2019/08/07

Resource allocation problems with interpersonal
welfare comparison:
An axiomatization of the impartial Walras rule
Ryo-ichi Nagahisa (Kansai University)

1 The purpose• Applying extended sympathy approach toresource allocation problems• An axiomatization of the impartial Walras
Rule1

2 Nagahisa (1991)• The Walras rule is the unique rule satisfying
individual rationality, Pareto optimality,nondiscrimination, and local independence• The same axiomatization leads to theimpartial Walras rule, not to the Walras rule ifinterpersonal comparisons of welfare areadmitted.

23 Extended sympathy approach• Extended preference; Arrow (1961), Suppes(1966), Sen (1970).
(x, i) <E (y, j)
Being in i ’s position in social state x is at least
as good as being at j ’s position in social state y3

4 Extended sympathy approach• From the viewpoint of impartial observer.
• Putting oneself in another shoes=⇒thepossibility of interpersonal comparisons ofwelfare• The departure from from Arrow’s impossibility
theorem: relaxing IIA• The utilitarian rule and the leximin rule,Hammond(1976), Roberts(1980a,b), Nagahisaand Suga(1996).
45 Application to resource
allocation problems• To justify market mechanism by extended
sympathy approach.
• Adam Smith problem: public interest vs.
private interest

56 Exchange economies
N = {1, 2, ..., n} : the set of agents
L = {1, 2, ..., l} : the set of private goodslR+：consumption setlzi = (zi1 , ..., zil ) ∈ R+: agent i’s consumptionnlz = (z1 , ..., zn ) ∈ R+ : an allocationlωi ∈ R++: i’s initial endowmentnlω = (ω1 , ..., ωn ) ∈ R++ : the endowment proﬁleX
X
An allocation z is feasible ifzi =ωii∈N6i∈N

7 Exchange economiesl.
<i : i’s preference on R+<= (<i )i∈N : a (preference) proﬁleD : the domain, the set of proﬁles.

7n

8 D =L∪Ql∆l := {p ∈ R+:lP

pi = 1},i=1lint.∆l := {p ∈ R++:lP

pi = 1}.

i=1p• < = (<i )i∈N ∈L⇐⇒ ∃p ∈ int.∆l s. t. ∀i,lx <i y ⇐⇒ px ≥ py ∀x, y ∈ R+.
p8n

9 D = L∪Qnz}|{n• Q =Q×···×Q• <i ∈Q ifl• <i is continuous and convex on R+andl
continuously diﬀerentiable on R++ .

• x y implies x <i y, and if in additionlx ∈ R++, this implies x ≻i y.
lll• ∀ x ∈ R++, {y ∈ R+: y <i x} ⊂ R++.
9

10 Allocationsz ∈ Z is• Pareto optimal ⇐⇒ No feasible allocation z ′such that zi′ <i zi ∀i and zi′ ≻i zi , ∃i
• individually rational ⇐⇒ zi <i ωi , ∀ i• a Walrasian allocation ⇐⇒ ∃p ∈ int.∆l s. t.
l∀ i ∈ N , zi <i xi ∀xi ∈ R+
s. t. pxi ≤ pωi .

10

11 Allocations• P O(<) :the set of Pareto optimal allocations• IR(<) : the set of individual rational
allocations• W (<) : the set of Walrasian allocations.

1112 Extended Preferences
A hypothetically existing ethical observercompares the welfare of diﬀerent persons from asocial point of view while respecting (orsympathizing with) their subjective preferences.

12

13 Extended Preferences• Given <∈ D, an extended preference <E
generated from < is a complete and transitivelbinary relation on R+× N.
(x, i) <E (y, j)• ”being agent i with consumption x is at least
as well oﬀ as being agent j with consumptiony.”13

14 Extended Preferences• The axiom of identity:lx <i y ⇐⇒ (x, i) <E (y, i) ∀x, y ∈ R+ ∀i.

1415 Extended Preferences• Example 1l
Let <∈ D and p ∈ int.∆ be given.
(x, i) <E(p) (y, j) ⇐⇒ min{pq : q ∼i x} ≥min{pq : q ∼j y}.
• Example 2pp
If < ∈ L, then <E(p) reduces to a simple
form as follows.
p(x, i) <E(p) (y, j) ⇐⇒ px ≥ py.
15

j(x,i)≳E(p)(y,j)

yxi

16 Suppes criterion
Π: the set of permutation on N .
Given z and π ∈ Π, z π be s. t. ziπ = zπ(i) , ∀i.

16

17Suppes criterion• z is at least as just as z ′ for <E if and only if′∃π ∈ Π s. t. (zi , i) <E (zπ(i) , π(i)) ∀i• z is more just than z ′ for <E if and only if zis at least as just as z ′ with′(zi , i) ≻E (zπ(i), π(i)) ∃i• z is equally as just as z ′ for <E if and only if′∃π ∈ Π s. t. (zi , i) ∼E (zπ(i) , π(i)) ∀i.
17

(X,1)∼E(p)(Y,2)(X,2)∼E(p)(Y,1)

Y2

X is equally as just as Y for ≳E(p)

X2

X1
Y1

18 Suppes criterion
A feasible allocation z is Suppes equitable for<E if and only if there are no feasible′allocations z that are more just than z for <E .

18

19 Suppes criterion• SE(<E ): the set of Suppes equitableallocations for <E .
• IM R(<E ) : the set of impartial rationalallocations that are at least as just as ω for<E .
• SE(<E ) ⊂ P O(<) and IR(<) ⊂ IM R(<E )for all <E . Note SE(<E(p) ) ̸= ∅.

19

20 Rules
f : D −→ Z : a rule.
<∈ D

extend preferences−→fD(<) −→ Z

D(<): a nonempty subset of <Ef<∈ D −→ Z (Nagahisa1991)
20

21 Rules•∪ D(<): the extended domain.

<∈D• D(<) satisﬁespppD.1. ∀ < ∈ L, then D(< ) = {<E }.
D.2.∀ <∈ D, ∀ <E ∈ D(<), (x, i) <E (0, j)l∀i, j ∈ N ∀x ∈ R+.
D.3.∀ <∈ D, ∀p ∈ int.∆l , <E(p) ∈ D(<).
D(<) is well deﬁned.
21

22 Walras rule vs. impartial
Walras ruleW<∈ D−→

W (<, ω)<∈ D−→∪ W (<, ω π )IWπ∈ΠW (<, ω π ) : The set of Walrasian allocations
when all agents’ endowment are pemuted
through π:
If ωi is equal across all i, then W = IW .
22

23 Axioms
This paper
Suppes Nondiscrimination
Suppes Equity
Impartial Rationality
Extended Local Independence⇐⇒⇐⇒⇐⇒⇐⇒

23Nagahisa (1991)
Utility Nondiscrimination
Pareto Optimality
Individual Rationality
Local Independence24

Suppes Nondiscrimination (SN)Utility ndiscrimination (ND:Nagahisa1991)• A rule f satisﬁes SN if∀ <∈ D, and ∀ z, z ′ ∈ Z, z is equally as justas z ′ ∀ <E ∈ D(<),
then z ∈ f (<) ⇐⇒ z ′ ∈ f (<).
• A rule f satisﬁes UN if∀ <∈ D, and ∀ z, z ′ ∈ Z, zi ∼i zi′ ∀i,then z ∈ f (<) ⇐⇒ z ′ ∈ f (<).

24

25

Suppes Equity (SE), Pareto Optimality (PO:Nagahisa1991)• A rule f satisﬁes SE if
Sf (<) ⊂SE(<E ) ∀ <∈ D.
<E ∈D(<)• A rule f satisﬁes PO iff (<) ⊂ P O(<) ∀ <∈ D25

26

Impartial Rationality (IMR), Individual Rationality (IR:Nagahisa1991)• A rule f satisﬁes IMR if
Sf (<) ⊂IM R(<E ) ∀ <∈ D.
<E ∈D(<)• A rule f satisﬁes IR if f (<) ⊂ IR(<) ∀<∈ D.

26

27

Local Independence (LI:Nagahisa1991)• A rule f satisﬁes LI ifnl
∀ <, <′ ∈ D, ∀ z ∈ Z ∩ R++,p(<i , zi ) = p(<′i , zi ) ∀ i,then z ∈ f (<) ⇐⇒ z ∈ f (<′ ).
• p(<i , zi ) : the supporting price of <i at zi .

27

28

Extended Local Independence (ELI)• <E is essentially identical to <′E aroundnlz ∈ R+
if there exist functions ui ,llεi : R+−→ R s. t. ∀ i, j ∈ N ∀x, y ∈ R+,(x, i) <E (y, j) ⇐⇒ ui (x) ≥ uj (y),(x, i) <′E (y, j) ⇐⇒ ui (x) + εi (x) ≥uj (y) + εj (y),εi (x)where εi (zi ) = 0 and ||x−zi || −→ 0 as
xi −→ zi , and the same property holds for εj .
28

29

Extended Local Independence (ELI)• A rule f satisﬁes ELI if ∀ <∈ D, ∀nlz ∈ Z ∩ R++, if there exists ∃ <E ∈ D(<),∃p ∈ int.∆l s. t. <E is essentially identical top<E around z, then z ∈ f (<) ⇐⇒ z ∈ f (<p ).
• ELI is equivalent to LI if PO is satisﬁed(Lemma 5).

29•i•j30 Results
Theorem 1
Assume D.1, D.2, and D.3. The impartial
Walras rule is the unique rule satisfying SN,IMR, SE, and ELI.

30

O2
•y•

ω-1

ω••

x

O1
If every has the same linear preference r, the image of rules satisfying
SN and IMR coincide with the red lines.

•O2

y
ω-1

•ω•
•

x

O1

IW⊂f: x and y are impartial Walrasian allocations.
Lemma shows x∈f(≳). ELI implies x∈f(≳). Thus IW⊂f.

O2
ω or ω-1••

O1
f⊂IW:xSuppose x∈f(≳) and x∉IW(≳)

contradiction.

ELI implies x∈f(≳), a31 Interpretation 1.”ﬁctional”
social choiceA social choice problem where no one knows
who has which initial endowment.
Rawls’s veil of ignorance (Rawls 1971).

31

32 Interpretation 2.”ﬁctional”
social choice
A social choice problem in which everyone
expects and/or imagines to own other agent’s
initial endowment with equal probability.
Putting oneself in another’s shoes of Hare(1981) and Harsanyi (1955).

32

33 Axiomatization of the equalincome Walras rule
It is semantically the same as the situation
where everyone has the same initial endowment.
In this case, the impartial Walras rule isequivalent to the equal income Walras rulestudied by Thomson (1988), Nagahisa and Suh(1995), Maniquiet (1996), and Toda (2004).
33

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