Volume 30, Issue 2
A characterization of the maximin social ordering
Kaname Miyagishima
Hitotsubashi UniversityAbstract
This note shows that the maximin social ordering, which is wellknown in the social choice theory, is characterized byHammond Equity, Continuity, and Weak Pareto Principle.

The author thanks Professor Richard McLean for helpful comments. The author also thanks Jun Honda and Nozomu Muto for helpful
discussions.
Citation: Kaname Miyagishima, (2010) ''A characterization of the maximin social ordering'', Economics Bulletin, Vol. 30 no.2 pp. 12781282.
Submitted: Mar 01 2010. Published: May 07, 2010.

1. Introduction
This note provides an axiomatic characterization of the weighted maximin social ordering, which compares utility vectors based on the least
weighted utilities of the utility vectors. When the weights are symmetric,
this ordering becomes the maximin social orderings. We introduce axioms
named α-Hammond Equity, Continuity, and Weak Pareto Principle. We
follow the strategy of proof by Fleurbaey (2005, Theorem 3), who gives acharacterization of the Pazner-Schmeidler social ordering in the model ofexchange economy. Note that the Pazner-Schmeidler social ordering is akind of maximin social ordering. 1
As long as we know, there are few studies to characterize the maximinsocial ordering over utility vectors. Strasnick (1976) characterizes the social
ordering by using axioms named Anonimity, Nonnegativity and Unanimity.
2Bosmans and Ooghe (2006) prove that the social ordering is axiomatized by Anonymity, Hammond Equity, Continuity, and Weak Pareto. In
contrast, our result implies that, when the weights are symmetric, the maximin social ordering is characterized by Hammond Equity, Continuity, andWeak Pareto. Hence, our characterization does not need Anonymity.
The remaining part of this note is as follows. Section 2 gives notationand definitions. Section 3 provides our characterization result.

2. Notation
Let N = {1, ..., n} be the set of individuals. R and N are, respectively,
the sets of real numbers and natural numbers. X N = Rn denotes the ndimensional utlity space.
A social ranking over utility vectors is denoted by R. For any two
utility vectors u, v ∈ X N , [uRv] is interpreted as “u is socially at least as
good as v.” Symmetric and asymmetric parts of R are denoted by I and P ,respectively. A binary relation is a quasi-ordering if it satisfies reflexivity
and transitivity. A binary relation is an ordering if it satisfies completenessand transitivity.
We define the α-maximin social ordering.
1

There are many maximin types of social ordering in economic environments, becauseof various ways of interpersonal comparison.
2
He used also an axiom Neutrality, though this is known to be redundant.

1Definition: Given a vector of weight α = (α1 , ..., αn ) ∈ Rn++ , a social
ranking RM (α) is the α-maximin social ordering defined as follows:
For any u, v ∈ X N ,uRM (α)v ⇐⇒ min{αi ui } ≥ min{αi vi }.
i∈Ni∈N

This social ordering compares utility vectors, u and v, based on the least
weighted utilities, mini∈N {αi ui } and mini∈N {αi vi }. Note that, when α =(1, ..., 1), this soical ordering becomes the maximin social ordering.
We introduce the axioms to characterize the maximin social ordering.
Weak Pareto: For all u, v ∈ X N , if ui > vi for all i ∈ N , then uP v.
α-Hammond Equity: Given α = (α1 , ..., αn ) ∈ Rn++ , for all u, v ∈ X N ,if αi vi > αi ui > αj uj > αj vj for some i, j ∈ N , and uk = vk for all
k 6= i, j, then uRv.
Continuity: For all u ∈ X N , if a sequence of vectors (v k )k∈N converges tov ∈ X N and uRv k (resp. v k Ru) holds for all k ∈ N, then uRv (resp.
vRu).
Weak Pareto requires that, if all agents are better in one situation uthan another v, the former should be socially prefered to the latter.
α-Hammond Equity is a modified version of Hammond Equity proposed
by Hammond (1976). This axiom insists that a reduction of inequality inweighted utilities between two individuals should be socially accepted. Note
that, when α = (1, ..., 1), the axiom becomes Hammond Equity.
Continuity requires social orderings to be continuous.
3. Characterization
Theorem Suppose that R is a quasi-ordering. Then, R satisfies α-Hammond
Equity, Weak Pareto and Continuity if and only if R = RM (α).
Proof. It is obvious that α-weighted maximin social ordering satisfies theaxioms in the Theorem. We show the converse result. Suppose that a social
quasi-ordering R satisfies the axioms. We first prove that, for any ulitity
vectors u, v ∈ X N
min{αi ui } > min{αi vi } =⇒ uP v.
i∈Ni∈N2(1)We first show that one can go from v to u through a sequence of utility
vectors z 1 , ..., z T such that z 1 = v, z T = u, and for all t = 1, ..., T − 1, either(Case 1) zit+1 > zit for all i ∈ N , or(Case 2) for two agents i and j,
αi zit > αi zit+1 > αj zjt+1 > αj zjt ,and for all other agents k, zkt+1 > zkt .
We prove this fact.3 Let m be an agent such that αm vm = mini {αi vi }.
©
ªDefine S = i|αi vi > αm vm and M = mini∈S {αi vi }. Let ² > 0 be suchthatno´1³min M, min{αi ui } − αm vm .
²<i∈NnLet T = |S| + 2 and s be a bijection from {1, ..., |S|} to S. At every stept = 1, ..., T − 2, let(a) αi zit+1 = αm vm + (t + 1)² for i = s(t) ∈ S,t+1(b) αk zkt+1 = zkt + ² for all k 6= i. (In particular, αm zm= αm vm + t².)
tt+1
For t = 1, ..., T − 2, the step from z to z
corresponds to (Case 2)
with i = s(t) and j = m, since
t
t+1
αm zm
< αm zm= αm vm + t² < zit+1 = αm vm + (t + 1)² < zit ,(2)where the last inequality is derived fromαm vm + (t + 1)² < αm vm +t+1
(M − αm vm ) ≤ M ≤ αi zit .
nThe last step from z T −1 to z T = u corresponds to (Case 1). This is because,
for all i,
αi ziT −1 ≤ αm vm + (T − 1)² < αm vm + n² < min{αi ui },iwhere the last inequality follows the assumption regarding ² above.
Now we prove (1). For all steps t = 1, ..., T − 2, let ²0 be such thatt+1t
αi zit+1 − ²0 > αm zm− ²0 > αm zm,where i = s(t). By (2) and α-Hammond Equity,t+1t(zit+1 − ²0 , zm− ²0 , z−im)Rz t .
3

The proof is essentially due to Fleurbaey (2005, proof of Theorem 3, Step 1).

3

By Weak Pareto,t+1tz t+1 P (zit+1 − ²0 , zm− ²0 , z−im).

By transitivity, z t+1 P z t . Moreover, by Weak Pareto, z T = uP z T −1 . By
transitivity, uP v. Thus, (1) has been shown.
From (1) and the usual argument of Continuity, we can easily show that,for any u, v ∈ X N ,min αi ui = min αi vi =⇒ uIv.
i∈Ni∈N(3)By (1) and (3), we have completed the proof. ¤The axioms in the Theorem are clearly independent.

References[1] Bosmans, K., and Ooghe, E., (2006), “A Characterizaion of Maximin,”Mimeo, Katholieke Universiteit Leuven.
[2] Fleurbaey, M., (2005), “The Pazner-Schmeidler Social Ordering: A Defense,” Review of Economic Design, 9, pp. 145-166.
[3] Hammond, P., J., (1976), “Equity, Arrow s Conditions, and Rawls
difference principle,” Econometrica 44, pp. 793-803.
[4] Strasnick, S., (1976), “Social Choice and the Derivation of Rawls’s Difference Principle,” Journal of Philosophy, 73-4, pp. 85-99.

4