Fair criteria for social decisions under uncertainty

Accepted Manuscript Fair criteria for social decisions under uncertainty Kaname Miyagishima

PII: DOI: Reference:

S0304-4068(18)30122-8 https://doi.org/10.1016/j.jmateco.2018.10.005 MATECO 2278

To appear in:

Journal of Mathematical Economics

Received date : 14 March 2018 Revised date : 27 October 2018 Accepted date : 31 October 2018 Please cite this article as: Miyagishima K., Fair criteria for social decisions under uncertainty. Journal of Mathematical Economics (2018), https://doi.org/10.1016/j.jmateco.2018.10.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript Click here to view linked References

Fair Criteria for Social Decisions under Uncertainty∗ Kaname Miyagishima† Aoyama Gakuin University

This version: August 10, 2018

Abstract In a simple model where agents have ordinal and interpersonally noncomparable subjective expected utility preferences over uncertain future incomes, we analyze the implications of equity, efficiency, separability, and social rationality. Our efficiency conditions are fairly weak, because there are criticisms on the standard ex ante Pareto principle in the litererature. Our social welfare criteria from the axioms satisfy ex ante equity, but violate Statewise Dominance, often referred to as "the minimal criterion" of rationality under uncertainty.

∗

This research was started while I visited Princeton University. I am especially grateful to Marc Fleur-

baey for invaluable discussions and suggestions as well as kind hospitality at the university. I also thank Walter Bossert, Susumu Cato, Kazuhiko Hashimoto, Laurence Kranich, Antonin Macé, Daisuke Nakajima, Shmuel Nitzan, Marcus Pivato, Yasuhiro Shirata, William Thomson, Toyotaka Sakai, Norihito Sakamoto, Kotaro Suzumura, Stéphane Zuber, participants at Conference on Economic Design in 2015, and two anonymous referees and Simon Grant for helpful comments and discussions. I greatly appreciate Koichi Suga for his hospitality at Waseda University. The financial supports by JSPS and the Institute of Economic Research at Aoyama Gakuin University are gratefully acknowledged. † Email: [email protected]

1

1

Introduction

Although welfare economics has provided various social criteria, there is still disagreement concerning which should be used to evaluate social situations under risk and uncertainty.1 A major reason for this disagreement is tension between equity, efficiency, and social rationality. In this paper, we study implications of these principles in a simple model where agents’ future monetary payoffs (called acts) are uncertain, and preferences are represented by ordinal and interpersonally noncomparable subjective expected utility functions, following the fair social welfare function approach (Fleurbaey and Maniquet, 2011). We first study social criteria satisfying weak equity and efficiency requirements as well as axioms of separability and invariance of preference changes. In particular, we consider fairly weak efficiency axioms because, as discussed in Section 6, it has been often argued in the literature that the standard ex ante Pareto principle is not compelling. Despite these criticisms, this principle is considered important in many fields of economics. We examine what kind of efficiency principle can be obtained from our axioms. One of our efficiency conditions is inspired by Pareto for Equal Risk (Fleurbaey, 2010; Fleurbaey and Zuber, 2017). Another efficiency axiom is Social Monotonicity, which states that increases in all agents’ incomes in every state should imply a social improvement. We consider two equity axioms. The first one, Transfer Principle, requires that for a pair of agents with the same preference, if one agent has more income in each state than the other, then the inequality should be reduced by a transfer. Another one, Dominance Averse Transfer, insists that any transfer should be socially accepted. The former equity is weaker than the latter. We also introduce separability requiring that agents who are indifferent on a social judgment should not affect the judgment (Fleming, 1952), and an independence of risk preferences whenever riskless allocations are compared (Chambers and Echenique, 2012). A main contribution of this paper is as follows. We show that the separability, the efficiency principles above, and Transfer Principle together imply Ex Ante Pareto (Lemma 1), and thus are incompatible with Dominance Averse Transfer as shown by Fleurbaey 1

Mongin and Pivato (2016) and Weymark (1991) provide comprehensive surveys of the literature.

2

and Trannoy (2003). This result is striking because it uncovers a tension between equity, efficiency, and separability in our environment. Lemma 1 would also provide a normative justification to use Ex Ante Pareto if the separability is accepted. Using this result, we derive a maximin social criterion based on certainty equivalents (Theorem 1). By the result above, if Dominance Averse Transfer is considered more compelling than Ex Ante Pareto, the separability axiom should be weakened. Then, our next step is to consider a combination of weaker separability termed Well-off Separability (Fleurbaey and Maniquet, 2011, Axiom 5.3), Dominance Averse Transfer, and two efficiency principles. Well-off Separability requires that an irrelevant agent should not affect social judgements if the agent is unambiguously better off than some other agent. This separability respects information on the worse-off agents which would be important for egalitarian social evaluations. Two other Pareto conditions are introduced. The first is Pareto for Riskless Acts, which requires that agents should not be forced to take risks if they do not wish to. The second is Pareto for Consensual Risk-taking, which states that if each agent can move from a riskless situation to a risky situation and this risk-taking behavior is supported by all agents, the risk-taking is also socially supported. Using these axioms and the independence axiom, we derive another form of maximin criterion satisfying Dominance Averse Transfer (Theorem 3). This is another main contribution of this paper. We also show some impossibilities under Statewise Dominance, often referred to as the "minimal" rationality under uncertainty. Theorem 5 shows that an efficiency axiom weaker than Unanimity Pareto Principle proposed by Gayer et al. (2014) is not compatible with a weak ex post equity axiom, Weak Certainty Inequality Aversion, under Statewise Dominance. Theorem 6 implies that a large class of social criteria, including expected equally distributed equivalent criteria (Fleurbaey and Zuber, 2017), does not satisfy Weak Certainty Inequality Aversion under Well-off Separability. Then, we argue that our criteria satisfy ex ante equity but violate Statewise Dominance. The remainder of the paper is organized as follows. In Section 2, we present the model. Section 3 analyzes the implications of the first set of axioms including the separability, Transfer Principle, and Pareto for Equal Risk. In Section 4, we consider social criteria satisfying Well-off Separability and Dominance Averse Transfer. In Section 5, we discuss

3

Statewise Dominance and ex ante equity. In Section 6, the related literature is introduced. Section 7 offers concluding remarks. All proofs and the independence of axioms are shown in the appendix.

2

The Model

Let N be an infinite set of possible agents. N is the family of finite subsets of N such that for each N ∈ N , |N | ≥ 2. S = {s1 , ..., sm } is the finite set of states with m ≥ 2. We denote

by xis ∈ R+ the amount of money agent i receives under state s ∈ S. An act of agent i

is denoted by xi = (xis )s∈S ∈ RS+ , which is a vector of state-contingent monetary payoffs. Let X = RS+ be the set of acts. x = (xs )s∈S ∈ X is called a constant act if xs = xs′ for all

¯ be the set of constant acts. For each x ∈ X, ¯ the value of money in every s, s′ ∈ S. Let X

state is denoted by x. An allocation is denoted by xN = (xi )i∈N ∈ X N for each N ∈ N .

Ri is agent i’s preference relation over X, with the strict part Pi and the indifference

part Ii . A binary relation is an ordering if it is complete and transitive. Let R denote a set of preferences represented by subjective expected utility functions. More specifically, for

each Ri ∈ R, there exists a subjective expected utility function such that, for all xi , y i ∈ X, xi Ri y i ⇐⇒

∑ s∈S

pis ui (xis ) ≥

∑

pis ui (yis ),

s∈S

where pi is a probability distribution over S and ui : R+ → R is a continuous and increasing function. For convenience, we express a pair of agent i’s belief and utility function as (pi , ui ).2 Given xj ∈ X and Rj ∈ R, I(xj , Rj ) = {z ∈ X|zIj xj } is the indifference set,

L(xj , Rj ) = {z ∈ X|xj Rj z} is the lower-contour set, ˚ L(xj , Rj ) = {z ∈ X|xj Pj z} is

the strict lower-contour set, U (xj , Rj ) = {z ∈ X|zRj xj } is the upper-contour set, and

˚(xj , Rj ) = {z ∈ X|zPj xj } is the strict upper-contour set. Let 1 = (1, . . . , 1) ∈ X. ¯ Given U

N ′ ⊂ N , let us denote by (xN ′ , y N \N ′ ) an allocation such that each i ∈ N ′ has xi and each j ∈ N \N ′ has y j . 2

Our results do not depend on the domain restriction: The results hold on broader domains where

preference orderings satisfy, for instance, only monotonicity and continuity. The proofs are upon request.

4

A social ordering function (SOF), R, is a mapping that for every preference profile determines a complete and transitive binary relation over the set of allocations. The domain ∪ is denoted by D = N ∈N RN . Given a preference profile RN ∈ D, R(RN ) is a social ordering

over X N . In addition, let P (RN ) and I(RN ) be the strict and indifference parts of R(RN ), respectively.

3

Ex Ante Pareto, Transfer Principle, and Separability

In this section, we introduce several axioms and provide the first characterization. The first axiom is the standard ex ante Pareto condition. Ex Ante Pareto. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N , if xi Pi x′i for all i ∈ N , then xN P (RN )x′N .

This axiom implies that a unanimous improvement in terms of ex ante preferences should be socially preferred. We also introduce an equity axiom. Dominance Averse Transfer. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N , if there exist j, k such that xi = x′i for all i ̸= j, k, and for all ∆ ∈ RS++ such that

∆ = λ(xj − xk ) for some λ ∈ R++ ,

[ ] xj = x′j − ∆ ≥ xk = x′k + ∆ ⇒ xN R(RN )x′N . This axiom states that for two agents, if one has more income in every state than the other, a transfer in each state to reduce the inequality is acceptable. This axiom could be interpreted as an ex post equity condition because ex post inequality is reduced in each state. As shown by Fleurbaey and Trannoy (2003), there exists no social ordering satisfying Ex Ante Pareto and Dominance Averse Transfer. Thus, in order to have some possibility results, we must weaken at least one of these conditions. In what follows, we introduce weak and compelling versions of efficiency and equity, and elaborate on what orderings would be obtained by combining other axioms. 5

We introduce an efficiency requirement relevant to Pareto for Equal Risk first developed by Fleurbaey (2010) in a model of interpersonally comparable utility, and considered by Fleurbaey and Zuber (2017) in a model of economic environments with noncomparable utility. Weak Pareto for Equal Risk. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N such that Ri = Rj , xi = xj , x′i = x′j for all i, j ∈ N , if xi Pi x′i for all i ∈ N , then xN P (RN )x′N .

This axiom requires that an ex ante unanimous agreement should be judged as a social improvement only when all agents have an equal preference and acts in the allocations. In this situation, because all agents are subject to the same conditions and will have the same consequence, there would be no concern that the agreement is inconsistent with any equity conditions. Moreover, this requirement avoids the problem of spurious unanimity because the agents have the same outcome ex post. Since this axiom considers the situations where all agents have the same preference, Weak Pareto for Equal Risk is weaker than the Pareto for Equal Risk condition of Fleurbaey and Zuber (2017). We introduce another efficiency principle. Social Monotonicity. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N , if xi ≫ x′i for all i ∈ N , then xN P (RN )x′N .

According to this axiom, increases in all agents’ future incomes should be socially preferred. This requirement is clearly compelling. The next axiom is an equity principle weaker than Dominance Averse Transfer. Transfer Principle. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N , if there exist j, k

such that Rj = Rk , and xi = x′i for all i ̸= j, k, then for all ∆ ∈ RS++ such that ∆ = λ(xj − xk ) for some λ ∈ R++ ,

[ ] xj = x′j − ∆ ≥ xk = x′k + ∆ =⇒ xN R(RN )x′N . This requirement insists that among two agents with the same preference, if one has more income in every state than the other, a redistribution to decrease the inequality should be 6

socially acceptable.

3

The next invariance axiom was essentially introduced by Chambers and Echenique (2012). Invariance to Risk Attitudes and Beliefs for Constant Acts (IRBC). For all N ∈ ′ ¯N, N , all RN , RN ∈ D, and all xN , x′N ∈ X

′ xN R(RN )x′N ⇐⇒ xN R(RN )x′N .

This axiom claims that social judgments over allocations of constant acts should be invariant of risk preferences and beliefs. The idea is that as long as riskless outcomes are compared, agents’ risk preferences and beliefs are irrelevant for the comparisons. This axiom could also be justified from the strategic point of view that social decisions over certain outcomes should be robust to agents’ misreporting of their risk preferences. Next, we introduce two forms of separability. Separability. For all N ∈ N such that |N | ≥ 3, all RN ∈ D and all xN , x′N ∈ X N , if xi = x′i for some i ∈ N , then

xN R(RN )x′N ⇐⇒ xN \{i} R(RN \{i} )x′N \{i} . Weak Separability. For all N ∈ N such that |N | ≥ 3, all RN ∈ D and all xN , x′N ∈ X N , if xi = x′i for some i ∈ N , then

xN P (RN )x′N =⇒ xN \{i} P (RN \{i} )x′N \{i} . These axioms require that if one agent has the same act under two allocations, the social ranking over the two allocations should be invariant of excluding this indifferent agent. Clearly, Separability is stronger than Weak Separability. As discussed below, these requirements have different implications. Based on the axioms described above, we derive a social welfare criterion based on certainty equivalence. Given xi ∈ X and Ri ∈ R, let C(xi , Ri ) = inf{c ∈ R+ |(c, · · · , c)Ri xi }, 3

Note that the form of transfer in this requirement is proportional to the agents acts. This is because

the usual transfer principle is strong and close to maximin, as argued by Bosmans et al. (2017).

7

which is the certainty equivalence of xi with respect to Ri . Then, we obtain the following result. Theorem 1. Suppose that an SOF R satisfies Weak Pareto for Equal Risk, Social Monotonicity, Transfer Principle, IRBC, and Weak Separability. Then, for all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N ,

min C(xi , Ri ) > min C(x′i , Ri ) =⇒ xN P (RN )x′N . i∈N

i∈N

We offer remarks on this result. As is standard in the literature on fair social ordering, the characterization is partial. For instance, if a standard continuity condition is also required, we immediately have a full characterization of the certainty equivalence maximin ordering, RC , which is defined below: For all N ∈ N , all RN ∈ D, all xN , x′N ∈ X N , xN RC (RN )x′N ⇐⇒ min C(xi , Ri ) ≥ min C(x′i , Ri ). i∈N

i∈N

Note that RC does not satisfy Separability (considered in Lemma 3 and Theorem 2), and hence we just require Weak Separability here so as not to exclude the maximin social criterion. RC was characterized by Fleurbaey and Maniquet (2011, Section 6.2) and Ertemel (2016) in related models. A difference between their researches and ours is that they used standard ex ante Pareto principles, while we use much weaker efficiency conditions. The proof of Theorem 1 makes use of the following two lemmas, which are interesting in their own right. The first lemma reveals a tension between equity, efficiency, and separability under risk and uncertainty: Weak Separability in addition to the fairly weak axioms of equity and efficiency yields Ex Ante Pareto, which is incompatible with Dominance Averse Transfer (Fleurbaey and Trannoy, 2003).4 Lemma 1. Weak Pareto for Equal Risk, Social Monotonicity, Transfer Principle, and Weak Separability together imply Ex Ante Pareto. This result is relevant to tensions between equity, efficiency, and separability under uncertainty: If we require that form of separability in addition to those fairly weak conditions of equity and efficiency, we should accept Ex Ante Pareto and thus give up Dominance 4

Note that RC satisfies Ex Ante Pareto and violates Dominance Averse Transfer.

8

Averse Transfer by Fleurbaey and Trannoy (2003, Proposition 5). Lemma 1 could be a normative justification to adopt Ex Ante Pareto rather than Dominance Averse Transfer if Weak Separability is accepted. The next lemma establishes infinite ex post inequality aversion, which is captured by the next axiom. ¯ N , if there Certainty Inequality Aversion. For all N ∈ N , all RN ∈ D and xN , x′N ∈ X exist j, k such that xi = x′i for all i ̸= j, k ∈ N , then

[ ′ xj > xj > xk > x′k ] ⇒ xN R(RN )x′N . Lemma 2. Weak Pareto for Equal Risk, Social Monotonicity, Transfer Principle, Weak Separability, and IRBC together imply Certainty Inequality Aversion. Then, we can derive the conclusion of Theorem 1 from Certainty Inequality Aversion and Ex Ante Pareto applying the lemmas above. If we require Separability instead of Weak Separability, the following result is obtained. Lemma 3. If an SOF R satisfies Weak Pareto for Equal Risk, Transfer Principle, and Separability, it also satisfies Social Monotonicity. Then, since Separability implies Weak Separability, the conclusions of Lemmas 1 and 2 are obtained by replacing Social Monotonicity and Weak Separability with Separability. Hence, we have the following theorem by the same discussion as above. Theorem 2. Suppose that an SOF R satisfies Weak Pareto for Equal Risk, Transfer Principle, IRBC, and Separability. Then, for all N ∈ N , all RN ∈ RN and all xN , x′N ∈ X N ,

min C(xi , Ri ) > min C(x′i , Ri ) =⇒ xN P (RN )x′N . i∈N

i∈N

From Lemma 3, the axioms imply Social Monotonicity. Since Separability is stronger than Weak Separability, the desired result follows from Theorem 1.

9

We introduce a social criterion satisfying all of the axioms in Theorem 2. This criterion is the certainty equivalence leximin ordering RLC : For all N ∈ N , all RN ∈ D, all xN ,

x′N ∈ X N ,

( ) ( ) xN RLC (RN )x′N ⇐⇒ C(xi , Ri ) i∈N ≥lex C(x′i , Ri ) i∈N ,

5 where ≥lex is the usual lexicographic ordering over RN +.

As shown above, the separability principles and the quite weak conditions of equity and

efficiency imply Ex Ante Pareto, which is incompatible with Dominance Averse Transfer. However, if Dominance Averse Transfer is considered to be more reasonable than Ex Ante Pareto in the society, Lemma 1 is an undesirable result. Then, our next problem is to consider social orderings satisfying Dominance Averse Transfer. In the next section, we relax the separability principle, and derive a certain social ordering from the stronger equity conditions including Dominance Averse Transfer and another two Pareto conditions.

4

Dominance Averse Transfer and Relaxing Separability

In this section, we introduce a weaker separability condition and two Pareto conditions, and characterize a social ordering satisfying Dominance Averse Transfer. First, we introduce the weaker separability condition. Well-off Separability. For all N ∈ N such that |N | ≥ 3, all RN ∈ D, and all xN , x′N ∈ X N , if Ri = Rj , xi Pi xj , x′i Pi x′j , and xi = x′i for some i, j ∈ N , then xN P (RN )x′N =⇒ xN \{i} P (RN \{i} )x′N \{i} . In an egalitarian society, information about disadvantaged individuals would be important for social decision making. This axiom captures the idea and requires that an irrelevant agent should not affect the social ordering only when s/he is unambiguously better off than another. 5

Theorem 2 does not provide a direct characterization of RLC because it does not say anything when

mini∈N C(xi , Ri ) = mini∈N C(x′i , Ri ). It remains for future research to obtain a full characterization of RLC .

10

Next, we introduce two efficiency conditions. The first Pareto axiom requires that agents’ preferences for riskless situations be respected. Pareto for Riskless Acts. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N such that

¯ M , if xi Pi x′ for all i ∈ M and xj = x′ for all there exists M ⊆ N with xM ∈ X i j j ∈ N \M , then xN R(RN )x′N .

This axiom insists that if a subgroup of agents prefer riskless acts to risky acts, such preferences should be socially supported. In other words, this requirement says that agents can avoid risks if they want to. For instance, suppose that Ann does not want to buy a stock because she expects the price to decrease, whereas Bob does not want to sell short because he thinks that the price will increase. Then, it would be unreasonable to force Ann to buy the stock and make Bob sell short. Note that if the agents in M have constant acts also in x′M , the axiom still makes sense because xM ≫ x′M . We also introduce the second Pareto condition.

Pareto for Consensual Risk-taking. For all N ∈ N , all RN ∈ D, and all xN ∈ X N , ¯ N , if xi Pj x′ for all i, j ∈ N , then xN P (RN )x′ . x′N ∈ X i N

This axiom requires that if each agent’s (potential) risk-taking behavior is supported by all agents’ preferences, such a risky situation should be socially preferred. Note that if xN is also a riskless allocation, the axiom remains reasonable because xN ≫ x′N . Note also

that this axiom is weaker than Sprumont’s (2012) Consensus.6 Pareto for Consensual Risktaking avoids the problem of spurious unanimity caused by different beliefs, because each agent thinks that all individuals’ risk-takings are beneficial according to his/her preference. This axiom may be criticized that an allocation with unequal outcomes can be socially preferred to another allocation with equal outcomes. However, we subsequently show that the axiom is consistent with strong equity conditions such as Dominance Averse Transfer and Certainty Inequality Aversion. From the axioms described above, we derive another maximin social choice criterion. 6

Consensus says that for all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N , if xi Pj x′i for all i, j ∈ N , then

xN P (RN )x′N .

11

Theorem 3. Suppose that an SOF R satisfies Pareto for Consensual Risk-taking, Pareto for Riskless Acts, IRBC, Well-off Separability, and Dominance Averse Transfer. Then, for all N ∈ N , all RN ∈ RN and all xN , x′N ∈ X N , min min C(xi , Rj ) > min min C(x′i , Rj ) =⇒ xN P (RN )x′N .7 i∈N j∈N

i∈N j∈N

This social criterion firstly evaluates each agents’ act using the lowest certainty equivalence, and then compares allocations by the minimum values of the lowest certainty equivalences. Next, we consider a stronger equity condition, which requires that if there is income inequality in every state between two agents, reducing such inequality should be socially weakly preferable. Dominance Aversion. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N , if there exist j, k ∈ N such that xi = x′i for all i ̸= j, k,

[ ′ ] xj ≫ xj ≥ xk ≫ x′k ⇒ xN R(RN )x′N . Then, we obtain the following result, which is remarkable since a few axioms of equity and efficiency lead to a characterization of RM . Theorem 4. Suppose that an SOF R satisfies Pareto for Consensual Risk-taking, Pareto for Riskless Acts, and Dominance Aversion. Then, for all N ∈ N , all RN ∈ RN and all xN , x′N ∈ X N ,

min min C(xi , Rj ) > min min C(x′i , Rj ) =⇒ xN P (RN )x′N . i∈N j∈N

i∈N j∈N

If we additionally require a continuity property in Theorems 3 and 4, we can characterize another maximin ordering RM defined below. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ XN ,

xN RM (RN )x′N ⇐⇒ min min C(xi , Rj ) ≥ min min C(x′i , Rj ). i∈N j∈N

i∈N j∈N

The results in this section are related to those of Sprumont (2012). In a deterministic environment, Sprumont (2012, Theorem 1) characterized a class of social orderings called 7

It is not difficult to show another characterization replacing Well-off Separability with Replication

Invariance, which requires that replicating the economy should not affect the social rankings.

12

consensual Rawlsian orderings using Dominance Aversion and Consensus introduced above. Each consensual Rawlsian ordering evaluates allocations based on worst bundles with respect to a social evaluation ordering R∗ over commodity bundles. The ordering R∗ is assumed to agree with unanimous judgments in the sense that for any two bundles x and y, xP ∗ y if xPi y for all i ∈ N . Note that there are as many consensual Rawlsian orderings as the number of social evaluation orderings over commodity bundles. Although Spru-

mont’s results are remarkable, it is difficult to determine which social evaluation ordering should be adopted. Our social ordering uses simple criteria for interpersonal comparison based on certainty equivalence. Another difference between Sprumont’s (2012) results and ours is that he used a single-profile framework, while we have provided a multi-profile theorem (Theorem 3) using the weaker equity condition (Dominance Averse Transfer) and independence axiom (IRBC).8

5

Statewise Dominance

So far we have only required that social criteria should satisfy completeness and transitivity as social rationality. In this section, we consider a well-known social rationality condition called Statewise Dominance, and show that this axiom conflicts with ex post equity, efficiency, and separability in our environment. The argument is not new, but it is worth discussing inconsistency between Statewise Dominance and these three principles in the context of our environment. First, we formally introduce Statewise Dominance. For each x ∈ X and each s ∈ S, let

¯ be such that xs′ (s) = xs′′ (s) = xs for all s′ , s′′ ∈ S. This is the constant act where x(s) ∈ X ¯ N is similarly defined. the consequence of act x in state s is obtained for sure. xN (s) ∈ X

Statewise Dominance. For all N ∈ N , all RN ∈ D and all xN , x′N ∈ X N , if xN (s)R(RN )x′N (s) for all s ∈ S, then xN R(RN )x′N .

This axiom states that if every consequence of an allocation is weakly socially better than that of another allocation, the former allocation is socially weakly preferred to the latter. If 8

Sakamoto (2018) develops a formally similar rule refining the consensus approach in the problem of the

indexing dilemma in the context of an ordinal interpersonal comparison of individual situations.

13

the axiom is violated, society may choose an allocation that could result in a worse outcome. We also consider an efficiency requirement. Pareto for Risk with Equivalent Value. For all N ∈ N , all RN ∈ D, and all xN ∈ X N

¯ N such that Ri = Rj , xi Ii xj , and x′ = x′ for all i, j ∈ N , if xi Pi x′ for and x′N ∈ X i j i

all i ∈ N , then xN P (RN )x′N .

This axiom requires that if all agents with the same expected utility functions prefer possibly uncertain acts to riskless acts and the acts in each allocation are equally valuable to the agents, then the possibly uncertain allocation should be socially preferred to the riskless allocation. This requirement is weaker than Pareto for Consensual Risk-taking, and Unanimity Pareto Principle advocated by Gayer et al. (2014).9 This condition also avoids the problem of spurious unanimity because all agents are assumed to have the same preferences. Lastly, we introduce an ex post equity condition. Weak Certainty Inequality Aversion. For all N ∈ N , all RN ∈ D, there exists b > 0

¯ N , if there exist j, k such such that, for all ϵ ∈ [0, b], all t ∈ R++ , and all xN , x′N ∈ X that Rj = Rk and xi = x′i for all i ̸= j, k ∈ N ,

[ xj = x′j − (t + ϵ) ≥ xk = t > x′k = 0] =⇒ xN R(RN )x′N . This axiom considers income transfers under certainty which are "leaky" by a limited amount b, and requires that these transfers to the agent with no income should be accepted. Note that the transfer is between agents with the same preference. This is clearly much weaker than Certainty Inequality Aversion, and satisfied not only by RC and RM but also a broader class of SOFs (especially when the limitation b is very small). We consider this axiom to show a large class of SOFs including RC and RM cannot be compatible with Statewise Dominance. The reader can notice that the impossibilities below can be obtained even if b is very small. 9

( ) Unanimity Pareto Principle requires that for all N ∈ N , all RN ∈ D with (pi , ui ) i∈N such that

pi = pj for all i, j ∈ N , and all xN , x′N ∈ X N , if xi Pj x′i for all i, j ∈ N , then xN P (RN )x′N .

14

We now obtain the following impossibility theorem.10 Theorem 5. There exists no SOF satisfying Pareto for Risk with Equivalent Value, Weak Certainty Inequality Aversion, and Statewise Dominance. Since RC and RM satisfies Pareto for Risk with Equivalent Value and Weak Certainty Inequality Aversion, we can see from this result that the systems of axioms in Theorem 1 and 3 are incompatible with Statewise Dominance. If agents are risk averse and have nonconvex preferences, it is straightforward to obtain a stronger impossibility result by replacing Weak Certainty Inequality Aversion with a weaker ex post Pigou-Dalton transfer principle. A crucial point is that, if an SOF satisfies Pareto for Risk with Equivalent Value, an uncertain allocation in which agents have different outcomes is preferred to a riskless allocation where agents have equal outcomes, which leads to incompatibility with Statewise Dominance and Weak Certainty Inequality Aversion. One way to avoid this problem is to restrict the Pareto principle to the case in which all agents have equal risk such as Pareto for Equal Risk (Fleurbaey 2010; Fleurbaey and Zuber, 2017). However, if we additionally require Weak Separability as well as Social Monotonicity and Transfer Principle, we have Ex Ante Pareto (Lemma 1), and thus it could be difficult to satisfy Statewise Dominance. Even with Well-off Separability, we have a difficulty as shown below. Theorem 6. There exists no SOF satisfying Weak Pareto for Equal Risk, Social Monotonicity, Weak Certainty Inequality Aversion, Well-off Separability, and Statewise Dominance. The proof shows that if Well-off Separability is combined with Weak Pareto for Equal Risk, we have an implication similar to Pareto for Risk with Equivalent Value discussed above. Thus, we should relax Well-off Separability if rationality is more compelling than separability. 10

The example in the proof is similar to the one provided by Fleurbaey and Voorhoeve (2013) to show

that Statewise Dominance and Ex Ante Pareto conflict with ex post equity. Our discussion is intended to clarify that in our environment, the conflict remains even if Ex Ante Pareto is relaxed to Pareto for Risk with Equivalent Value.

15

The reader can notice that in the proof, Social Monotonicity is applied only under certainty. This means that expected equally distributed equivalent criteria (Fleurbaey and Zuber, 2017), which are derived from Weak Pareto for Equal Risk and Social Monotonicity only under certainty, cannot satisfy Weak Certainty Inequality Aversion under Well-off Separability. These impossibilities above are relevant to the results by Gajdos and Tallon (2002). They consider a resource allocation problem under uncertainty, and show that there exist no feasible allocations satisfying ex ante Pareto efficiency and ex post no-envy if agents’ beliefs are different or there are aggregate risks on initial endowments (otherwise, there exists such a feasible allocation). In contrast, Theorem 5 shows that even if agents share the same belief, Pareto for Risk with Equivalent Value (much weaker than ex ante Pareto) and Weak Certainty Inequality Aversion (which can be applied only when the agent receiving the transfer has no income) are incompatible under Statewise Dominance. Moreover, Theorem 6 implies that we have another impossibility under Well-off Separability even if the efficiency axiom is further relaxed. RC and RM violate Statewise Dominance, but satisfy the following ex ante equity condition. Ex Ante Inequality Reduction. For all N ∈ N , all RN ∈ D, and all xN , x′N ∈ X N , if there exist j, k such that xi = x′i for all i ̸= j, k ∈ N ,

[ ] U (x′i , Ri ) ⊂ U (xi , Ri ) ⊆ U (xj , Rj ) ⊂ U (x′j , Rj ) =⇒ xN R(RN )x′N . This axiom requires that reducing inequality in terms of ex ante preferences should be socially acceptable. It can be checked, using the usual example (Diamond, 1967), that this requirement is not compatible with Statewise Dominance. Thus, the criteria derived from our axioms respect ex ante equity rather than dynamic consistency. This is relevant to Grant’s (1995) example showing that there is a tension between Statewise Dominance and ex ante equity.11

11

See Mongin and Pivato (2016, section 24.5).

16

6

Related Literature

The pathbreaking work is Harsanyi’s (1955) aggregation theorem stating that, in the context of risk, if a social welfare function satisfies the expected utility condition and Ex Ante Pareto, then it should be an affine combination of the agents’ expected utility functions. It has often been pointed out that Harsanyi’s social welfare function is not compatible with equity properties. For instance, Diamond (1967) criticizes that it does not satisfy ex ante equity because of the expected utility condition. Neither can ex post equity be satisfied: Ex Ante Pareto is not compatible with ex post equity under Statewise Dominance (Adler and Sanchirico, 2006; Fleurbaey and Voorhoeve, 2013). Fleurbaey (2010) proposes a class of expected equally distributed equivalent (EEDE) orderings, and characterizes the class by Pareto for Equal Risk, a Pareto condition under no risk, and Statewise Dominance. He derives two leximin criteria belonging to the class using additional conditions of equity and efficiency. He also shows that among EEDE orderings, the utilitarian social ordering is the only one satisfying separability motivated by independence of the utility of the dead (Blackorby et al., 2005), and thus it cannot be compatible with equity properties (see also Fleurbaey et al. 2015). Fleurbaey and Zuber (2013) consider separability only under certainty, and derive an ex post equitable EEDE criterion. In contrast, our results show that separability is consistent with ex ante equity, but not compatible with rationality even in the form of Statewise Dominance. As mentioned in the introduction, there are some criticisms on Ex Ante Pareto. For instance, as argued by Hammond (1981), decision making under risk and uncertainty is quite difficult for individuals because of misperceptions of probability, overconfidence, and other heuristics. Then, it is not compelling to fully respect agents’ ex ante preferences, and a degree of paternalism may be justified. For example, mandatory social insurance programs are partly based on this idea. Another criticism is that if agents have different beliefs, ex ante unanimity may be spurious (Mongin, 2015).12 Thus, we just require weaker Pareto conditions and investigate the implications.13 12 13

See also Gayer et al. (2014). Recent contributions on this issue include Gilboa et al. (2014), Danan et al., (2015), Zuber (2016),

Hayashi and Lombardi (2017), and Qu (2017).

17

We do not assume interpersonal comparability of utility, because there are criticisms that the assumption is strong. We have derived resource-based criteria for interpersonal comparison from our axioms. By adopting this approach, we can clarify ways of comparing agents’ levels of well-being and the value judgments behind the interpersonal comparisons. The domain of expected utility functions is standard in the literature, but any experimental results show that individuals’ preferences typically violate the independence axiom of von Neumann and Morgenstern (vNM) or Savage’s sure-thing principle. Our results do not depend on the domain restriction, and could be obtained in much larger domains. Miyagishima (2018) derives a social welfare criterion using axioms of ex post equity, efficiency, and Statewise Dominance. The social criterion compares allocations by the statewise minimum incomes evaluated by the certainty equivalences. This criterion was considered by Fleurbaey and Zuber (2017): Under the situation the planner and agents share the same belief, they characterized the criterion using the expected utility condition, supposing that there exists an agent who is always the most risk averse. This criterion also satisfies Weak Separability under certainty.

7

Concluding Remarks

In this paper, we studied welfare criteria for social decisions under uncertainty. The results are summarized on the table above. Two criteria have been derived from the principles of equity, efficiency and separability. Our results show that separability principles (Separability and Weak Separability) have the strong implication that when those are combined with requirements of equity and efficiency (Weak Pareto for Equal Risk, Social Monotonicity, and Transfer Principle), we must have Ex Ante Pareto and thus give up Dominance Averse Transfer. Hence, to obtain a social ordering satisfying Dominance Averse Transfer, we have introduced Well-off Separability and derived another maximin criterion. Our interpersonal comparison is based on certainty equivalence, which is reasonable in the context of risk and uncertainty. We have also argued that our social criteria are not consistent with Statewise

18

Theorems

Criterion

Theorem1

mini∈N C(xi , Ri )

Theorem 2

mini∈N C(xi , Ri )

Theorem 3

mini∈N minj∈N C(xi , Rj )

Theorem 4

mini∈N minj∈N C(xi , Rj )

Theorem 5

None

Theorem 6

None

Axioms Weak Pareto for Equal Risk, IRBC Transfer Principle, Weak Separability, Social Monotonicity Weak Pareto for Equal Risk, IRBC, Transfer Principle, Separability Pareto for Riskless Acts, Pareto for Consensual Risk-taking, Dominance Averse Transfer, IRBC, Well-off Separability Pareto for Riskless Acts, Pareto for Consensual Risk-taking, Dominance Aversion Pareto for Risk with Equivalent Value, Statewise Dominance, Weak Certainty Inequality Aversion Weak Pareto for Equal Risk, Social Monotonicity, Weak Certainty Inequality Aversion, Well-off Separability, Statewise Dominance

Dominance, but satisfy ex ante equity such as Ex Ante Inequality Reduction.14 Our results show that separability conditions, in the form of Separability or Well-off Separability, can be compatible with ex ante equity or Ex Ante Pareto, but incompatible with rationality conditions, as shown by Lemma 1 and Theorem 6. If we place more importance on principles of equity and dynamic consistency such as Dominance Averse Transfer and Statewise Dominance, other forms of separability are required. This issue has been considered in the environment of expected utility (Fleurbaey and Zuber, 2013; Fleurbaey et al., 2015). In this paper, we have considered individual preferences over state-contingent monetary outcomes. In general, however, state-contingent outcomes are multidimensional (e.g., Fleurbaey and Zuber, 2017). On this point, it would be straightforward to extend our results to the general case with minor modifications. We also emphasize that our results do not depend on the domain restriction to the subjective expected utility functions: The same results can be obtained in the domains of various preferences including non-expected 14

Epstein and Segal (1992) characterize quadratic social welfare functions using an ex ante equity condi-

tion.

19

utility functions. We focused on the implications of the axioms rather than optimal allocations or public policies to implement the allocations. In future research, we intend to model social insurance systems, such as unemployment and disability insurance using modified versions of our social welfare criteria in the relevant models.

8

Appendix

Proof of Lemma 1. Let xN , x′N be such that xi Pi x′i for all i ∈ N . For each i ∈ N , let x∗i ∈ I(xi , Ri ) be such that x∗i ≫ x′i .

A roadmap of the proof is as follows: By Weak Separability, we can consider economies

composed of each agent i ∈ N and her copies with the same preference. Then, using

Transfer Principle, Pareto for Equal Risk, Social Monotonicity, and Weak Separability, we can show that in each economy, it is socially more desirable for agent i to have xi − ϵi 1 than

x′i , where ϵi > 0 is sufficiently small. Using Weak Separability, these results can be combined

and the copies of the agents in N are eliminated: Thus, we have (xi − ϵi 1)i∈N P (RN )x′N . Social Monotonicity and transitivity imply the desired result.

!′ #$& = #$( ) %% ) #$ )′

#$% *(#$% , -$ )

Figure 1: N 1 = {1, 1a , 1b }

20

*(#$ , -$ ) !

¯ \N such that R1a = R1 = R1 . Now we give the formal proof. First, consider 1a , 1b ∈ N b

This step is illustrated by Figure 1. Let us denote N 1 = {1, 1a , 1b } and x1a = x1b = x∗1 . Define y = x′1 +2(x∗1 −x′1 )/3 and y N 1 = (y 1 , y 1a , y 1b ) = (y, y, y). By Transfer Principle, we

have y N 1 R(RN 1 )(x′1 , x1a , x1b ). Let y ′N 1 = (x1 − (n + 1)ϵ1 1, x1 − (n + 1)ϵ1 1, x1 − (n + 1)ϵ1 1), where ϵ1 > 0 is small enough that [x1 − (n + 1)ϵ1 1]P1 y. Then, by Weak Pareto for Equal Risk, we obtain y ′N 1 P (RN 1 )y N 1 .

Next, define y ′′ ∈ X such that x∗1 ≫ y ′′ and y ′′ P1 y ′ . Denote y ′′N 1 = (x1 − (n +

1)ϵ1 1, y ′′ , y ′′ ). Since (y ′′ , y ′′ )P (R{1a ,1b } )(x1 − (n + 1)ϵ1 1, x1 − (n + 1)ϵ1 1) by Weak Pareto for Equal Risk, we obtain y ′′N 1 R(RN 1 )y ′N 1 from Weak Separability. Transitivity implies y ′′N 1 P (RN 1 )(x′1 , x1a , x1b ). Here we consider N 1+ = {1, 2, · · · , n, 1a , 1b }. From y ′′N 1 P (RN 1 )(x′1 , x1a , x1b ) and Weak

Separability, we have

(x1 − (n + 1)ϵ1 1, x′2 , · · · , x′n , y ′′ , y ′′ )R(RN 1+ )(x′1 , x′2 , · · · , x′n , x1a , x1b ), and Social Monotonicity implies (x1 − nϵ1 1, x′2 + ϵ∗ 1, · · · , x′n + ϵ∗ 1, x1a , x1b )P (RN 1+ )(x1 − (n + 1)ϵ1 1, x′2 , · · · , x′n , y ′′ , y ′′ ), where ϵ∗ is small enough that x∗i ≫ x′i + nϵ∗ 1 for all i ̸= 1. By transitivity and Weak Separability, we have

(x1 − nϵ1 1, x′2 + ϵ∗ 1, · · · , x′n + ϵ∗ 1)P (RN )x′N . Similarly, considering N 2 = {2, 2a , 2b } and x2a = x2b = x∗2 , we can show (x2 − nϵ2 1, x2a − ϵ′ 1, x2b − ϵ′ 1)P (RN 2 )(x′2 + ϵ∗ 1, x2a , x2b ), where ϵ2 > 0 is defined similarly to ϵ1 and ϵ′ > 0 is sufficiently small. Let N 2+ = {1, · · · , n, 2a , 2b }. Consider the following allocations. z N 2+ = (x1 − nϵ1 1, x′2 + ϵ∗ 1, x′3 + ϵ∗ 1, · · · , x′n + ϵ∗ 1, x2a , x2b ), z ′N 2+ = (x1 − nϵ1 1, x2 − nϵ2 1, x′3 + ϵ∗ 1, · · · , x′n + ϵ∗ 1, x2a − ϵ′ 1, x2b − ϵ′ 1), z ′′N 2+ = (x1 − (n − 1)ϵ1 1, x2 − (n − 1)ϵ2 1, x′3 + 2ϵ∗ 1, · · · , x′n + 2ϵ∗ 1, x2a , x2b ). 21

Applying Weak Separability to the last social ranking, we obtain z ′N 2+ R(RN 2+ )z N 2+ . Moreover, by Social Monotonicity, we have z ′′N 2+ P (RN 2+ )z ′N 2+ . z ′′N 2+ P (RN 2+ )z N 2+ follows from transitivity. Then, Weak Separability yields z ′′N P (RN )z N , where z N = (x1 − nϵ1 1, x′2 + ϵ∗ 1, x′3 + ϵ∗ 1, · · · , x′n + ϵ∗ 1), z ′′N = (x1 − (n − 1)ϵ1 1, x2 − (n − 1)ϵ2 1, x′3 + 2ϵ∗ 1, · · · , x′n + 2ϵ∗ 1). Remember z N = (x1 − nϵ1 1, x′2 + ϵ∗ 1, · · · , x′n + ϵ∗ 1)P (RN )x′N . Then, by transitivity again, we obtain z ′′N = (x1 − (n − 1)ϵ1 1, x2 − (n − 1)ϵ2 1, x′3 + 2ϵ∗ 1, · · · , x′n + 2ϵ∗ 1)P (RN )x′N . By repeating the same procedure, we have (x1 − ϵ1 1, x2 − ϵ2 1, · · · , xn − ϵn 1)P (RN )x′N . The desired result can be obtained from Social Monotonicity and transitivity. □ ¯ N be allocations such that x′ > xj > xk > x′ and Proof of Lemma 2. Let xN , x′N ∈ X j k

xi = x′i for all i ̸= j, k. Since these allocations are composed of constant acts, we can invoke

IRBC to arbitrarily modify the preferences. Let Rj and Rk be such that Rj = Rk = R0

represented by an expected utility function with (p0 ; u0 ) defined below. We construct a situation illustrated in Figure 2. Then, applying Transfer Principle, Ex-ante Pareto (Lemma 1), and transitivity, we obtain the desired result. (p0 ; u0 ) is constructed as follows. Let us denote a, b, c, α, β, γ, δ, ϵ1 , ϵ2 , ϵ3 ∈ R++ as pa-

rameters such that

αxk + β(a − xk ) = mαx′k + ϵ1 ,

(1)

αxk + β(a − xk ) + γ(b − a) = mαxk − ϵ2 ,

(2)

αxk + β(a − xk ) + γ(b − a) + δ(c − b) = m[αxk + β(x′j − xk )] + ϵ3 ,

(3)

22

#′

'"

'(" = '(%

!" !"&

'%

!%& !%

#

Figure 2: Proof of Lemma 2 where a > xk , c − a = x′j − x′k , b =

c+a , 2

and ϵ1 , ϵ2 , ϵ3 are arbitrarily close to 0. By simple

calculations, we have

mαx′k − αxk + ϵ1 , a − xk mα(xk − x′k ) − ϵ1 − ϵ2 , γ= b−a β(x′j − xk ) − ϵ2 + ϵ3 δ= . c−b β=

Note that

1 1 b − a = (x′j − x′k ), c − b = (x′j − x′k ). 2 2 For α, β, γ, δ to be positive, we can set, for instance, a = 2xk , α =

ϵ1 . 2|mx′′k − xk |

Define u0 : R+ → R+ as follows.    αx for x ∈ [0, xk ],        αxk + β(x − xk ) for x ∈ (xk , a], u0 (x) =    αxk + β(a − xk ) + γ(x − a) for x ∈ (a, b],       αx + β(a − x ) + γ(b − a) + δ(x − b) for x > b, k k 23

Let us also define p0s = 1/m for all s ∈ S.15 It is straightforward to check that this is consistent with conditions (1) to (3) above. Note that under (p0 ; u0 ), it follows u0 (a) u0 (b) u0 (c) > u0 (x′k ), < u0 (xk ), > u0 (x′j ). m m m Let RN ∈ D denote the preference profile where all agents have R0 .

Consider N ∗ = {j, k}. Let y k = (ϵ + x′k /πs1 , 0, · · · , 0). By the construction of R0 ,

˚(x′j , R0 ) such that y j ≫ y k and y ˆ = if ϵ is sufficiently small, there exists y j ∈ RS++ ∩ U

y k + (y j − y k )/2 ∈ ˚ L(xk , R0 ) (y j can be chosen sufficiently close to (c, 0, · · · , 0)). Define

ˆ ). By Lemma 1, we can apply Ex Ante Pareto to obtain y, y y N ∗ = (y j , y k ), and y ′N ∗ = (ˆ

y N ∗ P (RN ∗ )(x′j , x′k ). Transfer Principle implies y ′N ∗ R(RN ∗ )y N ∗ . Again by Ex Ante Pareto, (xj , xk )P (RN ∗ )y ′N ∗ . It follows from transitivity that (xj , xk )P (RN ∗ )(x′j , x′k ). Since xi = x′i for all i ̸= j, k, Weak Separability implies xN R(RN )x′N . Applying IRBC

to adjust the preference profile, we have the desired result. □

Proof of Theorem 1. Let xN and x′N be allocations such that mini∈N C(xi , Ri ) > mini∈N C(x′i , Ri ). ¯ N as Without loss of generality, suppose C(x′1 , R1 ) = mini∈N C(x′i , Ri ). Define y N ∈ X ( ) y i = C(x′i , Ri ) + ϵ 1 for all i ∈ N , where ϵ > 0 is small enough that { } ′ ′ C(x1 , R1 ) + 3ϵ < min min C(xi , Ri ), min C(xi , Ri ) . i̸=1

i∈N

From Lemma 1, we can apply Ex Ante Pareto to obtain y N P (RN )x′N . ( ) ( ) Next, let y ′N ∈ X N be such that y ′1 = C(x′1 , R1 ) + 2ϵ 1 and y ′i = C(x′1 , R1 ) + 3ϵ 1

for all i ̸= 1. Applying Certainty Inequality Aversion (Lemma 2) repeatedly, we have

y ′N R(RN )y N . It follows from Ex Ante Pareto (Lemma 1) that xN P (RN )y ′N . Transitivity implies xN P (RN )x′N as sought. □ Proof of Lemma 3. Let xN , x′N be such that xN ≫ x′N . Let N = {1, 2, · · · , n}. First,

we consider, for each i ∈ N , {i, id } such that id ̸∈ N and Ri = Rid . Let xid = xi and

y i = y id = x′i + 21 (xi − xid ). By Transfer Principle, (y i , y id )R(R{i,id } )(x′i , xid ). Weak

Pareto for Equal Risk implies (xi , xid )P (R{i,i′ } )(y i , y id ). From transitivity, we obtain (xi , xid )P (R{i,id } )(x′i , xid ) for each i ∈ N. 15

Remember that m is the cardinality of S.

24

(4)

Next, we show (x1 , x2 )P (R{1,2} )(x′1 , x′2 ). Since (x1 , x1d )P (R{1,1d } )(x′1 , x1d ) by (1), repeated applications of Separability imply (x1 , x1d , x′2 , x2d )P (R{1,1′ } )(x′1 , x1d , x′2 , x2d ), and thus (x1 , x′2 , x2d )P (R{1,2,2d } )(x′1 , x′2 , x2d ).

(5)

Moreover, (4) for i = 2 and Separability imply (x1 , x2 , x2d )R(R{1,2,2d } )(x1 , x′2 , x2d ).

(6)

It follows from (5), (6), transitivity and Separability that (x1 , x2 )P (R{1,2} )(x′1 , x′2 ).

(7)

Next, we prove (x1 , x2 , x3 )P (R{1,2,3} )(x′1 , x′2 , x′3 ). By (7) and Separability, we have (x1 , x2 , x′3 , x3d )P (R{1,2,3,3d } )(x′1 , x′2 , x′3 , x3d ). Moreover, (4) for i = 3 and Separability implies (x1 , x2 , x3 , x3d )P (R{1,2,,3,3d } )(x1 , x2 , x′3 , x3d ). It follows from transitivity and Separability that (x1 , x2 , x3 )P (R{1,2,3} )(x′1 , x′2 , x′3 ). By repeating this procedure, we have the desired result. □ To prove Theorem 3, we use the lemma below, which has a similar implication to Certainty Inequality Aversion. Lemma 4. Suppose that an SOF R satisfies Pareto for Riskless Acts, Pareto for Consensual Risk-taking, Dominance Averse Transfer, and IRBC. Then, for all N ∈ N , all ¯ N , if there exist j, k such that xi ≫ x′ for all i ̸= j, k, RN ∈ D and all xN , x′N ∈ X i [ ′ xj > xj > xk > x′k ] ⇒ xN P (RN )x′N .16 16

It is straightforward to obtain a stronger result using Transfer Principle rather than Dominance Averse

Transfer.

25

Proof. The proof is similar to that of Lemma 2. Suppose that x′j > xj > xk > x′k and xi > x′i for all i ̸= j, k. Since xN and x′N are riskless allocations, RN can be arbitrarily

modified using IRBC. Then, assume that RN is such that Ri = R0 for all i ∈ N , where R0 is the same as defined in the proof of Lemma 2. Let y N , y ′N ∈ X N be such that y j , y ′j , y k , y ′k are as introduced in the proof of Lemma 2, and for all i ̸= j, k, ¯ and yi = y ′ = xi + ϵ, y i , y ′i ∈ X, i where ϵ > 0 is small enough that xi > x′i + ϵ and xk − (n − 2)ϵ1Pk y ′k . By similar

arguments to Lemma 2, we can obtain y N P (RN )x′N from Pareto for Consensual Risktaking and y ′N P (RN )y N from Dominance Averse Transfer. It follows from transitivity that y ′N P (RN )x′N . ¯ N be such that Let y ′′N ∈ X yk′′ = xk − (n − 2)ϵ, yj′′ = xj , yi′′ = yi′ for all i ̸= j, k. Noting that y ′′k Pk y ′k , we obtain y ′′N R(RN )y ′N from Pareto for Riskless Acts. Repeated applications of Dominance Averse Transfer (transfers from all i ̸= j, k to k) imply xN R(RN )y ′′N . xN P (RN )x′N follows from transitivity. By modifying RN appropriately using IRBC, we obtain the desired result. □ Proof of Theorem 3. Let xN and x′N be allocations such that xN P M (RN )x′N . We show xN P (RN )x′N . Without loss of generality, let 1 ∈ N be such that min C(x′1 , Rj ) = min min C(x′i , Rj ). j∈N

i∈N j∈N

The proof is divided into two cases depending on whether or not C(x′1 , R1 ) = minj∈N C(x′1 , Rj ). Case 1. Let C(x′1 , R1 ) = minj∈N C(x′1 , Rj ). This case is illustrated by Figure 3 (with the ¯ N as ˆN , z ˆ ′N ∈ X assumption that C(x2 , R2 ) = mini∈N minj∈N C(xi , Rj )). We introduce z

ˆ k Rk x′k and follows: For all k ̸= 1, z

C(x′1 , R1 ) = min C(x′1 , Rj ) < zˆ1 < zˆ1′ < zˆk′ < min min C(xi , Rj ) < zˆk . j∈N

i∈N j∈N

26

!′ (&̂

#

%$(′̂

#&

($̂ )(#$% , ,$ )

!

Figure 3: Theorem 3; Case 1

ˆ N R(RM )x′N . Applying Lemma 4 repeatedly, we can By Pareto for Riskless Acts, we have z ˆ ′N P (RM )ˆ ˆ ′N P (RM )x′N . show z z N . Transitivity implies z ˆ i′ for all i, j ∈ N . Thus, by Pareto for By the definition of RM , we can see xi Pj z

Consensual Risk-taking, we have xN P (RN )ˆ z ′N . By transitivity, we obtain xN P (RM )x′N as desired. Case 2. Suppose that C(x′1 , R1 ) ̸= minj∈N C(x′1 , Rj ). Let agent i∗ (i∗ ̸= 1) be such that

C(x′1 , Ri∗ ) = minj∈N C(x′1 , Rj ).

Figure 4 illustrates the proof of this case with i∗ = 2. In this case, x′1 is the worst-off in x′N but evaluated by R2 , and thus we cannot directly apply the argument of Case 1. The proof proceeds as follows. First, by Pareto for Riskless Act, agent 2’s act can be moved from x′2 to y ≫ x′1 . Then, applying Well-off Separability repeatedly, we can introduce sufficiently

many copies of agent 1 with x′1 + ϵ1 for sufficiently small ϵ > 0. Using Dominance Averse Transfer repeatedly, all agents can move to y ′ such that } { ′ ′ C(y , R2 ) < min min min C(xi , Rj ), C(x1 + ϵ1, R1 ) , i∈N j∈N

so that we can later eliminate the copies of agent 1 using Well-off Separability. Here, agent 2 becomes the worst-off in the allocation and C(y ′ , R2 ) = minj∈N C(y ′ , Rj ), and thus we 27

!′ ' '%

#

%$-

((#$% , +& )

#&

! Figure 4: Theorem 3; Case 2

can apply the same argument as Case 1 above. The formal proof is as follows. Let y N ∈ X N be such that y 1 = x′1 , y i = y k for all

¯ ∩U ˚(x′ , Rj ) and y j ≫ x′ for all j ̸= 1. By Pareto for Riskless i, k ∈ N \{1}, and y j ∈ X j 1 Acts, we have y N R(RN )x′N .

¯ \N be as follows: Define ∆∗ = y i∗ − y 1 . Let ϵ > 0 and Q ⊂ N (i) y i∗ − ϵ|Q|∆∗ = y 1 + 12ϵ∆∗ ; (ii) minj∈N C(y 1 + 12ϵ∆∗ , Rj ) < mini∈N minj∈N C(xi , Rj ); (iii) Rj = R1 for all j ∈ Q. Without loss of generality, suppose that C(y 1 + 11ϵ∆∗ , R2 ) = minj∈N C(y 1 + 11ϵ∆∗ , Rj ). ˆ M = (y N , x∗Q ) and x ˆ ′M = (x′N , x∗Q ), Denote x∗ = y 1 + 11ϵ∆∗ and M = N ∪ Q. Define y ˆ M R(RM )ˆ where x∗Q = (x∗ , · · · , x∗ ) ∈ X Q . By Well-off Separability, we have y x′M . Let us introduce y ′M , y ′′M ∈ X M such that

y ′2 = y ′i = y 1 + 12ϵ∆∗ for all i ∈ Q, y ′k = y k for all other k. y ′′1 = y ′′2 = y 1 + 6ϵ∆∗ , y ′′k = y ′k for all other k.

28

By repeated applications of Dominance Averse Transfer (transferring from agent 2 to all i ∈ Q), we have y ′M R(RM )ˆ y M . Again by Dominance Averse Transfer (transferring from agent 2 to 1), we obtain y ′′M R(RM )y ′M . Then, y ′′M R(RM )ˆ x′M by transitivity. ¯ M as follows. Define z M , z ′M ∈ X z i = y 2 for all i ∈ Q ∪ {1}, z 2 = C(y 1 + 7ϵ∆∗ , R2 )1, z k = y ′k = y 2 for all other k. [ ] [ ] z ′2 = C(y 1 + 7ϵ∆∗ , R2 ) + ϵ′ 1, z ′k = C(y 1 + 7ϵ∆∗ , R2 ) + 2ϵ′ 1 for all k ̸= 2,

where ϵ′ > 0 is sufficiently small so that C(y 1 +7ϵ∆∗ , R2 )+4ϵ′ < C(y 1 +11ϵ∆∗ , R2 ). Pareto for Riskless Acts implies z M R(RM )y ′′M . Applying Lemma 3 repeatedly, z ′M P (RM )z M . x′M . From transitivity, we obtain z ′M P (RM )ˆ Let z ′′M ∈ X M be such that [ ] z ′′k = C(y 1 + 7ϵ∆∗ , R2 ) + 3ϵ′ 1 for all k ∈ N, z ′′i = x∗ = y 1 + 11ϵ∆∗ for all i ∈ Q.

Then, since C(y 1 + 11ϵ∆∗ , R2 ) = minj∈N C(y 1 + 11ϵ∆∗ , Rj ), by the definition of RM and condition (ii) above, we can see z ′′i Pj z ′i for all i, j ∈ M . Thus, by Pareto for Consensual

Risk-taking, we have z ′′M P (RN )z ′M . By transitivity, we obtain z ′′M P (RM )ˆ x′M . From Ri = R1 for all i ∈ Q, x∗ P1 x′1 , and x∗ P1 z ′′1 , Well-off Separability yields z ′′N P (RM )x′N . Similarly,

by the definition of RM and condition (ii), we can see xi Pj z ′′i for all i, j ∈ M , and thus

xN P (RN )z ′′N from Pareto for Consensual Risk-taking. Transitivity implies xN P (RN )x′N as sought. □

Proof of Theorem 4. Let xN and x′N be allocations such that xN P M (RN )x′N . The goal is to show xN P (RN )x′N . As in the proof of Theorem 3, let 1 ∈ N be such that minj∈N C(x′1 , Rj ) = mini∈N minj∈N C(x′i , Rj ), and the proof is divided into two cases depending on whether or not C(x′1 , R1 ) = minj∈N C(x′1 , Rj ). Case 1. Let C(x′1 , R1 ) = minj∈N C(x′1 , Rj ). This case exactly resembles Case 1 in the proof of Theorem 3, and thus we can safely omit it. Case 2. Let C(x′1 , R1 ) ̸= minj∈N C(x′1 , Rj ). Without loss of generality, let 2 ∈ N be such 29

that C(x′1 , R2 ) = min min C(x′i , Rj ). i∈N j∈N

¯ ∩U ˚(x′j , Rj ) and y j ≫ x′1 for all j ̸= 1. By Pareto Define y N ∈ X N as y 1 = x′1 , and y j ∈ X

for Riskless Acts, we have y N R(RN )x′N .

Let us also introduce z N , z ′N , z ′′N ∈ X N as follows. y 1 ≪ z 1 ≪ z 2 ≪ y 2 , min C(z 2 , Rj ) < min min C(xi , Rj ), z k = y k for all k ̸= 1, 2. j∈N

i∈N j∈N

¯ min C(z 2 , Rj ) < min C(z ′ , Rj ) < min min C(xi , Rj ), z ′ = z k for all k ̸= 1, 2. z ′1 = y 2 , z ′2 ∈ X, 2 k j∈N

j∈N

i∈N j∈N

¯ N , min C(z ′ , Rj ) < min C(z ′′ , Rj ) < min C(z ′′ , Rj ) < min min C(xi , Rj ) for all k ̸= 2. z ′′N ∈ X 2 k 2 j∈N

j∈N

j∈N

i∈N j∈N

By Dominance Aversion, we have z N R(RN )y N . Pareto for Riskless Acts implies z ′N R(RN )z N . By applying Dominance Aversion repeatedly, z ′′N R(RN )z ′N . From transitivity, we obtain z ′′N R(RN )x′N . Here, by the definition of RM and min min C(xi , Rj ) > min C(z ′′k , Rj ) for all k ∈ N i∈N j∈N

j∈N

we can see xi Pj z ′′i for all i, j ∈ N . Thus, by Pareto for Consensual Risk-taking, we have xN P (RN )z ′′N . By transitivity, we obtain xN P (RN )x′N as sought. □

Proof of Theorem 5. Let us consider S = {a, b}, N = {1, 2} and R1 = R2 = R∗ ∈ R

represented by an expected utility function 12 xia + 12 xib , where xia and xib are incomes of agent i in states a and b, respectively. Let b be the upper bound of income transfers for this profile in Weak Certainty Inequality Averse. By taking sufficiently small ϵ ∈ (0, b], Weak Certainty Inequality Aversion implies (

) ( ) (15 − ϵ, 15 − ϵ), (15 − ϵ, 15 − ϵ) R(RN ) (30, 30), (0, 0) , ( ) ( ) (15 − ϵ, 15 − ϵ), (15 − ϵ, 15 − ϵ) R(RN ) (0, 0), (30, 30) .

Thus, Statewise Dominance implies (

) ( ) (15 − ϵ, 15 − ϵ), (15 − ϵ, 15 − ϵ) R(RN ) (30, 0), (0, 30) . 30

(8)

However, it follows from Pareto for Risk with Equivalent Value that (

) ( ) (30, 0), (0, 30) P (RN ) (15 − ϵ, 15 − ϵ), (15 − ϵ, 15 − ϵ) .

(9)

(8) and (9) together imply a contradiction. □

Proof of Theorem 6. Let us consider S = {a, b}, N = {1, 2, 3, 4} ∈ N and Ri = R∗ ∈ R

for all i ∈ N represented by an expected utility function 12 xia + 12 xib , where xia and xib are incomes of agent i in states a and b, respectively.

We first introduce allocations v N , wN , xN , xN , y N , z N such that v 1 = v 2 = (30, 0), v 3 = v 4 = (0, 30 + a); w1 = w2 = (0, 30), w3 = w4 = (0, 30 + a); x1 = x2 = (30, 0), x3 = x4 = (30 + a, 0); y 1 = y 2 = (0, 30), y 3 = y 4 = (30 + a, 0); z 1 = z 2 = z 3 = z 4 = (15 − a′ , 15 − a′ ), where a, a′ ∈ R++ are sufficiently small and a < a′ . By completeness, we consider two cases: (i) y N R(RN )xN ; (ii) xN P (RN )y N .

(i) Suppose y N R(RN )xN . For a sufficiently small ϵ > 0, Weak Pareto for Equal Risk implies

(

) (30 − ϵ, 0), (30 − ϵ, 0), (30 − ϵ, 0), (30 − ϵ, 0) P z N .

Moreover, 0I(RN )0 by completeness, and by Social Monotonicity, (

) (30, 30), (30, 30), (30 + a, 30 + a), (30 + a, 30 + a) ( ) P (RN ) (30 − ϵ, 30 − ϵ), (30 − ϵ, 30 − ϵ), (30 − ϵ, 30 − ϵ), (30 − ϵ, 30 − ϵ) ,

(10)

and then Statewise Dominance imply

( ) xN R(RN ) (30 − ϵ, 0), (30 − ϵ, 0), (30 − ϵ, 0), (30 − ϵ, 0) ,

and hence we have y N P (RN )z N by transitivity. It is straightforward, however, to obtain z N P (RN )y N using Statewise Dominance and Weak Certainty Inequality Aversion, by the same argument as in Theorem 5 given a and a′ are sufficiently small. 31

(ii) Suppose xN P (RN )y N . We have x{1,2} P (R{1,2} )y {1,2} by Well-off Separability. By

Weak Separability again, we obtain v N R(RN )wN . For a sufficiently small ϵ > 0, Weak Pareto for Equal Risk yields ( ) (0, 30 − ϵ), (0, 30 − ϵ), (0, 30 − ϵ), (0, 30 − ϵ) P z N .

As in (i) above, by 0I(RN )0, (10), and Statewise Dominance, we have wN P (RN )z N . Then, we obtain v N P (RN )z N by transitivity. Again by the same argument as in Theorem 5, we can show z N P (RN )v N , which is a contradiction. □ Next, we show the independence of the axioms below. First, the following examples show none of the axioms in Theorem 1 is redundant. Dropping Weak Pareto for Equal Risk. Consider social ordering R1 defined as follows: For all xN , x′N ∈ X N , xN R1 (RN )x′N if and only if mini∈N mins∈S xis ≥ mini∈N mins∈S x′is .

Dropping Social Monotonicity. Consider social ordering R2 defined as follows: For all xN , x′N ∈ X N , (1) if |N | ≤ 3, xN R2 (RN )x′N if and only if mini∈N C(xi , Ri ) ≥ mini∈N C(x′i , Ri ); (2) if |N | > 3, xN R2 (RN )x′N if and only if mini∈N C(xi , Ri ) ≥ mini∈N C(x′i , Ri ) when

Ri = Rj for all i, j ∈ N ; and xN I 2 (RN )x′N otherwise.

Dropping Transfer Principle. xN , x′N ∈ X N , xN R3 (RN )x′N

Consider social ordering R3 defined as follows: For all ∑ ∑ if and only if i∈N C(xi , Ri ) ≥ i∈N C(x′i , Ri ).

Dropping Weak Separability. Consider RM .

Dropping IRBC. Let A(x, Ri ) = inf{c ∈ R+ |(2c, c, · · · , c)Ri x} for each x ∈ X and Ri ∈ R.

Consider social ordering R4 defined as follows: Then, for all xN , x′N ∈ X N , xN R4 (RN )x′N if and only if mini∈N A(xi , Ri ) ≥ mini∈N A(x′i , Ri ).

Now we show the independence of the axioms in Theorem 2. To drop Transfer Principle, Separability, and IRBC, we can use R3 , RM , and R4 defined above, respectively. Dropping Weak Pareto for Equal Risk. Consider social ordering R5 defined as follows: For all xN , x′N ∈ X N , xN I 5 (RN )x′N . The following examples show none of the axioms in Theorem 3 is redundant. Dropping Pareto for Riskless Acts. Consider social ordering R1 above. 32

Dropping Pareto for Consensual Risk-taking. Consider social ordering R6 defined as follows: For all xN , x′N ∈ X N , xN R6 (RN )x′N if and only if mini∈N maxj∈N C(xi , Rj ) ≥

mini∈N maxj∈N C(x′i , Rj ).

Dropping Dominance Averse Transfer. Consider RC . Dropping IRBC. Remember A(x, Ri ) = inf{c ∈ R+ |(2c, c, · · · , c)Ri x} for each x ∈ X and

Ri ∈ R. Consider social ordering R7 defined as follows: Then, for all xN , x′N ∈ X N , ( ) ( ) (1) xN P 7 (RN )x′N if κ(xi , RN ) i∈N >lex κ(x′i , RN ) i∈N , where ≥lex is the usual lexico( ) ( ) graphic ordering; (2) if κ(xi , RN ) i∈N =lex κ(x′i , RN ) i∈N , then xN R7 (RN )x′N if and

only if mini∈N A(xi , Ri ) ≥ mini∈N A(x′i , Ri ).

Dropping WOS. Consider social ordering R8 defined as follows: For all xN , x′N ∈ X N ,

(1) If mini∈N κ(xi , RN ) > mini∈N κ(x′i , RN ), then xN P 8 (RN )x′N ; (2) if mini∈N κ(xi , RN ) = mini∈N κ(x′i , RN ), then xN R8 (RN )x′N if and only if |{i|C(xi , Ri ) = minj∈N κ(xj , RN )}| ≥ |{i|C(x′i , Ri ) = minj∈N κ(x′j , RN )}|.

The axioms in Theorem 4 are obviously independent.

References [1] Adler, M.D., Sanchirico, C.W. (2006). "Inequality and uncertainty: Theory and legal applications." University of Pennsylvania Law Review, 155, 279-377. [2] Blackorby, C., Bossert, W., Donaldson, D. (2005). Population Issues in Social Choice Theory, Welfare Economics and Ethics. Econometric Society Monograph, Cambridge University Press. [3] Bosmans, K., Decancq, K., Ooghe, E., (2017). "Who’s afraid of aggregating money metrics?" Forthcoming in it Theoretical Economics. [4] Chambers, C.P., Echenique, E. (2012). "When does aggregation reduce risk aversion?" Games and Economics Behavior, 76, 582-595. [5] Diamond, P. (1967). "Cardinal welfare, individual ethics, and interpersonal comparisons of utility: Comment." Journal of Political Economy, 75, 765-766. 33

[6] Danan, E., Gajdos, T., Hill, B., Tallon, J.-M. (2015). "Robust social decisions." American Economic Review, forthcoming. [7] Epstein, L.G., Segal, U. (1992). "Quadratic social welfare functions." Journal of Political Economy, 100, 691-712. [8] Ertemel, S., (2016). "Welfare egalitarianism under uncertainty." Available at https://dl.dropboxusercontent.com/u/9401843/Welfare.pdf. [9] Fleming, M. (1952). "A cardinal concept of welfare." Quarterly Journal of Economics 66, 366-384. [10] Fleurbaey, M. (2010). "Assessing risky social situations." Journal of Political Economy, 118-4, 649-680. [11] Fleurbaey, M., Gajdos, T., Zuber, S. (2015). “Social rationality, separability, and equity under uncertainty." Mathematical Social Sciences, 73, 13-22. [12] Fleurbaey, M., Maniquet, F. (2011). A Theory of Fairness and Social Welfare. Econometric Society Monograph, Cambridge University Press. [13] Fleurbaey, M., Trannoy, A., (2003). "The impossibility of a Paretian egalitarian." Social Choice and Welfare, 21-2, 243-263. [14] Fleurbaey, M., Voorhoeve, A. (2013). "Decide as you would with full information. An argument against ex ante Pareto." In N. Eyal, S.A. Hurst, O.F. Norheim, D. Wikler eds., Inequalities in Health. Concepts, Measures, and Ethics, Oxford University Press. [15] Fleurbaey, M., Zuber, S. (2013). "Inequality aversion and separability in social risk evaluation." Economic Theory, 54, 675-692. [16] Fleurbaey, M., Zuber, S. (2017). "Fair management of social risk." Journal of Economic Theory, 169, 666-706. [17] Gajdos, T., and Tallon, J.-M., (2002) "Fairness under Uncertainty." Economics Bulletin, 4-18, 1-7. 34

[18] Gayer, G., Gilboa, I., Samuelson, L., Schmeidler, D., (2014). "Pareto efficiency with different beliefs." Journal of Legal Studies, 43, S151-S171. [19] Grant, S., (1991). "Subjective Probability Without Monotonicity: or How Machina’s Mom May Also be Probabilistically Sophisticated." Econonometrica, 63-1, 159-189. [20] Gilboa, I., Samuelson, L., Schmeidler, D. (2014). "No betting Pareto dominance." Econometrica, 82, 1405-1442. [21] Hammond, H. (1981). "Ex-ante and ex-post welfare optimality under uncertainty." Economica, 48, 235-50. [22] Harsanyi, J. (1955). "Cardinal welfare, individualistic ethics and interpersonal comparisons of utility." Journal of Political Economy, 63, 309-321. [23] Hayashi, T., Lombardi, M. (2016). "Fair social decision under uncertainty and responsibility for beliefs." Forthcoming in Economic Theory. [24] Miyagishima, K., (2017). "Efficiency, equity, and social rationality under uncertainty." Available at https://sites.google.com/site/kanamemiagishimashp/eg [25] Mongin, P., (2016). "Spurious unanimity and the Pareto Principle." Economics and Philosophy, 32-3, 511-532. [26] Mongin, P., Pivato, M. (2015). "Ranking multidimensional alternatives and uncertain prospects." Journal of Economic Theory, 157, 146-171. [27] Mongin, P., Pivato, M. (2016). "Social preference and social welfare under risk and uncertainty." In M. Adler, M. Fleurbaey eds., Oxford Handbook of Well-Being and Public Policy, Oxford, University Press. [28] Qu, X., (2017). "Separate aggregation of beliefs and values under ambiguity." Economic Theory, 63-2, 503-519. [29] Sakamoto, N., (2018). "Equity criteria based on the dominance principle and individual preferences: Refinements of the consensus approach." Working paper. 35

[30] Sprumont, Y., (2012). "Resource egalitarianism with a dash of efficiency." Journal of Economic Theory, 147, 1602-1613. [31] Weymark, J. (1991). "A reconsideration of the Harsanyi-Sen debate on utilitarianism." In J. Elster, J. E. Roemer eds., Interpersonal Comparisons of Well-Being. Cambridge, Cambridge University Press. [32] Zuber, S. (2016). "Harsanyi’s theorem without the sure-thing principle: On the consistent aggregation of Monotonic Bernoullian and Archimedean preferences." Journal of Mathematical Economics, 63, 78-83.

36