plasticity, monotonicity, and …...plasticity, monotonicity, and implementability juan carlos...

PLASTICITY, MONOTONICITY, AND IMPLEMENTABILITY

JUAN CARLOS CARBAJAL AND RUDOLF MULLER

Abstract. Consider a setting in which agents have quasilinear utilities overmoney and social alternatives. The set of alternatives can be finite or infinite.A domain D of admissible valuation functions of an agent is called a revenuemonotonicity domain if every weakly monotone allocation rule is truthfully im-plementable (in dominant strategies) and satisfies revenue equivalence. We intro-duce the notion of plasticity and prove that every plasticity domain is a revenuemonotonicity domain. Our proof is elementary and does not rely on strenuousadditional machinery. We also show that various economic environments, withcountable or uncountable allocations, satisfy domain plasticity. Our applicationsinclude public and private supply of divisible public goods, multi-unit auction-likeenvironments with increasing valuations, allocation problems with single-peakedvaluations, and allocation problems with externalities.

JEL Classification Numbers: C72, D70, D82.

Keywords: Weak Monotonicity, Cyclic Monotonicity, Truthful Implementabil-ity, Revenue Equivalence, Plasticity, Public Goods, Multi-Unit Auctions, Single-Peaked Domains, Contracts with Externalities.

1. Introduction

In this paper, we consider allocation problems in a mechanism design setting inwhich agents have quasilinear utilities over monetary transfers and social alterna-tives. A valuation for an agent, which fully captures his relative preferences overalternatives in this private values setting, is a real function defined on the allocationset and is the agent’s private information. We treat the set of admissible valuationfunctions —the domain of preference— as the primitives of our design problem.We study truthful (dominant strategy) implementation of direct revelation mech-anisms, which are composed of an allocation rule mapping profiles of admissiblevaluations onto the allocation set, and an additional payment rule mapping profilesof valuations into monetary transfers. Our purpose is to contribute to the theoreti-cal mechanism design program in its identification of settings where “well-behavedmechanisms” exist —i.e., mechanisms in which the allocation rule can be said tobe truthfully implementable if it satisfies a system of constraints independent ofpayments, and for which any incentive compatible payment rule is expressible interm of the allocation rule alone.

In his famous auction design paper, Myerson (1981) showed that in single param-eter settings one can replace the incentive compatibility constraints with a simple

Date: February 2014.We would like to thank Andrew McLennan, Wojciech Olszewski, Jean-Charles Rochet, Rabee

Tourky, Rakesh Vohra and John Weymark for very fruitful discussions and comments. We alsothank audiences at the 2013 CMSS Workshop in Auckland and the 2013 Dagstuhl Workshop onElectronic Markets and Auctions. The first author gratefully acknowledges financial support fromthe Australian Research Council.

1

2 CARBAJAL AND MULLER

monotonicity requirement on the allocation rule. Moreover, once a monotone al-location rule was obtained, it could be used to completely recover the incentivecompatible price system (up to an affine transformation). Myerson’s monotonicitycondition does not extend easily to multi-dimensional settings. However, it hasbeen known, since Rochet (1987) seminal contribution, that a cyclic monotonicitycondition on an allocation rule is equivalent to its truthful implementability in everyquasilinear environment. In other words, Rochet taught us that an allocation ruleis truthfully implementable if, and only if, it is cyclically monotone —that is to say,its corresponding allocation graph contains no cycle of negative length. One majorcontribution of our investigation is finding sufficient attributes on the domain ofvaluations to ensure that weak monotonicity —2-cycle monotonicity— is not onlynecessary but also sufficient for the truthful implementability of allocation rules. Asit turns out, these properties also guarantee that every implementable allocationrule satisfies the revenue equivalence property. We illustrate the breadth of ourresults with some important economic applications.

Section 2 contains the formal details of our model. For notational convenience,we deal only with one agent whose domain of admissible valuations, denoted byD, is a subset of the space of real-valued functions defined on the allocation setA. For most of our results, we do not impose any a priori restriction on A, whichcan be finite or infinite. The domain D is called a revenue monotonicity domain ifevery weakly monotone allocation rule is truthfully implementable and satisfies therevenue equivalence property. In Section 3 we introduce the notion of plasticity ofa domain. Our main theorem shows that every weak plasticity domain is a revenueequivalence domain. Our proofs are elementary and do not rely on strenuous addi-tional assumptions on the allocation set or any additional machinery. We believethat by getting rid of unnecessary assumptions, our results deepen our understand-ing of truthful implementability and revenue equivalence. For instance, we obtainan analogue of the Mirrlees representation of the indirect utility function, which isvalid in any plasticity domain, without relying on the standard envelope theoremformulation that requires a parameterization of the preference domain in terms ofan auxiliary type space.

Our notion of domain plasticity rests on the possibility of distorting one or twovaluations around one or two alternatives, respectively, in such a way that the valueof each of the aforementioned alternatives is enhanced, relative to the value of allother allocations, by the resulting admissible valuations. Definition 2 provides theformal requirements. To illustrate the demands of plasticity, we can say that given avaluation vx in the domain D and an alternative x in the allocation set A, it shouldbe possible to find an admissible valuation v in D such that the value differencebetween x and any other alternative a corresponding to v is strictly larger than thevalue difference between x and a corresponding to the original valuation vx. In theillustration of Figure 1, this is accomplished by letting the value of x under v beequal to the value of x under vx, and then letting the value of all other alternativesunder v be below the corresponding value under vx. Plasticity requires this typeof distortions to occur, under certain circumstances, for pairs of valuations andalternatives.

The importance of domain plasticity is that it allows us to conclude that the sumof the lengths between any pair of alternatives in the allocation graph defined byany weakly monotone allocation rule is exactly zero (Proposition 3). From this we

PLASTICITY MONOTONICITY IMPLEMENTABILITY 3

x zy

vx

v

vyvz

v

Figure 1. Distortions around one and two alternatives for A ⊆ R

deduce several relevant consequences. First, if the allocation rule is implementable,then it must satisfied the revenue equivalence property. This is a direct outcome ofthe sum of the lengths between pairs of alternatives being equal to zero, a fact thatwe then combine with the results in Heydenreich, Muller, Uetz, and Vohra (2009).Second, if an allocation rule is 3-cycle monotone then it must be cyclically monotone.Indeed, under plasticity one can transform the sum of the lengths in a 4-cycle toan equivalent sum of lengths of two adjacent 3-cycles, which have non-negativelength due to 3-cycle monotonicity. Applying this argument recursively yields tothe desired conclusion. Third, we deduce that for any weakly monotone allocationrule, if there exists a 3-cycle with negative length in the allocation graph associatedto it, then the reverse 3-cycle must have strictly positive length (Proposition 4).

The final step is to show that when D is a plasticity domain, no weakly monotoneallocation rule can generate a positive 3-cycle (Proposition 5). Our main result,Theorem 3, now falls in place immediately: if D is a plasticity domain, then everyweakly monotone allocation rule is truthfully implementable and satisfies revenueequivalence.

Our definition of plasticity does not impose a priori assumptions on the cardinalityof the allocation set. Thus, it is possible to apply our results to various settingsthat are not covered by previous work. In particular, Ashlagi, Braverman, Hassidim,and Monderer (2010) show that every finite-valued, weakly monotone allocation ruledefined on a domain with a convex closure is cyclically monotone.1 Moreover, theyshow that if the closure of a domain (of dimension 2 or higher) is not convex, thenthere is a finite-valued weakly monotone allocation rule that is not implementable.Thus, while Ashlagi, Braverman, Hassidim, and Monderer (2010) provide a fullcharacterization of weak monotonicity domains (which we do not), their effectiveallocation set is finite. To the best of our knowledge, our paper is the first to offersufficient conditions for a revenue monotonicity domain that can be applied to bothfinite and infinite allocation sets.

1For previous related sufficient conditions with finite allocation sets, see Saks and Yu (2005),Bikhchandani, Chatterji, Lavi, Mu’alem, Nisan, and Sen (2006b), Archer and Kleinberg (2008),and Mishra, Pramanik, and Roy (2013).


Our interest in allocation sets with high cardinality is undoubtedly theoreticalbut also stems from practical considerations. Indeed, various applications of mech-anism design to public good provision and resource allocation problems assumethat the allocation set is an interval of the real line or other convex subset of anEuclidean space. Moreover, in some standard multi-dimensional auction and mech-anism design problems where revenue maximization is the objective, one cannot apriori assert the cardinality of the effective allocation set. In particular, Manelliand Vincent (2007) present examples in the context of a multi-good monopolistwhere the optimal allocation rule entails randomization over outcomes and is notfinite-valued.2

In Section 5 we explore applications of our main theorems. Section 5.1 containsapplications with uncountable allocation sets and is motivated by the public goodprovision literature. We are able to prove that domain plasticity is satisfied if the al-location set is an interval of the real line and the domain of valuations consists of allcontinuous, non-negative, strictly increasing functions. We also show that plasticityholds when the allocation set is a compact manifold and the domain of valuations isrestricted to the set of all smooth valuations. The settings of Section 5.2 resemblethose considered in the multi-unit auction design literature or in other allocationproblems with indivisible goods. We show that domain plasticity is satisfied whenthe allocation set is a countable ordered set and the domain of valuations consistsof all increasing valuations on A. Because plasticity requires distortions aroundtwo alternatives, single-peaked domains are not plasticity. However, Mishra, Pra-manik, and Roy (2013) have shown that single-peaked domains on a finite tree arerevenue monotonicity domains. We pursue a modification of plasticity to a localversion that requires distortions around one alternative alone, and with this we areable to show that single-peaked domains and truncated domains are local plasticitydomains. In Section 5.3 we extend the allocation set to be the product space oftwo subsets of the real line (at least one of which is finite). We do so in order tomodel situations in which externalities are present —e.g., an upstream monopolistssells an essential input to two downstream competitors. So, for instance, the payofffunction of a firm may be increasing in its first component (the amount of inputs itaccesses) but decreasing in its second component (the amount of inputs received byits competitor). We show that in these models of contracts with externalities, do-main plasticity is present, thus every weakly monotone allocation rule is truthfullyimplementable and satisfies revenue equivalence.

Section 6 offers a few final remarks.

2. Preliminaries

The setting we consider in this paper is reminiscent of the models studied inBikhchandani, Chatterji, Lavi, Mu’alem, Nisan, and Sen (2006b), Chung and Ol-szewski (2007), Heydenreich, Muller, Uetz, and Vohra (2009), Ashlagi, Braverman,Hassidim, and Monderer (2010), among others. For simplicity of notation, we dealwith single-agent allocation environments. The results derived here will remainunchanged in multi-agent environments, when the solution concept employed isdominant strategy implementation. Thus, one can interpret our statements and

2See also Figueroa and Skreta (2009).


results as to be made for a given agent, having the profile of valuations (types) ofall other agents fixed.3

There is a set A of alternatives or allocations, which we refer to as the allocationset. Most of our theoretical results are valid for countable or uncountable alloca-tion sets —we shall mention explicitly when countability, or any other restriction, isimposed on A. Our agent has quasi-linear preferences over elements of A and mon-etary transfers, so that the utility he derives from alternative a ∈ A and paymentρ ∈ R is given by

v(a) − ρ.

The function v : A→ R is called a valuation and is the agent’s private information.Let D ⊆ RA denote the set of admissible valuations.4

An allocation rule is a mapping

f : D → A.

For each alternative a ∈ A, denote by f−1(a) ⊆ D the subset of admissible valua-tions that select a under the allocation rule f ; i.e.,

f−1(a) = {v ∈ D : f(v) = a}.Except when explicitly mentioned, we restrict the analysis to surjective allocationrules, so that f−1(a) 6= ∅, for all a ∈ A. Throughout the paper, D is called theallocation domain, or domain for short. In our applications, we use the implicitunderstanding that properties ascribed to D may refer to both the allocation setA and the set of admissible valuations defined on A. For example, a domain Dmay be a subset of Rm when the allocation set has cardinality |A| = m <∞, or itmay be a subset of the space of continuous or smooth functions defined on a metric(or topological) space, and so forth. This object constitutes the primitives of ourmodel. A domain D ⊆ RA coupled with an allocation rule f : D → A forms anallocation problem, which we denote by (D, f).

The designer observes D but the actual realization of valuations is private infor-mation of the agent. Thus, monetary transfers are in general required to inducetruthful revelation. The allocation rule f : D → A is said to be truthfully imple-mentable (in dominant strategies) if there is a payment rule π : D → A such that

v(f(v)) − π(v) ≥ v(f(w)) − π(w), for all v, w ∈ D.

Suppose that f is truthfully implementable by π. By the Taxation Principle, forany two valuations v, w ∈ f−1(a), for any alternative a ∈ A, it must be the case thatπ(v) = π(w) —else the agent will find a profitable misreport. Thus, we associatethe incentive compatible payment rule π with a real-valued function p : A → Rdefined by

p(a) ≡ π(v), for all v ∈ f−1(a), for all a ∈ A.

Without risk of confusion, we refer to p : A → R (associated to π) as a non-linearprice scheme that truthfully implements f . Say that f satisfies revenue equivalenceif for all price schemes p, q : A→ R that implement f , one has

p(a) − p(b) = q(a) − q(b), for all a, b ∈ A.

3The presence of more than one agent in the economy will, nonetheless, become important forcomparisons between the notion of plasticity in this paper and related notion previously exploitedin Carbajal, McLennan, and Tourky (2013).4Following standard notation, we denote by RA the set of all real-valued functions defined on A.


It will be convenient to introduce here the following additional notation anddefinitions. For any pair of alternatives a, b ∈ A, for any admissible valuationv ∈ D, let 4v(a, b) represent the value difference between a and b under v; i.e.,

4v(a, b) ≡ v(a) − v(b).

A k-path in the allocation set A is a finite sequence of alternatives {a1, a2, . . . , ak},where ai belongs to A for each i = 1, . . . , k. A k-cycle in A is a (k + 1)-pathwith the additional provision that ak+1 = a1. The allocation rule f : D → A issaid to be cyclically monotone if for every positive integer k ≥ 2, for every k-cycle{a1, a2, . . . , ak, ak+1 = a1} in A, and for all vi ∈ f−1(ai), i = 1, . . . , k,∑k

i=14vi(ai, ai+1) ≥ 0. (1)

The allocation rule f is said to be weakly monotone when Equation 1 holds for all2-cycles in A. Immediately from these definitions, cyclic monotonicity implies weakmonotonicity for all (D, f). The converse of this observation is however not true inevery allocation problem.5

The relevance of cyclic monotonicity comes from the fact that, as Rochet (1987)demonstrated —see also Rockafellar (1970)— it is always equivalent to truthfulimplementability.

Theorem 1 (Rochet). For every domain D, the allocation rule f : D → A istruthfully implementable if, and only if, it is cyclically monotone.

Given f : D → A, define for all pairs of alternatives a, b ∈ A, a 6= b,

`f (a, b) ≡ inf {4v(a, b) : v ∈ f−1(a)}. (2)

Heydenreich, Muller, Uetz, and Vohra (2009) interpret this as the length fromnode a to node b induced by f in the allocation graph defined by A. When noconfusion arises, we refer to this object as the f -length from a and b. We also let`f (a, a) ≡ 0 for every a ∈ A. If f is weakly monotone, then `f (a, b) > −∞ for allalternatives a, b ∈ A. One can readily deduce that an allocation rule f : D → A iscyclically monotone if, and only if, for every positive integer k ≥ 2, every k-cycle{a1, a2, . . . , ak, ak+1 = a1} in A has non negative f -length; i.e.,∑k

i=1`f (ai, ai+1) ≥ 0. (3)

A similar statement, with the obvious adjustment to Equation 3 to consider only2-cycles in A, holds for weakly monotone allocation rules (see also Proposition 1below). The f -distance from a to b is defined by

distf (a, b) ≡ inf{∑k−1

i=1`f (ai, ai+1) : {a = a1, . . . , ak = b}, k ≥ 2

},

where the infimum is taken over all paths in A connecting a to b. Heydenreich,Muller, Uetz, and Vohra (2009) provide a characterization of revenue equivalencein terms of the f -distance between pairs of alternatives.6

5See Example 1 below. Other counter examples can be found in Bikhchandani, Chatterji, Lavi,Mu’alem, Nisan, and Sen (2006a) and Archer and Kleinberg (2008).6See also Kos and Messner (2013).


Theorem 2 (Heydenreich et al). In every allocation problem (D, f), the allocationrule f is truthfully implementable and satisfies revenue equivalence if, and only if,

distf (x, y) + distf (y, x) = 0 for all x, y ∈ A.

Proof. Let x, y ∈ A, x 6= y, be arbitrary alternatives. If distf (x, y)+distf (y, x) < 0,then readily there are paths {x = a1, a2, . . . , ak = y} and {y = b1, b2, . . . , bj = x}for which the inequality∑k−1

i=1`f (ai, ai+1) +

∑j−1

i=1`f (bi, bi+1) < 0

is satisfied. This expression contradicts the cyclic monotonicity of f . If on the otherhand distf (x, y) + distf (y, x) ≥ 0, then for any k-cycle {a1, a2, . . . , ak, ak+1 = a1} inA, letting x = a1 = ak+1 and y = aj for some 2 ≤ j ≤ k, we obtain that∑k

i=1`f (ai, ai+1) =

∑j−1

i=1`f (ai, ai+1) +

∑k

i=j`f (ai, ai+1)

≥ distf (x, y) + distf (y, x) = 0.

It follows that f is cyclically monotone, therefore implementable. The fact thatdistf (x, y) + distf (y, x) = 0 if and only if revenue equivalence is satisfied followsfrom Theorem 1 in Heydenreich, Muller, Uetz, and Vohra (2009). �

3. plasticity

The basic elements of our model are the allocation set and the set of admissiblevaluations. Because they are treated as primitives, we shall try to impose as fewrestrictions on them as possible to accomplish our objectives. Namely, we want tostudy conditions on the domain D under which it can be shown that every weaklymonotone allocation rule is implementable and satisfies revenue equivalence. In thissection we introduce and discuss the important notion of domain plasticity. Wealso provide some illustrative examples. Section 4 presents the main consequencesof weak monotonicity in plasticity domains. In Section 5 we provide some relevanteconomic applications.

Definition 1. A domain D is called a revenue monotonicity domain if every weaklymonotone allocation rule f : D → A is truthfully implementable and satisfies rev-enue equivalence.

We begin with the following result, which has been stated in different forms invarious places before. Its proof is included for completeness.

Proposition 1. For any allocation problem (D, f), the following holds:

(1) If f is weakly monotone, then for all alternatives x, y ∈ A, x 6= y, for alladmissible valuations vx ∈ f−1(x), vy ∈ f−1(y), one has

4vx(x, y) ≥ `f (x, y) ≥ − `f (y, x) ≥ 4vy(x, y).

(2) If f is truthfully implementable by the price scheme p : A→ R, then for allalternatives x, y ∈ A, x 6= y, one has

`f (x, y) ≥ p(x) − p(y) ≥ − `f (y, x).


Proof. (1) Fix x, y ∈ A, x 6= y. The weak monotonicity of f implies that

4vx(x, y) ≥ 4vy(x, y)

holds for any pair of valuations vx ∈ f−1(x) and vy ∈ f−1(y). Taking the infimumon the left-hand side of this inequality over all valuations in f−1(x), we obtain`f (x, y) ≥ 4vy(x, y). Taking the supremum on the right-hand side of the lastinequality over all valuations in f−1(y), we obtain `f (x, y) ≥ −`f (y, x).

(2) If p : A→ R implements f , then for all x, y in A, x 6= y, for all vx ∈ f−1(x),vy ∈ f−1(y), we have

vx(x) − p(x) ≥ vx(y) − p(y) and vy(y) − p(y) ≥ vy(x) − p(x).

Rearranging and combining these two expressions, we obtain

4vx(x, y) ≥ p(x) − p(y) ≥ 4vy(x, y).

Our desired expression now follows from part (1). �

From Proposition 1 one observes that the f -length of any 2-cycle {a, b, a} gener-ated by an implementable allocation rule f is non negative; i.e.:

`f (a, b) + `f (b, a) ≥ 0.

Immediately, a sufficient condition for revenue equivalence to hold is that the f -length of any 2-cycle {a, b, a} in A is exactly zero. We state our next result in termsof a weaker sufficient condition.

Proposition 2. Consider any allocation problem (D, f) where the allocation rulef : D → A is truthfully implementable. If for all x, y ∈ A there exists a finite path{x = a1, a2, . . . , ak = y} such that

`f (ai, ai+1) + `f (ai+1, ai) = 0 for all i = 1, . . . , k − 1,

then f satisfies revenue equivalence.

Proof. If every consecutive 2-cycle {ai, ai+1, ai} in the path {x = a1, a2, . . . , ak = y}connecting x to y has zero f -length, then one concludes that

0 =∑k−1

i=1`f (ai, ai+1) +

∑k−1

i=1`f (ai+1, ai) ≥ distf (x, y) + distf (y, x) ≥ 0,

where the last inequality is required because f is truthfully implementable. Conse-quently, f satisfies revenue equivalence by means of Theorem 2. �

Suijs (1996) characterizes revenue equivalence for efficient allocation rules map-ping into a finite allocation set in terms of the uniqueness of Groves payments. Itis not difficult to see that his result does not depend on properties of the allocationrules other than implementability. In particular, Suijs’s (1996) Theorem 3.2 can beaccommodated to show that for a finite allocation set A, a sufficient and necessarycondition for revenue equivalence to be satisfied in the allocation problem (D, f)is that for any pair x, y ∈ A, there exists a path {x = a1, . . . , ak = y} such that`f (ai, ai+1) + `f (ai+1, ai) = 0, for all i = 1, . . . , k − 1, where 2 ≤ k ≤ |A|.

We want to emphasize two points in relation to this issue. First, the necessity ofthe aforementioned condition for revenue equivalence is lost when the allocation setis infinite, as one may not be able to bound the number of nodes in the allocation setA needed to connect pairs of alternatives —we illustrate this point with Example 5in Appendix B. Second, intuition may suggest that the f -length of every 2-cycle


{a, b, a} in the allocation set A adding up to zero would serve as a sufficient conditionto ascribe truthful implementability to f . This turns out to be false, even when thecardinality of A is finite, because one may still obtain a cycle of negative length.The following example provides an illustration.

Example 1. The set of alternatives is A = {x, y, z}. A valuation here is a vectorw = (w(x), w(y), w(z)) ∈ R3. Let Dα, Dβ and Dγ be subsets of R3 defined asfollows:

Dα ={

(14 + αx, 8, 2 + αz) | 0 < αx < 1, αx < αz < 1 + αx},

Dβ ={

(15 + βx, 10 + βy, 6) | 0 < βy < 1, βy < βx < 1 + βy},

Dγ ={

(10, 2 + γy, γz − 1) | 0 < γz < 1, γz < γy < 1 + γz}.

The domain D of admissible valuations is D = Dα ∪Dβ ∪Dγ.Let f : D → A be an allocation rule that chooses x whenever the declared val-

uation lies in Dα, alternative y when the valuation is announced from Dβ, and zwhen the declared valuation lies in Dγ. Readily, we have that

`f (x, y) = 6 = − `f (y, x),

`f (y, z) = 4 = − `f (z, y),

`f (z, x) = −11 = − `f (x, z).

The f -length of every 2-cycle in the allocation graph of (D, f) is zero, and thus weakmonotonicity is satisfied. But the 3-cycle {x, y, z, x} has negative length, thereforef fails to be truthfully implementable. �

A reasonable conjecture derived from the previous example is that what suffices toobtain truthful implementability is ensuring that both 2-cycles and 3-cycles havezero length. With this in mind, we introduce the important notion of domainplasticity. Henceforth, a given statement about an interval I of the real line is saidto hold for essentially all δ in I if it holds for all but finitely many elements of I.

Definition 2. A domain D is called a plasticity domain when the following condi-tions are satisfied.

(P1) For all x ∈ A, all w ∈ D, and all ε > 0, there exists an admissible valuationv ∈ D such that for every alternative a ∈ A \ {x},

0 < 4v(x, a) − 4w(x, a) < ε.

(P2) For all x, y ∈ A, x 6= y, all vx, vy ∈ D, and essentially all δ ∈ R satisfying

4vx(x, y) > δ > 4vy(x, y),

there exists an admissible valuation v ∈ D such that 4v(x, y) = δ, and forevery alternative a ∈ A \ {x, y},

either 0 < 4v(x, a) − 4vx(x, a) or 0 < 4v(y, a) − 4vy(y, a).

(P3) For all distinct alternatives x, y, z ∈ A, for all vx, vy ∈ D, and essentially allδ ∈ R and ε > 0 satisfying

4vx(x, y) > δ + ε > δ − ε > 4vy(x, y),


there exist an admissible valuation v ∈ D such that for every other alterna-tive a ∈ A \ {x, y},

either 0 < 4v(x, a) − 4vx(x, a) or 0 < 4v(y, a) − 4vy(y, a).

Moreover, v can be chosen to satisfy

either[4v(x, y) = δ − ε and 4v(x, z) − 4vx(x, z) < ε

]or

[4v(x, y) = δ + ε and 4v(y, z) − 4vy(y, z) < ε

].

To gain intuition on the requirements demanded by Definition 2, note that ifcondition (P1) is satisfied then any valuation in D can be distorted around a givenalternative to obtain a new admissible valuation that enhances the value of suchalternative over everything else, relative to the original valuation. Now let x, y ∈ Abe two distinct allocations and vx, vy two admissible valuations. Suppose that theappreciation for x versus y is larger at vx than at vy; that is 4vx(x, y) > 4vy(x, y).Under condition (P2), there must exist a valuation v ∈ D that lowers the valuedifference between x to y relative to vx, thus making y more attractive when thecomparison takes place with the values initially obtained under vx, while at the sametime increases the value difference between x and y relative to vy, thus making xmore attractive when the comparison is with the values initially obtained under vy.In addition, v is such that, for any other alternative a ∈ A\{x, y}, it either enhancesthe value of x versus to a relative to vx, or it enhances the value of y versus to arelative to vy. Under (P3), in addition, it is possible to restrict the penalty incurredby alternative z 6= x, y under v to an arbitrarily small amount. Maintaining thedistinction between (P2) and (P3) will clarify the role each condition plays in ourresults.7

Example 2. Let A = {a1, a2, . . .} be a countable set ordered according to the relationam � an if and only if m > n, for any positive integers n,m. For instance, an mayrepresent the number of units of a good to be allocated. Let D ⊆ RA be the setof all strictly increasing, non-negative valuations defined on the totally ordered set(A,�). That is, for v ∈ D we have v(a) > v(b) if and only if a � b, for any a, b ∈ A.We claim that D is a plasticity domain.

To simplify notation, for any B ⊆ A we let 1B denote the indicator functionassigning value 1 to all b ∈ B and 0 to all a ∈ A \ B. Additionally, for any twofunctions w, w ∈ RA, let w ∨ w denote the pointwise maximum:

w ∨ w(a) = max{w(a), w(a)}, for all a ∈ A.

We show that a stronger version of (P3) is in place, from where condition (P2) fol-lows. An obvious modification of our argument can be used to verify that condition(P1) holds as well.

Fix distinct alternatives x, y ∈ A. Assume without loss of generality that x � y(otherwise exchange the roles of x and y in the following argument). Fix vx, vy ∈ Dand δ ∈ R such that 4vx(x, y) > δ > 4vy(x, y) > 0. Let ε > 0 be arbitrarily

7The notion of domain plasticity introduced here bears resemblance with the notion of flexibilitythat Carbajal, McLennan, and Tourky (2013) used to study Roberts’ Theorem. We discuss theirrelation in Section 6.


chosen. Now let ξx, ξy > 0 be two real numbers satisfying

ξx − ξy = δ − vx(x) + vy(y).

Construct two functions vx = vx + ξx and vy = vy + ξy. Clearly, they belong to D.We notice that vx(x)−vy(y) = δ > 0, furthermore vx(x)−vy(x) = δ−4vy(x, y) > 0and vy(y)− vx(y) = 4vx(x, y)− δ > 0. Therefore, for sufficiently small 0 < ξ < ε,the function v defined on A by

v(a) = vx ∨ vy(a) + ξ 1{x,y}(a)

is strictly increasing and non-negative, thus admissible. We show that it is ourdesired function.

By construction, one has 4v(x, y) = vx(x)− vy(y) = vx(x) + ξx− vy(y)− ξy = δ.For any other a ∈ A, either v(a) = vx(a) or v(a) = vy(a) is satisfied. Thus, weobtain that either

4v(x, a) − 4vx(x, a) = 4vx(x, a) + ξ − 4vx(x, a) = ξ or

4v(y, a) − 4vy(y, a) = 4vy(y, a) + ξ − 4vy(y, a) = ξ

is satisfied. This argument is valid for arbitrary 0 < ξ < ε, hence condition (P3) isin place, as desired. �

The setting in Example 2 lies outside the models studied in Ashlagi, Braverman,Hassidim, and Monderer (2010) —because the allocation set is countably infinite—and Mishra, Pramanik, and Roy (2013) —because, for instance, alternative a1 can-not be lifted to a top position, even if the allocation set were finite. The next twoexamples show plasticity domains with a finite and uncountable allocation set, re-spectively. The first one is adapted from an example in Vohra (2011) but preservesthe underlying economic structure.

Example 3. There are up to two different objects to allocate to a single agent. Thealternative set is A = {o, a, b, c}, where alternative o corresponds to the agent beingassigned no object, a and c denote the allocation of one of the two distinct objects,respectively, and b corresponds to the situation where the agent receives the bundlecontaining both objects.

A valuation v on A is represented by an ordered tuple

v = (v(o), v(a), v(b), v(c)).

Impose the following restrictions on the set of admissible valuations: (i) v(o) = 0;(ii) v(a) > 0 and v(c) > 0; and (iii) v(b) = max{v(a), v(c)}+γ, for some 0 < γ < γ,where γ > 0 is a fixed parameter. Let D ⊆ R4 contain all valuations that satisfyconditions (i) to (iii). Notice that it is not a convex domain. However, as we showin Appendix B, the domain D in this example is a plasticity domain. �

Example 4. Let A = [0, 1] and let D be the space of all continuous functions definedon A. In this case, D is a plasticity domain. In fact, one can show that the space ofall continuously differentiable functions defined on A = [0, 1] is a plasticity domain,cf. Corollary 3. Here we provide intuition for a key element in the arguments forthe continuous case.

For any alternative x ∈ A and any subinterval Bx of A that contains x and isopen (relative to A), there exists a continuous function µx mapping A to [0, 1] such


that µx(x) = 1, 0 < µx(b) < 1 for all b ∈ Bx \{x}, and µx(a) = 0 for all a ∈ A\Bx.In particular, we can define µx by the expression8

µx(a) = 1 − |a − x||a − x| + inf {|a− b| : b ∈ A \Bx}

.

Consequently, for any two alternatives x, y ∈ A and open disjoint neighborhoodsBx, By ⊂ A containing x and y, respectively, we can combine µx and µy in anadditive fashion to obtain a continuous function µ for which µ(x) = µ(y) = 1,0 < µ(b) < 1 for all b ∈ Bx ∪By, and µ(a) = 0 elsewhere —see Figure 1. �

Our definition of plasticity requires distortions of valuations around two alterna-tives, which means that domains with restrictions like single-peakedness lie outsideits scope.9 However, Mishra, Pramanik, and Roy (2013) have shown that generalizedsingle-peaked domains on a finite allocation set are revenue monotonicity domains.Inspired by their work, we modify plasticity to the related notion of local plasticity,requiring distortions around one alternative, to deal with this type of domains. Inaddition, local plasticity assumes that the allocation set A is at most countable andendowed a priori with a family E of two-alternative subsets of A called edges andrepresented by e = {a, b}, where a, b ∈ A, a 6= b. The graph G = (A,E) shall beinterpreted as being independent of the implementation problem.

Definition 3. A domain D is called a local plasticity domain when A is a countableset and there exists a family E of edges such that G = (A,E) is a connected graphand the following conditions are satisfied.

(P1) For all x ∈ A, all w ∈ D, and all ε > 0, there exists an admissible valuationv ∈ D such that for every alternative a ∈ A \ {x},

0 < 4v(x, a) − 4w(x, a) < ε.

(LP2) For all e = {x, y} ∈ E, all w ∈ D, and essentially all δ ∈ R satisfying

4w(x, y) > δ > 0,

there exists an admissible valuation v ∈ D such that 4v(x, y) = δ, and forevery alternative a ∈ A \ {x, y},

0 < 4v(x, a) − 4w(x, a).

(LP3) For all distinct x, y, z ∈ A with e = {x, y} ∈ E, all w ∈ D, and essentiallyall δ, ε > 0 satisfying

4w(x, y) > δ + ε > δ − ε > 0,

there exists an admissible valuation v ∈ D such that for every alternativea ∈ A \ {x, y},

0 < 4v(x, a) − 4w(x, a).

8Similar constructions have been used in the social choice literature before to prove, for instance,that the Gibbard-Satterthwaite Theorem holds on continuous domains over a metric space. SeeBarbera and Peleg (1990), and more recently Le Breton and Weymark (1999).9This is easy to see. Let A = {x, y, z} be such that x � y � z, and let vx, vz valuations withpeaks at x and z, respectively. Then 4vx(x, z) > 0 > 4vz(x, z). For δ = 0, any valuation vthat satisfies 4v(x, z) = 0 and decreases the value of alternative y relative to either vx or vz willviolate single-peakedness.


Moreover, v can be chosen to satisfy

4v(x, y) = δ − ε and 4v(x, z) < 4w(x, z) + ε.

In a local plasticity domain, distortions are required to happen around one al-ternative and one valuation. Given x ∈ A and valuation w ∈ D, it is possible tofind an admissible valuation v that increases the value of x relative to any otheralternative a at w, with the exception of alternative y whose value increases relativeto x. But y is restricted to be adjacent to x.

4. The Main Results

In this section we explore the consequences of weak monotonicity of an allocationrule in plasticity, and local plasticity, domains. Because the local version is weaker,we are able to derive weaker results.

4.1. Monotonicity under Plasticity

We start by observing that in a plasticity domain, the length of any 2-cycle{x, y, x} in the allocation set associated to a weakly monotone allocation rule isexactly zero.

Proposition 3. Let (D, f) be an allocation problem and f be weakly monotone. Ifthe domain D satisfies (P2), then one has

`f (x, y) + `f (y, x) = 0, for all x, y ∈ A.

Proof. Let x, y be arbitrary alternatives and assume, to obtain a contradiction,that `f (x, y) + `f (y, x) > 0. Note that the opposite strict inequality is ruled out byProposition 1(1). Let vx ∈ f−1(x) and vy ∈ f−1(y), and choose δ ∈ R to satisfy

4vx(x, y) ≥ `f (x, y) > δ > − `f (y, x) ≥ 4vy(x, y).

Using (P2), we assert the existence of v ∈ D such that 4v(x, y) = δ and for anya ∈ A, a 6= x, y, either 4v(x, a) > 4vx(x, a) or 4v(y, a) > 4vy(y, a) holds. Letf(v) = z, we claim that z ∈ {x, y}. Suppose the contrary holds. Then we havethat either 4v(z, x) < 4vx(z, x) or 4v(z, y) < 4vy(z, y) is satisfied, which yieldsa violation of the weak monotonicity of f . It follows that f(v) ∈ {x, y}, thusv ∈ f−1(x) ∪ f−1(y) and

`f (x, y) > 4v(x, y) = δ > − `f (y, x). (4)

But these strict inequalities contradict the definition of the f -length between alter-natives —refer to Eq. (2). �

Let f : D → A be such that the length of every 2-cycle {a, b, a} is exactly zero.Suppose in addition that the f -length of every 3-cycle in A is non-negative. Itfollows that the f -length of any 4-cycle can be converted into the sum of lengthsof two adjacent 3-cycles —see Figure 2 for an illustration. And because theselast are non negative, the larger cycle will also have non negative length. Appliedrecursively, this logic yields to the following result.

Proposition 4. Let (D, f) be an allocation problem, and suppose that `f (a, b) +`f (b, a) = 0 is satisfied for all alternatives a, b ∈ A. Then:

(1) For all x, y, z ∈ A, the 3-cycle {x, y, z, x} has negative f -length if, and onlyif, the reverse 3-cycle {x, z, y, x} has positive f -length.


(2) f is cyclically monotone if, and only if, for all x, y, z ∈ A, it is the case that`f (x, y) + `f (y, z) + `f (z, x) = 0.

Proof. (1) Assume without loss of generality that x, y, z are distinct alternatives inA and suppose that

`f (x, y) + `f (y, z) + `f (z, x) < 0. (5)

Multiplying Eq. (5) by minus one, rearranging, and using the fact that −`f (a, b) =`f (b, a) for all a, b ∈ A, we obtain

`f (x, z) + `f (z, y) + `f (y, x) > 0.

(2) Suppose that f is cyclically monotone, so that for any 3-cycle {x, y, z, x},we have `f (x, y) + `f (y, z) + `f (z, x) ≥ 0. If this inequality is strict, then by theprevious part of the proposition we conclude that the 3-cycle {x, z, y, x} has negativef -length, which is a contradiction.

Conversely, suppose that the length of any 3-cycle {x, y, z, x} is exactly zero. Weargue by induction that for every k-cycle {a1, a2, . . . , ak, ak+1 = a1} in the allocationgraph, k ≥ 2, Eq. (3) is satisfied:∑k

i=1`f (ai, ai+1) ≥ 0.

By assumption, for any x, y, z ∈ A, we have `f (x, y) + `f (y, x) = 0 and further`f (x, y) + `f (y, z) + `f (z, x) = 0 being satisfied, so that the desired inequality holdsfor all 2-cycles and 3-cycles in the allocation graph. Suppose that Eq. (3) holds for all3 ≤ k ≤ K−1; we now show that it remains true for K. Let {a1, a2, . . . , aK , aK+1 =a1} be an arbitrary K-cycle in A. We can write∑K

i=1`f (ai, ai+1) = `f (a1, a2) + . . .+ `f (aK−2, aK−1) + `f (aK−1, aK) + `f (aK , a1)

= `f (a1, a2) + . . .+ `f (aK−2, aK−1) + `f (aK−1, a1)

+ `f (a1, aK−1) + `f (aK−1, aK) + `f (aK , a1)

≥ 0,

where the second equality follows from the fact that `f (aK−1, a1)+`f (a1, aK−1) = 0,and the third follows from our induction step. �

An important consequence of Proposition 3 combined with Proposition 4 is that,under condition (P2), if one could verify that weak monotonicity implies that every3-cycle has zero f -length, then cyclically monotone would be obtained. Unfortu-nately, weak monotonicity does not necessarily imply non-negative 3-cycles under(P2), not even when the allocation set is finite. A counter example can be foundin the allocation problem studied in Example 1, where we obtained a 3-cycle withnegative length even though all 2-cycles in the allocation set have zero length. Itcan be shown, and we do this in Appendix B, that the domain in Example 1 satisfies(P2) but, crucially, misses (P3).

We shall use conditions (P1) and (P3) of Definition 2 to ensure that every 3-cycle in the allocation set has zero length when generated by a weakly monotoneallocation rule. The logic behind this step can be formulated along these lines.Suppose that f is weakly monotone and chooses x at valuation vx and y at valuationvy. Now suppose that the value difference between x and z at vx is sufficiently close,so that z was “almost” chosen by f at vx instead. If the domain D is a plasticity


a1

a2

a3

a4

f (a1 , a 3 ) + f (a3 , a 1 ) = 0

Figure 2. 4-cycle monotonicity from 3-cycle monotonicity

domain, we are sure to find an admissible valuation v such that f selects y at v,but for which (i) the value difference between x and y is small, and (ii) the valuedifference between x and z is sufficiently close to the value difference between thesealternatives under vx. In this case, because x was “almost” selected at v and zwas “almost” selected at vx, it follows that the same should happen for z at v.Thus, the value difference between y and z at v should not be much larger thanthe cumulative value difference between y and x at v, and between x and z at vx.If that is the case, then we have arrived, from weak monotonicity, to the equality`f (y, z) = `f (y, x) + `f (x, z).

Proposition 5. Suppose that D is a plasticity domain. For any weakly monotoneallocation rule f : D → A, one has that `f (x, y) + `f (y, z) + `f (z, x) = 0 is satisfiedfor all x, y, z ∈ A.

Proof. Let x, y, z ∈ A be given alternatives. If x = y = z, the result holds trivially,as `f (a, a) = 0 holds for all a ∈ A. When only two of the three given alternativesare different, the result follows from Proposition 3, since we can conclude that`f (a, b) + `f (b, a) = 0 for all a, b ∈ A, a 6= b, because f is weakly monotone and (P2)is present. Thus, assume x, y, z are distinct alternatives.

Fix ε′ > 0. We argue first that, since D is a plasticity domain, there are twovaluations vx ∈ f−1(x) and vy ∈ f−1(y) such that

`f (x, z) ≤ 4vx(x, z) < `f (x, z) + ε′, (6)

`f (y, z) ≤ 4vy(y, z) < `f (y, z) + ε′, (7)

and4vy(x, y) < `f (x, y) < 4vx(x, y). (8)

Indeed, for fixed ε′ > 0, we can find valuations wx ∈ f−1(x) and wy ∈ f−1(y) suchthat `f (x, z) ≤ 4wx(x, z) < `f (x, z) + ε′ and `f (y, z) ≤ 4wy(y, z) < `f (y, z) + ε′.Note that it follows 4wy(x, y) ≤ − `f (y, x) = `f (x, y) ≤ 4wx(x, y). Let εx =`f (x, z) + ε′ − 4wx(x, z). Using (P1), we obtain a valuation vx ∈ D such that4wx(x, a)+ εx > 4vx(x, a) > 4wx(x, a) for all a 6= x. Since f is weakly monotone,it follows that vx ∈ f−1(x). Moreover, we have that `f (x, y) ≤ 4wx(x, y) <4vx(x, y) and `f (x, z) ≤ 4wx(x, z) < 4vx(x, z) < 4wx(x, z) + εx = `f (x, z) + ε′.


This shows that vx ∈ f−1(x) satisfies Eq. (6) and the second inequality in Eq. (8).The same argument with εy = `f (y, z) + ε′ − 4wy(y, z) shows that we can find avy ∈ f−1(y) for which Eq. (7) and the first inequality in Eq. (8) are present.

Now let 0 < ε < ε′ be such that

4vy(x, y) < `f (x, y) − ε < `f (x, y) + ε < 4vx(x, y).

For δ = `f (x, y), by (P3) we deduce the existence of a valuation v ∈ D such thatfor every alternative a ∈ A, a 6= x, y, either 4v(x, a) > 4vx(x, a) or 4v(y, a) >4vy(y, a) holds. Observe this immediately implies, using previous arguments, thatf(v) ∈ {x, y}. Moreover, the valuation v can be selected to satisfy one of thefollowing conditions:[

4v(x, y) = δ − ε and 4v(x, z) < 4vx(x, z) + ε], (9)[

4v(x, y) = δ + ε and 4v(y, z) < 4vy(y, z) + ε]. (10)

Suppose that Eq. (9) is in place. Then, since we defined δ = `f (x, y), we shallconclude that f(v) = y. Thus, we obtain:

`f (y, z) ≤ 4v(y, z) = 4v(y, x) + 4v(x, z)

< − `f (x, y) + 4vx(x, z) + 2ε,

where the last inequality is obtained using condition Eq. (9). We combine Eq. (6)with the previous expression to deduce

`f (x, y) + `f (y, z) < 4vx(x, z) + 2ε < `f (x, z) + ε′ + 2ε,

therefore, as − `f (x, z) = `f (z, x),

`f (x, y) + `f (y, z) + `f (z, x) < ε′ + 2ε.

When Eq. (10) is instead in place, we have f(v) = x. Repeating the above argumentand employing Eq. (7) in this case, we obtain

`f (x, z) + `f (z, y) + `f (y, x) < ε′ + 2ε.

The preceding argument is valid for arbitrarily small 0 < ε < ε′. Henceforth, weconclude that for distinct x, y, z ∈ A,

either `f (x, y) + `f (y, z) + `f (z, x) ≤ 0

or `f (z, y) + `f (y, x) + `f (x, z) ≤ 0

is satisfied. In any event, exchanging the roles of y and z, we deduce that the f -length of the cycles {x, y, z, x} and {x, z, y, x} are both non-positive. Now supposethat `f (x, y) + `f (y, z) + `f (z, x) < 0. Then, using Proposition 4(1), we concludethat `f (x, z) + `f (z, y) + `f (y, x) > 0, which contradicts our previous findings. Itfollows that

`f (x, y) + `f (y, z) + `f (z, x) = 0. �

Combining Proposition 5 with part (2) of Proposition 4 and Proposition 2, ourmain result becomes evident.

Theorem 3. Every plasticity domain D is a revenue monotonicity domain.


An interesting observation is that a plasticity domain is sufficiently rich to allowreformulation of an allocation problem —i.e., finding an allocation rule f : D → Rand corresponding incentive payment rule π : D → R— as one where the objectiveis to find a well-defined price scheme to satisfy

p(a) − p(a0) = `f (a, a0) ≡ inf {v(a)− v(a0) : v ∈ f−1(a)} (11)

for a weakly monotone allocation rule, for a fixed alternative a0 ∈ A. In this dualformulation, Equation 11 plays the role of the Mirrlees representation of the priceschemes derived under various assumptions in parameterized mechanism designenvironments (e.g., differentiability of the valuation with respect to types). It maybe of use in applied design problems that —for instance when the objective ofthe designer can be stated in terms of revenue maximization— as it may facilitatewriting the optimization problem as a saddle-point problem.10

4.2. Monotonicity under Local Plasticity

As we mentioned, some interesting economic applications do not satisfy plasticity—for instance, the single-peaked domains studied by Mishra, Pramanik, and Roy(2013). When this happens, it may be possible that local plasticity is satisfied. Forthe remainder of this subsection, we restrict attention to a countable allocation setA endowed with a graph G = (A,E).

Proposition 6. Let (D, f) be an allocation problem and f be weakly monotone. Ifthe graph G = (A,E) is connected and the domain D satisfies (LP2), then for alledges e = {x, y} ∈ E one has `f (x, y) + `f (y, x) = 0.

Proof. Suppose we find an edge e = {x, y} ∈ E such that `f (x, y) + `f (x, y) > 0.Without loss of generality, assume `f (x, y) > 0. Let vx ∈ f−1(x) and choose δ ∈ Rsuch that

4vx(x, y) ≥ `f (x, y) > δ > max {0,− `f (y, x)}.(LP2) grants the existence of v ∈ D such that 4v(x, y) = δ and 4v(x, a) >4vx(x, a), for all a 6= x, y. This last inequality and the weak monotonicity of fallows us to conclude that f(v) ∈ {x, y}. This implies that v ∈ f−1(x)∪f−1(y) and`f (x, y) > 4v(x, y) = δ > − `f (y, x), a contradiction. �

The above proposition is weaker than Proposition 3 as it only ensures that the2-cycles formed by all adjacent pairs {x, y} ∈ E have zero length. Because of that,we also derive a weaker analogue to Proposition 4.

Proposition 7. For any given allocation problem (D, f), suppose that G = (A,E)is a connected graph and `f (x, y) + `f (y, x) = 0 is satisfied for all e = {x, y} in E.If `f (x, z) ≥ `f (x, y) + `f (y, z) for all {x, y} ∈ E, all z 6∈ {x, y}, then f is cyclicallymonotone.

Proof. Let {b1, b2, . . . , bm} be a path in A such that ei = {bi, bi+1} ∈ E, for alli = 1, . . . ,m − 1. Then immediately from the assumptions of the proposition onehas that

`f (b1, bm) ≥∑m−1

i=1`f (bi, bi+1). (12)

Let {a1, a2, . . . , ak, ak+1 = a1} be an arbitrary cycle in A. We show that itsf -length is non negative. Because the graph G = (A,E) is connected, for each

10We would like to thank Jean-Charles Rochet for this observation.


i = 1, . . . , k − 1, we can find a path {ai = bi1, bi2, . . . , b

imi

= ai+1} connecting ai toai+1 for which {bij, bij+1} ∈ E, for all j = 1, . . . ,mi − 1. Using Eq. (12) we concludethat

`f (ai, ai+1) ≥∑mi−1

j=1`f (b

ij, b

ij+1), for all i = 1, . . . , k − 1.

Notice now that the path

{ a1 = b11, . . . , b1m1

= a2 = b21, . . . , b2m2

= a3 = b31, . . . , bk−11 , . . . , bk−1mk−1

= ak }

is composed of adjacent elements and connects a1 to ak. But this means the reversepath is formed of adjacent elements and connects ak to a1 = ak+1. Consequently,using Eq. (12) again and the fact that by assumption `f (x, y) + `f (y, x) = 0 for alladjacent {x, y} ∈ E, we obtain∑k

i=1`f (ai, ai+1) ≥

∑k−1

i=1

∑mi−1

j=1

[`f (b

ij, b

ij+1) + `f (b

ij+1, b

ij)]

= 0. �

Proposition 8. Suppose that G = (A,E) is a connected graph and D is a localplasticity domain. If f : D → A is weakly monotone and for all {x, y} ∈ E one has`f (x, y) 6= 0, then for every z ∈ A \ {x, y}, `f (x, z) ≥ `f (x, y) + `f (y, z).

Proof. Let e = {x, y} ∈ E and assume that `f (x, y) > 0. When `f (x, y) < 0, we have`f (y, x) = − `f (x, y) > 0, in which case simply exchange the roles of alternativesx and y. Fix arbitrarily ε′ > 0. Using (P1), we assert the existence of a valuationvx ∈ f−1(x) such that

`f (x, z) ≤ 4vx(x, z) < `f (x, z) + ε′. (13)

Write δ = `f (x, y) > 0, and let 0 < ε < ε′ be such that 0 < δ − ε < δ + ε <4vx(x, y). Using (LP3) we deduce that there is a valuation v ∈ D such that forevery alternative a ∈ A, a 6∈ {x, y}, 0 < 4v(x, a) −4vx(x, a). This implies, usingprevious arguments, that f(v) ∈ {x, y}. Moreover, the valuation v can be selected tosatisfy 4v(x, y) = δ− ε and 4v(x, z) < 4vx(x, z) + ε. As δ = `f (x, y) = − `f (y, x),it shall be concluded that f(v) = y. Consequently,

`f (y, z) ≤ 4v(y, z) = 4v(y, x) + 4v(x, z) < − `f (x, y) + 4vx(x, z) + 2ε.

Combining this last expression with Eq. (13) obtains

`f (x, y) + `f (y, z) < 4vx(x, z) + 2ε < `f (x, z) + ε′ + 2ε,

and therefore

`f (x, y) + `f (y, z) < `f (x, z) + ε′ + 2ε.

The preceding argument is valid for arbitrarily small 0 < ε < ε′. Henceforth, weconclude that for e = {x, y} ∈ E and z ∈ A \ e, `f (x, y) + `f (y, z) ≤ `f (x, z). �

Combining Proposition 8 with Proposition 7 and Proposition 2, we accommodateour main result to local plasticity domains.

Theorem 4. If the domain D is local plasticity and f : D → A is weakly monotoneand such that every adjacent pair {x, y} ∈ E has non-zero f -length, then f istruthfully implementable and satisfies revenue equivalence.


Here, as in the case of plasticity domains, a Mirrlees representation of the incen-tive price scheme can be obtained based on the f -length. In particular, in settingswhere the allocation set A = {a0, a1, a2, . . . , } is fully ordered and every edge in Ecan be written as ei = {ai, ai−1}, for i = 1, 2, . . ., a marginal pricing rule emerges.Given a weakly monotone allocation rule f such that `f (ai, ai−1) 6= 0, the pricescheme p : A→ R implements f if and only if for every positive integer k,

p(ak) − p(a0) =∑k

i=1`f (ai, ai−1) = p(ak−1) + `f (ak, ak−1).

4.3. Contractions

Given a domain D and a surjective allocation rule f : D → A, for any C ⊆ Dlet f |C denote the restriction of f to C. The allocation problem (C, g) is called acontraction of (D, f) in case C ⊆ D and g : C → A is a surjective allocation rulesatisfying g = f |C . When this happens, we say that the allocation problem (D, f)admits a contraction on C. We finish this section with the following observation,which will play a role in a few applications of Section 5.

Proposition 9. Let (D, f) be a given allocation problem and (C, g) a contractionof (D, f). The following two statements hold:

(1) If f is weakly monotone, then so is g.(2) If in addition C is a plasticity domain, then for all a, b ∈ A, `g(a, b) =

`f (a, b).

Proof. (1) Let f : D → A satisfy weak monotonicity. Because (C, g) is a contractionof (D, f), for all a ∈ A one has that g−1(a) = {v ∈ C : g(v) = a} ⊆ {v ∈ D : f(v) =a} = f−1(a). Therefore, for all pairs of alternatives x, y ∈ A, it follows from thedefinition of the length between two alternatives given in Eq. (2) that

`g(x, y) = inf {4v(x, y) : v ∈ g−1(x)}≥ inf {4v(x, y) : v ∈ f−1(x)} = `f (x, y).

Exchanging the roles of x and y, we obtain the relation

`g(x, y) + `g(y, x) ≥ `f (x, y) + `f (y, x) ≥ 0,

as desired.

(2) Additionally, if C is a plasticity domain, then since g : C → A is weaklymonotone by the previous step, it follows from Proposition 3 that `g(x, y)+`g(y, x) =0 for all x, y ∈ A. Henceforth, the relations

− `g(y, x) ≤ − `f (y, x) ≤ `f (x, y) ≤ `g(x, y)

imply that `f (x, y) = `g(x, y) when it is noted that − `g(y, x) = `g(x, y). �

5. Applications

We now describe different economic applications of our main theorems on revenuemonotonicity domains. The proofs of all results derived in this section are gatheredin Appendix A.

5.1. Implementation on Large Allocation Sets

The problem public good provision under quasilinear preferences and asymmetricinformation has received much attention in the mechanism design literature, with


a substantial part of the work focusing on efficiency (public provision11) and, toa lesser extend, revenue maximization (private provision12). In models of pollu-tion rights allocation —e.g., Dasgupta, Hammond, and Maskin (1980), Montero(2008)— and network resource allocation —e.g., Kelly, Maulloo, and Tan (1998)Johari and Tsitsiklis (2004)— it is reasonable to assume that the public good isdivisible, so that for instance the set of alternatives is A = [0, 1] or some otherconvex subset of the real line.

It has been known for some time that if the admissible domain of valuations is thespace of continuous (or the space of differentiable functions) on a metric space, thenefficient allocation rules can be implemented uniquely using VCG transfers —thusrevenue equivalence holds for efficient allocation rules, c.f. Green and Laffont (1977).Our results in this section demonstrate that under different configurations on theallocation environment, revenue equivalence holds for all implementable allocationrules. Moreover, one can replace the incentive compatibility conditions with weakmonotonicity even when the allocation set has the cardinality of the continuum.

Corollary 1. Let A = [a, a] be a compact interval of the real line and D1 ⊆ RA bethe space of all non-negative, continuous, strictly increasing real-valued functionson A whose left- and right-derivatives are well-defined. Then every weakly mono-tone allocation rule f : D1 → A is truthfully implementable and satisfies revenueequivalence.

The proof of Corollary 1 relies heavily on the fact that the domain of valuationsadmits only strictly increasing functions. The next corollary shows that this resultextends to the case of weakly increasing functions as well, provided the allocationproblem at hand admits an appropriate contraction.

Corollary 2. Let A = [a, a] be a compact interval of the real line and D2 ⊆ RA

be the space of all non-negative, continuous, weakly increasing real-valued functionson A whose left- and right-derivatives are well-defined. Let f : D2 → A be weaklymonotone and suppose that (D2, f) admits a contraction on D1 ⊂ D2. Then f istruthfully implementable and satisfies revenue equivalence.

The next corollary extends the setting of Example 4 to include the space ofsmooth valuations on a compact set as a plasticity domain.

Corollary 3. For any 1 ≤ r <∞, let A be a compact C r manifold and D3 ⊆ RA bethe space of all C r real-valued functions on A. Then every weakly monotone alloca-tion rule f : D3 → A is truthfully implementable and satisfies revenue equivalence.

5.2. Implementation on Small Allocation Sets

In Example 2 we argued that if A = {a1, a2, . . . , } is a countable set ordered bythe complete binary relation am � an if and only if m > n, and if D is the set ofall strictly increasing, non negative valuations defined on A, then D is a plasticitydomain. Using Proposition 9, we obtain the following result.

11The economic literature on this topic is extensive. Early contributions, which follow closelythe model we consider in this paper, include Green and Laffont (1977), and Laffont and Maskin(1980).12Among others, see Guth and Hellwig (1986) and more recently Csapo and Muller (2013).


Corollary 4. Let A = {a1, a2, . . .} be a countable ordered set such that am � an ifand only if m > n, for any two natural numbers m,n. Let D4 ⊆ RA be the spaceof all non-negative, weakly increasing real-valued functions on A. Let f : D4 → Abe weakly monotone and suppose that (D4, f) admits a contraction on D ⊂ D4,where D is the set of all non-negative, strictly increasing functions on A. Then fis truthfully implementable and satisfies revenue equivalence.

Corollary 4 above uses the fact that the domain of strictly increasing valuations onA = {a1, a2, . . .} is a plasticity domain. But single-peaked preference domains arenot plasticity domains, irrespective of the cardinality of the allocation set. Thus werely on local plasticity to obtain truthful implementability and revenue equivalencein this case. Inspired by Mishra, Pramanik, and Roy (2013), we state our nextresult in terms of a generalization of single-peaked preferences. Let A be finite andG = (A,E). Say that G is a tree if it does not admit a any cycle, and moreover forany pair of alternatives x, y ∈ A, there exists a unique path {x = a1, a2, . . . , ak = y}such that ei = {ai, ai+1} belongs to E. A function w : A→ R is single-peaked withrespect to G if the set w(A) = {w(a) : a ∈ A} constitutes a linear order that satisfiesthe following requirement: given a∗ ∈ A such that w(a∗) > w(a) for all a ∈ A\{a∗},for all x ∈ A \ {a∗} and every alternative b 6= x in the unique path connecting x toa∗, one has that w(b) > w(x).

vxvy

x y

Figure 3. Single-peaked valuations on a finite allocation set

Corollary 5. Let A be finite and G = (A,E) be a tree. Let D5 ⊆ RA be the spaceof all non-negative, single-peaked valuations defined on A with respect to G. Letf : D5 → A be weakly monotone and such that `f (ai, ai+1) 6= 0 for all {ai, ai+1} ∈ E.Then f is truthfully implementable and satisfies revenue equivalence.

The above corollary does not include models of single-peak preferences on un-countable allocation sets that have been used in the social choice literature to studyallotment of divisible tasks among various participants —e.g., assigning assets todifferent creditors on a bankrupt procedure, as in Barbera, Jackson, and Neme(1997)— or the location of a public facility on a (physical or otherwise) network—e.g., choosing the location of a public library on a road network, as in Schummer


and Vohra (2002). However, as the proof of Corollary 5 makes clear, our result isextendable to strict single-peaked valuations on a countable ordered set, providedevery admissible valuation attains a finite-valued peak. Thus, we could study ap-proximate versions of the aforementioned models in settings that allow monetarytransfers among participants.

Additional applications involving local plasticity domains include, for example,truncated preference domains where agents’ reports include valuations for a strictsubset alternatives. This type of preferences has been used in the market designliterature —e.g., Abdulkadiroglu, Pathak, and Roth (2005)— to model situationswhere are asked to submit rankings of few, among the many, possible alterna-tives. For simplicity, we use the graph G = (A,E) to define truncated preferences.This accommodates environments where agents are allowed to report valuationson “nearby” alternatives —e.g., in a college housing allocation problem an agentis allowed to report a few options located in the same building or neighborhood.Extensions to other environments are easily obtained. Say that w : A → R is atruncated valuation if there exist edges ei = {bi, bi+1} ∈ E, i = 1, 2, . . . k, and aconstant κ such that κ = w(a) < max{w(b) : b ∈ ∪iei}, for all a ∈ A \ ∪iei.

Corollary 6. Let A be countable and G = (A,E) be a connected graph. Let D6 ⊆RA be the space of all truncated valuations defined on A. If f : D6 → A is weaklymonotone and such that `f (ai, ai+1) 6= 0 for all {ai, ai+1} ∈ E, then f is truthfullyimplementable and satisfies revenue equivalence.

5.3. Implementation on Mixed Allocation Sets

In some applications, an alternative a is modelled as a tuple (a1, . . . , an) in asubset of a n-dimensional Euclidean space, with ai ≥ 0 representing the amountof a given resource assigned to agent i. For example, ai may be the quantity ofan intermediate good received by firm i, who competes with the other firms in adownstream market. In such models of contracting with externalities —see Segal(1999) and references therein— an agent’s valuation may be increasing in ai anddecreasing in aj, for all j 6= i.

Here we show how that such settings can be encompassed in our framework. Toeasy exposition, for the remainder of the subsection we assume that the allocationset is A = A1 × A2, where both A1 and A2 are subsets of the real line. Moreover,we assume that A1 is either finite or a compact interval, and A2 is finite.13 Analternative a is an ordered pair a = (a1, a2), where a1 ∈ A1 and a2 ∈ A2. A valuationw : A1 ×A2 → R exhibits positive externalities on A1 and negative externalities onA2 if for all a2 ∈ A2, w(·, a2) is strictly increasing on A1, continuous with well-defined left- and right-derivatives whenever A1 is a compact interval, and for alla1 ∈ A1, w(a1, ·) is strictly decreasing on A2.

For example, in licensing models of technological innovations by an upstreammonopolist to n firms in a downstream oligopoly —e.g., Katz and Shapiro (1986),Kamien, Oren, and Tauman (1992)— one assumes that A1 = {0, 1} represents theaccess of a firm to the new technology. If all firms are (approximately) identi-cal, then profits will depend on the number, not the identity, of competitors who

13Additional work is required to accommodate the more general case where A1 and A2 are count-ably infinite ordered set, but there seems to be few additional economic applications covered undersuch assumption.


share license to the innovation. Thus, its natural to let A2 = 0, 1, 2, . . . , n − 1,assuming that v(a1, a2) is increasing in a1 and decreasing in a2. This setting isalso present in classic takeover models with “atomistic” stockholders —e.g., Gross-man and Hart (1980), Burkart, Gromb, and Panunzi (1998)— where v(a1, a2) rep-resents the expected value of the firm’s public shares for an agent who tendersa2 ∈ A2 = {0, 1, . . . , k} of his shares to a corporate raider when all other stockhold-ers are tendering a proportion a1 ∈ [0, 1] of the firm’s total shares. With a superiorraider, v is increasing in a1 and decreasing in a2.

Corollary 7. Let A = A1 × A2, where A1 ⊂ R is compact and A2 ⊂ R is finite.Let D7 ⊆ RA be the space of all valuations that exhibit positive externalities onA1 and negative externalities on A2. Then every weakly monotone allocation rulef : D → A is truthfully implementable and satisfies revenue equivalence.

Our last result does not depend on the sign of the externalities. It may be possiblethat valuations are strictly increasing in both arguments. This occurs, for instance,in models of diffusion of innovations of technology standards —e.g., Dybvig andSpatt (1983)— where the payoffs of an agent depend positively on the (aggregate)adoption choices of other agents.

6. Concluding Remarks

In this paper, we introduced the notion of plasticity of a domain in a mecha-nism design setting with quasilinear utilities. We showed that the requirementsimposed by plasticity are satisfied in a large variety of economic problems and havesurprisingly strong implications. In particular, we demonstrated that every plas-ticity domain is a revenue monotonicity domain. Our results are derived using asfew assumptions or extra machinery as possible. This, we believe, enhances ourunderstanding of truthful implementability and revenue equivalence.

Our notion of domain plasticity is related to the notion of domain flexibility thatwas used by Carbajal, McLennan, and Tourky (2013) to study Roberts’ Theorem.The two main differences are that (i) plasticity is an individual condition, insofarit requires distortions of individual valuations around certain alternatives, whileflexibility demands similar distortions for all agents; and (ii) plasticity requiresdistortions around one or two allocations, whereas flexibility demands distortionsaround one, two, and three allocations.

We leave for future research the connection of plasticity with Roberts’ Theorem.It seems also possible that an adaptation of plasticity to settings without quasilinearpreferences may be useful to study related problems of strategy-proofness in thesocial choice literature.

Appendix A. Ommitted Proofs

Proof of Corollary 1. By Theorem 3, it suffices to argue that D1 is a plasticitydomain. To show (P1), fix w ∈ D1, x ∈ A and ε > 0. Let dw(x−) and dw(x+)denote the left- and right-derivative of w at x, respectively. For simplicity, assumethat a < x < a. The proof for the end-point cases is easily adapted from thearguments below. Consider an affine function φ1 with slope equal to 2dw(x−)passing through x, and a second affine function φ2 with slope (1/2)dw(x+) passing


through x. Choose alternative b1 to be such that a ≤ b1 < x and to satisfy, for allb1 < a < x,

0 < w(a) − φ1(a) < w(b1) − φ1(b1) = ξ1 < ε.

In a similar fashion, choose alternative b2 such that x < b2 ≤ a and to satisfy, forall x < a < b2,

0 < w(a) − φ2(a) < w(b2) − φ2(b2) = ξ2 < ε.

Finally, construct the function v defined on A as follows:

v(a) =

w(a)− ξ1 : a ≤ a < b1,

φ1(a) : b1 ≤ a ≤ x,

φ2(a) : x ≤ a ≤ b2,

w(a)− ξ2 : b2 < a ≤ a.

Note that v is continuous, strictly increasing, has left- and right-derivatives, co-incides with w only at x and is everywhere else strictly below w. Moreover, it isreadily verified that for all a ∈ A, a 6= x, one has

ε > 4v(x, a) − 4w(x, a) > 0,

as desired. Because we only care about valuation differences, for κ > 0 sufficientlylarge, v = v + κ belongs to D1 and satisfies the required inequality. It follows that(P1) is in place.

We show that (P3) is also in place, from where (P2) will be obtained. Let x, y ∈ A,x 6= y, vx, vy ∈ D1, and δ ∈ R satisfy 4vx(x, y) > δ > 4vy(x, y). Let also ε > 0.Choose real numbers ξx, ξy > 0 so that

ξx − ξy = δ − vx(x) + vy(y).

Now consider the functions wx, wy ∈ D1 defined respectively by wx = vx + ξx andwy = vy + ξy. Notice that, by construction,

wx(x) − wy(y) = δ. (14)

Furthermore, we have

wx(x) − wy(x) = δ − vy(x, y) > 0, and

wy(y) − wx(y) = vx(x, y) − δ > 0.

From the arguments used in (P1), we deduce the existence of strictly increasing,continuous functions vx and vy, both admitting left- and right-derivatives every-where, such that vx ≤ wx with strict inequality everywhere except at x, vy ≤ wy

with strict inequality everywhere except at y, and further

ε > 4vx(x, a) − 4wx(x, a) > 0, for all a 6= x, and (15)

ε > 4vy(y, a) − 4wy(y, a) > 0, for all a 6= y. (16)

Now let v = vx∨ vy, noticing that v(x) = vx(x) = wx(x) and v(y) = vy(y) = wy(y).For sufficiently large κ > 0, it will follow that v = v + κ belongs to D1. UsingEq. (14) we obtain 4v(x, y) = 4v(x, y) = δ. Further, for all a 6= x, y such thatvx(a) ≥ vy(a), we have

4v(x, a) − 4vx(x, a) = 4v(x, a) − 4wx(x, a) = 4vx(x, a) − 4wx(x, a),


whereas for all a 6= x, y such that vx(a) < vy(a), we have

4v(y, a) − 4vy(y, a) = 4v(y, a) − 4wy(y, a) = 4vy(y, a) − 4wy(y, a).

Combining these two last expressions with Eq. (15) and Eq. (16) yields to thedesired conclusion. This completes the proof of the corollary. �

Proof of Corollary 2. By assumption, there exits a contraction (D1, g) of the allo-cation problem (D2, f), where g : D1 → A is surjective and satisfies g = f |D1 . Sincef is weakly monotone, it follows from part (1) of Proposition 9 that g is also weaklymonotone. We have showed in Corollary 1 that D1 is a plasticity domain, thereforefor all x, y, z ∈ A, one has

`g(x, y) + `g(y, z) + `g(z, x) = 0.

Using now part (2) of Proposition 9 we conclude that `f (a, b) = `g(a, b) for alla, b ∈ A, and therefore f : D2 → A is also truthfully implementable and satisfiesrevenue equivalence. �

Proof of Corollary 3. From Theorem 3, it suffices to show that D3 is a plasticitydomain. We begin with the following observation —for details, see Hirsch (1976).For any alternative x ∈ A and open neighborhood B of x, there exists a C r functionµ mapping A to [0, 1] that takes values µ(x) = 1, 0 < µ(b) < 1 for all b ∈ B \ {x},and µ(a) = 0 for all a ∈ A \B.

To show (P1), fix x ∈ A, w ∈ D and ε > 0. Choose a real number ξ such thatε > ξ > 0. Using our previous observation, let B be a neighborhood of x and λ bea C r function that attains values λ(x) = 0, 0 < λ(b) < 1 for all b ∈ B \ {x}, andλ(a) = 1 for all a ∈ A \B. Now consider the function v defined on A by

v(a) = w(a) − ξλ(a).

Clearly, v is a C r function, thus it belongs to D3. Moreover, for all a ∈ A, a 6= x,it follows that

0 < 4v(x, a) − 4w(x, a) = ξ λ(a) < ε.

This show that condition (P1) is satisfied.To show (P3), fix arbitrarily distinct alternatives x, y, z ∈ A, valuations vx, vy ∈

D3 and δ ∈ R satisfying 4vx(x, y) > δ > 4vy(x, y). Let ε > 0. Further, choosetwo real numbers ξx, ξy to satisfy

ξx − ξy = δ − vx(x) + vy(y),

and let wx = vx + ξx ∈ D3 and wy = vy + ξy ∈ D3. Let Bx and By be openneighborhoods of x and y, respectively, with disjoint closure, such that z is containedin Bx. Following our preliminary observation, there exist functions µx, µy ∈ D3 suchthat µx(x) = 1, 0 < µx(a) < 1 for all a ∈ Bx \{x}, and µx(a) = 0 for all a ∈ A\Bx,and similarly µy(y) = 1, 0 < µy(a) < 1 for all a ∈ By \ {y}, and µy(a) = 0 forall a ∈ A \ By. Let µ = µx + µy. Finally, let κ be a constant function satisfyingκ < wx ∧ wy, which is possible since A is compact. With these elements we nowconstruct a valuation v ∈ D3 as follows:

v(a) =

µ(a)wx(a) + (1− µ(a))κ : for all a ∈ Bx;

µ(a)wy(a) + (1− µ(a))κ : for all a ∈ By;

κ : for all a ∈ A \ (Bx ∪By).


By construction, v ∈ D3 and 4v(x, y) = wx(x)− wy(y) = δ.For all a ∈ Bx, a 6= x, we have that

4v(x, a) = wx(x) − µx(a)wx(a) − (1− µx(a))κ

> 4wx(x, a) = 4vx(x, a).

It is shown similarly that for all a ∈ By, a 6= y, 4v(y, a) > 4vy(y, a). For all a ∈A\(Bx∪By), one has immediately4v(x, a) = wx(x)−κ > 4wx(x, a) = 4vx(x, a),and similarly 4v(y, a) = vy(y)− κ > 4vy(y, a). We conclude by noticing that

4v(x, z) − 4vx(x, z) = (1− µx(z))(vx(z) + ξx − κ).

Because we can construct µx so that its value at z is arbitrarily close to 1, it followsthat 4v(x, z)−4vx(x, z) < ε. Thus, condition (P3) is satisfied. �

Proof of Corollary 4. The arguments exhibited in Example 2 show that the domainof non-negative, strictly increasing functions on A is a plasticity domain. Applynow Proposition 9 to obtain the desired conclusion. �

Proof of Corollary 5. From Theorem 4, it suffices to show that the domain D5 ofnon-negative, single-peaked valuations with respect to G, constitutes a local plas-ticity domain. We first point out that for any w ∈ D5 and any alternative x in A,the function w defined by

w(a) = w(a) + ξ 1{x}(a), for all a ∈ A,

preserves the linear order on the set w(A) = {w(a) : a ∈ A} when ξ ∈ R is suchthat |ξ| > 0 is sufficiently small. From this observation, (P1) is obtained readily.

To show that (LP3) is satisfied, fix {x, y} ∈ E, w ∈ D and δ ∈ R such that4w(x, y) > δ > 0. Note that this implies x is the immediate successor in the pathconnecting y to a∗, where this last alternative is the most valued alternative underw. Let ξ = 4w(x, y) − δ > 0 and choose ξ′ > 0 to be arbitrarily small. Considerthe valuation v defined by

v(a) = w(a) + ξ 1{y}(a) + ξ′1{x,y}(a), for all a ∈ A.

Note that 4v(x, y) = 4w(x, y) − ξ = δ > 0. This together with our previousobservation imply that v ∈ D5. Moreover, for all a ∈ A \ {x, y}, we see that4v(x, a) − 4w(x, a) = ξ′ > 0. This suffices for (LP3) to be present, hence thedomain D5 is a local plasticity domain. �

Proof of Corollary 6. It suffices to show that D6 is a local plasticity domain, whichcan be accomplished along the lines of the proof of Corollary 5 above. We omit thedetails. �

Proof of Corollary 7. By Theorem 3, it suffices to show that D7 is a plasticity do-main. Assume that A1 ⊂ R is a compact interval A2 ⊂ R is finite —the casewhere A1 is finite is readily adapted from our arguments below. We begin withthe following observation. Fix w ∈ D7 and a2, b2 ∈ A2 such that a2 < b2.Then the function a1 7→ w(a1, a2) − w(a1, b2) is strictly positive and continuouson A1, and as a consequence of the compactness of A1, its minimum φw(a2, b2) =


min{w(a1, a2) − w(a1, b2) : a1 ∈ A1} is also strictly positive. Since A2 is finite, weobtain that

min {φw(a2, b2) : a2, b2 ∈ A2, a2 < b2} ≡ φw > 0. (17)

With an alternative a2 ∈ A2 fixed, one has that for any ξ ∈ R such that |ξ| < φw,the function w defined on A by

w(a) =

{w(a1, a2) : for all a1 ∈ A1, for a2 ∈ A2,

w(a1, a2) + ξ : for all a1 ∈ A1, all a2 ∈ A2 \ {a2},

is strictly increasing and continuous on A1 and strictly decreasing on A2, thus itbelongs to D7.

To show that (P1) is satisfied, fix alternative x = (x1, x2) ∈ A1 × A2, valuationw ∈ D and arbitrary ε > 0. Choose 0 < ξ < min{ε, φw}, for φw as defined inEq. (17). The function w(·, x2) is strictly increasing and continuous on A1, withwell-defined left- and right-derivatives. Therefore, an application of the argumentsemployed to prove Corollary 1 provides us with an strictly increasing, continuousfunction v(·, x2) defined on A1 satisfying v(x1, x2) = w(x1, x2) and further

0 < 4v((x1, x2), (a1, x2)) − 4w((x1, x2), (a1, x2)) < ξ,

for all a1 ∈ A1, a1 6= x1. For any other a2 6= x2 in A2, define the function v(·, a2) onA1 by v(a1, a2) = w(a1, a2)− ξ. Put together, it follows that v defined on A1 × A2

belongs to D7. Moreover, for any a = (a1, a2) with a2 6= x2, we have that

4v((x1, x2), (a1, a2)) − 4w((x1, x2), (a1, a2)) = ξ.

Therefore, for every alternative a 6= x in A, one has 0 < 4v(x, a) −4w(x, a) < ε,as desired.

To show that (P3) is satisfied, fix alternatives x = (x1, x2) and y = (y1, y2) in A,with x 6= y, valuations vx, vy in D7, and δ ∈ R such that4vx(x, y) > δ > 4vy(x, y).Additionally, fix an arbitrary ε > 0 and let

0 < ξ < min{ε, φvx , φvy},for φvx and φvy as in Eq. (17), replacing w with vx and vy, respectively. It is withoutloss of generality that we assume vx and vy are such that vx(x)−vy(y) = δ, as whenwx ∈ D7 and ξx > 0, wx = wx + ξx belongs to D7 —see the arguments leading toEq. (14). Notice that this implies

vx(x1, x2) > vy(x1, x2) and vy(y1, y2) > vx(y1, y2).

From (P1), we obtain that there exist functions vx(·, x2), vy(·, y2) defined on A1,strictly increasing, continuous and with left- and right-derivatives, and such thatvx(x1, x2) = vx(x1, x2), v

y(y1, y2) = vy(y1, y2), and

0 < 4vx((x1, x2), (a1, x2)) − 4vx((x1, x2), (a1, x2)) < ξ, all a1 6= x1,

0 < 4vy((y1, y2), (a1, y2)) − 4vy((y1, y2), (a1, y2)) < ξ, all a1 6= y1.

Define now vx(·, a2) = vx(·, a2)− ξ on A1, for all a2 6= x2, and similarly vy(·, a2) =vy(·, a2)− ξ on A1, for all a2 6= y2. Finally, we define v on A as

v( · , a2) = vx( · , a2) ∨ vy( · , a2), for all a2 ∈ A2.

It is readily verified that v is our desired valuation, as it belongs to D7 and satisfiescondition (P3). Therefore, D7 is a plasticity domain. �


Appendix B. Examples

Example 1 in Section 3 shows that it is possible to construct an allocation problemfor which the length of every 2-cycle is zero, yet cyclic monotonicity does not follow.We pointed out that the problem in this setting is the lack of (P3), even when (P2)is present.

Example 1 (continued). Let A = {x, y, z} and D = Dα∪Dβ∪Dγ be as in Example 1.A valuation is written here as w = (w(x), w(y), w(z)) ∈ R3. Recall that 1B denotesthe indicator function assigning 1 to all b ∈ B ⊆ A and 0 to all a ∈ A \ B.Slightly abusing notation, we let 1B denote the vector representation of the indicatorfunction.

We claim that D satisfies condition (P1). Fix alternative x and suppose w ∈ Dα.Letting ξ > 0, define v = w + ξ 1{x}. For sufficiently small ξ, one obtains thatv ∈ Dα and 4v(x, a) − 4w(x, a) = ξ > 0, for a = y, z. A similar constructionworks when w ∈ Dβ. Finally, when w ∈ Dγ we define v = w − ξ 1{y,z}. Asbefore, for sufficiently small ξ > 0, we have v ∈ Dγ, and this construction yields to4v(x, a) −4w(x, a) = ξ > 0, for a = y, z. Because the roles of x, y and z can beexchanged, we conclude that D satisfies (P1).

It is more laborious to verify that D satisfies condition (P2) of Definition 2. Fixx, y ∈ A. We exhaust all possible cases.

Case 1. Suppose that vx = (14 +αxx, 8, 2 +αxz ) and vy = (14 +αyx, 8, 2 +αyz) belongto Dα. One has that the relation

4vx(x, y) = 6 + αxx > 6 + αyx = 4vy(x, y)

holds if and only if αxx > αyx. Let that be the case and let δ ∈ R be an arbitrarynumber satisfying 4vx(x, y) > δ > 4vy(x, y). Now choose ξ = 6 + αxx − δ, noticingthat 0 < ξ < αxx. Furthermore, select a real number ξ′ so that 0 < ξ < ξ′ andαxx − ξ < αxz − ξ′ < 1 + αxx − ξ are both satisfied. This is possible because we havethat ξ < αxx < αxz . With these elements, we define a valuation

v = vx − ξ 1{x} − ξ′ 1{z},

which belongs to Dα. Moreover, v is constructed so that4v(x, y) = 4vx(x, y)−ξ =δ and 4v(x, z)−4vx(x, z) = ξ′ − ξ > 0 are both satisfied.

Case 2. Suppose that vx = (14+αxx, 8, 2+αxz ) is in Dα and vy = (15+βyx, 10+βyy , 6)is in Dβ. The relation

4vx(x, y) = 6 + αxx > 5 + βyx − βyy = 4vy(x, y)

holds for all permissible values of αxx, βyx and βyy . Let δ be such that 4vx(x, y) >

δ > 4vy(x, y). When δ > 6, proceed as in Case 1 by selecting ξ = 6 + αxx − δand ξ′ to satisfy both 0 < ξ < ξ′ and αxx − ξ < αxz − ξ′ < 1 + αxx − ξ, and letv = vx − ξ 1{x} − ξ′1{z}. This is our desired valuation.

When δ < 6 instead, select ξ = δ − 5 − βyx + βyy > 0. Now choose ξ′, ξ′′ > 0 tosatisfy ξ′−ξ′′ = ξ, and further letting ξ′′ be sufficiently small so that 0 < βyy+ξ′′ < 1.Notice that by construction we have (βyx + ξ′) − (βyy + ξ′′) = δ − 5 > 0 and also(1+βyy+ξ′′)−(βyx+ξ′) = 6−δ > 0. It follows that 0 < βyy+ξ′′ < βyx+ξ′ < 1+βyy+ξ′′.With these choices, we now define a valuation

v = vy + ξ′ 1{x} + ξ′′1{y},


concluding from previous analysis that v ∈ Dβ. Also, 4v(x, y) = 4vy(x, y) + ξ = δand 4v(y, z)−4vy(y, z) = ξ′′ > 0, as desired.

Case 3. Suppose that vx = (14 + αxx, 8, 2 + αxz ) belongs to Dα and the valuationvy = (10, 2 + γyy , γ

yz − 1) to Dγ. It follows that the relation

4vx(x, y) = 6 + αxx > 8− γyy = 4vy(x, y)

holds if and only if 2 > γyy > 2 − αxx, for 0 < αxx < 1. For a number δ such that4vx(x, y) > δ > 4vy(x, y), choose ξ and ξ′ as in Case 1 to obtain a valuationv = vx − ξ 1{x} − ξ′1{z}, for which the desired conditions are indeed satisfied.

Case 4. When vx = (15 + βxx , 10 + βxy , 6) ∈ Dβ and vy = (14 +αyx, 8, 2 +αyz) ∈ Dα,we obtain the relation

4vx(x, y) = 5 + βxx − βxy > 6 + αyx = 4vy(x, y),

which never holds for permissible values of βxx , βxy and αyx.

Case 5. Suppose that vx = (15 + βxx , 10 + βxy , 6) and vy = (15 + βyx, 10 + βyy , 6) areboth in Dβ. The inequality

4vx(x, y) = 5 + βxx − βxy > 5 + βyx − βyy = 4vy(x, y)

now holds if and only if 0 < βyx − βyy < βxx − βxy < 1. Choose a real number ξ sothat ξ = δ − 5− βyx + βyy > 0, for δ ∈ R satisfying 4vx(x, y) > δ > 4vy(x, y). Letalso ξ′, ξ′′ > 0 be chosen as in Case 2 above, and define v = vy + ξ′1{x} + ξ′′1{y}.This valuation belongs to D and satisfies our requirements.

Case 6. When vx = (15 + βxx , 10 + βxy , 6) ∈ Dβ and vy = (10, 2 + γyy , γyz − 1) ∈ Dγ,

the inequality

4vx(x, y) = 5 + βxx − βxy > 8− γyy = 4vy(x, y)

never holds for admissible values of βxx , βxy and γyy .

Case 7. Let vx = (10, 2 + γxy , γxz − 1) ∈ Dγ and vy = (14 + αyx, 8, 2 + αyz) ∈ Dα.

Then we have that

4vx(x, y) = 8− γxy > 6 + αyx = 4vy(x, y),

which holds for all parameter values of αyx and γxy for which 2 > αyx + γxy . Nowlet δ be a real number such that 4vx(x, y) > δ > 4vy(x, y). When δ > 7, chooseξ = 8 − γxy − δ > 0, noticing that 0 < γxz < γxy < γxy + ξ < 1 + γxz , where the lastinequality follows because (1 + γxz ) − (γxy + ξ) = δ − 7 + γxz > 0. Define then thevaluation

v = vx + ξ 1{y} − ξ′1{z}.

For sufficiently small ξ′ > 0, this valuation belongs to Dγ. Moreover, 4v(x, y) =4vx(x, y)− ξ = δ, and 4v(x, z)−4vx(x, z) = ξ′ > 0.

When we have δ < 7 instead, choose ξ = δ − 6 − αyx, noticing that we obtain0 < αyx + ξ < 1. Additionally, choose a real number 0 < ξ′ < ξ satisfying αyx + ξ <αyz +ξ′ < 1+αyx+ξ, which is feasible as αyx < αyz < 1+αyx. Define now the valuation

v = vy + ξ 1{x} + ξ′1{z}.

By construction, v belongs in this case to Dα and satisfies 4v(x, y) = 4vy(x, y) +ξ = δ. Moreover, we obtain 4v(x, z)−4vx(x, z) = [(1 + αyx + ξ)− (αyz − ξ′)] + γxz .Because v ∈ Dα, it follows that the term in square brakes in this expression ispositive, thus we obtain 4v(x, z) > 4vx(x, z).


Case 8. Let vx = (10, 2 + γxy , γxz ) ∈ Dγ and vy = (15 + βyx, 10 + βyy , 6) ∈ Dβ. The

expression

4vx(x, y) = 8− γxy > 5 + βyx − βyy = 4vy(x, y),

holds for all admissible values of the parameters γxy , βyx and βyy . Let 4vx(x, y) >

δ > 4vy(x, y). When δ > 7, choose ξ = 8− γxy − δ and write

v = vx + ξ 1{y} − ξ′1{z}

for sufficiently small ξ′ > 0. It follows that v ∈ Dγ and4v(x, y) = 4vx(x, y)−ξ = δ.Moreover, 4v(x, z)−4vx(x, z) = ξ′ > 0.

When δ < 6 instead, select ξ = δ − 5 − βyx + βyy > 0. Now choose ξ′, ξ′′ > 0 tosatisfy ξ′− ξ′′ = ξ, and let ξ′′ be arbitrarily small so that 0 < βyy + ξ′′ < 1. It followsby construction that (βyx + ξ′)− (βyy + ξ′′) = δ−5 > 0 and (1+βyy + ξ′′)− (βyx + ξ′) =6− δ > 0, thus we have 0 < βyy + ξ′′ < βyx + ξ′ < 1 + βyy + ξ′′. With these parametervalues, define

v = vy + ξ′1{x} + ξ′′1{z}.

Notice that v ∈ Dβ. Moreover, we obtain 4v(x, y) = 4vy(x, y) + ξ = δ and4v(y, z)−4vy(y, z) = ξ′′ > 0, as desired.

When 6 < δ < 7, for some parameter values it may not be possible to distort vx

to reach a value difference at x and y equal to δ. This happens, in particular, whenγxz < 7 − δ. In this case, let αx = δ − 6, noticing 0 < αx < 1, and select further areal number αz such that αx < αz < 1 + αx. With these parameter values, define

v = (14 + αx, 8, 2 + αz) (18)

which belongs to Dα. Readily, 4v(x, y) = 6 + αx = δ, and

4v(x, z) − 4vx(x, z) = (1 + αx − αz) + γxz > 0,

where the last inequality follows from the fact that the expression in parentheses ispositive, as v ∈ Dα.

Case 9. Suppose that vx = (10, 2 + γxy , γxz − 1) and vy = (10, 2 + γyy , γ

yz − 1) belong

to Dγ. We see that the relation

4vx(x, y) = 8− γxy > 8− γyy = 4vy(x, y)

holds if and only if 2 > γyy > γxy > 0. Let that be the case and let δ be a realnumber such that 4vx(x, y) > δ > 4vy(x, y). If 7 < δ < 8, then we can generatean admissible valuation v meeting all required properties by distorting vx as inthe first part of Case 8. On the other hand, if 6 < δ < 7, then we construct anadmissible valuation v ∈ Dα as in the last part of Case 8 for which all the requiredproperties hold.

Cases 1 to 9 above show that (P2) is satisfied for x, y ∈ A. A similar analysisshows that (P2) is satisfied for x, z and y, z. Thus, the domain D = Dα ∪Dβ ∪Dγ

satisfies condition (P2) of Definition 2. However, (P3) is not satisfied. To see this,go to Case 8 where vx ∈ Dγ, v

y ∈ Dβ. Now let 6 < δ < 7, and choose parametervalues γxz , β

yy in the open unit interval such that 0 < 7 − δ − βyy and γxz < 7 − δ.

We stress that for this choice of γxz , there is no w ∈ Dβ ∪ Dγ with a difference4w(x, y) = δ. Thus, it must be v ∈ Dα constructed as above —see Equation 18.But in this case, for all permissible parameter values, we obtain

4v(x, z)−4vx(x, z) > γxz and 4v(y, z)−4vy(y, z) > 7− δ + βyy .


This last expression shows that condition (P3) is not satisfied. Hence, the domainof this example is not a plasticity domain. �

We come back to Example 3, adapted from Vohra (2011), that presents a nonconvex plasticity domain with a finite number of alternatives, and show that it is aplasticity domain.

Example 3 (continued). Recall that A = {o, a, b, c} and a valuation here is a vectorw = (w(o), w(a), w(b), w(c)) ∈ R4

+. Notice that for any set B ⊆ A \ {o}, for anyvaluation w ∈ D, and sufficiently small ξ > 0, w + ξ 1B and w − ξ 1B both belongto the domain D.

To obtain (P1), first consider alternative o ∈ A and valuation w ∈ D. Whenξ > 0 is chosen to be arbitrarily small, one has that v = w− ξ 1{a,b,c} belongs to Dand satisfies 4v(o, x)−4w(o, x) = ξ > 0, for all x ∈ {a, b, c}. Fix now alternativea ∈ A and valuation w in D, where w(b) = max{w(a), w(c)} + γ for 0 < γ < γ.Choose a real number 0 < ξa < γ and define

v = w + ξa1{a}.

Readily, v ∈ D and 4v(a, x) − 4w(a, x) = ξa > 0 for all x 6= a. As ξa > 0 canbe made arbitrarily small, one obtains (P1). Similar arguments can be applied toshow (P1) holds for alternatives b and c.

We show directly that (P3) is satisfied, exhausting all possible cases.

Case 1. Fix alternatives o, a in A, valuations vo, va in D, and δ ∈ R such that4va(a, o) > δ > 4vo(a, o). Note this implies δ > 0. If va(c) > va(a), then chooseξ = va(a)− δ > 0 and let ξ′ > 0 be sufficiently small. Then the valuation

v = va − ξ 1{a,b,c} − ξ′ 1{b,c}

belongs toD and satisfies4v(a, o) = δ. Moreover,4v(a, b)−4va(a, b) = 4v(a, c)−4va(a, c) = ξ′ > 0. Since ξ′ is arbitrarily close to zero, (P3) is satisfied.

If vo(c) > δ, then choose ξ = δ − vo(a) > 0 and let ξ′ > 0 be sufficiently small.The valuation

v = vo + ξ 1{a} − ξ′ 1{b,c}belongs to D by construction and is such that 4v(a, o) = δ. Moreover, we obtain4v(o, b)−4vo(o, b) = 4v(o, c)−4vo(o, c) = ξ′ > 0, as desired.

Finally, when va(c) ≤ va(a) and vo(c) ≤ δ, let ξ = va(a) − δ > 0 and ξ′ > 0arbitrarily close to zero. Now set

v = va − ξ 1{a,b} − ξ′1{b,c} + (vo − va)1{c}.By construction, v belongs to D and satisfies 4v(a, o) = δ. Further, 4v(a, b) −4va(a, b) = 4v(o, c)−4vo(o, c) = ξ′ > 0, as desired.

Case 2. Fix o, b ∈ A, valuations vo, vb ∈ D and δ ∈ R such that 4vb(b, o) > δ >4vo(b, o). Assume without loss of generality that vb(a) ≥ vb(c). When δ < γ, let

v = vo − ξ′1{a,c} + (δ − vo)1{b},for ξ′ > 0 sufficiently small. Note that 4v(b, a) = δ − vo(a) + ξ′ < γ when ξ′ isclose to zero, so that v ∈ D. Further, we have 4v(b, o) = δ, 4v(o, a)−4vo(o, a) =4v(o, c)−4vo(o, c) = ξ′ > 0, as desired.

When δ > γ instead, let ξ = vb(b)− δ and write

v = vb − ξ1{a,b} − ξ′1{a,c} + (vo − vb)1{c},


where ξ′ > 0 is chosen arbitrarily close to zero. Notice that v(a) = δ − γ − ξ′ >δ − γ − ξ′ > 0 and 0 < 4v(b, y) < γ for y such that v(y) = max{v(a), v(c)}. Itfollows that v ∈ D and further we obtain 4v(b, o) = δ, 4v(b, a) − 4vb(b, a) =4v(o, c)−4vo(o, c) = ξ′ > 0.

Case 3. When o, c ∈ A, vo, vc ∈ D, and δ ∈ R are such that 4vc(c, o) > δ >4vo(c, o), (P3) is proven using the arguments exhibited in Case 1.

Case 4. Let a, b ∈ A, va, vb ∈ D and δ ∈ R be such that 4vb(b, a) > δ > 4va(b, a).Choose ξ = 4vb(b, a)−δ > 0 and ξ′ > 0 sufficiently small. Now define the valuation

v = vb + ξ 1{a} + ξ′ 1{a,b}.

Notice that when vb(a) ≥ vb(c), we have 4v(b, a) = δ < 4vb(b, a) < γ. Whenvb(c) > vb(a), we have that either v(c) ≥ v(a) and thus4v(b, c) = 4vb(b, c)+ξ′ < γ,or v(a) ≥ v(c) and thus 4v(b, a) ≤ 4v(b, c) < γ. It follows that v ∈ D. Moreover,4v(b, a) = 4vb(b, a) − ξ = δ, and 4v(b, o) − 4vb(b, o) = 4v(b, c) − 4vb(b, c) =ξ′ > 0, as desired.

Case 5. Let now a, c ∈ A, va, vc ∈ D and δ ∈ R be such that 4va(a, c) > δ >4vc(a, c). When δ ≥ 0, choose ξ = 4va(a, c)− δ > 0 and ξ′ > 0 arbitrarily small.Now define the valuation

v = va + ξ 1{c} + ξ′ 1{a,c},

which belongs to D, as 4v(a, c) = 4va(a, c) − ξ = δ ≥ 0. Further, we obtain4v(a, b)−4va(a, b) = 4v(a, o)−4va(a, o) = ξ′ > 0. When δ < 0, write4vc(c, a) >−δ > 4va(c, a) and repeat the preceding argument.

Case 6. The case of b, c ∈ A, vb, vc ∈ D and δ ∈ R such that 4vb(b, c) > δ >4vc(b, c) is similar to Case 4, and therefore omitted.

Cases 1 to 6 exhaust all possible cases. Thus, from the analysis presented, weconclude that (P3) is in place, and therefore so is (P2). It follows that this domainis a plasticity domain. �

The next example, adapted from Heydenreich, Muller, Uetz, and Vohra (2009),shows that the sufficient condition of Proposition 2 is not necessary for revenueequivalence when the allocation set is infinite.

Example 5. Let A = [0, 1] be the allocation set. For each x ∈ A, define the functionvx : A→ R by

vx(a) =

{0 : a ≥ x,

a− x : a < x.

A common interpretation is that x ∈ A represents the minimal amount of a divisiblegood demanded by the agent when endowed with valuation vx. The agent expe-riences zero disutility when his demand is met (i.e., when the chosen alternativea ≥ x), otherwise he experiences a linear disutility. Let D = {vx : x ∈ A} be theallocation domain.

Consider f : D → A defined by f(va) = a, for all a ∈ A. It is well knownthat this particular allocation rule is implementable but does not satisfy revenueequivalence.14 Therefore, it will be impossible to find a directed path between anytwo distinct alternatives a, b, for which the f -length of every consecutive 2-cycle

14The earliest reference to this type of examples that we are aware of is Holmstrom (1979).


is equal to zero. To see this, note for every x ∈ A, f−1(x) = {vx}. Also, for alla, b ∈ A, a 6= b, one has `f (a, b) + `f (b, a) = |a− b| > 0.

Consider instead the allocation rule g : V → A defined by g(va) = a/2, for alla ∈ A.15 In this case, of course, g−1(a/2) = {va}. It is also well understood that gis implementable and satisfies the revenue equivalence property. We want to shownow that there exists no positive integer K that bounds the number of nodes neededto connect pairs of alternatives in the allocation set. Let x, y ∈ A, x 6= y, be suchthat y/2 ≤ x ≤ 2y. In this case one immediately observes that:

− `g(y2, x2

)= (x− y)/2 = `g

(x2, y2

).

Let now x, y ∈ A, x 6= y be such that x < y/2. One has:

− `g(y2, x2

)= (x− y)/2 < −x/2 = `g

(x2, y2

).

One can find a finite path {x/2 = a0, a1, . . . , ak = y/2} such that for each i =0, 1, . . . , k − 1, ai/2 ≤ ai+1 ≤ 2ai is satisfied. However, there is no positive integerK <∞ that uniformly bounds the cardinality of the paths that are used to connectall pairs of alternatives in A. For instance, let y = 1 and x = 1/m, where m is apositive integer. One can show without much difficulty that the number of nodesk needed to connect x/2 to y/2 is dependent on m by the following relationship:m ≤ 2k−1. �

References

Abdulkadiroglu, A., P. A. Pathak, and A. E. Roth (2005): “The NewYork City High School Match,” American Economic Review, 95(2), 364–367.

Archer, A., and R. Kleinberg (2008): “Truthful germs are contagious: a local-to-global characterization of truthfulness,” in 10th ACM Conference on ElectronicCommerce.

Ashlagi, I., M. Braverman, A. Hassidim, and D. Monderer (2010):“Monotonicity and implementability,” Econometrica, 78(5), 1749–1772.

Barbera, S., M. O. Jackson, and A. Neme (1997): “Strategy-proof allotmentrules,” Games and Economic Behavior, 18(1), 1–21.

Barbera, S., and B. Peleg (1990): “Strategy-proof voting schemes with con-tinuous preferences,” Social Choice and Welfare, 7(1), 31–38.

Bikhchandani, S., S. Chatterji, R. Lavi, A. Mu’alem, N. Nisan, and

A. Sen (2006a): “Supplement to Weak monotonicity characterizes deterministicdominant-strategy implementation,” Econometrica Supplement Material.

(2006b): “Weak monotonicity characterizes deterministic dominant-strategy implementation,” Econometrica, 74(4), 1109–1132.

Burkart, M., D. Gromb, and F. Panunzi (1998): “Why higher takeover pre-mia protect minority shareholders,” Journal of Political Economy, 106(1), 172–204.

Carbajal, J. C., A. McLennan, and R. Tourky (2013): “Truthful implemen-tation and preference aggregation in restricted domains,” Journal of EconomicTheory, 148(3), 1074–1101.

15Here the allocation rule under consideration is not surjective, but this is inconsequential for thepoint the example is illustrating.


Chung, K.-S., and W. Olszewski (2007): “A non-differentiable approach torevenue equivalence,” Theoretical Economics, 2(4), 469–487.

Csapo, G., and R. Muller (2013): “Optimal mechanism design for the privatesupply of a public good,” Games and Economic Behavior, 80, 229–242.

Dasgupta, P., P. Hammond, and E. Maskin (1980): “On imperfect informa-tion and optimal pollution control,” Review of Economic Studies, 47(5), 857–860.

Dybvig, P. H., and C. S. Spatt (1983): “Adoption externalities as publicgoods,” Journal of Public Economics, 20(2), 231–247.

Figueroa, N., and V. Skreta (2009): “A note on optimal allocation mecha-nisms,” Economics Letters, 102(3), 169–173.

Green, J. R., and J.-J. Laffont (1977): “Characterization of satisfactory mech-anisms for the revelation of preferences for public goods,” Econometrica, 45(2),427–438.

Grossman, S. J., and O. D. Hart (1980): “Takeover bids, the free-rider prob-lem, and the theory of the corporation,” Bell Journal of Economics, 11(1), 42–64.

Guth, W., and M. Hellwig (1986): “The private supply of a public good,”Journal of Economics, 5(S1), 121–159.

Heydenreich, B., R. Muller, M. Uetz, and R. Vohra (2009): “Character-ization of revenue equivalence,” Econometrica, 77(1), 307–316.

Hirsch, M. W. (1976): Differential Topology, Graduate Texts in Mathematics.Springer-Verlag, New York.

Holmstrom, B. (1979): “Groves’ scheme on restricted domains,” Econometrica,47(5), 1137–1144.

Johari, R., and J. N. Tsitsiklis (2004): “Efficiency loss in a network resourceallocation game,” Mathematics of Operations Research, 29(3), 407–435.

Kamien, M. I., S. S. Oren, and Y. Tauman (1992): “Optimal licensing ofcost-reducing innovation,” Journal of Mathematical Economics, 21(5), 483–508.

Katz, M. L., and C. Shapiro (1986): “How to license intangible property,”Quarterly Journal of Economics, 101(3), 567–590.

Kelly, F. P., A. K. Maulloo, and D. K. H. Tan (1998): “Rate controlfor communication networks: shadow prices, proportional fairness and stability,”Journal of the Operational Research Society, 49(3), 237–252.

Kos, N., and M. Messner (2013): “Extremal incentive compatible transfers,”Journal of Economic Theory, 148(1), 134–164.

Laffont, J.-J., and E. Maskin (1980): “A differential approach to dominantstrategy mechanisms,” Econometrica, 48(6), 1507–1520.

Le Breton, M., and J. A. Weymark (1999): “Strategy-proof social choice withcontinuous separable preferences,” Journal of Mathematical Economics, 32(1),47—85.

Manelli, A. M., and D. R. Vincent (2007): “Multidimensional mechanismdesign: revenue maximization and the multiple-good monopoly,” Journal of Eco-nomic Theory, 137(1), 153–185.

Mishra, D., A. Pramanik, and S. Roy (2013): “Implementation in multidi-mensional domains with ordinal restrictions,” Discussion paper, Indian StatisticalInstitute.

Montero, J.-P. (2008): “A simple auction mechanism for the optimal allocationof the commons,” American Economic Review, 98(1), 496–518.


Myerson, R. B. (1981): “Optimal auction design,” Mathematics of OperationsResearch, 6(1), 58–73.

Rochet, J.-C. (1987): “A necessary and sufficient condition for rationalizabilityin a quasi-linear context,” Journal of Mathematical Economics, 16(2), 191–200.

Rockafellar, R. T. (1970): Convex Analysis. Princeton University Press,Princeton, NJ.

Saks, M., and L. Yu (2005): “Weak monotonicity suffices for truthfulness onconvex domains,” in 7th ACM Conference on Electronic Commerce.

Schummer, J., and R. V. Vohra (2002): “Strategy-proof location on a net-work,” Journal of Economic Theory, 104(2), 405–428.

Segal, I. (1999): “Contracting with externalities,” Quarterly Journal of Econom-ics, 114(2), 337–388.

Suijs, J. (1996): “On incentive compatibility and budget balancedness in publicdecision making,” Economic Design, 2(1), 193–209.

Vohra, R. V. (2011): Mechanism Design: a Linear Programming Approach,Econometric Society Monographs. Cambridge University Press.

School of Economics, University of New South WalesE-mail address: [email protected]

Department of Quantitative Economics, Maastricht UniversityE-mail address: [email protected]

plasticity, monotonicity, and …...plasticity, monotonicity, and implementability juan carlos...

Documents