Title

Egalitarian equivalence and implementability
imply equal division

Author(s)Hayashi, Takashi

Citation

大阪大学経済学. 2023, 73(2・3), p. 8-15

Version Type VoRURL

https://doi.org/10.18910/93330

rights
Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/Osaka University

Vol. 73 Nos. 2・3

OSAKA ECONOMIC PAPERS

December 2023

Egalitarian equivalence and implementability
imply equal division＊
Takashi Hayashi †Abstract
This paper shows that any Nash-implementable social choice correspondence satisfying
egalitarian equivalence must select equal division when the number of individuals is greater than
the number of goods. It also shows that any strategy-proof and non-bossy social choice function
satisfying egalitarian equivalence must select equal division, under the same assumption.
JEL Classification: D61, D63, D71, D82
Keywords: Egalitarian equivalence, Nash implementability, strategy-proofness, non-bossiness

1 Introduction
Egalitarian equivalence, a concept of equity proposed by Pazner and Schmeidler [23], requires that anallocation should be associated with some common reference bundle which is equally preferable as
his/her own consumption bundle for each individual. It is seen as an ordinal counterpart of the conceptof welfare egalitarianism and allows a wider set of allocations than equal division.1 It is compatible
with efficiency, in contrast to equal division that is typically inefficient. It is compatible with efficiency
even in economies in which envy-freeness, another prominent concept of equity (Foley [10]), fails to
be compatible with efficiency (Pazner and Schmeidler [22], Varian [32], Maniquet [14]), as shown for
example in Fleurbaey and Maniquet [8, 9]. Moreover, it is compatible with or characterized by various
solidarity conditions (Moulin [16], Dutta and Vohra [6], Fleurbaey and Maniquet [8, 9]).
An egalitarian-equivalent solution is manipulable in general, however. To illustrate, consider
that there are two individuals, A and B, and two private goods 1 and 2. Consider for simplicity thatreference bundles are taken from the 45-degree line

, and we represent eachindividual ’s preference by, forwheremaximizing the egalitarian social welfare function＊†1. Then,yields an egalitarian-equivalent andI thank François Maniquet for comments to the previous version.
Professor in Economics, Adam Smith Business School, University of Glasgow, Email: Takashi.Hayashi@glasgow.
ac.uk
See Moulin [17, 18] and Thomson [31] for comprehensive illustrations.

December 2023

Egalitarian equivalence and implementability imply equal division－9－efficient allocation, uniquely up to Pareto-indifference.2
Suppose that A and B have preferences with constant marginal rates of substitution, where A’s MRS, B’s

is 2 and B’s MRS is 1. Then A’s preference is represented in the form
preference is represented in the form. Suppose there are 2 units of Good

1 and 1 unit of Good 2. Then the solution yieldsand.

Now suppose that A misreports his/her preference, saying that MRS is still constant but itis, whereis positive but close to 0. Then the representation to be used is changed to, and the allocation is changed toand. Hence A gains by misreporting his/her preference.
One might think that the above problem is rather due to efficiency, since we know that dictatorshipor something close is the only solution satisfying strategy-proofness and efficiency (see Gibbard[11] and Satterthwaite [24] for the results on the abstract domain, and Zhou [34], Serizawa [26],Serizawa and Weymark [27] for the results in exchange economies). If we give up efficiency, there isa straightforward solution that is strategy-proof (also Nash-implementable) and egalitarian-equivalent:
equal division.
But can we have anything nicer, because equal division is typically severely inefficient? In otherwords, is there an implementable/non-manipulable selection of egalitarian-equivalent allocations thatstill leaves a room for other nice features? The current paper examines this question.3
We show that any Nash-implementable social choice correspondence satisfying egalitarianequivalence must select equal division, when the number of individuals is greater than the numberof goods. Also, we show that any strategy-proof and non-bossy social choice function satisfying
egalitarian equivalence must select equal division, under the same assumption.4 This contrasts to andis parallel to the argument by Fleurbaey and Maniquet [7], who showed that envy-freeness is rather
a consequence of implementability plus a mild horizontal equity condition. Also it is parallel to thefinding by Thomson [29] that an equilibrium allocation in the game of reporting preference underegalitarian equivalence must weakly Pareto-dominate the equal division.
Our results tell that we must accept severe inefficiency if we insist on egalitarian equivalence
under the condition of implementability. Or, our results explain at least why a multi-stage mechanism
234

This is a special case of -egalitrian-equivalent solution, which is a refinement of egalitarian equivalence, inwhich reference points are taken from the line spanned by vector . It allows us to avoid the problem with generalegalitarian-equivalent and efficient allocation that somebody’s consumption bundle is dominated by another’s. See
Thomson [31].
Thomson [29] instead considered a positive question about what the equilibrium consequence of adopting a particularclass of equity criteria including egalitarian equivalence is, by considering a game of reporting preferences in which
the agents play a generalized version of Nash equilibrium.
In the domain of allocating indivisible objects with monetary transfers, Ohseto [21] shows that there is no strategyproof, budgets-balanced and egalitarian-equivalent mechanism, and also that there is no Nash-implementable andegalitarian-equivalent mechanism, which may be attributed to the fact that there is no such thing as equal division inthat domain. In this sense we provide a tighter characterization in the domain of exchange economies.

－ 10 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

inducing subgame-perfection (Moore and Repullo [19]) is necessary for implementing egalitarianequivalent and efficient allocations, as in Crawford [3] and Demange [5], or why a specific domain
has to be considered for implementation with normal-form mechanisms as in Chun and Mutuswami [2],Yengin [33].
Finally, let us note two reservations. One is that we assume the number of individuals is greaterthan the number of goods. It is left as an open question whether we still obtain the same theorem orthere is a counterexample when the commodity space is richer compared to the number of individuals.
Second is that we assume the domain of all the complete, transitive, continuous, convex and strongly
monotone preferences. It excludes Leontief-type preferences and includes all the strongly monotone
preferences arbitrarily closer to them. On the other hand, there are studies since Nicoló [20], in which
positive results are obtained in the restricted domain of Leontief-type preferences.Ghodsi, Zaharia,Hindman, Konwinski, Shenker and Stoica [12], Li and Xue [13] show that egalitarian equivalenceand efficiency are compatible with implementability in the domain of Leontief-type preferences. Notealso that such possibility results are not obtained in the domain of linear preferences (Schummer [25]),
which suggests the importance of Leontief-type preferences. It is left as an open question whether westill obtain the same theorem or there is a counterexample when we consider the domain of all thecomplete, transitive, continuous, convex and weakly monotone preferences.
2 Nash-implementable social choice correspondences
There are
There arebe the social endowment.5

private goods in the economy, and letindividuals. Each individual’s consumption set is the non-negative quadrant

be the set of preference relations over. Let, which are complete, transitive, continuous, convex andstrongly monotone, although further domain restriction is possible following Fleurbaey and Maniquet[7].6,For eachand, letand,.

be the set of feasible allocations. Then, letLetdenote a social choice correspondence.

The primary normative criterion we consider is Egalitarian Equivalence (Pazner and Schmeidler[23]).
Egalitarian Equivalence: For all

andthere issuch thatfor all.
56

Our arguments can be readily extended to production economies and economies with public goods, as long as we candefine equal division as an element of feasible set.
Preference relation
overis strongly monotone iffor allandfor at least one
imply.

December 2023

Egalitarian equivalence and implementability imply equal division－ 11 －

It is immediate to see that Egalitarian Equivalence is stronger than Equal Treatment of Equals,
which states that individuals with identical preferences must be treated equally in welfare.
Equal Treatment of Equals: For all
it holds

andand, ifthen for all

for all.

.

Envy-Freeness is another prominent concept of equity.
Envy-Freeness: For allandit holds

We examine the consequence of imposing Egalitarian Equivalence under an implementability
condition. Here we consider Monotonicity (Maskin [15]), which is necessary and sufficient for
implementability in Nash equilibria in the current economic environment.
Monotonicity: For all
then,and, iffor all.

The proposition below has been shown by Fleurbaey and Maniquet [7].
Proposition 1 If a social choice correspondence satisfies Monotonicity and Equal Treatment ofEquals, then it satisfies Envy-Freeness.
Since Egalitarian Equivalence immediately implies Equal Treatment of Equals, we obtain thefollowing proposition.7
Proposition 2 If a social choice correspondence satisfies Monotonicity and Egalitarian Equivalence,then it satisfies Envy-Freeness.
A social choice correspondence
there iswithTheorem 1 Suppose

is said to select equal division if for all

such thatfor alland.

. Then, if a social choice correspondence satisfies Monotonicity andEgalitarian Equivalence then it must select equal division.
Proof. Let

be any social choice correspondence satisfying Monotonicity and EgalitarianEquivalence.

7

This contrasts to the result by Thomson [30] that egalitarian equivalence and envy-freeness are generally incompatible
in the problem of allocating indivisible objects with monetary transfers.

－ 12 －OSAKA ECONOMIC PAPERS

Pick any. Suppose that

Case 1: Suppose there exist

with. Since

so that,.

and.

.

By Monotonicity, we have,.

.

be such that

Since

such

and
is sufficiently close to

By Monotonicity, we haveLet

is not equal division, there isis envy-free it holds

so thatThen one can takeby takingis not an equal division.

. Note that sincethatVol. 73 Nos. 2・3

cannot be an egalitarian equivalent allocation sinceis the only

possible candidate for reference point.
Case 2: Suppose we havefor all.

is envy-free and preferences are strongly monotone, for all

Then, since

it holds.
, it holds

Since.
, letFor eachbe such thatandis sufficiently.

close to, but it cannot be an egalitarian equivalent allocation since

By Monotonicity,.
Remark 1 When, Envy-Freeness implies Egalitarian Equivalence, because if an allocation

is not egalitarian-equivalent it must be the case that one’s corresponding indifference surface isstrictly above the other’s corresponding indifference surface, which must cause an envy. Since thecorrespondence that selects the entire set of envy-free allocations is monotonic, the above theorem
does not hold when.
,

We do not have a proof of the theorem or a counterexample, unfortunately, for the casein which the commodity space is richer compared to the set of individuals. We leave it as an openquestion.
The above result implies that under the restriction of Nash-implementability any social choicecorrespondence satisfying egalitarian equivalence is Pareto-dominated by another one, for example
the (constrained) Walrasian solution in which the initial endowments are taken to be the equal division, regardless of which allocation is selected from the prescribed set.
3 Strategy-proof and non-bossy social choice functions
Let

denote a social choice function.

We consider the following two non-manipulability conditions. One is that reporting own true
preference truthfully is a dominant strategy under the direct mechanism.
Strategy-Proofness: For all,and, it holds.

December 2023

Egalitarian equivalence and implementability imply equal division－ 13 －The other is that if one cannot change own consumption allocation by reporting different preferenceof own, it cannot change others’ allocations either, since otherwise another individual has an incentive
to bribe him to report a different preference.
Non-Bossiness: For all,and, ifthen.
The proposition below has been shown by Fleurbaey and Maniquet [7].
Proposition 3 If a social choice function satisfies Strategy-Proofness, Non-Bossiness and Equal
Treatment of Equals, then it satisfies Envy-Freeness.
Since Egalitarian Equivalence immediately implies Equal Treatment of Equals, we obtain thefollowing proposition.
Proposition 4 If a social choice correspondence satisfies Strategy-Proofness, Non-Bossiness andEgalitarian Equivalence, then it satisfies Envy-Freeness.
Theorem 2 Suppose. Then, if a social choice function satisfies Strategy-Proofness, Non-Bossiness and Egalitarian Equivalence then it must select equal division.
Proof. Let

be any social choice function satisfying Strategy-Proofness, Non-Bossiness andEgalitarian Equivalence.
By following the standard argument (see Dasgupta, Hammond and Maskin [4] for example), one
can establish a version of monotonicity property: For all
for all, thenand, if.

Under the monotonicity property, we can prove the assertion in the same way as in the previoustheorem.
Remark 2 When, Envy-Freeness implies Egalitarian Equivalence. Hence any strategy-proof rule satisfying Envy-Freeness serves as a counterexample to the theorem, as Non-Bossiness isvacuously met as far as resource constraint is met with equality. For example, when we draw a line
passing through the equal division point which lies on a hyperplane with some positive normal vector,
the individuals have single-peaked preferences over the line and we can apply the uniform rule there(Sprumont [28]).
We do not have a proof of the theorem or a counterexample, unfortunately, for the casein which the commodity space is richer compared to the number of goods.
The above result implies that under the restriction of strategy-proofness and non-bossiness any
social choice function satisfying egalitarian equivalence is Pareto-dominated by another one, for－ 14 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

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