Monotonic extensions on economic domains

Rev. Econ. Design 4, 13–33 (1999)

c© Springer-Verlag 1999

Monotonic extensions on economic domains

William Thomson

Department of Economics, University of Rochester, Rochester, NY 14627, USA(e-mail: [email protected])

Received: 21 September 1996 / Accepted: 17 August 1998

Abstract. The property of “monotonicity” is necessary, and in many contexts,sufficient, for a solution to be Nash implementable (Maskin 1977). In this paper,we follow Sen (1995) and evaluate the extent to which a solution may fail mono-tonicity by identifying the minimal way in which it has to be enlarged so as tosatisfy the property. We establish a general result relating the “minimal mono-tonic extensions” of the intersection and the union of a family of solutions to theminimal monotonic extensions of the members of the family. We then calculatethe minimal monotonic extensions of several solutions in a variety of contexts,such as classical exchange economies, with either individual endowments or asocial endowment, economies with public goods, and one-commodity economiesin which preferences are single-peaked. For some of the examples, very littleis needed to recover monotonicity, but for others, the required enlargement isquite considerable, to the point that the distributional objective embodied in thesolution has to be given up altogether.

JEL classification: C7, D50, D63, D78

Key words: Maskin-monotonicity, Nash-implementation, minimal monotonic ex-tension, fair allocation

1 Introduction

A “solution” is a mapping that associates with each economy in some admissibledomain a non-empty subset of its feasible set. Solutions are mathematical rep-resentations of the objectives of a social planner. They are typically required to

This is the revision of an April 1993 mimeo entitled “Monotonic extensions”. Support from NSF undergrant SES 9212557 and the comments of L. Corchon and F. Maniquet are gratefully acknowledged. Iam particularly endebted to Eiichi Miyagawa for providing a complete proof for one of the examplesof Sect. 3, and to a referee and an associate editor for their very detailed comments.

14 W. Thomson

select efficient allocations, but also to satisfy some distributional requirement orparticipation constraint. In order to attain the allocations selected by a solution,information about the agents’ preferences is necessary in almost all interestingcases. The theory of implementation (see Moore 1992, and Corchon 1996, forsurveys) was developed to deal with the strategic problems due to the fact thatpreferences are private information and that an agent often gains by unilateralmisrepresentation. For some solutions however this problem can be circumventedby confronting agents with well-chosen “game forms”: a game form consists ofa list of “strategy spaces”, one for each agent, and an “outcome function”; eachagent is supposed to choose and announce an element of his strategy space, andthe outcome function associates with every one of the possible resulting profilesof strategies a feasible allocation. A solution is “(Nash)-implementable” if thereexists a game form such that for each admissible economy the set of Nash equi-librium outcomes of the game form played in that economy coincides with theset of outcomes that the solution would select for it.

A property was shown to be necessary, and in many contexts, sufficient, fora solution to be implementable (Maskin 1977). It says that the desirability ofan allocation should be unaffected by changes in preferences that improve itsposition relative to other allocations. Unfortunately, many interesting solutionsare not monotonic and therefore not implementable. Here, we follow Sen (1995)who proposed a method of evaluating the extent to which a solution may fail tobe monotonic. Specifically, we look for the minimal way in which the solutionhas to be enlarged so as to satisfy the property. For each solution, there is indeeda “minimal monotonic” solution that contains it. In his work on the subject,Sen limited his attention to the Arrovian model of abstract social choice. Ourmain objective here is to study the concept of minimal monotonic extension inconcretely specified economic models.

First however, we establish a general result relating the minimal monotonicextensions of the intersection and the union of a family of solutions to the minimalmonotonic extensions of the members of the family. Then we turn to examplesand calculate the minimal monotonic extensions of several solutions in a varietyof contexts. We consider classical exchange economies, with either individualendowments or a social endowment, economies with public goods, and one-commodity economies in which preferences are single-peaked. We chose theexamples so as to illustrate the wide range of possibilities. For some of them, verylittle is needed to recover monotonicity, but for others, the required enlargementis quite considerable, to the point that the distributional objective embodied in thesolution has to be given up altogether. However, efficiency is generally preserved,since under minor assumptions on preferences, the solution that associates witheach economy its set of efficient allocations is monotonic.

In each of the examples we specify domains so as to make the analysisas simple as possible. This typically required assumptions eliminating boundarysituations. The Walrasian solution only violates monotonicity on the boundary(under convexity of preferences) and of course when discussing it, we did notmake any such assumptions.

Monotonic extensions on economic domains 15

We conclude this introduction by noting that when a solution is not mono-tonic, an approximation from below might also be considered. Indeed, if a so-lution contains at least one monotonic solution, there is a “largest monotonic”solution that it contains. When the solution to be implemented has been chosenon the basis of one-economy requirements, any of the allocations it selects foreach economy is as good as any other, and replacing the solution by a subsolutionis certainly more appealing than replacing it by a supersolution. Unfortunately,many non-monotonic solutions do not have monotonic subsolutions. Moreover,we are often interested in a solution because it satisfies criteria that relate theallocations that it chooses for some economy to the allocations that it choosesfor other economies in its domain of definition. When such relational conditionsplay a role, an approximation from below is not any more meaningful than anapproximation from above.1

2 Minimal monotonic extensions: Definition and basic properties

We start with a general statement of the problem. LetA be a set of feasible alter-natives andR a class of preference relations defined onA. Let N = 1, . . . ,nbe a set of agents whose preferences belong toR. Given i ∈ N , Ri ∈ R,and a ∈ A, let L(Ri ,a) = b ∈ A: a Ri b be the lower contour set ofRi ata. A solution is a correspondenceϕ: Rn → A that associates with each profileR = (Ri )i ∈N ∈ Rn a non-empty subset ofA. Given i ∈ N , Ri ∈ R, anda ∈ A,we say thatR′

i ∈ R is obtained fromRi by a monotonic transformation ata ifL(Ri ,a) ⊆ L(R′

i ,a). Let MT(Ri ,a) ⊆ R denote the class of all suchR′i . Given

R,R′ ∈ Rn such that for alli ∈ N , R′i ∈ MT(Ri ,a), we write R′ ∈ MT(R,a).

The following property of solutions is fundamental: it says that if an alterna-tive is selected for some profile of preferences, and preferences change in sucha way that the alternative does not fall in anybody’s estimation relative to anyother feasible alternative, then it is still selected for the new profile.

1 Thomson (1996) discusses what it means that the object of our study should be solutions insteadof allocations. One cannot in general say that all the allocations that a solution selects for someeconomy are equally desirable. If this were the case, a solution would be as good as any of itssubsolutions. In fact, by speaking of allocations as being desirable, one suggests that the objectof choice are allocations whereas in general it is in the solutions in which we are interested. Thislanguage would make sense only if the allocations that are chosen for an economy are chosen becausethey maximize a preference relation over the feasible set that the planner would have. Then of course,any two maximal elements would be equivalent. A circumstance in which this would be appropriate iswhen the solution is defined by imposing “one-economy” conditions. Then, its domain of definitioncould in fact consist of a single economy. In general, however a solution has been arrived at byimposing a combination of one-economy conditions and multi-economy conditions.

To illustrate the difference under discussion, note that when the domain of economies underconsideration changes, an allocation that might have been selected initially for some economy thatbelongs to both domains may not be selected anymore. This is because the properties required ofsolutions on the two domains may have different implications concerning the values they take for aneconomy that belong to both domains.

16 W. Thomson

(Maskin)-monotonicity: (Maskin 1977) For allR, R′ ∈ Rn and alla ∈ ϕ(R),if R′ ∈ MT(R,a), thena ∈ ϕ(R′).

It is easy to check that an arbitrary intersection ofmonotonicsolutions, ifwell-defined, that is, if non-empty for each economy in its domain, ismonotonic.Moreover, the solution that associates with each economy its feasible set ismonotonic. Therefore, the intersection of all themonotonicsolutions that containa given solutionϕ is a well-definedmonotonicsolution that containsϕ, andit is the smallest solution with these properties. These observations suggest thefollowing definition:

Minimal monotonic extension: (Sen 1995) Givenϕ: Rn → A, the minimalmonotonic extension ofϕ, denoted bymme(ϕ), is defined by

mme(ϕ) = ∩ψ:ψ ⊇ ϕ,ψ is monotonic.Our first lemma, although straigthforward, will be of great help in our calcu-

lations:

Lemma 1 Let ϕ: Rn → A. For all R ∈ Rn, mme(ϕ)(R) = a ∈ A: there existsR′ ∈ Rn such that a∈ ϕ(R′) and R∈ MT(R′,a).

Proof. Let R ∈ RN andϕ∗(R) = a ∈ A: there existsR′ ∈ Rn such thata ∈ϕ(R′) andR ∈ MT(R′,a). This solution obviously containsϕ (take R′ = R).Also, anymonotonicsolution containingϕ containsϕ∗. Finally,ϕ∗ is monotonic.Indeed, letR ∈ Rn, a ∈ ϕ∗(R), andR′′ ∈ MT(R,a). To show thata ∈ ϕ∗(R′′),observe that by definition ofϕ∗, there existsR′ ∈ Rn such thata ∈ ϕ(R′)and R ∈ MT(R′,a). But sinceR′′ ∈ MT(R,a) and R ∈ MT(R′,a), we haveR′′ ∈ MT(R′,a), so that indeeda ∈ ϕ∗(R′′).

Sen studies the concept ofminimal monotonic extensionin the context ofabstract social choice when the number of alternatives is finite and under the“unrestricted domain” assumption. However, the case of concretely specifiedeconomic models seems to have been left open. The purpose of Sect. 3 is toexamine such models. We consider several examples of commonly discussedsolutions and calculate theirminimal monotonic extensions.

The notation is as follows. Let ∈ N be the number of goods. Preferencerelations are defined overR`

+. As before, for eachi ∈ N , Ri denotes agenti ’s preference relation. LetPi be the strict preference relation associated withRi and Ii the corresponding indifference relation. LetRcl be the domain of“classical”, that is, continuous, convex, and monotone (this means thatzi > z′

i

implieszi Pi z′i )2 preferences. LetRcl ,B be the subdomain ofRcl of preferences

such that no indifference surface containing a positive point meets any of thecoordinate subspaces.3 We refer to this boundary condition as “condition B”:

2 Vector inequalities: givena, b ∈ R`, a = b meansak ≥ bk for all k ; a > b meansa = b and

a /= b; a > b meansak > bk for all k.3 Cobb–Douglas preferences satisfy this property and so do Leontieff preferences. Linear prefer-

ences do not.


Condition B: A preference relationRi defined onR`+ satisfiescondition B iff

for all zi ∈ R`++ and allz′

i ∈ R`+, zi Ii z′

i implies z′i ∈ R

`++.

Each agenti ∈ N is endowed with a vectorωi ∈ R`+ of goods. We refer to

the listω = (ωi )i ∈N as theinitial allocation. Endowments are given once and forall, and therefore aneconomyis described by a listR = (Ri )i ∈N of preferencerelations. Thefeasible setis Z = z ∈ R

`n+ :

∑N zi =

∑N ωi . When we discuss

the problem of fair division, we specify a social endowmentΩ ∈ R`++ instead

of individual endowments. Each individuali ’s preference relationRi over hisconsumption spaceR`

+ induces a preference relation over the feasible set in theusual way.4 If z, z′ ∈ Z are such that for alli ∈ N , zi Ii z′

i , we write z I z′.Before we turn to the examples, we present a lemma that describes a simple

relationship between theminimal monotonic extensionsof two solutions and thoseof their intersection and union. The lemma is an exact counterpart of a resultconcerning a certain property of “consistency” of solutions (Thomson 1994b).To prove the strict inclusion that it states, we find it convenient to consider thefollowing domain of “quasi-linear” economies: preferences are now defined overR × R

`−1+ , the consumption of the first good being unrestricted in sign; they are

continuous, convex, and they admit representations that are separably additivein the first good and linear in that good. This means that there is a functionvi : R

`−1+ → R such that, denoting byxi ∈ R agenti ’s consumption of the first

good andyi ∈ R`−1+ the vector of his consumptions of the remaining goods, his

preference relation can be represented by the function assigning to the bundle(xi , yi ) the valueui (xi , yi ) = xi +vi (yi ). In a quasi-linear economy, if an allocationis efficient, then so is any other allocation obtained by arbitrary redistributionsof the first good. LetRql be the class of all such preference relations. We arenow ready for the statement of the lemma.

Lemma 2 Given two solutionsϕ andϕ′,

mme(ϕ ∪ ϕ′) = mme(ϕ) ∪ mme(ϕ′).

Also, ifϕ ∩ ϕ′ is a well-defined solution,

mme(ϕ ∩ ϕ′) ⊆ mme(ϕ) ∩ mme(ϕ′).

This inclusion may be strict.5

Proof. The proofs of the equality and of the inclusion are identical to their coun-terparts pertaining to consistency, and we refer the reader to Thomson (1994b) fordetails. Indeed, the only property of consistency used there is that it is preservedunder arbitrary intersections and unions.

4 Given z = (zi )i ∈N and z′ = (z′i )i ∈N ∈ Z , we say that agenti finds z as desirable asz′ if and

only if zi Ri z′i .

5 The result holds for any domain. The same statements also apply to arbitrary, finite or not,unions and intersections.

18 W. Thomson

Fig. 1a,b. Example showing that theminimal monotonic extensionof the intersection of a family ofsolutions may be strictly contained in the intersection of theminimal monotonic extensionsof theelements of the family (Lemma 2).a Definition of the solution.b By “flattening” his indifferencecurve through his endowment, agent 1 can push the chosen allocation in a direction that is favorableto him

To show that the inclusion may be strict, we construct an example. Let` = 2andRt

ql ⊆ Rql be the class of quasi-linear preferences defined onR×R+ suchthat at every pointzi , there is a unique line of support to the upper-contour setat zi (this implies that at every pointzi on thex-axis, the only line of supportto the indifference curve passing throughzi is horizontal, so that indifferencecurves reach thex-axis tangentially.) LetW denote the Walrasian solution. Given


R ∈ (Rtql )

2 and z∗ ∈ W(R), let z1(R, z∗) be the individually rational6 andefficient allocation that differs fromz∗ only in the distribution of thex-goodand that is worst for agent 1 (Fig. 1a). Now, givenk ∈ [0,1], let ϕk(R) =kz1(R, z∗) + (1 − k)z∗: z∗ ∈ W(R) ∪ W(R).7 If there is a unique Walrasianallocation, the generic case,ϕk(R) contains at most two points, one of which isthat allocation.

Under our domain assumptions, the Walrasian solutionW is monotonic(moreon this issue in Sect. 3.2). However,ϕk is not monotonicexcept fork = 0, inwhich case it coincides withW. Finally, we show that theminimal monotonicextensionof ϕk is the solutionψk defined byψk(R) = ∪z∗∈W(R)[z∗, kz1(R, z∗) +(1−k)z∗]. Therefore, in order to obtainmonotonicity, and when there is a uniqueWalrasian allocation, say, we need to add to the two points that the solutionwould have selected, the whole interval joining them. To prove this, it sufficesto observe that (i)ψk is monotonicand that (ii) by Lemma 1, for allR ∈ (Rt

ql )2

and allz′ ∈ ψk(R), there isz∗ ∈ W(R) such thatz′ ∈ [z∗, kz1(R, z∗) + (1−k)z∗]and there isR′ ∈ (Rt

ql )2 such thatz′ ∈ ϕk(R′) andR ∈ MT(R′, z′). Indeed, let

z ∈ [z1(R, z∗), z∗] be such thatz′ = kz + (1 − k)z∗. A relation R′1 in Rt

ql canbe defined by specifying one upper-contour set. For upper-contour set ofR′

1 atz, choose the intersection of the upper-contour set atz1(R, z∗) for the relationR1 with the half-space above the line normal top passing through that point,wherep are prices supportingz∗. This construction is illustrated in Fig. 1b (thedashed line). Now, givenk, k′ ∈ ]0,1], with k /= k′, we haveϕk ∩ϕk′

= W, andby monotonicityof the Walrasian solution on the domain under consideration,we obtainmme(ϕk ∩ ϕk′

) = W. However,mme(ϕk) ∩ mme(ϕk′) = ψk ∩ ψk′

=ψmink,k′ /= W.

3 Applications

In order to illustrate the notion ofminimal monotonic extension, we start witha private good model, in economies with individual endowments first, then ineconomies with a social endowment. We discuss two fundamental solutions, thePareto solution and the Walrasian solution. Then we turn to various solutionsto the problem of fair division. Our final example pertains to one-commodityeconomies with single-peaked preferences. Our motivation in this selection ofexamples was to illustrate the wide range of possibilities: for some of the solu-tions, a very minor enlargement is needed to recovermonotonicity; for others, therequired enlargement is considerable, to the point that the distributional objectiveembodied in the solution is essentially lost; the remaining cases fall somewherein between. For applications of the notion ofminimal monotonic extensionstoclasses of matching problems, see Kara and Sonmez (1996, 1997).

6 An allocation is individual rational if it weakly Pareto-dominates the initial allocation.7 The setϕk (R) consists of the Walrasian allocations and the allocations that are obtained as linear

combinations, with weights 1− k and k, of a Walrasian allocationz∗ and the worst individuallyrational allocation for agent 1 at which the distribution of the second good is the same as atz∗.

20 W. Thomson

3.1 The Pareto solution

The Pareto solution is the solution that associates with each economy its set offeasible allocations such that there is no other feasible allocation that all agentsfind at least as desirable and at least one agent prefers.

Pareto solution, P : Given R ∈ Rn, P(R) = z ∈ Z : there is noz′ ∈ Z suchthat for all i ∈ N , z′

i Ri zi , and for somei ∈ N , z′i Pi zi .

The fact that this solution is notmonotonicon the classical domain is illus-trated in Fig. 2.8 In the economyR ∈ R2

cl it depicts, agent 1’s preferences canbe represented by the functionu1 defined byu1(z1) =

∑k z1k for all z1 ∈ R

2+,

and agent 2 has the same preferences. Letz = ((Ω1, Ω2/2), (0, Ω2/2)), and notethat z ∈ P(R). Now, let R′

1 ∈ Rcl be the preference relation that can be repre-sented by the functionu′

1 defined byu′1(z1) = z11 for all z1 ∈ R

2+, (now, agent

1 cares only about the first good), and letR′2 = R2. Let R′ = (R′

1,R′2). Note that

R′ ∈ MT(R, z) and thatz is Pareto-dominated inR′ by z′ = ((Ω1,0)(0, Ω2)),proving the claim.

This lack ofmonotonicityof the Pareto solution is due to the fact that pref-erences are not strictly monotone (this expression means thatzi ≥ z′

i implieszi Pi z′

i ). When the domain is required to contain only strictly monotone prefer-ences, the Pareto solution coincides with the solution that picks all the feasibleallocations to which there is no other feasible allocation that all agents prefer,the “weak Pareto solution”:

Weak Pareto solution,Pw : Given R ∈ Rn, Pw(R) = z ∈ Z : there is noz′ ∈Z such that for alli ∈ N , z′

i Pi zi .This solution ismonotonicon the classical domain. We state without proof

the following result, which relates the Pareto and weak Pareto solutions.

Theorem 1 On Rncl , the minimal monotonic extension of the Pareto solution is

the weak Pareto solution.

A consequence of this result and of Lemma 2 together is that on the classicaldomain, theminimal monotonic extensionof a subsolution of the Pareto solutionis a subsolution of the weak Pareto solution.

3.2 The Walrasian solution

Our next example is the Walrasian solution. Hurwicz et al. (1995) noted, andthis fact is now well-known, that the Walrasian solution is notmonotoniconthe classical domain. A violation ofmonotonicityis illustrated in Fig. 3a foran economyR ∈ R2

cl . We havez ∈ W(R) with supporting pricesp ∈ ∆`−1,

8 This is a point worth clarifying since it is a common misconception that the Pareto solution ismonotonicon this domain.


Fig. 2. The Pareto solution is notmonotonic. On the classical domain violations ofmonotonicityoccur on the boundary of the feasible set

R′1 ∈ MT(R1, z), andR′

2 = R2, but z /∈ W(R′). Indeed,z1 fails to maximizeR′1 in

agent 1’s budget set at pricesp, these prices being the only candidate equilibriumprices. Under convexity of preferences, violations ofmonotonicitycan only occurat boundary Walrasian allocations. Figure 3b shows that if preferences are notconvex, violations can also occur in the interior of the feasible set. There,z ∈W(R), R′

1 ∈ MT(R1, z), andR′2 = R2, but z /∈ W(R′), since at the only prices at

which z1 is a local maximizer ofR′1 on the resulting budget set, the pointz′

1 isaffordable by agent 1 andz′

1 P′1 z1.

Hurwicz et al. defined the concept of the “constrained Walrasian” solution byhaving each agent maximize his preferences in the intersection of his Walrasianbudget set with the projection of the feasible set onto his consumption set, andshowed that it ismonotonicand that it contains the Walrasian solution. The resultfollowing the definition provides a formal justification for the introduction of thissolution in terms of the concept of aminimal monotonic extension.

Constrained Walrasian solution, Wc: GivenR ∈ Rn, Wc(R) = z ∈ Z : thereexistsp ∈ ∆`−1 such that for alli ∈ N ,pzi ≤ pωi , and for all i ∈ N and allz′

i ∈ R` such that [pz′

i ≤ pωi , and for somez′−i ∈ R

`(n−1)+ , (z′

i , z′−i ) ∈ Z ], we

havezi Ri z′i .

Theorem 2 On Rncl , the minimal monotonic extension of the Walrasian solution

is the constrained Walrasian solution.

Proof. Given R ∈ Rncl and z ∈ Wc(R) with associated supporting price

p ∈ ∆`−1, it is trivial to constructR′ ∈ Rncl such thatz ∈ W(R′) and

R ∈ MT(R′, z). For instance, for alli ∈ N , let R′i be a linear preference relation

whose indifference surfaces are hyperplanes normal top. Then, we conclude byappealing to Lemma 1.

22 W. Thomson

Fig. 3. aThe Walrasian solution is notmonotonic. Violations ofmonotonicityoccur on the boundary ofthe feasible set. Itsminimal monotonic extensionis the constrained Walrasian solution (Theorem 3.2).b If preferences are not convex, violations ofmonotonicitymay occur in the interior of the Edgeworthbox


Another proof of this result, noted by Hurwicz et al. (1995), can be obtainedalong the lines of Hurwicz (1979). Incidentally, on the classical domain, theconstrained Walrasian solution is not a subsolution of the Pareto solution (on thisdomain, the Walrasian solution is). Indeed, in the example depicted in Fig. 2,z ∈ Wc(R′

1,R2) (the supporting prices are normal to the hyperplane of supportto agent 2’s indifference curve throughω) but z is not efficient. Any point ofthe segment [z, z′] is a constrained Walrasian allocation for (R′

1,R2), but onlyz′,which is Walrasian, is efficient.9

Note that on the domainRncl ,B, the Walrasian solution ismonotonicsince

the corner problem illustrated in Fig. 3a does not occur.At this point, it may also be useful to observe that on the classical domain, the

minimal monotonic extensionof any solution that satisfies the following minorcondition (Property P in Thomson 1983;non-discrimination between Pareto-indifferent allocationsin Gevers 1986), and selects allocations that weakly Paretodominate the initial allocation, contains the constrained Walrasian solution. Thecondition says that if an allocation is selected, then so is any allocation that isPareto-indifferent to it:

Pareto-indifference: For all R ∈ Rn and allz, z′ ∈ Z , if z ∈ ϕ(R) andz′ I z,thenz′ ∈ ϕ(R).

The same statement can be made about solutions satisfying the next condi-tion (Thomson 1987b), which says that if the initial allocation is efficient, thenthe solution selects all the allocations that are Pareto-indifferent to it (perhapsothers):10

Property α: For all R ∈ Rn, if ω ∈ P(R), thenϕ(R) ⊇ z ∈ Z : z I ω.

For public good economies, a similarly defined extension of the Lindahl solu-tion, known as the constrained Lindahl solution, can be given a similar justifica-tion. Also, the ratio equilibrium solution (Kaneko 1977), and recently introducedsolutions such as the balanced linear cost share equilibrium solution (Mas-Colelland Silvestre 1989), are notmonotonic, but theirminimal monotonic extensionscan be easily obtained by requiring maximization of each agent’s preferencesover the intersection of his budget set as specified in the original definitions ofthese solutions, with the projection of the feasible set onto his consumption space(Diamantaras 1993).

9 However, a constrained Walrasian allocation that is not Walrasian may be efficient. In Fig. 2,if R1 is replaced by a linear preference map with steeper indifference curves,z remains constrainedWalrasian without being Walrasian, but now it is efficient.

10 In the first case, the conclusion follows from the fact that on the classical domain anymonotonicsubsolution of the solution that selects allocations that Pareto-dominate the initial allocation andsatisfiesPareto-indifferencecontains the constrained Walrasian solution. In the second case, it followsfrom the fact that anymonotonicsolution satisfyingPropertyα defined next contains the constrainedWalrasian solution.

24 W. Thomson

3.3 The essential no-envy solution

The next example shows that the extension required to obtainmonotonicitycanbring about a considerable change in a solution. It pertains to a generalizationof the no-envy solution, the solution that selects for each economy its set ofallocations such that no agent would prefer someone else’s bundle to his own.In private good economies with non-convex preferences, and even if all otherstandard properties of preferences are maintained, there may be no envy-freeand efficient allocations (Varian 1974). To remedy this difficulty, Vohra (1991)suggested to select the allocations such that for each agent, there is a Pareto-indifferent allocation at which he envies no-one. He proved the non-emptinessof this solution under very general conditions. LetR be the class of preferencesdefined onR

`+ that are continuous onR+, strictly monotone onR`

++ and satisfycondition B (convexity is not assumed any longer). On this domain the Paretosolution ismonotonic.11

Essential no-envy solution,V : (Vohra 1991) GivenR ∈ Rn, V (R) = z ∈ Z :for all i ∈ N , there iszi ∈ Z such that (i)zi I z and (ii) for all j ∈ N , zi

i Ri zij .

A disadvantage of the essential no-envy solution as compared to the no-envysolution is that it is notmonotonic. We show next that theminimal monotonicextensionof its intersection with the Pareto solution can be easily calculated,at least in the two-agent case. The result is that this extension almost coincideswith the Pareto solution! On the domain considered in Theorem 3, the onlyPareto-efficient allocations that are excluded are the origins. LetP0 denote this“interior” Pareto solution andF denote the no-envy solution.

Theorem 3 On R2, the minimal monotonic extension of the essential no-envyand Pareto solution is the interior Pareto solution.

Proof. (Fig. 4) Letϕ = mme(V ∩ P). Let R = (R1,R2) ∈ R2 andz ∈ P0(R) begiven. If z ∈ F (R), and sinceϕ ⊇ V ∩ P ⊇ F ∩ P, thenz ∈ ϕ(R). Suppose nowthat z /∈ F (R) and to fix the ideas, that agent 1 envies agent 2 atz: z2 P1 z1.Sincez ∈ P(R), then agent 2 does not envy agent 1 atz (Varian 1974).12 Letπ: R

2 → R2 denote the symmetry operator with respect to the middle of the

Edgeworth box. LetC be agent 1’s indifference curve passing throughz, andC ′ = π(C). Now, selectz′ ∈ C such thatπ(z′) be belowC . By condition B,such az′ exists (any allocation on the curvi-linear segment connectinga to bwould do, wherea is the point of intersection ofC with π(C) to the southwestof z andb is the point of intersection ofC with the right side of the Edgeworthbox). Let R′

1 = R1 and R′2 ∈ R be such thatz, z′ ∈ P(R′) and R ∈ MT(R′, z).

Since atz′, which is Pareto-indifferent toz in R′, agent 1 does not envy agent

11 So is the Walrasian solution on the subdomain on which it is well-defined.12 Varian shows that at an efficient allocation, there is at least one agent that envies no-one. In our

two-person economy, if agent 2 envied agent 1, we could exchange their bundles and make themboth better off.


Fig. 4. Minimal monotonic extensionof the essential no-envy solution (Proof of Theorem 3).

2, and atz, agent 2 does not envy agent 1, we havez ∈ (V ∩ P)(R′). Sinceϕ ⊇ V ∩ P, z ∈ ϕ(R′). The proof concludes by appealing to Lemma 1.

Note that by contrast with the no-envy solution, Vohra’s solution satisfiesPareto-indifference. One could ask how much the no-envy solution, or its inter-section with the Pareto solution, differ from what could be called theirminimalPareto-indifferent extensions: Pareto-indifferencebeing closed under arbitraryintersections, this concept can be defined just as the notion ofminimal monotonicextensionwas. If preferences are strictly convex, then of course, the no-envy andPareto solution does satisfyPareto-indifference. It turns out that otherwise, wemay have to add points at which not only no-envy is violated, but even the weakerproperty of no-domination fails; this is the requirement that no agent should con-sume more of every good than any other agent (Thomson 1983; Moulin andThomson 1988). In fact, given anyε > 0, economies can be constructed forwhich one may have to add points at which some agent’s bundle multiplied bythe factorε may still dominate the bundle of some other agent (this is a violationof the property ofε-no-domination formulated for trades by Thomson 1987).Since the essential no-envy solution satisfiesPareto-indifference, one can deducefrom these observations that the considerable enlargement to which it has to besubjected so as to recovermonotonicityis at least in part due to the enlarge-ment of no-envy needed to obtainPareto-indifference. The minimal monotonicextensionof the Pareto-indifferentextension of the no-envy and Pareto solutioncan be easily determined, on the classical domain whenn = 2, by adapting theargument of Theorem 3.

26 W. Thomson

3.4 Egalitarian-equivalence andΩ-egalitarianism

Next we consider another solution that has played an important role in the theoryof fair allocation. Say that an allocation is “egalitarian-equivalent” if there existsa “reference bundle” such that every agent is indifferent between his assignedconsumption bundle and the reference bundle.

Egalitarian-equivalence solution,E: (Pazner and Schmeidler 1978) GivenR ∈Rn, E(R) = z ∈ Z : there existsz0 ∈ R

`+ such that for alli ∈ N , zi Ii z0.

It is easy to see that the egalitarian-equivalence solution is notmonotonicandthat neither is its intersection with the Pareto solution, with which Theorem 4 isconcerned. For that result, we will find it convenient to consider the classR∗

of preferences that are continuous, convex, strictly monotonic inR`++, and satisfy

conditionB (This class includes Cobb–Douglas preferences):

Theorem 4 On R∗n, the minimal monotonic extension of the egalitarian-equi-valence and Pareto solution is the interior Pareto solution.

Fig. 5. Minimal monotonic extensionof the egalitarian-equivalence solution (Proof of Theorem 4)

Proof. (Fig. 5) First, note that ifR ∈ R∗n andz ∈ (E ∩ P)(R), thenz ∈ P0(R).Let ϕ = mme(E ∩ P). Now, let R ∈ R∗n andz ∈ P0(R). Also, let p ∈ ∆`−1 beprices supportingz. (They exist by convexity of preferences.) Sincez ∈ P0(R),thenp > 0. Let z0 ∈ ∩N L(Ri , zi ) be such thatpz0 ≥ maxi ∈N pzi . For eachi ∈ N ,let R′

i ∈ R∗ be such that (i)Ri ∈ MT(R′i , zi ), (ii) z′

i Ii z0, and (iii) p supportsthe upper-contour set atzi , z′

i ∈ R`+: z′

i R′i zi . Note thatz ∈ (E ∩ P)(R′), with

reference bundlez0. The proof concludes by appealing to Lemma 1.

If conditionB is not imposed, the result is a little more difficult to state but theproof is essentially the same. Then,mme(E ∩P)(R) consists of all the allocationsz at which the intersection of all the lower contour sets contains at least one pointwhose value at some prices supportingz is greater than the maximal value of


Fig. 6. a Definition of the Ω-egalitarian solution.b Its minimal monotonic extension(Proof ofTheorem 5).

any of the components ofz. Then, a profileR′ ∈ R∗n satisfying statements (i),(ii), and (iii) of the proof can easily be found.

A useful selection from egalitarian-equivalence is obtained by requiring thatthe reference bundle be proportional to the social endowment. Again, we willalso consider its intersection with the Pareto solution (Fig. 6a). It was suggestedby Pazner and Schmeidler and it has played an important role in a number ofrecent studies.

The Ω-egalitarian solution, EΩ : GivenR ∈ Rn, EΩ(R) = z ∈ Z : there existsλ ∈ R+ such that for alli ∈ N , zi Ii λΩ.

We refer to this solution as “egalitarian” since it involves equating numericalrepresentations of preferences but note that this differs from standard usage of

28 W. Thomson

this term where the utility functions capture notions of intensity of satisfaction.By contrast, our definition is ordinal.

We will need the following piece of notation: givenR ∈ R∗n, z ∈ P(R),and i ∈ N , let λ(Ri , zi ) ∈ R+ be such thatzi Ii λ(Ri , zi )Ω.

Theorem 5 On R∗n, the minimal monotonic extension of theΩ-egalitarian andPareto solution is the solutionϕ defined by:ϕ(R) = z ∈ P(R): there existsupporting prices p∈ ∆`−1 such thatmaxj ∈N pzj ≤ mini ∈N pλ(Ri , zi )Ω.

Proof. We begin with the proof thatϕ is monotonic. Let R ∈ R∗n andz ∈ ϕ(R),with p ∈ ∆`−1 as supporting prices. Also, letR′ ∈ R∗n ∩ MT(R, z). First, wenote thatz ∈ P(R′) with supporting pricesp. Also, for eachi ∈ N , λ(R′

i , zi ) ≥λ(Ri , zi ). Therefore, maxj ∈N pzj ≤ mini ∈N pλ(R′

i , zi )zi , so thatz ∈ ϕ(R′).Next, givenR ∈ R∗n and z ∈ ϕ(R), with p ∈ ∆`−1 as supporting prices,

we show that there isR′ ∈ R∗n such thatz ∈ (EΩ ∩ P)(R′) andR ∈ MT(R′, z).Indeed the only additional requirement is that that agenti ’s indifference curvethrough zi passes through the point [minj ∈N λ(Ri , zi )]Ω (Fig. 6b). Obviously,EΩ ∩ P ⊆ ϕ. The proof concludes by appealing to Lemma 1.

3.5 An equal-gains solution for economies with quasi-linear preferences

We now turn to an examination of the domain of quasi-linear economies andfor this domain, we focus on the solution that divides the gains from tradeequally among all agents. These gains can be unambiguously measured in termsof the good with respect to which the representations of preferences are linear.Let s(R) ∈ R+ be the gains from trade achievable inR ∈ Rn

ql . Using thenotation introduced when we first defined quasi-linearity of preferences,s(R) =max∑

ui (zi ):∑

zi =∑ωi −∑

ui (ωi ). The definition is illustrated in the two-person case in Fig. 7a. Note that the solution is a subsolution of the Paretosolution.

Equal-gains solution,G: For all R ∈ Rnql , G(R) = z ∈ Z : ui (zi ) = ui (ωi ) +

s(R)/n.For n = 2, theminimal monotonic extensionof the equal-gains solution can

be informally described as the solution that selects for each economy the set ofallocations at which each agent’s gain from his endowment is equal to at leasthalf of his gain at a constrained Walrasian allocation. This result is illustrated inFig. 7b by means of an exampleR ∈ R2

ql for which the Walrasian allocation isunique. The worst allocation for agent 1 inmme(G)(R) is half-way between theWalrasian allocation and the efficient allocation that is indifferent for him to hisendowment.

For n ≥ 2, the construction is as follows (I owe the complete argument forthat case to Eiichi Miyagawa). LetR ∈ Rn

ql , andz∗ be a constrained Walrasianallocation forR, with supporting pricesp ∈ ∆`−1. For all i ∈ N , let ai (R, z∗) ∈R be such that (ai (R, z∗), y∗

i ) Ii ωi , and Bi (R, z∗) = [ai (R, z∗), x∗i ]. Now, let


Fig. 7. a The definition of the equal-gains solution.b In the two-person case, itsminimal monotonicextensionconsists of half of the set of efficient allocations that Pareto-dominate equal division

ϕ(R) = z ∈ Z : there existz∗ ∈ Wc(R) and b ∈ ∏Bi (R, z∗) such thatxi =

bi +∑

(x∗i − bi )/n andy = y∗. Note thatϕ containsWc (for all i ∈ N , choose

bi = x∗i ) andϕ containsG (for all i ∈ N , choosebi = ai (R, z∗)).

Theorem 6 On Rnql , the minimal monotonic extension of the equal-gains solu-

tion is the solutionϕ.

30 W. Thomson

Proof. First, we show thatϕ is monotonic. Let R ∈ Rnql and z ∈ ϕ(R).

Let R′ ∈ MT(R, z). By definition of ϕ, there existsz∗ ∈ Wc(R) and b ∈∏Bi (R, z∗) such thatxi = bi +

∑(x∗

i −bi )/n andy = y∗. SinceWc is monotonic,z∗ ∈ Wc(R′). Also, note that for alli ∈ N , Bi (R′, z∗) ⊇ Bi (R, z∗). Therefore,b ∈ ∏

Bi (R′, z∗) and we are done.Finally, we show that for allR ∈ Rn

ql and allz ∈ ϕ(R), there existsR′ ∈ Rnql

such thatz ∈ G(R′) andR ∈ MT(R′, z). Let p be a price of support toz. For alli ∈ N , let R′

i ∈ Rql be the preference relation whose upper contour set atzi isthe intersection of the upper contour set ofRi at ωi and the upper contour set atzi of the linear preference relation whose indifference curves are all normal top. The proof concludes by Lemma 1.

Note that in the two-person case, the equal-gains solution coincides with theShapley value. Therefore, Theorem 6 tells us how much the Shapley value hasto be modified in that case so as to recovermonotonicity. It is an open questionwhether a simple formula exists for the Shapley value when there are more thantwo agents.

3.6 Two solutions to the problem of fair divisionin economies with single-peaked preferences

Finally, we consider the problem of fair division in the one-commodity case(` = 1) when preferences are single-peaked (Sprumont 1991). LetRsp be theclass of continuous preference relationsRi satisfying the following property:there is a numberp(Ri ) ∈ R+ such that for allzi , z′

i ∈ R+, if z′i < zi ≤ p(Ri ), or

p(Ri ) ≤ zi < z′i , thenzi Pi z′

i . This number is agenti ’s peak amount. LetΩ ∈ R+

denote the total amount to be allocated. Here too, the social endowment is keptfixed and ignored in the notation, so that aneconomyis a listR = (Ri )i ∈N ∈ Rn

sp.The feasible set is defined asZ = z ∈ R

n+:

∑zi = Ω.

Our first example of a solution is the “proportional solution”, which allocatesthe commodity proportionally to the peak amounts.

Proportional solution, Pro: Given R ∈ Rnsp, z = Pro(R) if z ∈ Z and there

existsλ ∈ R+ such that for alli ∈ N , zi = λp(Ri ); if no suchλ exists,z =(Ω/n, . . . , Ω/n).

The solution defined below is based on comparing distances from peakamountsunit for unit as opposed to proportionally. It selects the allocation atwhich all agents’ consumptions are equally far from their peak amounts exceptwhen boundary problems occur, in which case those agents whose consumptionswould be negative are given zero instead (Thomson 1994):

Equal-distance solution,D : Given R ∈ Rnsp, z = D(R) if z ∈ Z and (i) when

Ω ≤ ∑p(Ri ), there existsd ≥ 0 such that for alli ∈ N , zi = max0,p(Ri )−d,

and (ii) when∑

p(Ri ) ≤ Ω, there existsd ≥ 0 such that for alli ∈ N , zi =p(Ri ) + d.


Note that both solutions are selections from the Pareto solution. Neither ismonotonic. To describe theirminimal monotonic extensions, we need to intro-duce the solution that selects all the Pareto-efficient allocations except for thoseallocations at which one agent receives the whole endowment (if it is efficient).Let us refer to this solution as the strong Pareto solution and denote it byP∗. Forconvenience we will consider the classRsp,++ of preferences having a positivepeak amount.

Theorem 7 On Rnsp,++, the minimal monotonic extension of the proportional so-

lution contains the strong Pareto solution. OnRnsp, the minimal monotonic ex-

tension of the equal-distance solution contains the strong Pareto solution.

Proof. Let R ∈ Rnsp,++ andx ∈ P∗(R). First, we assume thatΩ ≤ ∑

p(Ri ). Letλ > 1 be such thatλmaxxi ≤ Ω. For eachi ∈ N , let R′

i ∈ Rsp,++ be such thatp(R′

i ) = λxi andΩ P′i xi . Note thatx = Pro(R′) and thatR ∈ MT(R′, x). The

argument for the caseΩ ≥ ∑p(Ri ) is similar and we omit it. The proof for the

equal-distance solution is similar.

4 Concluding comments

We applied the notion ofminimal monotonic extensionto a variety of economicmodels and showed that theminimal monotonic extensionsof a number of solu-tions can be easily calculated. We chose the examples so as to illustrate the widerange of possibilities: in some cases the enlargement needed to obtainmono-tonicity is in general quite small (the Walrasian solution, the Pareto solution).In some other cases, it is often considerable (the essential no-envy solution, theegalitarian-equivalence solution). The extensions needed to recovermonotonicityof other examples fall somewhere in between (theΩ-egalitarian solution). There-fore, the cost of implementability can be very low or very high depending on theparticular solution that is being considered. It would be interesting to identifygeneral properties of solutions under which each of these cases occurs.

Gevers (1986) defined a property closely related toMaskin-monotonicity:it says that if z is chosen by a solution for some profile of preferences andpreferences change in such a way that for every agent,z does not fall withrespect to any other allocation in the space over which his preferences are defined,then z is still chosen for the new profile. AnyMaskin-monotonicsolution isGevers-monotonic. The difference withMaskin-monotonicitylies in the fact thatin economic models, preferences are defined over a set that is not the feasible set,and thatMaskin-monotonicityonly pays attention to the way preferences changeover the feasible set.Gevers-monotonicitywas considered by Nagahisa (1994)and Maniquet (1994). The property is also closed under arbitrary intersectionsso that we could define the concept of theminimal Gevers-monotonic extensionof a solution similarly to the way we defined itsminimal Maskin-monotonicextension. On the classical domain, the Pareto solution and the Walrasian solutionareGevers-monotonic.

32 W. Thomson

Finally, we return to the possibility discussed in the introduction, of restrictinga non-monotonic solution in order to obtain the property, instead of enlargingit. Here, we would of course like to restrict it in a minimal way. That thiscan be done is a consequence of the fact thatmonotonicity is preserved underarbitrary unions. Therefore, if a solution contains at least onemonotonicsolution,it has a maximalmonotonicsubsolution, which is simply the union of all ofits monotonicsubsolutions.13 As examples of application, we can show thatin the two-person case, themaximal monotonic subsolutionof the egalitarian-equivalence and Pareto solution is the no-envy and Pareto solution. In the case ofthree or more agents, the egalitarian-equivalence and Pareto solution contains nomonotonicsubsolution, and therefore it has nomaximal monotonic subsolution.We explained in the introduction the conceptual reasons why such an operationshould not be seen as necessary preferable to the enlargements that we havestudied.

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