On the resource allocation problems withinterpersonal comparisons of welfare:
Extended preference approach revisited
Saturday, August 26, 2023

Hiromi Murakami, Kyoto City University of Arts

Ryo-Ichi Nagahisa, Kansai University
数理経済学会方法論分科会・大阪大学方法論研究会：経済学理論と「社会科学のためのリアリズム (RFSS)」＠大阪大学豊中キャンパス

Introduction• This paper introduces the extended preference approachstudied in the literature of social welfare functionals into thepure exchange economy model.
• We show that the leximin criterion, the equal income Walras
rule, and price indicator types of extended preferences aretied up together; any two of the three imply the rest. Thistied-up relation derives an axiomatization of the equalincome Walras rule using the leximin fairness.
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Exchange Economies 1/3• The notation for vector inequalities： ≥, >, ≫• The set of real numbers：𝑅ℓ•∆ ≔𝑝 ∈ 𝑅+ℓℓ
σℓ𝑖=1 𝑝𝑖 = 1}, 𝑖𝑛𝑡. ∆ℓ ≔ 𝑝 ∈ 𝑅++
σℓ𝑖=1 𝑝𝑖 = 1},• The set of agents：𝑁 = {1,2, … , 𝑛}
• The set of private goods：𝐿 = {1,2, … , ℓ}

• All agents have the same consumption set：𝑅+ℓ3

Exchange Economies 2/3• Agent 𝑖’s consumption：𝑧𝑖 = 𝑧𝑖1 , … , 𝑧𝑖ℓ ∈ 𝑅+ℓ• Allocation：𝑧 = (𝑧1 , … , 𝑧𝑛 ) ∈ 𝑅+𝑛ℓℓ• Total endowment of the economy：Ω ∈ 𝑅++• 𝑧 is feasible if σ𝑖∈𝑁 𝑧𝑖 ≤ Ω. 𝑍 be the set of all feasible 𝑧.
• Let ≽𝑖 be agent 𝑖’s preference on 𝑅+ℓ and continuous, convex,
and monotonic on 𝑅+ℓ . 𝑄 is the set of all ≽𝑖 .

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Exchange Economies 3/3• A list ≽ = ≽𝑖 𝑖∈𝑁 ∈ 𝑄 𝑛 , is called a profile.

• A feasible 𝑧 is Pareto optimal if there are no other feasible
ones 𝑧′ with 𝑧𝑖′ ≻𝑖 𝑧𝑖 for all 𝑖 ∈ 𝑁. [𝑃𝑂(≽), 𝑃𝑂(≽, 𝑝)]↑supporting price• A feasible allocation 𝑧 is an equal income Walrasian
allocation if there exists some 𝑝 ∈ 𝑖𝑛𝑡. ∆ℓ such that 𝑧𝑖 ≽𝑖 𝑥
Ωℓfor all 𝑥 ∈ 𝑅+ with 𝑝𝑥 ≤ 𝑝 𝑛 . [𝐸𝐼𝑊(≽), 𝐸𝐼𝑊(≽, 𝑝)]↑equilibrium price5Extended Preferences 1/3• The notion of extended preferences is based on theprinciple of extended sympathy mentioned by Arrow (1963)
and initiated by Suppes (1966) and Sen (1970).
• The basic idea is that a hypothetically existing ethicalobserver compares the welfare of different persons from asocial point of view while respecting (or sympathizing with)
their subjective preferences.

6Extended Preferences 2/3• An extended preference ≽𝐸 created from ≽ ∈ 𝑄 𝑛 is acomplete and transitive binary relation on 𝑅+ℓ × 𝑁.
◆ 𝑥, 𝑖 ≽𝐸 (𝑦, 𝑗) ：being agent 𝑖 with consumption 𝑥 is atleast as well off as being agent 𝑗 with consumption 𝑦.
• A set of continuous utility functions 𝑢𝑖 𝑖∈𝑁 represents ≽𝐸
if 𝑥, 𝑖 ≽𝐸 (𝑦, 𝑗) ⟺ 𝑢𝑖 (𝑥) ≥ 𝑢𝑗 (𝑦) for all 𝑖, 𝑗 ∈ 𝑁 and all𝑥, 𝑦 ∈ 𝑅+ℓ .

7Extended Preferences 3/3𝐸(≽) be the set of ≽𝐸 satisfying E.1, E.2 and E.3.
⚫E.1. For any 𝑖 ∈ 𝑁, 𝑥 ≽𝑖 𝑦 ⟺ 𝑥, 𝑖 ≽𝐸 𝑦, 𝑖 for all 𝑥, 𝑦 ∈ 𝑅+ℓ .
⚫E.2.

𝑥, 𝑖 ≽𝐸 0, 𝑗 for all 𝑖, 𝑗 ∈ 𝑁 and all 𝑥 ∈ 𝑅+ℓ .

⚫E.3. ≽𝑖 has representation.
✓E.2 reflects our intuition that as long as other conditions areequal, people without wealth are the most miserable in theworld. Note that E.2 implies 0, 𝑖 ∼𝐸 0, 𝑗 for all 𝑖, 𝑗 ∈ 𝑁.
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Leximin Criterion 1/3
The leximin criterion, a lexicographic extension of the differrence principle of Rawls(1971).
◆THE VEIL OF IGNORANCE: First of all, no one knows his place in society, his classposition or social status... The persons in the original position have noinformation as to which generation they belong. (Sec.24)◆SECOND PRINCIPLE: Social and economic inequalities are to be arranged so thatthey are both: (a)to the greatest benefit of the least advantaged, consistent withthe just savings principle, and (b)attached to offices and positions open to all
under conditions of fair equality of opportunity. (Sec.46)

9Leximin Criterion 2/3
Given an allocation 𝑧, we arrange all 𝑧𝑖 , 𝑖 in ascending orders.t. (𝑧𝑖1 , 𝑖1 ) ≼𝐸 (𝑧𝑖2, 𝑖2 ) ≼𝐸 ⋯ ≼𝐸 (𝑧𝑖𝑛−1, 𝑖𝑛−1 ) ≼𝐸 (𝑧𝑖𝑛 , 𝑖𝑛 ) ,where tie is broken arbitrarily.
≼𝐸 ⋯ ≼𝐸(𝑧𝑖1 , 𝑖1 )≼𝐸 ⋯ ≼𝐸(𝑧𝑖𝑘, 𝑖𝑘 )(𝑧𝑖𝑛 , 𝑖𝑛 )The agent 𝑖𝑘 (𝑘 = 1, … 𝑛) is the 𝑘-th worst off agent in 𝑧.
Denote 𝑖𝑘 by 𝑖𝑘 𝑧 . Let us consider lexicographic order ≥𝐿(𝐸) .

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Leximin Criterion 3/3• A feasible 𝑧 is leximin equitable
if there are no other feasible 𝑧 ′with 𝑧 >𝐿(𝐸) 𝑧 ′ . [𝐿𝑀𝐸(≽)]≼𝐸 ⋯ ≼𝐸(𝑧𝑖1 , 𝑖1 (𝑧))≼𝐸 ⋯ ≼𝐸(𝑧𝑖𝑘 , 𝑖𝑘 (𝑧))(𝑧𝑖𝑛 , 𝑖𝑛 (𝑧))~𝐸 ⋯ ~𝐸 ⋯ ≻𝐸
≼𝐸 ⋯ ≼𝐸(𝑧 ′ 𝑖1 , 𝑖1 (𝑧 ′ ))≼𝐸 ⋯ ≼𝐸(𝑧 ′ 𝑖𝑘 , 𝑖𝑘 (𝑧 ′ )) (𝑧 ′ 𝑖𝑛 , 𝑖𝑛 (𝑧 ′ ))𝑧 >𝐿(𝐸) 𝑧 ′ ⟺ ∃𝑘 ∈ 1, … , 𝑛 s.t.
(𝑧𝑖𝜏 𝑧 , 𝑖𝜏 (𝑧))~𝐸 (𝑧 ′ 𝑖𝜏 𝑧 ′ , 𝑖𝜏 (𝑧′)) ∀𝜏 ∈ {1, … , 𝑘 − 1}and (𝑧𝑖𝑘 𝑧 , 𝑖𝑘 (𝑧)) ≻𝐸 (𝑧 ′ 𝑖𝑘 𝑧 ′ , 𝑖𝑘 (𝑧′))11Price indicator type
An extended preference ≽𝐸 is a price indicator type at 𝑧
and 𝑝 if 𝑧 ∈ 𝑃𝑂 ≽, 𝑝 , and for any 𝑖, 𝑗 ∈ 𝑁 and any 𝑥 ∈ 𝑅+ℓ ,𝑥, 𝑗 ≽𝐸 𝑧𝑖 , 𝑖 implies 𝑝𝑥 ≥ 𝑝𝑧𝑖 , and 𝑥, 𝑗 ≻𝐸 𝑧𝑖 , 𝑖 implies
𝑝𝑥 > 𝑝𝑧𝑖 .
✓The welfare ranking at 𝑧 is completely determined with 𝑝.
✓Example: 𝑥, 𝑖 ≽𝐸 𝑦, 𝑗 ⟺ min{𝑝𝑞: 𝑞 ∼𝑖 𝑥} ≥ min{𝑝𝑞: 𝑞 ∼𝑗 𝑦}
is a price indicator type at 𝑧 and 𝑝 if 𝑧 ∈ 𝑃𝑂 ≽, 𝑝 .

12Lemma 4
Take 𝑧 ∈ 𝑍, 𝑝 ∈ 𝑖𝑛𝑡. ∆ℓ , ≽ ∈ 𝑄 𝑛 , and ≽𝐸 ∈ 𝐸(≽)arbitraly.
Any two of the three statements below imply the rest.
Therefore, they are equivalent to each other.
(1) 𝑧 ∈ 𝐸𝐼𝑊 ≽, 𝑝(2) 𝑧 ∈ 𝐿𝑀𝐸 ≽𝐸(3) ≽𝐸 is a price indicator type at 𝑝 and 𝑧.

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How to extend preferences 1/2
Given ≽ ∈ 𝑄 𝑛 and 𝑧 ∈ 𝑍, construct 𝐷 ≽, 𝑧 ⊂ 𝐸 ≽ according
to D.1 and D.2.
D.1 If ≽= ≽, … , ≽ ∈ 𝑄 𝑛 , 𝐷 ≽, 𝑧 = {≽𝐸 } for any 𝑧 ∈ 𝑍,
where ≽𝐸 satisfies 𝑥, 𝑖 ≽𝐸 𝑦, 𝑗 ⟺ 𝑥 ≽ 𝑦 for any 𝑖, 𝑗 ∈ 𝑁and 𝑥, 𝑦 ∈ 𝑅+ℓ .
𝑛ℓ
D.2 For any smooth ≽, ≽′ ∈ 𝑄 𝑛 and any 𝑧 ∈ 𝑍 ∩ 𝑅++, if forevery 𝑖, MRS between any two goods at 𝑧𝑖 remain unchanged
across from ≽𝑖 to ≽′𝑖 , then welfare rankings at 𝑧 are same. 14

How to extend preferences 2/2
D.1 is a self-evident truth, requiring that if every agent has
the same preference, that preference should be the only extended
preference.
D.2 is a requirement of informational economization in making
interpersonal comparisons of welfare, which claims that thewelfare ranking at an allocation be determined only by local
information of individual preferences around the allocation,independently from preferences about allocations far away.
➢cf. Local Independence condition in Nagahisa(1991)

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Social choice rules 1/2
Let 𝐷 be the domain, a nonempty subset of 𝑄𝑛 .
A rule 𝐹 is a mapping that associates with each ≽ ∈ 𝐷 a nonempty subset of 𝑍. It decides 𝐹(≽) comparing the welfare ofdifferent agents. (It uses some extended preferences createdfrom ≽, but not necessarily all.)Let us call ‫≽(ڂ‬,𝑧)∈𝐷×𝑍 𝐷(≽, 𝑧) the extended domain.

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Social choice rules 2/2
1. A rule 𝐹 satisfies Leximin Justification Possibility 𝐿𝑀𝐽𝑃 if𝐹 ≽ ⊂ {𝑧 ∈ 𝑍: 𝑧 ∈ ‫≽(𝐷∈𝐸≽ڂ‬,𝑧) 𝐿𝑀𝐸(≽𝐸 )} for any ≽ ∈ 𝐷.
✓If we want to select 𝑧, at least one extended preference must
make it leximin equitable.
2. A rule 𝐹 meets Leximin All Unaminity 𝐿𝑀𝐴𝑈
if {𝑧 ∈ 𝑍: 𝑧 ∈ ‫≽(𝐷∈𝐸≽ځ‬,𝑧) 𝐿𝑀𝐸(≽𝐸 )} ⊂ 𝐹(≽) for any ≽ ∈ 𝐷.
✓If 𝑧 becomes leximin equitable for all extended preferences,then we must select it.
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Theorem 2𝐿𝑀𝐸 ∃ ≽ = 𝑧 ∈ 𝑍: 𝑧 ∈ ‫≽ 𝐷∈𝐸≽ڂ‬,𝑧 𝐿𝑀𝐸 ≽𝐸𝐿𝑀𝐸 ∀ ≽ = {𝑧 ∈ 𝑍: 𝑧 ∈ ‫≽(𝐷∈𝐸≽ځ‬,𝑧) 𝐿𝑀𝐸(≽𝐸 )}Let 𝐹 and 𝐷 be a rule and an extended domain, respectively.
Suppose that 𝐿𝑀𝐽𝑃 and 𝐿𝑀𝐴𝑈 are defined on 𝐷. The statements
below are equivalent.

(1) There exits some 𝐷 such that 𝐹 is the only rule satisfying 𝐿𝑀𝐽𝑃and 𝐿𝑀𝐴𝑈.
(2) There exits some 𝐷 such that 𝐿𝑀𝐸 ∀ ≽ = 𝐹 ≽ = 𝐿𝑀𝐸 ∃ ≽for all ≽ ∈ 𝐷.
𝑧𝑖 ≠ 0 for all 𝑖 and all 𝑧 ∈ 𝐹 ≽(3) 𝐹 satisfies Pareto Optimality and Non-Zero Rationality.

18Theorem 3
Let 𝐷 be an extended domain. The statements below areequivalent.
(1) 𝐸𝐼𝑊 is the only rule satisfying 𝐿𝑀𝐽𝑃 and 𝐿𝑀𝐴𝑈 defined on 𝐷.
(2) 𝐷 consists of price indicator types.

19Summing up results
The leximin criterion (LM), equal income Walras rule (EIW),
and price indicator type of extended preferences (PIT) are
inseparably tied together.
(1) LM and PIT imply EIW.
(2) LM and EIW imply PIT.

(3) EIW and PIT imply LM.
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Conclusion 1/2
Hammond (1991) stated the following about interpersonal
comparisons of utilities (ICUs):
But if it is easier to think what is a good social welfare ordering, rather
than how to make ICUs, why should we not start with the ordering andhave it reveal the ICUs, instead of starting with ICUs and trying toderive a social ordering? Especially if it is not at all clear anyway how toincorporate ICUs into a social ordering even if we believe we have madesecurely founded and ethically relevant interpersonal comparisons ofboth utility levels and utility differences.

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Conclusion 2/2
Though only dealing with exchange economies, our result
answers Hammond’s question. The fact that people accepted
the rules corresponds to the fact that people agreed withmaking welfare comparisons based on the rules because eachperson receives their utility according to the rules.
According to Hammond’s argument, accepting EIW and theleximin fairness as valuable is equivalent to regarding welfare
comparisons of D.1 and D.2 as correct.
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References• Arrow K.J. (1963) Social Choice and Individual Values. New York: John Wiley, Second
Edition,1963.
• Hammond J.P. (1991) Interpersonal comparisons of utility: Why and how they are andshould be made. In: Elster J and Roemer JE (eds) Interpersonal Comparisons of Well-Being,200-254.
• Nagahisa R. (1991) A local independence condition for characterization of Walrasian
allocations rule. J. Econ. Theory 54:106-123.
• Rawls J. (1971) A Theory of Justice. Cambridge, Mass: Harvard University Press, andOxford: Clarendon Press• Sen A.K. (1970, 1972), Interpersonal Aggregation and Partial Comparability, and A
Correction, Econometrica 38: 393-409 and 40: 959-960• Suppes P. (1966) Some formal models of grading principles Synthese 6: 284-306; reprintedin P. Suppes, Studies in the Methodology and Foundations of Science. Dordrecht.

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