(i) ) for every permutation π, the lottery Π(b 1,..., b I ) is a lottery between socially-indifferent outcomes. The requirement that (b 1,..., b I ) Π(b 1,..., b I ) therefore states that society should be indifferent between a distribution (b 1,..., b I ) and a lottery between distributions that are equally as desirable as (b 1,..., b I ). Note that the impartiality axiom postulates the existence of an impartial base; it does not require that every base is an impartial base. As I establish in the following sections, requiring the impartiality of every base is generally inconsistent with the Pareto principle. The choice of an impartial base is related to the philosophical discussion on the appropriate benchmark for interpersonal comparability. This is often identified with the equalisandum, the object that society should strive to equalize across individuals. For example, Rawls [1971] argues for the equalization primary goods, an index of income, wealth and various social liberties (see also Roemer [1996], chapter 5). In contrast, Sen [1980] argues that society s objective should be to equalize capabilities, or functioning: a handicapped person may require more income to achieve the same level of mobility as an able-bodied person. Sen argues that, rather than striving to equalize income (or a richer index of primary goods), society should strive to equalize capabilities, although this may require an unequal distribution of primary goods. It will be established shortly that, under the assumption that all individuals are risk averse, the impartial base can be interpreted as the equalisandum. Thus, the different social preferences of Rawls and Sen can be represented by different choices of an impartial base. 8 8 The level of functioning is typically thought of as an individualized function of the 13

14 The second axiom is a standard Pareto condition, requiring consistency between individual and social preferences: Axiom (Pareto). 1. For each pair of distributions d, d D such that d i d for every i, the social ranking satisfies d d; and 2. For each pair of distributions d, d D such that d i d for every i, the social ranking satisfies d d. This Pareto condition relates individual and social preferences through the requirement that, if a distribution is weakly preferred by all individuals, then it should also be weakly preferred by the social ranking; further, if all individuals are indifferent between two distributions, then the social ranking should be indifferent between them as well. 9 The following representation theorem provides an axiomatic foundation for the extrapolated social welfare function. Theorem 2. Social preferences can be represented by an extrapolated social welfare function if and only if they satisfy the impartiality axiom and the Pareto axiom. The proof establishes that, if B is an impartial base and social preferences are consistent with the Pareto condition, then social preferences can be represented by the extrapolated social welfare function e B ; conversely, allocation of primary goods. This can be incorporated in this framework by specifying the set of goods, J, as including only functionings and not the primary goods from which they are derived. This setup embodies the assumption that individuals are indifferent between two allocations of primary goods that deliver the same levels of functioning. An alternative approach is to specify J as including both primary goods and functioning. In this case, the mapping between the two is a restriction on the feasible set of allocations. For example, the observation that, for a given individual, certain income is required in order to achieve a certain level of mobility is a restriction on the feasible set of allocations: an allocation in which the individual is assigned low income but high mobility is simply not feasible. This second approach allows individuals to have preferences over different bundles of primary goods that deliver the same level of functioning. 9 Note that this Pareto requirement is weaker than the standard Pareto condition, requiring that if (a) d i d for all i, and (b) d k d for some k, then d d. 14

15 if e B represents social preferences, then B is an impartial base and social preferences are consistent with the Pareto axiom. I refer to social preferences that can be represented by e B as the social preferences extrapolated from B (or extrapolated social preferences). Note that, although the definition of an impartial base requires that B is a base for the preference profile { i } I i=1 { }, the extrapolated social welfare function e B is the social welfare function defined in Theorem 1 using only the fact that B is a base for the individual preference profile { i } I i=1. The proof that e B represents social preferences when B is an impartial base uses repeated applications of the Pareto axiom and the impartiality axiom. By the Pareto condition, as d i e 0 i,b (d) for all i, it follows that d (e 0 1,B (d),..., e0 I,B (d)). As B is an impartial base, it follows that d (e 0 1,B (d),..., e0 I,B (d)) Π(e0 1,B (d),..., e0 I,B (d)). Similarly, as e1 i,b (d) i Π(e 0 1,B (d),..., e0 I,B (d)) for all i (by the construction in Theorem 1), an additional application of the Pareto condition implies that d (e 1 1,B (d),..., e1 I,B (d)); as (e 1 1,B (d),..., e1 I,B (d)) BI, an additional application of the impartiality condition then implies that d Π(e 1 1,B (d),..., e1 I,B (d)). Repeated alternating applications of the Pareto condition and the impartiality axiom imply that d (e n 1,B (d),..., en I,B (d)) for all n. Since, as n, en i,b (d) converges to e B (d) for every i, it follows that d e B (d). 10 The following corollary formalizes the sense in which the impartial base is the equalisandum. Corollary 3. Assume that social preferences satisfy the Pareto axiom and that there exists an impartial base of the linear form B = {λ a λ R + } for some a A. If all individuals are risk averse, then, for every (λ 1,..., λ I ) R I +, it holds that (λ 1 a,..., λ I a) (( I i=1 λ i/i) a,..., ( I i=1 λ i/i) a). The above corollary states that, under the assumption that all individuals are risk averse and the impartial base is linear, an equitable distribution of 10 The complete proof is provided in the appendix. 15

16 impartial-base elements is optimal (note that if B is not linear, averaging base elements does not necessarily result in a base element). The optimality of an equitable distribution is a feature of the extrapolated social welfare function that is not shared with Harsanyi s social welfare function Properties This section discusses the compatibility of the extrapolated social welfare function with several properties emphasized in the literature. Binary independence of irrelevant alternatives. In Arrow [1950], Binary Independence of Irrelevant Alternatives is an axiom which requires the social ranking of every two distributions d and d to depend only on the individual rankings of d and d. To be precise, consider two distinct individual preference profiles, ( 1,..., I ) and ( 1,..., I ), and two distributions d, d D, such that d i d if and only if d i d. Let and denote the corresponding social preferences. Binary independence of irrelevant alternatives requires that d d if and only if d d. In other words, the fact that individuals i and i may disagree about the rankings of distributions other than d and d should have no bearing on the social ranking of d and d. This condition is violated by extrapolated social preferences. To see this, note that, by Theorem 1, computing the extrapolated social welfare function e B requires both mapping individual allocations to their base equivalents and evaluating individual preferences over random permutations. Each of these steps depends on individual preferences for distributions other than d and d, which are, strictly speaking, irrelevant alternatives. 11 In Harsanyi s social welfare function, the conditions for the optimality of an equitable distribution are quite stringent: an equitable distribution is always optimal only when individual risk preferences are identical. Harsanyi s social welfare function has the implication that when aggregate consumption is low, it is better to allocate disproportionately more to relatively risk-averse individuals, while when aggregate consumption is high, it is better to allocate more to relatively risk-loving individuals. 16

17 To illustrate, consider a simple example in which there is only one good and individual preferences are identical. Let there be two distributions of deterministic allocations, d = (c 1,..., c I ) and d = (c 1,..., c I ), where c i, c i R +. Note that the individual rankings of d and d do not depend on individual risk preferences: regardless of risk preferences, each individual prefers the distribution that allocates him with more consumption. However, by Corollary 1, the social ranking of the two distributions depends on the ranking of the lotteries Π(d ) and Π(d), using common risk preferences. Obviously, different risk preferences may imply different rankings of these two lotteries. This example illustrates that, similar to Harsanyi [1955], the ranking of two distributions of deterministic allocations may depend on individual risk preferences. Whether or not individual risk preferences should have any relevance for the social ranking of distributions of deterministic allocations has been a topic of some discussion in the early literature on social choice (see Pattanaik [1968] and references therein). Independence. The independence axiom, which is part of the axiomatic foundation of expected utility theory, can be stated as follows. Given three distributions, d, d, d D, and α (0, 1), it holds that d d if and only if a lottery that delivers d with probability α and d with probability 1 α is preferable to a lottery that delivers d with probability α and d with probability 1 α. Extrapolated social preferences do not necessarily satisfy the independence axiom. Intuitively, the violation of the independence axiom reflects a social preference for allocating more risk to relatively more risk-loving individuals. To illustrate, consider an economy with two individuals, 1 and 2, where individual 1 is risk neutral and individual 2 is risk averse, and assume that the impartiality requirement is satisfied with respect to the base of deterministic allocations, C. Thus, society is indifferent between allocating an indivisible good to individual 1 or to individual 2 ((1, 0) (0, 1)). Consider 17

18 the four lotteries described below: (1, 0) with prob. 1/2 d = (0, 0) with prob. 1/2 (1, 0) with prob. 1/2 d = (1, 1) with prob. 1/2 (0, 1) with prob. 1/2 d = (0, 0) with prob. 1/2 (0, 1) with prob. 1/2 d = (1, 1) with prob. 1/2 (2) (3) The lotteries d and d are equiprobable mixtures of (1, 0) and (0, 1) with an event (0, 0), in which neither individual is allocated the good, and the lotteries d and d are the corresponding equiprobable mixtures with an event (1, 1), in which both individuals are allocated the good. Note that, as (1, 0) (0, 1), independence requires that d d and that d d, regardless of the risk preferences of individuals 1 and 2. However, intuitively, if individual 1 tolerates risk better than individual 2, it may be sensible to prefer distributions in which the risk is born by individual 1 rather than by individual 2. Indeed, the extrapolated social preferences satisfy this property in this example: Lemma 1. If are extrapolated social preferences and C is an impartial base, then d d and d d. As this example illustrates, the extrapolated social preferences do not systematically prefer distributions that are more favorable to either individual. When choosing between d and d, society prefers the distribution that is more favorable to the risk neutral individual, but when choosing between d and d, society prefers the distribution that is more favorable to the risk averse individual. Compound independence. Segal [1990] distinguishes between mixture independence, which is the property discussed above, and compound independence. In Segal [1990], preferences over two-stage lotteries are said to 18

19 satisfy compound independence if, for every two-stage lottery, replacing a contingent (second-stage) lottery with a preferred lottery yields a preferable two-stage lottery. A random permutation Π({(a 1 (s),..., a I (s), p(s))} S s=1) can be viewed as a two-stage lottery, in which the first stage determines the permutation (π) and the second stage determines the state (s). The Pareto condition implies some form of compound independence. If, for some d D, a A and i, it holds that [d] i a (where is the consensus-ranking), then Π([d] 1,..., [d] I ) Π([d] 1,..., [d] i 1, a, [d] i+1,..., [d] I ). As Π(d) i Π([d] 1,..., [d] I ) for every i, if social preferences satisfy the Pareto condition, then this implies that Π(d) Π([d] 1,..., [d] i 1, a, [d] i+1,..., [d] I ). Quasi-concavity. Diamond [1967], Ben-Porath et al. [1997], Epstein and Segal [1992], Grant et al. [2010] and Chew and Sagi [2012] argue that social preferences should be consistent with ex-ante fairness - an equal allocation of lottery tickets should be preferable to a deterministic allocation that induces the same final distribution. In the case of the above example, this would imply that (0, 1) (a, a) and (1, 0) (a, a), where a is a lottery ticket which delivers the good with probability 1/2 (and nothing otherwise). Extrapolated social preferences will not necessarily exhibit quasi-concavity. This is illustrated by the following lemma: Lemma 2. Let be extrapolated social preferences, and assume that individual preferences satisfy the expected utility axioms. 1. Assume that the certainty base is impartial. Then, (0, 1) (a, a). 2. Assume that the base {λ a λ R + {0}} is impartial, and that individual 1 is risk neutral. (a) If individual 2 is risk averse, then (a, a) (0, 1). (b) If individual 2 is risk loving, then (0, 1) (a, a). 19

20 The first part of the lemma states that, when the certainty base is impartial, extrapolated social preferences are indifferent between the allocations (0, 1) and (a, a). The second part of the lemma considers the case in which the set {λ a λ R + {0}} is an impartial base, and individual 1 is risk neutral. In this case, the social ranking of (a, a) and (0, 1) depends on the risk preferences of individual 2. If individual 2 is risk averse, then the deterministic allocation is preferable. Quasi-concavity can be achieved only when individual 2 is risk-loving. Ex-post fairness. Ben-Porath et al. [1997], Fleurbaey [2010] and Chew and Sagi [2012] suggest a social preference for ex-post fairness, or limiting ex-post inequality. In the above example, this would require that an equiprobable lottery between (1, 1) and (0, 0) is preferable to the allocation (a, a). Given the specification of individual preferences, the requirement of expost fairness is inconsistent with the Pareto principle: as all individuals are assumed to be indifferent between the allocation (a, a) and an equiprobable lottery between (1, 1) and (0, 0), society must be indifferent as well. 12 Separability. Separability is a feature of utilitarian social welfare functions, which amounts to the following property: for any d, d D, a, a A and i, ([d ] 1,..., [d ] i 1, a, [d ] i+1,..., [d ] I ) ([d] 1,..., [d] i 1, a, [d] i+1,..., [d] I ) ([d ] 1,..., [d ] i 1, a, [d ] i+1,..., [d ] I ) ([d] 1,..., [d] i 1, a, [d] i+1,..., [d] I ) In other words, if d is preferred over d when individual i s allocation is fixed at some a, then d should be preferred over d when individual i s allocation is fixed at any a. 12 See Fleurbaey [2010] for further discussion on the compatibility of a social preference for ex-post fairness with the Pareto principle. 20

21 Extrapolated social preferences may violate separability. This can easily be seen using Corollary 2: if there is an individual with maximin risk preferences, then the extrapolated social preferences rank allocations by their least desirable base-equivalents. Thus, for a sufficiently undesirable a such that ˆb i (a) is less preferred than all elements of {ˆb k (d ), ˆb k (d)}, it follows that ([d ] 1,..., [d ] i 1, a, [d ] i+1,..., [d ] I ) ([d] 1,..., [d] i 1, a, [d] i+1,..., [d] I ). Of course, this is not necessarily true for every choice of a, in violation of the separability requirement. The separability requirement may also be violated when all individual preferences can be represented by expected utility functions. For a numerical example, see Appendix B. Personal responsibility. Recent philosophical approaches to distributive justice have emphasized the idea of personal responsibility (see Roemer and Trannoy [2016] for a review). Personal responsibility captures the idea that while inequality due to pure luck-of-the-draw should be eliminated, inequality due to individual choices should be acceptable. For example, income inequality should be acceptable insofar as it reflects differences in effort rather than differences in opportunities. Extrapolated social preferences can be made consistent with the notion of personal responsibility, provided that the set of goods (J) is specified appropriately. If consumption is specified as the only good (and leisure is not taken into account), then allocations will be ranked only by the distribution of consumption. However, if the good space is specified to include both consumption and leisure, then any base would consist of joint allocations of consumption and leisure. It is straightforward to see that, in this case, the optimal feasible allocation would not necessarily assign individuals with equal consumption; rather, because of the Pareto principle, the optimal allocation would respect heterogeneous individual preferences for leisure. 21

22 6 The every-base ranking Theorems 1 and 2 illustrate a procedure for extrapolating complete social preferences given an impartial base. However, they do not provide any guidance as to how to determine which base should be considered impartial. A natural question is to what extent the combination of the impartiality axiom and the Pareto axiom allows for the ranking of distributions without knowing which base is impartial. Formally, let B denote the social ranking represented by the extrapolated social welfare function e B. The every-base ranking,, is defined as follows: d d if and only if for every base B, d B d, and d d if and only if for every base B, d B d. Theorem 3. The every-base ranking,, is complete if and only if preferences are identical. This theorem states that, unless preferences are identical, there exist some distributions d, d D and some bases B, B A such that d B d but d B d. This theorem therefore highlights the importance of the philosophical debate regarding the appropriate equalisandum. A corollary of Theorem 3 is that requiring social preferences to be consistent with the impartiality of two different bases might be inconsistent with the Pareto condition. This point can be illustrated with the following simple example, in which the economy consists of a single good (J = 1) and two individuals (I = 2). Individual 1 is risk neutral and individual 2 has maximin risk preferences. Let C = R + denote the certainty base, and let L denote a linear base consisting of lotteries of the form λ l, where l is a lottery that delivers one unit of the good with probability 1/2 and n units of the good otherwise. Lemma 3. For n sufficiently large, (0.5 l, 1.5 l) L (0.5 l, 1.5 l). (l, l) but (l, l) C In this example, an equal distribution of lotteries may imply a highly 22

23 unequal distribution of certainty equivalents. A social preference for equalizing certainty equivalents would be inconsistent with a social preference for equalizing lottery tickets. This stylized example suggests that the objective of equalizing income and the objective of ex-ante fairness may be mutually inconsistent in the presence of heterogeneous risk preferences. The choice of an impartial base is analogous, in some ways, to the choice of utility weights in Harsanyi [1955]. Harsanyi s social ranking depends on the choice of individual utility weights, but the theory provides no guidance as to how they should be chosen. Similarly, the extrapolated social preferences depend on the choice of an impartial base, but the theory does not provide any guidance as to how it should be chosen. However, there are two reasons to believe that the problem of selecting an impartial base may be more approachable than the problem of selecting utility weights. First, Corollary 3 provides a direct link between the philosophical equalisandum and the choice of an impartial base. While the appropriate equalisandum has been the subject of active philosophical debate, little progress has been made on the question of how to appropriately normalize utility functions in order to make them interpersonally comparable. 13 This perhaps suggests that characterizing the right impartial base is an easier philosophical problem than characterizing the right utility weights. Second, there is a technical sense in which the ambiguity of the impartial base is less consequential than the ambiguity of utility representations in Harsanyi s case. To formalize this, it is useful to define the following sets: H = {(d, d) D 2 i θ i u i (d ) i θ i u i (d) for all (θ 1,..., θ I ) R I +} (4) P = {(d, d) D 2 d i d for all i} (5) 13 A notable exception is Argenziano and Gilboa [2017], who establish the existence of interpersonally comparable utility representations when individual preferences are allowed small violations of transitivity. 23

24 B = {(d, d) D 2 d d} (6) The set H consists of all distribution pairs that can be ranked according to Harsanyi s criterion, without taking a stance on the individual utility weights (θ i > 0). The set P consists of all distribution pairs that can be Pareto ranked. Finally, the set B consists of all distribution pairs that can be ranked according to the every-base ranking. Lemma H = P B. 2. There are cases in which P B; for example, if there are at least two individuals with the same preferences. The first part of the lemma establishes that the set of distributions that can be ranked according to Harsanyi s criterion without taking a stance on individual utility weights coincides with the set of distributions that can be Pareto ranked. Further, if two distributions are Pareto ranked, then they can also be ranked according to the every-base ranking. The second part of the lemma establishes that the converse is not necessarily true: there are some instances in which two distributions can be ranked according to the every-base ranking, even though they cannot be Pareto ranked. Thus, while the every-base ranking is generally a partial ranking, under some circumstances, it is broader than the Pareto ranking. This implies that, even without taking a stance on which base should be considered impartial, Theorems 1 and 2 may provide some guidance as to how to rank distributions that are not Pareto ranked. 7 Conclusion The construction of extrapolated social welfare functions shares the same basic intuition with Harsanyi [1955] and Vickrey [1960]. If there is a single good that is valued similarly by all individuals, then basic fairness requires 24

25 society to be indifferent between any two distributions of the good that differ only in the names of the individuals receiving each allocation. A natural way to express this indifference is by requiring society to be indifferent between a given distribution and a lottery that gives each individual the same chance of obtaining each allocation in the distribution. One problem is that there are potentially many different social welfare functions that are consistent with this impartiality requirement. For example, any symmetric social welfare function that satisfies the independence axiom will also satisfy this impartiality condition. Thus, on its own, this basic intuition does not uniquely pin down a social preference ranking. Things become even more complicated when there are multiple goods (including lotteries over goods), and preferences are not identical across individuals. Then, it becomes important to know which individual receives which allocation, because allocations are valued differentially by individuals with different preferences. The construction of extrapolated social welfare functions begins by recovering the intuition from the simplest case by mapping individual allocations to their base equivalents. A base represents a good that is valued similarly by all individuals. The simplest example is the certainty base. While individuals may have different risk preferences, their preferences are identical over the subset of deterministic allocations. Restricting attention to the distributions of certainty equivalents allows for the formulation of an impartiality requirement that is analogous to the single-good case. To uniquely pin down a complete social ranking, the impartiality requirement is then combined with a standard Pareto condition. The Pareto condition allows for the social ranking of arbitrary distributions based on their base equivalents. By construction, each individual is indifferent between an allocation and its base equivalent. A standard Pareto condition implies that social preferences should be indifferent between them as well. I show that the combination of the Pareto condition and the impartiality 25

26 requirement uniquely pins down a social ranking, which can be represented by the extrapolated social welfare function. The extrapolated social welfare function is the limit of an iterative process, in which each individual is repeatedly asked to evaluate distributions of base equivalents under the veil of ignorance. Given a distribution of base elements, each individual is asked to report his base equivalent for the equiprobable lottery among the distribution s allocations. These base equivalents constitute a new distribution of base elements, with which the process repeats. By considering an equiprobable lottery over the base equivalents from the previous round, individuals are repeatedly incorporating each other s perspectives on the desirability of the underlying distribution. Remarkably, this process converges to a single agreement point, which can be thought of as the equally-distributed base equivalent of the underlying distribution. It is worth emphasizing that extrapolated social preferences are unique only up to a choice of base. Different bases may imply different social rankings. The ethical question of which base should be used for making interpersonal comparisons of welfare is related to the philosophical debate on the appropriate choice of an equalisandum, the object that society should strive to equalize across individuals. However, even without choosing a base, the theory is able to generate a partial social ranking of distributions that is broader than the Pareto ranking. Further research is required in order to understand how broad this partial ranking is. References Rossella Argenziano and Itzhak Gilboa. Foundations of weighted utilitarianism Kenneth J. Arrow. A difficulty in the concept of social welfare. Journal of Political Economy, 58(4): ,