Implementation in partial equilibrium Takashi Hayashi and Michele Lombardiy July 10, 2015

Consider a society with a …nite number of sectors (social issues or commodities). In a partial equilibrium mechanism a sector authority aims to elicit agents’ preference rankings for outcomes at hand, presuming separability of preferences, while such presumption is false in general and such isolated rankings are artifacts. Therefore, its participants are required to behave as if they had separable preferences. This paper studies what can be implemented if we take such misspeci…cation as a given constraint. Speci…cally, in our implementation model there are several sector authorities, agents are constrained to submit their rankings to each sector authority separately and, moreover, sector authorities cannot communicate with each other. When a social choice rule (SCR) can be Nash implemented by a product set of partial equilibrium mechanisms, we say that it can be implemented in partial equilibrium. We identify necessary conditions for SCRs to be implemented in partial equilibrium and show that they are also su¢ cient under mild auxiliary conditions. Thus, the implementation in partial equilibrium of SCRs is examined in several environments, mainly in auction and matching environments.

JEL classi…cation: C72; D71. Keywords: Nash implementation, non-separable preferences, partial equilibrium.

We are grateful to Salvador Barberà, William Thomson and audiences at the Workshop on Social Choice and Mechanism Design (Manchester, 2015), for useful comments and suggestions. The usual caveat applies. y Adam Smith Business School, University of Glasgow, Glasgow, G12 8QQ, United Kingdom. Emails: [email protected] (Hayashi) and [email protected] (Lombardi).

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Introduction To understand how to address a single social issue, economists have used the method-

ology of partial equilibrium mechanism design. This methodology isolates items to be allocated as well as people’s preferences for those items from the rest of the world, under a ceteris paribus (all else equal) assumption. Because of such isolation, it has provided exact mechanisms and algorithms on how to govern individual behavior so as to achieve desirable objectives and has proved capable of handling a wide variety of issues, not only economic but political and legal. The prominently successful cases are auction and matching, such as auction houses, labor market for medical interns and school admissions. The ceteris paribus (all else equal) assumption, however, cannot be true in general, since people’s preferences are generally non-separable. For example, which school one would like to be admitted depends on where she lives and, moreover, where she would like to live depends on which school she could be admitted to. When the central authority of one social issue assumes that each of its participants has a single preference ranking for the social issue at hand and requires participants to report their rankings, it is forcing its participants to behave as if their preferences were separable, while such rankings are artifacts. However, if we change something in the school admission program, it will have a general equilibrium e¤ect, such as changes in the housing market and how people choose where to live, etc. Likewise, if we change something in the house auction rule, it will have a general equilibrium e¤ect on how people consume other goods related to the auctioned house and, moreover, will a¤ect bidders’ willingness to pay, and so on. In this paper, we ask the following questions: What do we lose by ignoring such general equilibrium e¤ects? If we take the misspeci…cation described above as a given institutional constraint, how does it restrict the set of viable social arrangements? Furthermore, even if people’s preferences are separable and every individual has a single conditional preference for each social issue, that methodology hinders our understanding of how to assign priorities to individuals. However, priorities are eventually established in reality. It is likely, for example, that one individual should be prioritized

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in the school admission and another should be prioritized in the housing allocation, whereas the actual decisions go in the opposite directions. This leads to ine¢ ciency at a general equilibrium level. This paper answers the above questions by studying what social choice rules (SCR) can be Nash implemented in a society in which there is a …nite number of social issues or sectors, agents are constrained to submit their preferences to sector authorities separately and sector authorities are unable to communicate with each other, for example, due to misspeci…cation by the designer or due to technical/institutional constraints. Simply put, our Nash implementation problem consists of designing partial equilibrium mechanisms, one for each sector, with the property that the Nash equilibrium outcomes of their product set coincides with the recommendations of a given SCR. If this can be done, we say that a SCR can be implemented in partial equilibrium. Therefore, we answer the above questions by using a positive economic approach. We show that SCRs that can be implemented in partial equilibrium satisfy (Maskin) monotonicity, a decomposability condition and a decomposable (Maskin) monotonicity condition. In addition, if a SCR satis…es the above properties and some other mild auxiliary conditions, reminiscent of Maskin’s Theorem (Maskin, 1999), and if there are at least three agents, then the SCR can be implemented in partial equilibrium. The ideas are actually quite intuitive: (i) under separability, a SCR that can be implemented in partial equilibrium must induce one-dimensional SCRs, one for each sector, each of which depends only on conditional preferences; (ii) such sector-speci…c SCRs must satisfy the standard invariance condition due to Maskin, and that condition is also su¢ cient for constructing an implementing partial equilibrium mechanism under mild additional conditions; and …nally (iii) given a list of implementing partial equilibrium mechanisms, its extension to general preferences is obtained by simply letting each agent submit preferences to each sector authority separately, as if she had separable preferences. We also provide a characterization of what can be implemented when we start with a list of sector-speci…c SCRs de…ned over conditional preferences. We provide a natural extension of sector-speci…c SCRs, based on the as if idea, and show that it is the smallest SCR that can be implemented in partial equilibrium. 2

Our analysis shows that the positive nature of partial equilibrium mechanisms to require agents to behave as if they had separable preferences imposes constraints not only on what kinds of outcomes a sector authority can achieve, but also, and most importantly, on what kinds of outcomes the society as a whole can achieve. Section 2 provides motivating examples while section 3 presents the theoretical framework and outlines the basic model, with necessary conditions presented in section 4. Section 5 presents our characterization result, with its implications discussed in section 6. Section 7 concludes.

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Motivating examples To illustrate our points we discuss two prominent cases of partial equilibrium mech-

anism design: matching and auction.

2.1

Matching

In a matching problem, each involved agent is required to submit a preference ranking over mates or items. Moreover, the assignment authority solves that problem in isolation from other matching problems in which agents can also be involved as participants. For example, in a school choice program, parents of a student submit a strict ranking over schools to an education authority, which decides which school each student will attend, independently from other authorities’decisions and after having taken into consideration the assignment priorities of schools. However, parental preferences over schools are typically not independent of the decision made by other assignment authorities such as the housing authority. Examples below illustrate the problems arising when agents’ preferences are assumed to be separable, while they are not, and assignment authorities’decisions are not coordinated. The economy consists of two indivisible and non-homogeneous types of items, type 1 and type 2, and two agents, agent A and agent B. An agent is indexed with the subscripted letter i and a type is indexed with the superscripted letter s. Each agent starts with some initial bundle of items. Let esi denote item of type s owned by agent i. The set of items of type s is denoted by X s = fesA ; esB g. 3

We imagine that a new distribution of items of type s, or sector s allocation, is proposed by sector s assignment authority. A sector s allocation xs = (xsA ; xsB ) is a list of items of type s that is consistent with the initial endowments of sector s. An allocation is a list of bundles of items x = ((x1A ; x2A ) ; (x1B ; x2B )) that is consistent with the initial endowments. Our interpretation is that sector 1 authority proposes school seat allocations to agents, where each agent already owns one school seat, and that sector 2 authority proposes house allocations to agents, where each agent already owns one house. Basically, this is a model of barter exchange in which only items of the same type can be traded. Suppose that preferences of agent i are represented by an ordering Ri de…ned on the set of bundles X 1 X 2 . As noted in the previous section, in the ‘usual’case, preferences for items of type 1 will depend upon the consumption of items of type 2. The case in which such dependence does not occur is that in which preferences are separable. To be precise, suppose sector 2 authority has assigned the item x2i to agent i. We de…ne the conditional ordering, Ri1 (x2i ), on X 1 of agent i induced by the ordering Ri by: e1A Ri1 x2i e1B ()

e1A ; x2i Ri e1B ; x2i .

Likewise, one can de…ne the conditional ordering, Ri2 (x1i ), on X 2 given that sector 1 authority has assigned item x1i to agent i. Agent i’s ordering Ri is separable in X 1 if the conditional orderings on X 1 are identical, that is, Ri1 e2B = Ri1 e2A . We say that the ordering Ri is separable if that ordering is separable in X 1 as well as in X 2 . We subsequently indicate the conditional ordering on X s induced by an ordering Ri separable in X s by Ris . Let (RA ; RB ) be a pro…le of agents’ orderings that represent agents’ preferences. We say that an allocation x is Pareto dominated for (RA ; RB ) if some other allocation y would make at least one agent better o¤ without hurting the other agent, that is, (yi1 ; yi2 ) Ri (x1i ; x2i ) for all i, with at least one of the preferences being strict. An allocation is Pareto e¢ cient for (RA ; RB ) if it is not Pareto dominated by any other allocation. 4

An allocation x is individually rational for (RA ; RB ) if it leaves each agent i as well o¤ as her endowment, that is, (x1i ; x2i ) Ri (e1i ; e2i ) for all i. An allocation is a core allocation for (RA ; RB ) if it is Pareto e¢ cient and individually rational. Let us suppose that the objective of sector authorities is to propose sector s allocations that are sector-wise core allocations. To be precise, suppose that sector 2 authority has proposed (x1B ; x2B ) as a sector 2 allocation. We say that sector 1 allocation x1 = (x1A ; x1B ) is a sector 1 core allocation for the pro…le of conditional orderings 1 1 1 1 (RA (x2A ) ; RB (x2B )) if x1 is individually rational and Pareto e¢ cient for (RA (x2A ) ; RB (x2B )).

Then, an allocation x = ((x1A ; x2A ) ; (x1B ; x2B )) is a sector-wise core allocation if x1 = 1 1 (x2B )) (x2A ) ; RB (x1A ; x1B ) is a sector 1 core allocation for the pro…le of conditional orderings (RA

and x2 = (x2A ; x2B ) is a sector 2 core allocation for the pro…le of conditional orderings 2 2 (RA (x1A ) ; RB (x1B )). The idea behind this de…nition is that each agent is supposed to

submit her preferences for items of type s to a sector s authority as if her preferences were separable. Examples 1 and 2 below show that when sector authorities make non-coordinated decisions and each sector s authority considers it optimal to assign a sector s core allocation to agents, then sector-wise core allocations are not necessarily Pareto e¢ cient according to agents’preferences. This is regardless of whether agents’preferences are separable. Example 1. Suppose that agents A and B’s separable strict orderings on X 1

X 2 are

as follows: for A :

(e1B ; e2A )RA (e1B ; e2B )RA (e1A ; e2A )RA (e1A ; e2B )

for B :

(e1B ; e2A )RB (e1A ; e2A )RB (e1B ; e2B )RB (e1A ; e2B ),

where we say agent i “strictly prefers x to y according to Ri ” if “xRi y”. One can check that the conditional strict orderings of agent i are: for items of type 1 : e1B Ri1 e1A for items of type 2 : e2A Ri2 e2B .

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The unique core allocation for the pro…le (RA ; RB ) is the allocation x given by x1A ; x2A = (e1B ; e2B ) and x1B ; x2B = (e1A ; e2A ). Clearly, no single agent will want to block x because every agent strictly prefers (x1i ; x2i ) to her initial endowment (e1i ; e2i ) according to Ri . Moreover, the allocation x is Pareto e¢ cient. Sector 1 allocation y 1 given by yA1 ; yB1 = e1A ; e1B 1 1 is the unique sector 1 core allocation for the pro…le of conditional orderings (RA ; RB ).

This is so because the move from e1B to e1A is a bad deal for agent B. Moreover, neither agent would block y 1 because each agent keeps her initial item of type 1. Likewise, one can check that sector 2 allocation y 2 given by: yA2 ; yB2 = e2A ; e2B 2 2 is the unique sector 2 core allocation for (RA ; RB ). Thus, a sector-wise core allocation

is the allocation y given by yA1 ; yA2 ; yB1 ; yB2

.

However, y is not a Pareto e¢ cient allocation for (RA ; RB ). This is so because the move from (yi1 ; yi2 ) to (x1i ; x2i ) is a good deal for both agents. In short, if sector authorities proposed the sector-wise core allocation y and agents could freely barter exchange items, we would not expect barter exchange to lead to the allocation y. Example 2. Suppose that agents A and B’s non-separable strict orderings on X 1 are as follows: for A :

(e1B ; e2B )RA (e1B ; e2A )RA (e1A ; e2A )RA (e1A ; e2B )

for B :

(e1A ; e2A )RB (e1B ; e2A )RB (e1B ; e2B )RB (e1A ; e2B ). 6

X2

One can check that the conditional strict orderings of agent A are 1 1 eA for items of type 1 : e1B RA 2 e1A e2B for items of type 2, given e1A : e2A RA 2 for items of type 2, given e1B : e2B RA e1B e2A ,

and the conditional strict orderings of agent B are 1 for items of type 1, given e2A : e1A RB e2A e1B 1 for items of type 1, given e2B : e1B RB e2B e1A 2 2 for items of type 2 : e2A RB eB .

As in the preceding example, the unique core allocation for the pro…le (RA ; RB ) is the allocation x given by x1A ; x2A = (e1B ; e2B ) and x1B ; x2B = (e1A ; e2A ). Indeed, each agent i likes the bundle (x1i ; x2i ) as much as she likes her endowment (e1i ; e2i ). Moreover, each agent receives her top ranked bundle. Since x is Pareto e¢ cient and since x would not be blocked by either agent, x is a core allocation for (RA ; RB ). Sector 1 no-trade allocation y 1 = yA1 ; yB1 = e1A ; e1B 1 1 is a sector 1 core allocation for the pro…le of conditional orderings (RA ; RB (e2B ))

provided that there is no trade in sector 2. Indeed, given that agent i consumes her own endowment e1i , neither agent would block y 1 provided that there is no trade in sector 2. Moreover, provided that there is no trade in sector 2, the move from 1 yB1 = e1B to e1A is a bad trade for B according to her conditional ordering RB (e2B ). 1 1 Then, y 1 is Pareto e¢ cient for (RA ; RB (e2B )). Reasoning such as the one just used

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shows that sector 2 no-trade allocation y 2 given by yA2 ; yB2 = e2A ; e2B 2 2 is a sector 2 core allocation for the pro…le of conditional orderings (RA (e1A ) ; RB )

provided that there is no trade in sector 1. Thus, a sector-wise core allocation is the no-trade allocation y given by yA1 ; yA2 ; yB1 ; yB2

.

However, y is not a Pareto e¢ cient allocation for (RA ; RB ). This is so because x will make both agents better o¤ than they are at the no-trade allocation y. In other words, if sector authorities proposed the allocation y and agents could barter exchange items, we would not expect barter exchange to lead the economy to the allocation y. We conclude by noting that the x allocation is also a sector-wise core allocation.1 A common feature of the examples above is that the only Pareto e¢ cient allocation for the pro…le of agents’orderings was the core allocation x. Moreover, one can easily check that the x allocation is still the only core allocation even though we treat each agent i’s endowment (e1i ; e2i ) as a single commodity. One then may wonder whether the Pareto ine¢ ciency of sector-wise core allocations can be circumvented by considering core allocations computed as if each agent’s endowment were a single commodity. The answer is no. We prove this fact by means of the following example where agent i still views her endowment e1i of type 1 and her endowment e2i of type 2 as separated items. Example 3. Suppose that agents A and B’s non-separable strict orderings on X 1

X2

are as follows: for A :

(e1B ; e2A )RA (e1A ; e2B )RA (e1B ; e2B )RA (e1A ; e2A )

for B :

(e1A ; e2B )RB (e1B ; e2A )RB (e1A ; e2A )RB (e1B ; e2B ).

1. The sector 1 allocation x1 = x1A ; x1B recommended by x is a sector 1 core allocation for 2 2 e2A and the sector 2 allocation x2 = x2A ; x2B is a sector 2 core allocation for RA e1B ; RB .

1 1 RA ; RB

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One can check that the conditional strict orderings of agent i are for items of type 1, given e2A : e1B Ri1 e2A e1A for items of type 1, given e2B : e1A Ri1 e2B e1B for items of type 2, given e1A : e2B Ri2 e1A e2A for items of type 2, given e1B : e2A Ri2 e1B e2B . The unique Pareto e¢ cient allocation for the pro…le (RA ; RB ) is the allocation z given by zA1 ; zA2 = (e1B ; e2A ) and zB1 ; zB2 = (e1A ; e2B ). The reason is that each agent receives her top ranked bundle according to Ri . Consider the allocation x given, as above, by x1A ; x2A = (e1B ; e2B ) and x1B ; x2B = (e1A ; e2A ). Let us treat each agent’s endowment as a single commodity. Clearly, no agent will want to block x because each agent i likes (x1i ; x2i ) as much as she likes (e1i ; e2i ) according to Ri . Moreover, x is Pareto e¢ cient for (RA ; RB ). This is so because the move from x to the no-trade allocation is a bad deal for each agent i according to Ri . Therefore, since x is individually rational and Pareto e¢ cient for (RA ; RB ) provided that each agent’s endowment is treated as a single commodity, x is in the “core”of this economy. However, both agents would be better o¤ with z than they would be under the “core” allocation x, since (zA1 ; zA2 ) RA (x1A ; x2A ) and (zB1 ; zB2 ) RB (x1B ; x2B ). In short, if authorities proposed the allocation x and agents could act on their own, they would exchange items so as to arrive at the allocation z. Consider the allocation y given by yA1 ; yA2 = (e1A ; e2B ) and yB1 ; yB2 = (e1B ; e2A ). Reasoning like that used in the preceding example shows that sector 1 allocation 1 1 y 1 = (yA1 ; yB1 ) is a sector 1 core allocation for (RA (yA2 ) ; RB (yB2 )) and that sector 2

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2 2 allocation y 2 = (yA2 ; yB2 ) is a sector 2 core allocation for (RA (yA1 ) ; RB (yB1 )). There-

fore, the allocation y is a sector-wise core allocation for this economy. However, y is not a Pareto e¢ cient allocation for (RA ; RB ). This is so because z will make both agents better o¤ than they are at the allocation y. In short, if agents could freely barter exchange items, they would rearrange them so as to arrive at the allocation z. We conclude by noting that the z allocation is a sector-wise core allocation.

2.2

Auction

In auctions and, more generally, in social decision problems with income transfers, a social decision is a pair (d; t), where d denotes a pure social decision and t denotes a vector of income transfers across agents, which may either add up to zero or to a nonpositive number depending on the situation. The task of the central designer is to elicit agents’true preferences for “pure” social decisions, such as preferences for scales of a public project or for an exclusive licence and to specify income transfers across agents as a function of the elicited preferences. With few exceptions, the analysis of a social decision problem is typically performed by isolating that problem from others. Needless to say, such simpli…cation can be purchased only at the cost of realism. However, what is left unclear is the extent to which that simpli…cation limits the practical relevance of the analysis. We clarify that such simpli…cation restricts the relevance of the analysis to problems where the values of pure social decisions are small relative to agents’total wealths. In other words, it is limited to pure social decisions for which income e¤ects are minor. To this end, consider two non-identical social decision problems with income transfers. Let us refer to each of them as the social decision problem of sector s = 1; 2. Given that the noun “income” echoes the existence of some kind of mechanism that operates in the rest of the economy but sector authorities do not know what type of mechanism(s) is at work, we subsequently assume that income transfers are made by means of some physical good, which we name commodity money. Let I denote the set of agents.

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Let Ds denote the set of pure social decisions in sector s = 1; 2. Let

(1)

T =

(

t 2 [ t; 1)n :

X i2I

ti

)

0

denote the set of closed transfers, where the real number t > 0 denotes some predetermined upper-bound for payments. Let ei denote the initial endowment of commodity money of agent i 2 I, which is assumed to be ei

2t. Therefore, let X s = Ds T denote

the set of outcomes of sector s = 1; 2 where its elements are denoted by xs = (ds ; ts ). Suppose that agent i’s ordering Ri of outcomes in X 1 a utility function ui : X 1

X 2 can be represented by

X 2 ! R+ of the form ui x1 ; x2 = Ui (d1 ; d2 ; t1i + t2i + ei ),

(2) where Ui : D1

D2

R+ ! R+ is strictly increasing in its third argument, that is, for

all d1 2 D1 , all d2 2 D2 and all a; b

0,

a > b =) Ui d1 ; d2 ; a > Ui d1 ; d2 ; b . The source of the limited practical relevance of that analysis can be identi…ed in the standard assumption that agent i’s ordering Ri represents separable preferences of agent i 2 I. The reason is that if agent i’s preferences are separable and represented by an ordering Ri , which has a utility representation of the form given in (2), and if agent i’s willingness to paynaccept is well de…ned,2 then her preferences have a quasi-linear representation in the commodity money. One way to show it consists in proving that each conditional ordering Ris induced by Ri has a quasi-linear representation in the commodity money. We show it below by focussing on the conditional ordering Ri1 of sector 1, given that the arguments for the other conditional ordering Ri2 are entirely symmetric. One can easily see that the fact that Ui is strictly increasing in its third argument as2. In the sense that for any two decisions ds and d^s of sector s there exists a decision (dsC ; tsC ) of the other sector sC and two income transfers, say ts and t^s , of sectors s such that agent i …nds (ds ; dsC ; tsi + tsi C + ei ) and d^s ; dsC ; t^si + tsi C + ei equally good according to Ui .

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sures that more commodity money is better than less according to agent i’s conditional ordering Ri1 . Furthermore, the assumption that agent i’s willingness to paynaccept is well de…ned assures that no matter how much better the pure social decision d^1 is than d1 , according to her conditional ordering Ri1 , some amount of commodity money compensates her for getting d1 instead of d^1 . Therefore, to see that the conditional ordering Ri1 induced by Ri has a quasi-linear utility representation in the commodity money, we are left to show that Ri1 exhibits no income e¤ects. In other words, the conditional ordering Ri1 needs to satisfy the property that the trade-o¤s between pure social decisions and commodity money do not change with equal increases in the commodity money. Formally: For all d1 and d^1 in D1 and all income transfers t1 , t~1 , t^1 and t1 in T such that q = t~1i

(3)

t1i = t1i

t^1i ,

it holds that (d1 ; t1i + q)Ri1 (d^1 ; t^1i + q) () (d1 ; t1i )Ri1 (d^1 ; t^1i ). Then, consider any two pure social decisions of sector 1, say d1 and d^1 , and any four income transfers in T , say t1 , t~1 , t^1 and t1 , such that (3) holds. The separability requirement implies that for any two outcomes (d2 ; t2 ) and (d2 ; t^2 ) of sector 2 such that agent i’s income transfer is q at t2 and zero at t^2 , it holds that (d1 ; t1i )Ri1 (d^1 ; t^1i )

() Ui (d^1 ; d2 ; t^1i + t2i + ei )

()

Ui (d1 ; d2 ; t1i + t21 + ei )

()

Ui (d1 ; d2 ; t1i + q + t^2i + ei )

()

(d1 ; t1i + q)Ri1 (d^1 ; t^1i + q).

Ui (d^1 ; d2 ; t^1i + q + t^2i + ei )

Thus, the conditional ordering Ri1 satis…es the property of no income e¤ect, and so we conclude that agent i’s separable ordering Ri has a quasi-linear utility representation in the commodity money. Therefore, the exercise to isolate a particular sector from the rest of the economy implicitly relies on the assumption that income e¤ects on that sector are minor, meaning 12

that where income e¤ects are large those isolations fail and economic design recommendations based on them are misleading (see also Vives 1987; Hayashi, 2013).

3

The basic framework We consider a …nite set of agents indexed by i 2 I = f1;

of elementary sectors indexed by s 2 S = f1;

; ng and a …nite set

; `g. The set of outcomes of sector

s available to agents is represented by X s , with xs as a typical element. X s is called sector s outcome space. We assume that the set of outcomes available to agents is the product space X=

Y

X s.

s2S

To economize on notation, for any sector s, write sC for the complement of s in S. Thus, (xs ; xsC ) is an outcome of X, where it is understood that xsC is an element of the Y X s . Unless stated otherwise, the same notational convention product space X sC = s2sC

will be followed for any pro…le of items.

In the usual fashion, agent i’s preferences over X are given by a complete and transitive binary relation, subsequently an ordering, Ri on X. The corresponding strict and indi¤erence relations are denoted by P (Ri ) and I (Ri ), respectively. The condition of separability of preferences that must hold if the isolation of sector s decision problem from others is legitimate can be formulated as follows. For each xsC , we de…ne the s conditional ordering, Ris (xsC ), on X s by for all y s ; z s 2 X s : y s Ris (xsC ) z s () (y s ; xsC ) Ri (z s ; xsC ) . We say that the ordering Ri is separable in X s (over X sC ) if and only if for all xsC ; y sC 2 X sC : Ris (xsC ) = Ris (y sC ) . In other words, Ri is separable in X s if the agent i’s preferences over outcomes of X s are independent of outcomes chosen from X sC . Again, to save writing, for any separable ordering Ri in X s , write Ris for the s conditional ordering. The ordering Ri is separable 13

provided that for each sector s the ordering Ri is separable in X s . We assume that the central designer does not know agent i’s true preferences. Thus, write R (X) for the set of orderings on X, Rsep (X) for the set of separable orderings

on X, Ri for the domain of (allowable) orderings on X for agent i and Rsep for the i domain of (allowable) separable orderings on X for agent i. We assume, however, that there is complete information among the agents in I.

This implies that the central designer knows Ri and Rsep i . Moreover, given that any separable ordering Ri induces ` agent i’s conditional orderings, one for each s, the central designer also knows Dis , which is the set of the s conditional orderings on X s

induced by agent i’s domain Rsep i . In summary, the assumption of complete information implies that the central designer knows the domain of preferences for the set I, which is the product set of Ri ’s, that is, RI =

Y i2I

Ri ,

with R as a typical pro…le, and knows the domains Rsep and DIs , which are respectively I s s s the product set of Rsep i ’s and of Di ’s. A typical element of Di is denoted by R .

The goal of the central designer is to implement a SCR ' : RI

X where ' (R) is

non-empty for any R 2 RI . We shall refer to x 2 ' (R) as a '-optimal outcome at R. The central designer delegates the choice to agents according to a partial equilibrium mechanism which forces agents to behave as if they had separable orderings. Formally, for any sector s, the central designer delegates the choice to agents according to a partial equilibrium mechanism

s

= (Mis )i2N ; hs , where Mis is the strategy space of agent i

in sector s and hs : M s ! X s , the outcome function, assigns to every strategy pro…le ms 2 M s = a unique outcome in X s .

Y

Mis

i2I

A product set of partial equilibrium mechanisms

(Mi )i2N ; h is a mechanism, where Mi is the strategy space of agent i de…ned by Mi =

Y s2S

14

Mis

=

and h : M ! X, the outcome function, assigns to every strategy pro…le m2M =

Y

Mi

i2I

a unique outcome in X such that h (m) = (hs (ms ))s2S . A product set of partial equilibrium mechanisms

and a pro…le R 2 RI induce

a strategic game ( ; R). A strategy pro…le m 2 M is a Nash equilibrium (in pure strategies) of ( ; R) if for all i 2 I, it holds that for all mi 2 Mi : h (m) Ri h (mi ; m i ) , where, as usual, m

i

is the strategy pro…le of all agents except i such that (mi ; m i ) =

m. Write N E( ; R) for the set of Nash equilibrium pro…les of ( ; R). Likewise, any mechanism

s

together with the pro…le Rs 2 DIs de…nes a strategic game in s. A

strategy pro…le ms 2 M s is a Nash equilibrium (in pure strategies) of ( s ; Rs ) if for all i 2 I, it holds that

for all msi 2 Mis : h (ms ) Ris h msi ; ms i .

The following de…nition is then our formulation of the central designer’s (Nash)implementation problem. A SCR ' is (Nash-)implementable in partial equilibrium if there exists a product set of partial equilibrium mechanisms

such that for all R 2 RI ,

it holds that h (N E( ; R)) = ' (R) . The lemma below shows that the separability property implies that the set of Nash equilibrium strategy pro…les has a product structure. Lemma 1. Let

be a product set of partial equilibrium mechanisms. For all R 2 Rsep I , N E( ; R) =

Y s2S

15

N E( s ; Rs ),

where for all i 2 I and all s 2 S, Ris is the s conditional ordering induced by Ri . Proof. Let

be a product set of partial equilibrium mechanisms. Take any R 2 Rsep I .

For any i 2 I and any s 2 S, write Ris for the s conditional ordering induced by Ri . Consider any m 2 N E ( ; R). Thus, it follows that h (m) Ri h (mi ; m i ) for all mi 2 Mi . Fix any s 2 S and any i 2 I. Since Ri 2 Rsep i , it holds that for all msi 2 Mis : hs (ms ) Ris hs msi ; ms i . Since it holds for any i 2 I, we have that ms 2 N E ( s ; Rs ). Finally, given that the Q choice of s was arbitrary, we have that m 2 s2S N E( s ; Rs ). Q Consider any m 2 s2S N E( s ; Rs ). Thus, for all s 2 S and all i 2 I : hs (ms ) Ris hs msi ; ms i for all msi 2 Mis . Assume, to the contrary, that m 2 = N E ( ; R). Then, for at least one io 2 I and one mio 2 Mi0 , it holds that h (mio ; m

io ) P

(Rio ) h (m).

Since for sector 1, it holds that h1 m1 Ri1o h1 m1io ; m1 io , it follows from Rio 2 Rsep io that h (m) Rio h1 m1io ; m1 io ; (hs (ms ))s2Snf1g . Reasoning like that used in the preceding lines shows that for any s 2 Sn f1; `g, it holds that hp mpio ; mp io

p=1; ;s 1

; (hq (mq ))q=s;

;`

Rio

16

hp mpio ; mp io

p=1; ;s

; (hq (mq ))q=s+1;

;`

.

Likewise, for sector `, it holds that hp mpio ; mp io

p=1; ;` 1

; h` m`

Rio h (mio ; m

io ) .

Since Ri is transitive, it follows that h (m) Rio h (mio ; m in violation of h (mio ; m

4

io ) P

io ) ,

(Rio ) h (m). Thus, m 2 N E ( ; h).

Necessary conditions In this section, we discuss conditions that are necessary for the implementation in

partial equilibrium. We end the section by showing that no acceptable Pareto optimal SCR de…ned on the domain of separable orderings can be implemented in partial equilibrium. A condition that is central to the implementation of SCRs in Nash equilibrium is monotonicity (in the Maskin sense). This condition says that if an outcome x is 'optimal at the pro…le R and this x does not strictly fall in preference for anyone when the pro…le is changed to R0 , then x must remain a '-optimal outcome at R0 . Let us formalize that condition as follows. For any ordering Ri and outcome x, the weak lower contour set of Ri at x is de…ned by L (x; Ri ) = fx0 2 XjxRi x0 g. Therefore: De…nition 1. The SCR ' : RI

X is (Maskin) monotonic provided that for all

x 2 X and all R; R0 2 RI , if x 2 ' (R) and L(x; Ri )

L(x; Ri0 ) for all i 2 I, then

x 2 ' (R0 ). Theorem 1. The SCR ' : RI

X is monotonic if ' is implementable in partial

equilibrium. Proof. Suppose the SCR ' : RI

X is implementable in partial equilibrium.

Then, there exists a product set of partial equilibrium mechanisms

such that for all

R 2 RI , it holds ' (R) = h (N E( ; R)). For some pro…le R 2 RI consider x 2 ' (R). 17

Then, there exists m 2 N E ( ; R) such that h (m) = x and such that for all i 2 I, it holds that (4)

L (x; Ri ) ,

h (Mi ; m i )

where h (Mi ; m i ) is the set of outcomes that agent i can generate by varying her own strategy choice, keeping her opponents’actions …xed at m i . Consider the pro…le R0 2 RI such that for all i 2 I, it holds that L(h (m) ; Ri ) L(h (m) ; Ri0 ). It follows from (4) that for all i 2 I, it holds that h (Mi ; m i )

L(h (m) ; Ri0 ).

Therefore, m 2 N E ( ; R0 ). From the de…nition of implementability in partial equilibrium, we conclude that x 2 ' (R0 ). Thus, ' is monotonic. The relevance of implementation theory comes from the fact that it provides a theoretical construct within which to study the way in which a society shall trade o¤ agent preferences to achieve its goals. Unless the SCR is dictatorial, this involves a compromise. In light of Lemma 1, the second condition identi…es a property of how a SCR must handle the compromise across sectors where agents’preferences are separable. De…nition 2. The SCR ' : RI

X is decomposable provided that for all s 2 S, there

exists a (non-empty) correspondence 's : DIs X s with the following property: for Q s s s all R 2 Rsep I , ' (R) = s2S ' (R ), where for all i 2 I and all s 2 S, Ri is the s

conditional ordering induced by Ri .

This says that if a SCR is decomposable, then the sth dimension of the SCR depends only on the pro…les of conditional orderings of the sth sector. Di¤erently put, the SCR can be decomposed into the product of one-dimensional SCRs. Furthermore, it implies that the social objectives that a society or its representatives want to achieve can be decomposed in ‘small’social objectives, one for each sector. Therefore, to analyze the way in which the society should trade o¤ agent preferences for the sth sector to achieve its goal we can ignore consumption trade-o¤s across sectors and focus only on the pro…les of conditional orderings of sth sector. 18

Theorem 2. The SCR ' : RI

X is decomposable if ' is implementable in partial

equilibrium. Proof. Suppose ' : RI

X is implementable in partial equilibrium. Then, there

exists a product set of partial equilibrium mechanisms

such that for all R 2 RI , it

holds ' (R) = h (N E( ; R)). Furthermore, Lemma 1 implies h (N E( ; R)) =

Y s2S

and so, by the fact that (5)

hs (N E( s ; Rs )) for all R 2 Rsep I ,

implements ' in partial equilibrium, it holds that

'(R) =

Y s2S

hs (N E( s ; Rs )) for all R 2 Rsep I .

For all s 2 S, de…ne 's : DIs

X s by 's (Rs ) = hs (N E( s ; Rs )) for any Rs 2 DIs . Q We conclude from the de…nition of 's and (5) that '(R) = s2S 's (Rs ). Thus, ' is decomposable.

In the literature of strategy-proof social choice functions it has been shown that decomposability is implied by strategy-proofness where agents have separable preferences (as per Barberà et al., 1991; Le Breton and Sen, 1999). A natural question, then, is whether decomposability is implied by Nash implementation.3 The answer is no. We prove this by means of the following example. Example 4. There are two types of agents, say type A and type B, two sectors, say sector 1 and sector 2, and two distinct items per sector, say xs and y s . Consider a pro…le R where the separable strict orderings of types are for type A : (x1 ; x2 )RA (x1 ; y 2 )RA (y 1 ; x2 )RA (y 1 ; y 2 ) for type B : (y 1 ; y 2 )RB (x1 ; y 2 )RB (y 1 ; x2 )RB (x1 ; x2 ).

3. A SCR ' : RI X is Nash-implementable if there exists a mechanism all R 2 RI , ' (R) = h (N E ( ; R)).

19

(M; h) such that for

Furthermore, consider a pro…le R where the separable strict orderings of types are for type A : (x1 ; x2 )RA (y 1 ; x2 )RA (x1 ; y 2 )RA (y 1 ; y 2 ) for type B : (y 1 ; y 2 )RB (y 1 ; x2 )RB (x1 ; y 2 )RB (x1 ; x2 ). One can check that R and R induce the following conditional strict orderings: 1 1 for type A, sector 1 : x1 RA y 2 2 for type A, sector 2 : x2 RA y 1 1 for type B, sector 1 : y 1 RB x 2 2 for type B, sector 2 : y 2 RB x.

Suppose that there are three agents, where agents 1 and 2 are of type A and agent 3 is of type B. Furthermore, suppose that the pro…les R and R are the only allowable pro…les of separable orderings. Consider the SCR ' : R; R

X such that

'(R) = (x1 ; y 2 ); x1 ; x2

(6)

6= '(R) =

y 1 ; x2 ; x1 ; x2

.

This SCR is Maskin monotonic and satis…es the condition of no veto-power.4 Therefore, the SCR ' is Nash-implementable, according to Maskin’s Theorem (Maskin, 1999). Suppose that the SCR ' is decomposable. By construction, one has that the set of conditional orderings of sector 1 and sector 2 induced by R and R are 1 1 for type A : DA = RA

for type B :

1 1 DB = RB

2 2 and DA = RA 2 2 and DB = RB .

4. No veto-power says that if an outcome x is at the top of the preferences of all but possibly one of the agents, then x should be selected by the SCR '.

20

Decomposability implies that 1 1 1 ; RB ) ; RA ' (R) = '1 (RA

2 2 2 '2 (RA ; RA ; RB )=' R ,

in violation of (6). Thus, the SCR ' is not decomposable.

An equivalent statement of (Maskin) monotonicity stated above follows the reasoning that if x is '-optimal at R but not '-optimal at R0 , then the outcome x must have fallen strictly in someone’s ordering at the pro…le R0 in order to break the Nash equilibrium via some deviation. Therefore, there must exist some preference reversal if an equilibrium strategy pro…le at R is to be broken at R0 . When the new pro…le R0 satis…es the requirement of separability and the SCR ' is implementable in partial equilibrium, then the sth sector of the SCR depends only on the pro…les of conditional orderings of the sth sector. Therefore, a variant of monotonicity follows the reasoning that if x is '-optimal at R but not '-optimal at R0 and if R0 is a pro…le of separable orderings, then the outcome x must have fallen strictly in someone’s conditional ordering. Simply put, if an equilibrium strategy pro…le at R is to be broken at R0 , then the preference reversal must happen in one of the sectors. To introduce this variant of monotonicity, for any ordering Ri , outcome x and sector s, let the weak lower contour set of Ri and sector s at x be de…ned by Ls (x; Ri ) = f(y s ; xsC ) 2 XjxRi (y s ; xsC )g. Then: De…nition 3. The SCR ' : RI

X is decomposable (Maskin) monotonic provided

that for all x 2 X, all R 2 RI and all R0 2 Rsep I , if x 2 ' (R) and for all i 2 I : Ls (x; Ri )

Ls (x; Ri0 ) for all s 2 S, then x 2 ' (R0 ).

Theorem 3. The SCR ' : RI

X is decomposable monotonic if ' is implementable

in partial equilibrium. Proof. Suppose the SCR ' : RI

X is implementable in partial equilibrium.

Then, there exists a product set of partial equilibrium mechanisms

such that for all

R 2 RI , it holds ' (R) = h (N E( ; R)). For some pro…le R 2 RI consider x 2 ' (R). Then, there exists a Nash equilibrium strategy pro…le m 2 N E ( ; R) such that h (m) = 21

(hs (ms ))s2S = x. Moreover, for all i 2 I, it holds that the set of obtainable outcomes, that is, h (Mi ; m i ) = fh (m0i ; m i ) 2 Xjm0i 2 Mi g, is contained in L (x; Ri ). Consider any sector s and any agent i. Let hs Mis ; ms i ; (hs (ms ))s2sC =

hs msi ; ms i ; (hs (ms ))s2sC jmsi 2 Mis

be the set of outcomes that agent i can generate by varying his own strategy of sector s, keeping his own strategy choices and those of other agents for sector s di¤erent from s …xed at ms and keeping the strategy choices of other agents of sector s …xed at ms i . It follows from h (Mi ; m i )

L (h (m) ; Ri ) that hs Mis ; ms i ; (hs (ms ))s2sC

Ls (h (m) ; Ri ). Since agent i and sector s were arbitrary, it follows that (7)

for all i 2 I and all s 2 S : hs Mis ; ms i ; (hs (ms ))s2sC

Ls (h (m) ; Ri ) .

Consider the pro…le R 2 Rsep such that I for all i 2 I and all s 2 S : Ls (h (m) ; Ri )

Ls h (m) ; Ri .

Then, from (7), it follows that (8)

for all i 2 I and all s 2 S : hs Mis ; ms i ; (hs (ms ))s2sC

Ls h (m) ; Ri .

Given that R is a pro…le of separable orderings, let Ris be the s conditional ordering induced by Ri . Thus, from (8), we have that for all i 2 I and all s 2 S : hs Mis ; ms i

L hs (ms ) ; Ris ,

where L(hs (ms ) ; Ris ) is the weak lower contour set of Ris at hs (ms ). Then, for all s 2 S, ms is a Nash equilibrium strategy pro…le of that m 2 N E

s

; Rs , and so, by Lemma 1, it follows

; R . From the de…nition of implementability in partial equilibrium,

we conclude that x 2 ' R . Thus, ' is decomposable monotonic. Important properties of SCRs are as follows.

22

De…nition 4. The SCR ' : RI

X is Pareto optimal provided that for all R 2 RI

and all x 2 ' (R), there is no x0 2 X such that x0 Ri x for all i 2 I and x0 P (Rj ) x for some j 2 I. De…nition 5. The SCR ' : RI

X is dictatorial provided that there exists an agent

i 2 I such that for all R 2 RI and all x 2 X, x 2 ' (R) () xRi x0 for all x0 2 X. A social choice function (SCF) is a single-valued SCR. A SCF is strategy-proof if each agent does herself no good by misrepresenting her own ordering.5 Furthermore, a SCF is nonimposed if the set of outcomes is included in its range. A classic result due to Gibbard (1973) and Satterthwaite (1975) shows that a nonimposed, strategy-proof SCF de…ned on the domain of all possible linear orderings is dictatorial, provided that the unstructured …nite set of outcomes contains at least three outcomes.6 Using a framework similar to ours, Le Breton and Sen (1999) identify domain richness conditions that are su¢ cient for a nonimposed, strategy-proof SCF to be decomposable into one-dimensional strategy-proof SCFs. Therefore, where the …nite set of outcomes is a product set and each sector set contains at least three outcomes, the decomposability theorem of Le Breton and Sen implies that a nonimposed, strategy-proof SCF de…ned on the domain of all possible separable linear orderings can be decomposed into one-dimensional dictatorial SCFs. Given Le Breton and Sen’s negative result, one is forced to relax some of their assumptions in the hope of …nding more encouraging results. A requirement weaker than strategy-proofness is that of requiring truth-telling when the other agents are also telling the truth, that is, that of Nash equilibrium. We show below that the prospects for implementing in partial equilibrium a Pareto optimal SCR on an unrestricted domain 0 sep 5. The SCF ' : Rsep ! X is strategy-proof if for all i 2 I, all R 2 Rsep (X) I I , and all Ri 2 R sep 0 0 such that (Ri ; R i ) 2 RI , ' (R) Ri ' (Ri ; R i ). 6. A linear order R on X is a complete, transitive and antisymmetric (binary) relation. Barberà and Peleg (1990) show that the result of Gibbard (1973) and Satterthwaite (1975) holds true if one drops the assumption of universal domain of preferences and agents’ preferences are required to be continuous. The result of Gibbard-Satterthwaite is also basically robust to the consideration of SCRs, as per Barberà et al. (2001).

23

of separable orderings are quite bleak as well. The reason is that a decomposable, Pareto optimal SCR de…ned on Rsep I (X) is dictatorial, provided that each sector set contains at least two outcomes. This negative result is similar in spirit to a classic result due to Hurwicz and Schmeidler (1978) and Maskin (1999), which states that if a two-agent Pareto optimal SCR de…ned on the domain of all possible linear orderings is Nash implementable, then it is dictatorial. Furthermore, it is similar to a result due to Barberà et al. (1991), according to which there is no Pareto optimal, strategy-proof and non-dictatorial voting scheme de…ned on the domain of separable linear orderings. Theorem 4. Suppose n ' : Rsep I (X)

2. For any s 2 S, let jX s j

2. The SCR

X is dictatorial if ' is Pareto optimal and decomposable.

Proof. Let n ' : Rsep I (X)

2 and that `

2 and let `

2. For any s 2 S, let jX s j

2. Suppose the SCR

X is not dictatorial and that it is decomposable. We show that ' is

not Pareto optimal. Fix any two distinct agents i0 ; j0 2 I and any two distinct sectors s (i0 ) ; s (j0 ) 2 S. Suppose agent io is a dictator in sector s (io ), whereas agent jo is a dictator in sector s (jo ). Fix any xs(io ) ; y s(io ) 2 X s(io ) , with xs(io ) 6= y s(io ) , and any xs(jo ) ; y s(jo ) 2 X s(jo ) , with xs(jo ) 6= y s(jo ) . Let S = fs (io ) ; s (jo )g, and write SC for the complement of S in S. Consider any pro…le R 2 Rsep I (X) such that the conditional orderings of agents io and jo are, respectively, for sector s (io ): s(io )

xs(io ) I Rio

s(io )

y s(io ) P Rio

s(io )

P Rio

and s(i )

y s(io ) P Rjo o

s(i )

xs(io ) P Rjo o

s(i )

P Rjo o

,

and for sector s (jo ): s(jo )

xs(jo ) P Rio

s(jo )

y s(jo ) P Rio

s(jo )

P Rio

and s(jo )

xs(jo ) I Rjo

s(jo )

y s(jo ) P Rjo 24

s(jo )

P Rjo

.

Note that by the separability requirement it holds that for all z s(jo )C 2 X s(jo )C : xs(jo ) ; z s(jo )C P (Rio ) y s(jo ) ; z s(jo )C and for all z s(io )C 2 X

s(io )C

: y s(io ) ; z s(io )C P (Rjo ) xs(io ) ; z s(io )C .

Furthermore, suppose that R is such that for all i 2 In fio ; jo g, it holds that for all xSC 2 X SC : y s(io ) ; xs(jo ) ; xSC Ri xs(io ) ; y s(jo ) ; xSC . Since ' is decomposable and since, moreover, io is a dictator in sector s (io ) and jo is a dictator in sector s (jo ), we have that 's(io ) Rs(io ) = xs(io ) ; y s(io ) and 's(jo ) Rs(jo ) = xs(jo ) ; y s(jo ) . It follows that the SCR ' is not a Pareto optimal SCR for R given that Q for all i 2 I and all xSC 2 s2SC ' (Rs ), it holds that y s(io ) ; xs(jo ) ; xSC Ri xs(io ) ; y s(jo ) ; xSC , and, moreover, y s(io ) ; xs(jo ) ; xSC P (Rio ) xs(io ) ; y s(jo ) ; xSC and y s(io ) ; xs(jo ) ; xSC P (Rjo ) xs(io ) ; y s(jo ) ; xSC .

5

Su¢ cient conditions In implementation theory, it is Maskin’s Theorem (Maskin, 1999) that shows that

where the central designer faces at least three agents, a SCR is implementable in (purestrategies) Nash equilibrium if it is monotonic and it satis…es the auxiliary condition of no veto-power.7 7. Moore and Repullo (1990), Dutta and Sen (1991), Sjöström (1991) and Lombardi and Yoshihara (2013) re…ned Maskin’s Theorem by providing necessary and su¢ cient conditions for a SCR to be im-

25

In the abstract Arrovian domain, the condition of no veto-power says that if an outcome is at the top of the preferences of all agents but possibly one, then it should be chosen irrespective of the preferences of the remaining agent: that agent cannot veto it. The condition of no veto-power implies two conditions. First, it implies the condition of unanimity, which states that if an outcome is at the top of the preferences of all agents, then that outcome should be selected by the SCR. Thus, as a part of su¢ ciency, we require a variant of unanimity, which states that if all agents agree on which outcome is best for sector s, then this outcome should be chosen by the sth dimension of a decomposable SCR. De…nition 6. A decomposable SCR ' : RI

X satis…es unanimity provided that

s s s for all R 2 Rsep I , all s 2 S and all x 2 X , if X

L (xs ; Ris ) for all i 2 I, then

xs 2 's (Rs ). Second, the condition of no veto-power implies the condition of weak no veto-power, which states that if an outcome x is '-optimal at one pro…le R and if the pro…le change from R to R in a way that under the new pro…le an outcome y that was no better than x at Ri for some agent i is weakly preferred to all outcomes in the weak lower contour set of Ri at x according to the ordering Ri and this y is maximal for all other agents in the set X, then y should be a '-optimal outcome at R. As a part of su¢ ciency, we require the following adaptation of weak no veto-power to our implementation model. De…nition 7. A decomposable SCR ' : RI

X satis…es weak no veto-power provided

s s s s that for all R 2 Rsep I , all s 2 S and all x 2 X , if for some R 2 DI it holds that

xs 2 's Rs , y s 2 L xs ; Ris

L (y s ; Ris ) for some i 2 I and X s

L y s ; Rjs for

all j 2 In fig, then y s 2 's (Rs ). The main result of the section is also established with the aid of two domain conditions, Property A and Property B. Property A requires the following. For any arbitrary collection of admissible conditional orderings, one for each sector, there exists an admissible separable ordering on plementable in (pure strategies) Nash equilibrium. For an introduction to the theory of implementation see Jackson (2001) and Maskin and Sjöström (2002).

26

X, such that the induced conditional orderings over every sector coincides with the ones in the arbitrary collection. Formally: De…nition 8. The domain Ri R (X) satis…es Property A if, for all Ri1 ; ; Ri` 2 Q s s2S Di , there exists a separable ordering Ri 2 Ri such that the s conditional ordering Ris induced by Ri coincides with Ris , that is, Ris = Ris for all s 2 S.

Property B is central to implementation in partial equilibrium. It guarantees that one can always behave as if her preference was separable. Speci…cally, Property B says the following. For any ordering Ri and outcome x, there exists an admissible separable ordering Ri0 on X, such that the preferences change from Ri0 to Ri in a (Maskin) monotonic way around x (that is, whenever xRi0 x0 , one has that xRi x0 ) and, moreover, for any sector s and any outcome of the set X that di¤ers from x only for the the values of sector s, the outcome x must not strictly fall in preference for agent i when her preference changes from Ri to Ri0 . Formally: De…nition 9. The domain Ri

R (X) satis…es Property B if, for all Ri 2 Ri and all

x 2 X, there exists a separable ordering Ri 2 Ri such that (9)

for all s 2 S : Ls (x; Ri )

(10)

and L x; Ri

Ls (x; Ri ),

L (x; Ri ) .

Note that a separable ordering Ri that satis…es (9) and (10) also satis…es the following property: Ls (x; Ri ) = Ls (x; Ri ) for all s 2 S. De…nition 10. The domain RI satis…es Property A and Property B provided that for each i 2 I, the domain Ri satis…es Property A as well as Property B. Note that Property A imposes no restrictions of domains of interest. A discussion of the implications of Property B for the domain RI is provided in section 6. We now present our characterization result.

27

Theorem 5. Let RI satisfy Property A and Property B. If n

3 and ' : RI

X is

a SCR satisfying decomposability, monotonicity, decomposable monotonicity, weak no veto-power and unanimity, then it is implementable in partial equilibrium. Proof. Suppose RI satis…es Property A and Property B and that n the SCR ' : RI

3. Suppose

X is decomposable, monotonic, decomposable monotonic and it

satis…es unanimity and weak no veto-power. We …rst construct a canonical mechanism s

for each s 2 S. Then, we show that the constructed product set of partial equilibrium

mechanisms

= ( s )s2S implements the SCR ' in partial equilibrium.

For all i 2 I, all Ris 2 Dis and all xs 2 X s , the weak lower contour set of Ris at xs is de…ned by L (xs ; Ris ) = fy s 2 X s jxs Ris y s g. For each sector s 2 S and each agent i 2 I, de…ne the strategy space of sector s by Mis = DIs

Xs

Z+ ,

where DIs is the domain of pro…les of conditional orderings of sector s induced by Rsep and Z+ is the set of nonnegative integers. Denote the generic member of Mis as I

msi = (Rs )i ; (xs )i ; z i . Then, as usual, agent i chooses as a strategy a triple consisting i

of a pro…le of the s conditional orderings Rs in DIs , an outcome (xs )i from X s and Q s an integer z i in Z+ . Let M s = Mi and de…ne hs : M s ! X s as follows: For any i2I

ms 2 M s ,

Rule 1: If msi = Rs ; xs ; z i for all i 2 N and xs 2 's Rs , then hs (ms ) = xs . Rule 2: If for some i 2 N , msj = Rs ; xs ; z j for all j 2 N n fig, xs 2 's Rs and (Rs )i ; (xs )i 6= Rs ; xs , then

8 < (xs )i if (xs )i 2 L xs ; Rs i s s h (m ) = : xs otherwise.

Rule 3: Otherwise, hs (m) = (xs )i where i = max fi 2 N jz i = max fz 1 ; We show that

; z n gg.

= ( s )s2S implements the SCR ' in partial equilibrium. Therefore,

in the subsequent discussion we consider an arbitrary R 2 RI . 28

We …rst show that ' (R)

h (N E ( ; R)). Take an arbitrary x 2 ' (R), where

~ 2 Rsep such x = (xs )s2S . Given that RI satis…es Property B, there exists a pro…le R I that for each agent i 2 I, it holds that for all s 2 S : Ls (x; Ri )

(11)

~i Ls x; R

and that ~i L x; R

(12)

L (x; Ri ) .

Recall that we also write (xs ; xsC ) for x 2 X and that Ls (x; Ri0 ) = f(y s ; xsC ) 2 XjxRi0 (y s ; xsC )g for any Ri0 2 Ri . Since x 2 ' (R) and since, moreover, the SCR ' is decomposable monotonic and

~ 2 Rsep , it follows from (11) that x 2 ' R ~ . Furthermore, the decomposability of R I ~ = Q 's (R ~ s ) given that R ~ 2 Rsep , where R ~ is denotes the s ' implies that ' R I s2S ~ i and where R ~s = R ~ 1s ; conditional ordering induced by R

~ ns . ;R

~ s ; xs ; z i for all i 2 I. Then, the strategy pro…le Fix an arbitrary s 2 S. Let msi = R ms falls into Rule 1 and, therefore, hs (ms ) = xs . By the de…nition of the outcome function hs , any agent i who deviates from ms gets an outcome in hs Mis ; ms i ~ is , which is no better for her than xs . Therefore, ms is a Nash equilibrium of L xs ; R s

~s . ;R

~ 2 Rsep , it follows from Lemma 1 that Since the choice of s was arbitrary and R I the strategy pro…le m = (m1 ; equilibrium of

; mn ), where mi = (msi )s2S for all i 2 I, is a Nash

~ . Therefore, it holds that ;R

for all i 2 I : h (Mi ; m i ) where Mi =

Q

s2S

~i , L x; R

Mis and h (Mi ; m i ) = fh (m0i ; m i ) 2 Xjm0i 2 Mi g. By simply invoking

(12) we have that for all i 2 I : h (Mi ; m i )

L (x; Ri ) .

Therefore, m is a Nash equilibrium of ( ; R) such that h (m) = x. We conclude from

29

this that there is a Nash equilibrium of ( ; R) corresponding to each x 2 ' (R). To prove that h (N E ( ; R))

' (R), consider any strategy pro…le m 2 N E ( ; R).

Thus, for all i 2 I and all s 2 S, it holds that (13)

for all m ~ si 2 Mis : h (m) Ri hs m ~ si ; ms i ; (hs (ms ))s2sC .

~ 2 Rsep such that for all Given that RI satis…es Property B, there exists a pro…le R I i 2 I, it holds that (14)

for all s 2 S : Ls (h (m) ; Ri )

~i Ls h (m) ; R

and, moreover, ~i L h (m) ; R

(15)

L (h (m) ; Ri ) .

~ 2 Rsep , it holds that for all i 2 I Then, from (13) and (14) and from the fact that R and all s 2 S

~ s hs m hs (ms ) R ~ si ; ms i for all m ~ si 2 Mis , i

and so for all s 2 S : ms 2 N E

s

~s . ;R

~ . ;R

Hence, Lemma 1 implies that m 2 N E

~ . To To prove that h (m) 2 ' (R) we …rst need to show that h (m) 2 ' R

this end, for the given m 2 N E

~ , de…ne L1 (m) = fms jms falls into Rule 1 g, ;R

L2 (m) = fms jms falls into Rule 2 g and L3 (m) = fms jms falls into Rule 3 g. We proceed according to whether or not L1 (m) is an empty set. Case 1: L1 (m) is empty. Consider any ms 2 L2 (m), so that for some i 2 I, all j 2 In fig, xs 2 's Rs and

i

Rs ; (xs )i

Rs

j

; (xs )j

= Rs ; xs for

6= Rs ; xs . Each agent j 6= i can

deviate from msj and induce any outcome y s 2 X s by choosing z j high enough so as to win the integer game (that is, z j > z p for all p 2 In fjg). The fact that ms is a Nash equilibrium of

s

~s ;R

implies that X s 30

~ s . Consider agent L hs (ms ) ; R j

i. Agent i could deviate from msi and induce any outcome ws in L xs ; Ris n fxs g she wishes by changing msi into msi = Rs ; ws ; z s . To attain the outcome xs agent i can change msi into msi = Rs ; xs ; z s so as to induce Rule 1. Then, we have that hs (ms ) 2 L xs ; Ris follows that L xs ; Ris

hs Mis ; ms i . Since ms is a Nash equilibrium of

s

~ s , it ;R

~ s . Since ' is decomposable and xs 2 's Rs L hs (ms ) ; R i

~ 2 Rsep , weak no veto-power implies that hs (ms ) 2 's R ~s . and since R I

Consider any ms 2 L3 (m). Then, the de…nition of hs and the fact that ms is a Nash equilibrium of

s

~ s imply that X s ;R

~ js for all i 2 I. Again, since ' L hs (ms ) ; R

~ 2 Rsep , unanimity implies that hs (ms ) 2 's R ~s . is decomposable and R I

Given that L1 (m) is empty, we conclude that decomposability combined with the ~ s for all ms 2 L2 (m)[L3 (m) guarantees that h (m) 2 ' R ~ . fact that hs (ms ) 2 ' R Case 2: L1 (m) is not empty. First, suppose that ms 2 L1 (m) for all s 2 S. Thus Rule 1 applies to each ms , so that for each s 2 S,

i

Rs ; (xs )i

= Rs ; xs for all i 2 I and xs 2 's Rs . Given

such that the s that RI satis…es Property A, for each i 2 I there exists Ri 2 Rsep i

conditional ordering induced by Ri coincides with Ris . Therefore, there exists a pro…le R 2 Rsep such that the pro…le of s conditional orderings by the pro…le R coincides with I the pro…le announced by agents, that is, with Rs . Since xs 2 's Rs for all s 2 S and since, moreover, the SCR ' is decomposable, it follows that x 2 ' R , where x = (xs )s2S . Consider any s 2 S. Then, since each agent i 2 I can alter the current choice of sector s to an outcome in L xs ; Ris by a unilateral deviation, and ms is a Nash equilibrium of

s

~ s , we must have L xs ; Rs ;R i

~ s for all i 2 I. L xs ; R i

~ is are the s conditional orderings induced respectively by Ri Given that Ris and R ~ i , we have that L xs ; Ris and R

~ is is equivalent to Ls x; Ri L xs ; R

~i . Ls x; R

Therefore, for all i 2 I, it holds that for all s 2 S : Ls x; Ri

~i . Ls x; R

We conclude that decomposable monotonicity together with x 2 ' R ensures that ~ . x2' R

31

Finally, assume that L1 (m) is not an empty set and that for some s 2 S, ms 2 L2 (m) [ L3 (m). Consider any ms 2 L2 (m) [ L3 (m). Reasoning like that used in Case

~ s . But then, given that ms is a Nash equilibrium of 1 shows that hs (ms ) 2 's R s

~ s , it is also an equilibrium for each agent i 2 I to play ~ s and hs (ms ) 2 's R ;R

~ s ; hs (ms ) ; z i the strategy choice m ~ si = R

in sector s. Therefore, the corresponding

strategy pro…le m ~ s = (m ~ s1 ; ; m ~ sn ) falls into Rule 1 and hs (m ~ s ) = hs (ms ). Q s Let m ^ = (m ^ s) 2 M be a strategy pro…le such that the strategy pro…le m ^s 2 s2S

M s coincides with ms if ms 2 L1 (m) or else, m ^ s coincides with m ~ s . Since m ^ s is a Nash equilibrium of

s

~ s for each s 2 S, Lemma 1 implies that m ;R ^ 2 NE

~ . ;R

Furthermore, m ^ is such that Rule 1 applies to each m ^ s and hs (ms ) = hs (m ^ s ) for all s 2 S. Reasoning like that used in the …rst three paragraphs of Case 2 shows that ~ . h (m) ^ = h (m) 2 ' R

It remains to show that h (m) 2 ' (R). Because the SCR ' is monotonic, (15) and

~ imply that h (m) 2 ' (R). Thus, the product set of partial the fact that h (m) 2 ' R equilibrium mechanisms

6

implements the SCR ' in partial equilibrium.

Implications In this section, we brie‡y discuss the implications of Theorem 5. In sub-section 6.1, we follow Sen (1995) and Thomson (1999). Speci…cally, we con-

sider a sequence ('s )s2S of sector s (Maskin) monotonic SCRs where the domain of 's is the set of pro…les of orderings on X s , and we look for the minimal way in which that sequence has to be enlarged so as to satisfy (Maskin) monotonicity on the domain RI . In sub-section 6.2, we provide some examples of SCRs that are implementable in partial equilibrium. Moreover, given that Property A imposes no restrictions of domains of interest, we identify below a domain condition that is necessary for Property B. That domain condition is also su¢ cient provided that the set of outcomes X is …nite and the domain of (allowable) orderings on X for agent i includes the set of separable orderings on X. Finally, we reconsider the auction environment described in section 2.2 and show that for the class of orderings that can be represented by a utility function of the form given in (2), Property B restricts that domain to separable orderings. 32

6.1

Minimal monotonic extensions

Write Rsi (X s ) for the set of orderings on X s and RsI (X s ) for the product set of Rsi (X s )’s, that is,

RsI (X s ) =

Y i2I

Rsi (X s ) ,

with Rs as a typical pro…le. In this sub-section, we ask the following question: Given a sequence of (Maskin) monotonic SCRs ('s )s2S , where 's : RsI (X s )

X s , what does

their extension to the whole domain RI look like provided that each agent behaves as if she had separable preferences? To answer that question, we …rst provide an extension of the sequence of SCRs ('s )s2S over the whole domain RI and then show that it is the smallest decomposable, monotonic and decomposable monotonic extension. If it is assumed that the sequence of SCRs under consideration satisfy unanimity and the condition of weak no veto-power, then it is shown that our extension is the smallest extension on RI that is implementable in partial equilibrium. This is reminiscent of the idea of minimal monotonic extension due to Sen (1995) and Thomson (1999). We …rst establish some notation and de…nitions. For all Ri 2 Ri and x 2 X, recall that Ris (xsC ), on X s , denote the s conditional ordering, that is, for all y s ; z s 2 X s : y s Ris (xsC ) z s () (y s ; xsC ) Ri (z s ; xsC ) . To save notation, for any R 2 RI , x 2 X and s 2 S, write Rs (xsC ) for the pro…le of the s conditional orderings corresponding to the pro…le R, that is, Rs (xsC ) = (Ris (xsC ))i2I . For any given s 2 S, 's : RsI (X s )

X s is a sector s SCR that associates a non-empty

set 's (Rs ) of X s for every pro…le Rs . A central property which is crucial to Maskin’s Theorem for sector s is stated below. To introduce it, for any ordering Ris 2 Rsi (X s ) and outcome xs 2 X s , we write L (xs ; Ris ) for the weak lower contour set of Ris at xs , which can be de…ned by L (xs ; Ris ) = fy s 2 X s jxs Ris y s g. Then: De…nition 11. The SCR 's : RsI (X s )

X s is sector s (Maskin) monotonic provided

that for all xs 2 X s , all Rs ; Rs 2 RsI (X s ), if xs 2 's (Rs ) and L(xs ; Ris ) 33

L(x; Ris )

for all i 2 I, then xs 2 's Rs . We next de…ne our extension of the sequence of SCRs ('s )s2S over the domain RI . De…nition 12. Given any sequence ('s )s2S , an extension of ('s )s2S is a SCR ' : RI

X such that x 2 ' (R) () xs 2 's (Rs (xsC )) for all s 2 S.

The examples presented in sub-section 6.2, sector-wise core solution and sector-wise stable solution are all indeed extensions of sequences of sector s core solutions and of sector s stable solutions, respectively. Our …rst result is that the extension ' of ('s )s2S is a decomposable, monotonic and decomposable monotonic SCR provided that every 's is a sector s monotonic SCR. Theorem 6. For each s 2 S, let 's : RsI (X s ) extension ' : RI

X s be a sector s monotonic SCR. The

X of ('s )s2S is a decomposable, monotonic and decomposable

monotonic SCR. Proof. For each s 2 S, let 's : RsI (X s ) show that the extension ' : RI

X s be a sector s monotonic SCR. We

X of ('s )s2S is a decomposable, monotonic and

decomposable monotonic SCR. Since it is plain that ' is decomposable, we show that it is monotonic and decomposable monotonic. To show that ' is monotonic, for some pro…le R 2 RI consider x 2 ' (R). Furthermore, consider any pro…le R 2 RI such that for all i 2 I, it holds L(x; Ri )

~ i ). L(x; R

Then, for any i 2 I and any s 2 S, it also holds that for all y s 2 X s : (xs ; xsC )Ri (y s ; xsC ) =) (xs ; xsC )Ri (y s ; xsC ), so that for all i 2 I and all s 2 S : L(xs ; Ris (xsC ))

L(xs ; Ris (xsC )).

Moreover, by de…nition of ' , it follows that xs 2 's (Rs (xsC )) for all s 2 S. Then, for any given s 2 S, we established that xs 2 's (Rs (x s )) and L(xs ; Ris (x s )) 34

L(xs ; Ris (x s )) for all i 2 I. Given that 's is a sector s monotonic SCR, we have that xs 2 's Rs (xsC ) for all s 2 S, and so x 2 '

R . Thus, ' is monotonic.

To show that ' is decomposable monotonic, for some pro…le R 2 RI consider

x 2 ' (R). Furthermore, consider any pro…le R 2 Rsep such that for all i 2 I, it holds I that Ls (x; Ri ) x2'

Ls (x; Ri ) for all s 2 S. Reasoning like that used above shows that

R . Thus, ' is decomposable monotonic.

The next result asserts that the monotonic extension ' of the sequence of sector s monotonic SCRs ('s )s2S is its minimal monotonic extension. Speci…cally, given a sequence of sector s monotonic SCRs ('s )s2S , we show that any monotonic SCR

that

can be decomposed into the product of one-dimensional SCRs contains the monotonic extension ' provided that the sth component of

contains 's and that the domain

RI satis…es Property B. Note that the trivial SCR

that maps each pro…le of orderings

onto X is, of course, monotonic and decomposable and satis…es the requirement that s

's .8

Theorem 7. Let RI satisfy Property B. For each s 2 S, let 's : RsI (X s ) a sector s monotonic SCR. Let

: RI

X be any decomposable SCR such that

for all s 2 S, it holds that

s

('s )s2S . Then, if the SCR

is monotonic, then '

's . Let the SCR ' : RI

X be an extension of

.

Proof. Let RI satisfy Property B. For each s 2 S, let 's : RsI (X s ) sector s monotonic SCR. Let s 2 S, it holds that

s

Suppose that the SCR To show that '

: RI

X s be

X s be a

X be any decomposable SCR such that for all

's . Let the SCR ' : RI

X be an extension of ('s )s2S .

is a monotonic SCR. , for some pro…le R 2 RI consider x 2 ' (R). Then, xs 2

's (Rs (xsC )) for all s 2 S. We show that x 2

(R). Given that RI satis…es Property

B, there exists a pro…le R 2 Rsep such that for each i 2 I, it holds that I (16)

for all s 2 S : Ls (x; Ri )

Ls (x; Ri ),

8. Also, note that if and ~ are two decomposable SCRs such that for each s 2 S, it holds that s ;~ 's , then their intersection \ ~ is a decomposable SCR provided that it is a well de…ned SCR. s

35

and (17)

L (x; Ri ) .

L x; Ri

Consider any s 2 S. It follows from (16) and (17) that L (xs ; Ris (xsC )) = L xs ; Ris (xsC ) for all i 2 I. Since xs 2 's (Rs (xsC )) and 's is a sector s monotonic SCR, we have that xs 2 's Rs (xsC ) . Therefore, xs 2 's Rs (xsC ) for all s 2 S. Since by assumption s

's for all s 2 S and the SCR

(17) and the fact that the SCR

is decomposable, it holds that x 2

is monotonic, it follows that x 2

R . From

(R). Thus, ' is

the minimal monotonic extension of ('s )s2S .

6.2

Examples

The following examples give an idea of the range of applications covered by Theorem 5. Example 5 (Sector-wise core SCR). Consider ` kets with n

2 distinct barter exchange mar-

3 agents, where in each market s 2 S agents are allowed to exchange

items of the same type s. Each agent starts with some initial bundle of indivisible items, one for each type s 2 S. Let esi denote item of type s owned by agent i. Agents can consume exactly one item per type. The total of items of type s 2 S available is

X

esi .

i2I

The set of items of type s is denoted by X s . An allocation xs of items of type s 2 S is a list of items of type s consistent with the total initial available. That is, Ys =

(

xs jxsi 2 X s for all i 2 I and

X i2I

The set of allocations is given by Y =

Y s2S

36

Y s,

xsi =

X i2I

esi

)

for all s 2 S.

with x as a typical element. Consider any pro…le R 2 RI and allocation x 2 Y . The allocation xs is Pareto e¢ cient for the pro…le of the s conditional orderings Rs (xsC ) = (Ris (xsC ))i2I if it is not Pareto dominated by any other allocation y s . The allocation xs is individually rational for Rs (xsC ) if it leaves each agent i as well o¤ as her endowment of type s, that is, xsi Ris (xsC ) esi for all i 2 I. The allocation xs is a core allocation for Rs (xsC ) if it is Pareto e¢ cient and individually rational for Rs (xsC ). The SCR 'sC : RsI (Y s )

Y s is the sector s core SCR provided that for all

Rs 2 RsI (Y s ) and all xs 2 Y s , xs 2 'sC (Rs ) () xs is a core allocation for Rs . The SCR ' : RI

Y is the sector-wise core SCR provided that for all R 2 RI

and all x 2 Y : x 2 'SC (R) () xs 2 'sC (Rs (xsC )) for all s 2 S. It can be checked that the sector-wise core SCR is decomposable by construction and it satis…es sector s monotonicity, unanimity and the condition of weak no veto-power. Furthermore, in light of Theorem 6, 'SC also satis…es decomposable monotonicity and monotonicity. Thus, if RI on Y satis…es Property A and Property B, 'SC is implementable in partial equilibrium.

Example 6 (Sector-wise stable SCR). Consider ` (marriage) markets with n

2 distinct Gale-Shapley (1962)

3 agents. Consider the ordered triplet (M; W;

s

),

where: M is a …nite non-empty set of men, where a generic man is denoted by m. W is a …nite non-empty set of women, where a generic woman is denoted by w. s m

is a linear order for the man m in market s 2 S, i.e., a complete, transitive and

antisymmetric (binary) relation, on W [ fmg. 37

s w

is a linear order for the woman w on M [ fwg in market s 2 S.

s

is a pro…le of linear orders for men and women in market s 2 S. A market s 2 S matching

s

is a function

For all m 2 M ,

s

(m) 2 = W =)

For all w 2 W ,

s

(w) 2 = M =)

s s s

For all m 2 M and all w 2 W ,

s

: M [ W ! M [ W such that:

(m) = m.

(w) = w. s

(m) = w ()

(w) = m.

Let us denote the set of market s matchings on W [ M by Ms . Given a linear order

s m

of the man m, we extend it from W [ fmg to the set of matchings Ms in

the following way: for all

s

;

s

2 Ms :

s

s Rm

s

s

()

(m)

s m

s

(m) or

s

(m) =

s

(m) .

Simply put, a man’s preferences regarding alternative matchings correspond exactly s of a to his preferences regarding his mates at the two matchings. The ordering Rw

woman w is de…ned likewise. Let Rsi (Ms ) denote the set of market s orderings for agent i 2 M [ W and write RsI (Ms ) for the pro…le of market s orderings for women and men, with Rs as a typical element. A market s matching M,

s

is individually rational at Rs if for all i 2 W [

(i) Ris i. A market s matching

s

is blocked by a pair (m; w) under Rs if

s s wP (Rm ) (m) and mP (Rw ) (w). A market s matching

s

is market s stable at

Rs if it is individually rational at Rs and it is not blocked by any pair (m; w) under Rs . The SCR 'sSt : RsI (Ms ) Rs 2 RsI (Ms ) and all s

s

Ms is the market s stable SCR provided that for all

2 Ms ,

2 'sSt (Rs ) ()

s

is a market s stable matching at Rs .

Note that 'sSt is market s (Maskin) monotonic.

38

The SCR ' : RI R 2 RI and all

2

Q

s2S

Q

s2S s

Ms is the sector-wise stable SCR provided that for all

M :

2 'SSt (R) ()

s

2 'sSt (Rs (

sC

)) for all s 2 S.

Reasoning like that used in the preceding example shows that 'SSt is decomposable, monotonic, decomposable monotonic and, moreover, it satis…es unanimity and Q the condition of weak no veto-power. Thus, if RI on Ms satis…es Property A s2S

and Property B, 'SC is implementable in partial equilibrium.

A domain condition which is implied by Property B can be de…ned as follows: De…nition 13. The domain Ri

R (X) satis…es Property B if for all Ri 2 Ri and

all x; y 2 X it holds that xRi (y s ; xsC ) for all s 2 S =) xRi y. Property B is easier to check than Property B, and it rules out “too much complementarity.”The next result shows that Property B is necessary for Property B and that the two properties are equivalent where the sector s 2 S set X s is …nite and the domain Ri includes the set of separable orderings on X. Theorem 8. If Ri

R (X) satis…es Property B, then Ri satis…es Property B . The

converse is true provided that X s is …nite for all s 2 S and that Rsep (X)

Ri .

Proof. Consider any Ri 2 Ri and x; y 2 X such that xRi (y s ; xsC ) for all s 2 S.

Suppose that Ri satis…es Property B. Then, there exists a separable ordering Ri 2 Rsep i

such that (9) and (10) hold. Since, by hypothesis, xRi (y s ; xsC ), it follows from (9) that xRi (y s ; xsC ), and so the s conditional ordering is such that xs Ris y s . Given that Ri is a separable ordering, we have that xRi y 1 ; x1C ,

39

that for all s 2 Sn f1; `g : (y q )q=1;

;s 1

; (xq )q=s;

;`

Ri (y q )q=1;

;s

; (xq )q=s+1;

;`

and that y `C ; x` Ri y. Since Ri is transitive, it follows that xRi y. Given that (10) holds, we have that xRi y. Thus, Ri satis…es Property B . To show the converse, suppose X s is …nite for all s 2 S and that Rsep (X)

Ri .

Moreover, suppose that Ri satis…es Property B . Assume, to the contrary, that Property B is violated. Fix any R 2 Ri and x 2 X. For each s 2 S, …x a representation of the s conditional ordering Ris (xsC ), which is denoted by vis . Then, for any

> 0, let Ri be a separable ordering represented in the

form ui (y) =

X

exp (vis (y s )

vis (xs )).

s2S

For

su¢ ciently large it holds that xRi y =) xRi (y s ; xsC ) for all s 2 S.

This is because if xRi y but (y s ; xsC )P (Ri ) x for some s 2 S, then for the term exp (vis (y s )

su¢ ciently large

vis (xs )) becomes arbitrarily large, which leads to yP Ri x.

Fix any s 2 S. Suppose that xRi (y s ; xsC ) for some y s 2 X s . Then, vis (xs )

vis (y s )

given that xs Ris (xsC ) y s . We need to rule out the case that (y s ; xsC ) P Ri x to conclude that xRi (y s ; xsC ). Thus, suppose that (y s ; xsC ) P Ri x. By de…nition of ui , it must hold that ui (y s ; xsC ) > ui (x) or, equivalently, it must be the case that exp (vis (y s ) which is false given that vis (xs )

vis (xs )) > 1,

vis (y s ) and

> 0.

Suppose that there exists y 2 X such that xRi y but yP (Ri ) x. Since xRi y, then for

su¢ ciently large it holds that xRi (y s ; xsC ) for all s 2 S. Property B implies that 40

xRi y, which is a contradiction. Thus, Ri satis…es Property B . Example 7. In this example we provide an ordering Ri 2 Ri that violates Property B. To this end, let S = f1; 2g. Moreover, suppose that X s = fxs ; y s g, with xs 6= y s , for all s 2 S. Consider the following ordering Ri on X 1

X 2:

y 1 ; y 2 P (Ri ) (x1 ; x2 )P (Ri ) (y 1 ; x2 )I (Ri ) (x1 ; y 2 ). The ordering Ri violates Property B since (x1 ; x2 )P (Ri ) (y 1 ; x2 ), (x1 ; x2 )P (Ri ) (x1 ; y 2 ) but y 1 ; y 2 P (Ri ) (x1 ; x2 ). In light of Theorem 8, Ri violates Property B.

The above example also shows that Property B is indispensable for our su¢ ciency result, since its violation leaves room for pro…table deviations of agent i. The reason is that the second rule of the canonical mechanism of sector s is used to give incentives to whistle-blowers so as to rule out the possibility that a unanimously false announcement could constitute a Nash equilibrium of the mechanism. To be considered credible, the dissenter must have nothing to gain by untruthfully dissenting, that is, the dissenter’s announced outcome must not be strictly better for her according to the untruthful pro…le announced by the others. This incentive only works if the SCR is (Maskin) monotonic. However, if agents’preferences are not separable and they are required to act as if they were separable, agents can never announce the true environment. Then, once agents have made in each sector s a unanimously false announcement of their conditional orderings and unanimously announced the 's -optimal outcome xs at that pro…le, agent i could induce the ‘worst’outcome y s in each sector s by unilaterally deviating to the second rule, and according to the mechanism that deviation is credible. Now, if in each sector, no one of the other agents objects to agent i’s deviations, by unilaterally inducing the third rule, agent i attains the most preferred outcome y according to her true non-separable ordering.

41

Example 8. In this example we provide a preference domain which satis…es Property B. Let S = f1; 2g. Moreover, suppose that X s = fxs ; y s g, with xs 6= y s , for all s 2 S. De…ne Ri as follows: Ri 2 Ri if either Ri 2 Rsep (X) or if Ri 2 = Rsep (X) and for all x1 ; y 1 2 X 1 and x2 ; y 2 2 X 2 it holds that (18)

(x1 ; x2 )I (Ri ) (y 1 ; y 2 )P (Ri ) (y 1 ; x2 )Ri (x1 ; y 2 ).

One can check that if Ri satis…es (18), then it is not a separable ordering given that the sector 1 conditional ordering Ri1 (x2 ) di¤ers from Ri1 (y 2 ). As in sub-section 2.1, items of sector 1 can be viewed as school seats and items of sector 2 as houses. Suppose that houses x2 and y 2 are equally su¢ ciently close to respective schools x1 and y 1 . Therefore, an interpretation of (18) is that agent i strictly prefers the bundles that minimize the distance school-home to other available bundles and she …nds the bundles (x1 ; x2 ) and (y 1 ; y 2 ) equally good. Consider the following separable orderings: given (x1 ; x2 ) : (x1 ; x2 )P Ri (y 1 ; x2 )P Ri (x1 ; y 2 )P Ri (y 1 ; y 2 ) ~ i (x1 ; y 2 )P R ~ i (y 1 ; x2 )P R ~ i (x1 ; x2 ) given (y 1 ; y 2 ) : (y 1 ; y 2 )P R ^ i (y 1 ; y 2 )I R ^ i (x1 ; x2 )P R ^ i (x1 ; y 2 ) given (x1 ; y 2 ) : (y 1 ; x2 )P R given (y 1 ; x2 ) : (x1 ; y 2 )P (Ri0 ) (x1 ; x2 )I (Ri0 ) (y 1 ; y 2 )P (Ri0 ) (y 1 ; x2 ). One can check via Property B or directly using Property B that Ri satis…es Property B.

Below we reconsider the auction environment described in section 2.2 and show that for the class of orderings that can be represented by a utility function of the form given in (2), Property B restricts that domain to non-separable orderings which exibit complementarities between commodities via the commodity money. Example 9. Assume that preferences belonging to Ri are represented in the form given in (2).9 We show that Property B is equivalent to the following property: for all 9. To assure that agent i’s willingness to paynaccept is well de…ned, we also assume that Ui satis…es

42

d1 ; d1 2 D1 , d2 ; d2 2 D2 and t1 ; t2 2 T , if (19)

Ui (d1 ; d2 ; t1i + t2i + ei ) = Ui (d1 ; d2 ; t1i +

t1i + t2i + ei )

= Ui (d1 ; d2 ; t1i + t2i +

t2i + ei ),

then (20)

Ui (d1 ; d2 ; t1i +

t1i + t2i +

t2i + ei ) = Ui (d1 ; d2 ; t1i + t2i + ei ).

This means that there is no complementarity between pure social decisions in the two sectors but those together exhibit income e¤ects, and, therefore, the pure social decisions are not separable from each other because of them. Simply put, it means that there are no ‘direct’complementarities between pure social decisions but that, instead, the commodity money enables ‘indirect’ complementarities between them. To show that the above property is implied by Property B, pick any d1 ; d1 2 D1 , d2 ; d2 2 D2 and t1 ; t2 2 T . Take any

t1i and

t2i such that the equalities in (19)

hold. We need to show (20). Since agent i’s willingness to paynaccept is well de…ned, by assumption, there exists (21)

Ui (d1 ; d2 ; t1i +

t2i such that

t1i + t2i +

t2i + ei ) = Ui (d1 ; d2 ; t1i +

t1i + t2i + ei ),

and so, from (19), it follows that (22)

Ui (d1 ; d2 ; t1i +

t1i + t2i +

t2i + ei ) = Ui (d1 ; d2 ; t1i + t2i +

t2i + ei ).

Then, by applying Property B to the equalities (21) and (22), we have that (23)

Ui (d1 ; d2 ; t1i +

t1i + t2i +

t2i + ei ) = Ui (d1 ; d2 ; t1i +

t1i + t2i +

t2i + ei ),

the following property: For all d1 ; d1 2 D1 , all d2 ; d2 2 D2 , all t1 ; t2 2 T , there exist t1 ; t2 2 T such that Ui d1 ; d2 ; t1i + t2i + ei = Ui d1 ; d2 ; t1i + t2i + ei .

43

which, in turn, implies

t2i . Therefore, combining (19) and (21) with (23),

t2i =

we obtain (20). Thus, Ri satis…es the above property if it satis…es Property B. The converse is true, because the indi¤erence surface passing through (d1 ; d2 ; t1i + t2i + ei ) coincides exactly with the indi¤erence surface of the corresponding separable preference.

As a …nal application covered by Theorem 5, we consider the Marshallian SCR. Example 10 (Sector-wise Marshallian SCR). There are `

2 sectors. The task

of each sector s 2 S authority consists in allocating a single commodity with closed transfers. We assume that transfers are made by means of a commodity money, P which is used commonly across sectors. Let H = t 2 [ t; 1)n : i2I ti 0 de-

note the set of closed net transfers or trades, where t > 0 denotes some predetermined

upper-bound for payments. Then the set of outcomes of sector s 2 S is given by Xs = H

H, with (q s ; ts ) as a typical element. (q s ; ts ) is a pair of net trade of the

s commodity, q s , and closed net transfers of the commodity money, ts . We assume that there are at least n

3 agents and that each agent i 2 I is endowed with

an amount of commodity money, denoted by ei , which is assumed to be

`t. Let

! i 2 [t; 1)` denote the `-vector of initial endowment. Q s The SCR 'SM : RI X is the sector-wise Marshallian SCR provided s2S

that for all R 2 RI , (q; t) 2 'SM (R) if there exists a vector (p1 ; tsi =

; p` ) such that

ps qis for all i 2 I and all s 2 S and, moreover, for all i 2 I, all s 2 S and all

zis such that ps zis 2

(

tsi 2 [ t; 1) : tsi +

X

tsj

j6=i

0 for some ts i 2 [ t; 1)n

1

)

,

it holds that ui

! i + qi ; ei

ps qis

X

s2sC

ps qis

!

ui

!i +

(zis ; qisC ); ei

ps zis

X

s2sC

ps qis

!

.

Note that this SCR cannot be given in the form of minimal extension, in the sense that every sector s SCR is de…ned on the domain RsI (X). The reason is that 44

separable orderings in this environment are represented by the sum of quasi-linear functions, and so every sector s conditional ordering must be quasi-linear. If the sector s SCR was de…ned on RsI (X), the authority of sector s would realize that there are non-quasi-linear conditional orderings and therefore it could infer that some of agents have non-separable orderings, which is in con‡ict with our idea that every sector authority acts as if agents’orderings were separable. This rule is decomposable, because on the domain of separable preferences it induces a sector s Marshalian SCR 'sM : DIs

X s for each s 2 S, where DIs consists

of quasi-linear orderings: for all Rs 2 DIs , (q s ; ts ) 2 'sM (Rs ) if and only if there exists ps 2 R such that for all i 2 I: vis (! si + qis )

ps qis

vis (! si + zis )

n P for all zis such that ps zis 2 tsi 2 [ t; 1) : tsi + j6=i tsj

ps zis 0 for some ts i 2 [ t; 1)n

1

It is Maskin monotonic since it is coming from price-taking optimization under

o .

feasibility constraints. It is decomposable monotonic since it is coming from pricetaking optimization under the feasibility constraint in each sector, given that the allocations in other sectors are …xed. Finally, it satis…es the conditions of weak no veto-power and unanimity. Thus, if RI satis…es Property A and Property B, 'SM is implementable in partial equilibrium.

7

Concluding comments A product set of partial equilibrium mechanisms is a mechanism in which its par-

ticipants are constrained to submit their rankings to sector authorities separately and, moreover, sector authorities cannot communicate with each other, due to misspeci…cation by the central designer that preferences are separable or due to technical/institutional constraints. Therefore, a key property of a single partial equilibrium mechanism is that participants are required to behave as if they had separable preferences. We identify a set of necessary conditions for the implementation of SCRs via a product set of partial equilibrium mechanisms, that is, for the implementation in partial

45

equilibrium. Furthermore, under mild auxiliary conditions, reminiscent of Maskin’s Theorem (1999), we have also shown that they are su¢ cient for the implementation in partial equilibrium. We conclude by discussing future research directions. The …rst thing to come next will be to quantify how much we lose by the type of misspeci…cation considered in this paper. Theoretical, empirical and experimental studies will be helpful there. It is also worth investigating what can be implemented when an incomplete yet not negligible communication is allowed among sector authorities, while the central designer has to make some modeling choice about how sector authorities communicate. Another direction will be to study how we can improve the mechanism in a sector while keeping …xed the mechanisms in other sectors and given such change how we can improve the mechanism in another sector while keeping …xed those in other sectors, and so on. There is no obvious way do it, because under general equilibrium e¤ects it is not obvious whether or not a change regarded as an "improvement" from the point of view of partial equilibrium mechanism design is indeed an improvement. That research direction will answer the question of how we should change the partial equilibrium mechanism in an improving manner.

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