Decentralizability of Multi-Agency Contracting under Bayesian

Implementation�

Yu Cheny

Abstract

We examine the strategic equivalence between decentralized menu design and central-

ized mechanism design in general multi-agency games under Bayesian implementation. We

permit interdependent valuations, contract externalities, correlated types, and heterogen-

eous or di¤erent message sets of di¤erent agents. We identify that Bayesian menu design

is strategically equivalent to bilateral Bayesian mechanism design, which simpli�es collective

Bayesian mechanism design by ignoring relative information evaluation. Based on it, we

take advantage of interim-payo¤-equivalence to provide conditions on the primitives for the

full equivalence between collective mechanism design, bilateral mechanism design, and menu

design. We also discuss the approximation of centralization, the cases allowing primitive con-

straints across the contracts for di¤erent agents, and the inclusion of individual rationality

constraints.

Keywords: Bayesian Nash equilibrium, mechanism design, menu design, delegation principle,

interim-payo¤-equivalence

JEL Classi�cation: C72 D82 D86

�This paper is an extensive revision of Chapter II of my Ph.D. dissertation. I would like to thank FrankPage and Robert Becker for advice, encouragement and comments. For additional helpful remarks, suggestionand comments, thanks are also due to Alessandro Pavan, Johannes Horner, Haomiao Yu, Yongchao Zhang,Rongzhu Ke, Xiang Sun, Geo¤rey Woglom, Jun Zhang, Nian Yang, Jie Zheng, Zhiqi Chen, Mingjun Xiao, andthe participants/audiences at University of Queensland, Tsinghua University, Sun Yat-sen University, ShanghaiUniversity of Finance and Economics, APET annual meeting 2014 and SAET annual conference 2014. HoweverI am solely responsible for any errors.

ySchool of Economics, Nanjing University, 16 Jinyin Street, Nanjing, Jiangsu, China 210093. Email:yourchenyu@gmail.com.

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1 Introduction

The classic theory of mechanism design shows how one central planner can theoretically suc-

ceed in designing revelation mechanisms subject to incentive constraints to allocate resources

in a centralized way to deal with the economic agents with private, decentralized information

(knowledge). In practice, however, Friedrich August von Hayek�s warning about overcon�dence

of centralization in his in�uential paper (1945) is still relevant. Designing and implementing real

world revelation mechanisms seem unreasonably complicated, due to informational complexity,

limited information processing capacities of human beings, etc. Thus, as Hayek points out, "the

central planner will have to �nd some way or other in which the decisions depending on them can

be left to the �man on the spot.�" This implies that some form of decentralization in mechanism

design ought to be favorable.

This paper investigates when contracting is decentralizable in generalized multi-agency con-

tracting games under Bayesian implementation. Multi-agency denotes that a single principal

(assumed to be female) contracts with multiple agents (male). To solve asymmetric information

problem, contracting procedures can be either centralized via mechanism design or decentralized

via menu design. In centralized mechanism design, the principal o¤ers mechanisms, which are

mappings from the agents�reports to contracts for the agents, and has the agents report some

messages. After collecting the reports, she can specify the contracts for all agents. In decent-

ralized menu design, the principal can simplify the information communication for specifying

contracts to the agents. She instead o¤ers (joint) menus, which are simply sets of contract pro-

�les of the agents. The agents are entitled to directly select the contracts for themselves within

the pre-o¤ered menu. Contracting is decentralizable when the centralized mechanism design can

be equivalently implemented by the decentralized menu design, in other words, when the decent-

ralized menu design is strategically equivalent to the centralized mechanism design. Speci�cally,

if there is an optimal mechanism solving the mechanism design problem, there also exists an

optimal menu solving the menu design problem, and vice versa. Meanwhile, any solution to

either of these two problems brings the same (expected) payo¤ to the principal.

Decentralizability of contracting is of great signi�cance in multiagency situations, since menu

design is a "simpler" procedure and is more practical. Menu design procedure actually suggests

the feasibility of delegating to the agents the decision rights to specify contracts and that of sim-

plifying information communication required for the central designer to specify the contracts.

Meanwhile, just as McAfee and Schwartz (1994) pointed out, designing a complete and compre-

hensive (centralized) multilateral contract or mechanism might be indeed practically demanding,

and the associated costs of auditing and processing information might signi�cantly rise with the

number of parties involved. Therefore, decentralizability of multi-agency contracting will be

bene�cial for contracting practices in dealing with the increasing associated "transaction" costs

in modern society. This is also consistent with Hayek�s insight. Decentralized contracting would

be a better practical solution if it is strategically equivalent to centralization. Although many

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authors1 have investigated the decentralization of contracting in single-agency situations, there

are few comprehensive studies in this context under generalized multi-agency circumstances.

The multi-agency contracting games in our consideration are essentially two-stage one-shot

pure-strategy games. Since contracting games with pure strategies are more realistic in practice,

we focus on them in this paper. The principal�s strategy is to o¤er mechanisms or menus to

the agents. Each pre-o¤ered mechanism or joint menu de�nes a non-cooperative subgame for

all agents to play simultaneously. Bayesian Nash equilibrium (BNE) is the solution concept

used for such a subgame. In mechanism (respectively, menu) design, the principal seeks an

optimal mechanism (respectively, menu) from a class of mechanisms inducing a subgame for

the agents in which some particular reporting pro�le (respectively, contract selection pro�le) of

the agents is achieved as BNE. Such class of mechanisms (respectively, joint menus) are called

Bayesian mechanisms (respectively, Bayesian (joint) menus). Furthermore, the contracting

games over either Bayesian mechanisms or Bayesian menus are referred to as contracting games

under Bayesian implementation.

This paper establishes two major �ndings about the examination of the decentralizability of

multi-agency contracting. First, the delegation principle for Bayesian implementation identi�es

that Bayesian menu design is strategically equivalent to bilateral Bayesian incentive compatible

(BIC)2 mechanism design, which simpli�es the classic centralized collective BIC mechanism

design by ignoring relative information evaluation, even in the general multi-agency contracting

environment. Bilateral mechanisms associate each contract for an individual agent merely with

his individual report, whereas collective mechanisms associate each contract for an individual

agent with the joint reports of all agents. Although there has been a considerable literature

investigating bilateral mechanisms,3 our delegation principle implies that the greater signi�cance

of bilateral Bayesian mechanism is that it can indeed serve as a theoretical bridge from an

analytical perspective to connect decentralization and centralization in this context and compare

them. A by-product of the delegation principle is that we can further restrict attention to the

a particular class of menus with product structures out of the general menus in Bayesian menu

design problem, so that di¤erent agents can be allowed to separately choose the contracts within

the individual-speci�ed menus on their own accord.

Second, our analysis provides economically interesting conditions for the full equivalence un-

der which collective BIC mechanism design can make the principal as well o¤ as bilateral BIC

mechanism and Bayesian menu designs, although the former will clearly yield to the principal an

optimal (expected) payo¤ at least as large as the latter two do. In that case, we can substitute

collective mechanism design with bilateral mechanism design or menu design with no loss of

generality. Finer information structure with respect to Bayesian updated beliefs provides a pos-

1See Page(1992), Carlier (2001), Peters (2001), Martimort and Stole (2002), Page and Monteiro (2003), Car-mona and Fajardo(2007), Calzolari and Pavan (2008) among many others.

2Due to the well-known revelation principle, we can restrict attention to BIC mechanisms out of generalmechanisms.

3See Segal (1999), Han (2006), and Dequiedt and Martimort (2010) among many others.

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sibility for the full equivalence when the agents have no contract externalities and the principal

can separately draw welfare from contracts taken by di¤erent agents. In this vein, we �nd that

the collection of bilateral mechanisms interim-payo¤-equivalent to all collective BIC mechanisms

is exactly the collection of all bilateral BIC mechanisms in the quasi-separable environment.4

Then when the principal�s payo¤ is related with the agents�payo¤s in a linear or some particular

nonlinear form, the full equivalence can be established.

We also address several extensions based on our �ndings. (1) We show that the value of our

�ndings may lie in approximation of centralization, i.e., approximating collective BIC mechanism

design by bilateral BIC mechanism design or Bayesian menu design, even if the full equivalence

does not exactly hold. (2) We discuss the full equivalence under explicit primitive constraints

across the contracts for di¤erent agents. (3) It is not technically di¢ cult to incorporate the

individual rationality conditions in our results.

As a simpli�ed class of collective mechanism, bilateral mechanisms per se are still a pri-

ori adopted in some real-life situations due to antitrust laws, computational complexity, etc..

Dequiedt and Martimort (2015) mention the rationale of adopting bilateral mechanisms. For

instance, to discourage collusion, antitrust laws may forbid a monopolistic manufacturer to ad-

opt collective mechanisms where wholesale prices for each retailers depend on sales of others.

Moreover, when procurement is organized online, communication takes place between the buyer

and each seller independently.

Interim-payo¤-equivalence is an important concept and analytical tool in the Bayesian imple-

mentation literature. Recently, Manelli and Vincent (2010), and Gershkov et al. (2013) apply it

to examine the equivalence of Bayesian and dominant strategy implementation. In contrast, this

paper takes a new direction of the application of interim-payo¤-equivalence. The idea behind the

equivalence results of this paper is similar to theirs: a posteriori simpli�cation of multi-agency

contracting procedure can be o¤set by �ner a priori information (common knowledge) structure

in terms of Bayesian updated beliefs and certain speci�c contracting environment.

We consider a multi-agency environment in great generality of the pure adverse selection

model.5 The agents�types can be correlated. Contract sets for individual agents can be het-

erogeneous and permit some reasonable primitive constraints across the contracts for di¤erent

agents. Each agent�s (expected) payo¤ can depend not only on his own type and speci�ed con-

tract but also on those of the other agents. The �rst case is called information externalities

(or interdependent valuations), and the second case is called contract externalities. These two

points have attracted attention in a number of recent studies.6 Such features imply "full-blown

interdependence" among the agents. The message sets of di¤erent agents can be heterogeneous

4Quasi-separable environment is an extension of the separable environment introduced in Chung and Ely (2006)and is useful in many economic applications and previous studies.

5 It is not technically demanding to include moral hazard in this context. One can add one more part in themechanism as the action recommendation mechanism specifying action recommendations to the agents for anygiven their reports. See Kadan et al. (2014).

6See Jehiel et al. (1999), Jehiel and Moldovanu (2001) and Mezzetti(2004) among many others.

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or di¤erent. Moreover, the principal�s (expected) payo¤ is permitted to depend jointly on all

the agents�types and speci�ed contracts. Hence, the agents will behave strategically, and the

impacts of their respective asymmetric information will be interrelated. In addition, our model

is described in a general mathematical setting that is not necessarily restricted to �nite or vector

space structures.

Han (2006) also addresses the issue of decentralization in a mixed-strategy multi-principal

multi-agent "bilateral" environment with private valuations. All "bilateral" mechanisms the

principals can o¤er to each agent are restricted to be the functions from a single uniform message

set across the agents to an independent set of contracts7 available to that agent. Then in such an

environment with Perfect Bayesian Equilibrium, this particular "bilateral" mechanism design is

strategically equivalent to menu design. This paper extends Han�s model to a much more general

situation in multi-agency games and focus on pure strategies. We also takes into account the

related scenarios when the mechanisms are allowed to be "collective."

Moreover, Dequiedt and Martimort (2010) extend Han�s setting to allow message sets of

di¤erent agents to be heterogeneous or di¤erent subsets of Euclidean spaces and examine how

the principal acts opportunistically in a single-principal-two-agent bilateral vertical contracting

environment with private valuations and Bayesian implementation. But all agents� types be-

long to an identical unidimensional real closed interval. Although their study is not aimed at

a substantive discussion of the decentralizability of contracting in a generalized multi-agency

environment, it implies that the value of decentralization or "bilateral" centralization is worth

further exploring.

This paper focuses on the decentralizability problem with Bayesian implementation, while

Chen and Wu (2015) study the delegation principle for ex post implementation. The importance

of Bayesian implementation results from two reasons. First, the parties in a real world contract-

ing game may still have �ner information with respect to Bayesian updating. This is especially

true in the era of big data, in which having Bayesian updated beliefs as common knowledge

becomes more sensible. This will indeed provide more leeway for decentralizability than ex post

(or dominant-strategy) implementation. Second, (nontrivial) Bayesian implementation is more

likely to exist under the generalized models than ex post (or dominant-strategy) implementation.

2 Primitives

We consider a pure strategy multi-agency contracting game with one principal (short for PL) and

n agents indexed by i 2 N = f1; � � � ; ng.8 PL moves �rst. Then the agents follow simultaneouslyand behave non-cooperatively. Throughout this paper, the symbols B(X) is reserved for Borel�-algebra of a certain space X.

7 It implies the feasible contract sets of individual agents are not cross-constrained.8The model setup and major de�nitions in this paper basically follow Chen�s work (2012).

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Agent i (short for Ai) has some private payo¤ type �i 2 �i, where �i is a Borel space.9

We write � = (�i)i2N 2 � =nQi=1�i and ��j = (�i)i2Nnfjg 2 ��j =

nQi6=j�i.10 Let �i be a

probability measure de�ned on B(�i) and � be a probability measure on the associated productBorel �-algebra B(�). (�;B(�); �) is a probability measure space characterizing the commonprior over the agents� types, which are allowed to be correlated. Let ��i(�j�i) denote a condi-tional probability measure on �i over B(��i). Given �i, (��i;B(��i); ��i(�j�i)) is a probabilitymeasure space characterizing Ai�s interim belief about the other players�types after learning her

own type �i.11 For each i and each closed subset A of ��i, ��i(Aj�) is continuous on �i.12

The contract13 available to Ai is ki 2 Ki. The set of all possible joint contracts is given bythe (joint) contract set K =

nQi=1Ki. Its element is k = (ki)i2N . Write k�i = (kj)j2Nnfig. The set

K is assumed to be a compact metric space. Several typical examples �tting these assumptionson (joint) contract sets are shown below.14

Example 1 Finite contract sets: There are only �nitely many contracts in each Ki. K =nQi=1Ki

must be �nite and therefore a compact metric space.

Example 2 Product-price pairs in nonlinear pricing: Each buyer i is o¤ered a product-pricepair (xi; pi). xi is some product characteristics, such as quantity, quality, etc. pi is the price the

seller can charge for i. So the joint contract set can be

K = f(x1; � � � ; xn; p1; � � � ; pn) 2 Rn � Rn : 0 � xi � xi; 0 � pi � pig:

K is clearly a compact metric space.

Example 3 State-contingent contract sets: The state is ! 2 , where is a metric space. P isa probability measure over and B(). Assume all the contracts are outcome-contingent. If foreach i, (1) Ki is a (sequentially) compact subset of the collection of all (B();B(R))-measurablefunctions from to R for the topology of pointwise convergence on , (2) Ki contains noredundant contracts, that is, if for any two ki and k0i in Ki satisfying ki(!0) 6= k0i(!

0) for some

!0 2 , P (f! 2 : ki(!) 6= k0i(!)g) > 0, and (3) Ki is uniformly bounded, that is, all ki�sin Ki have the ranges contained in some closed bounded real interval, then Ki is compact andmetrizable for the topology of pointwise convergence by Proposition 1 in Tulcea (1973). So is K.

9Borel space is a Borel subset of a Polish space. A complete separable metric space is called Polish space. Afamiliar example of Polish space is any Euclidean space.10Note that � and ��i are also Borel spaces.11Note that ��i(�j�i) is not necessarily derived from the prior � on �. Moreover, we can allow heterogeneous

beliefs/priors across all parties. However, this will not change the main results of this paper.12One typical example for this assumption is that ��i has conditional density f(��ij�) which is continuous over

�i:13Some authors may also call it outcome, alternative, or allocation. But we use the term "contract" here to

literally take on richer meanings, for instance, outcome (state)-contingent contracts can also be considered as akind of contracts, and to be consistent with the term "contracting" games or procedures.14They are also provided in Chen and Wu (2015).

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Let vi : K � � ! R denote Ai�s payo¤ function de�ned over contract pro�les and type

pro�les. vi is continuous on K ��. Ai�s interim payo¤ function Vi : K ��i ! R is de�ned by

Vi(k; �i) =

Z��i

vi(k; �)��i(d��ij�i):

Let u : K�� ! R denote PL�s payo¤ function over contract and type pro�les. u is continuouson K and Borel-measurable on �. Moreover, u is �-integrably bounded, i.e. there exists a

�-integrable function U : � ! R such that for almost every � with respect to �, ju(k; �)j � Ufor all k 2 K.15

3 Mechanism Design and Menu Design16

3.1 Contracting Games over Bayesian Mechanisms

The classic contracting procedure is the centralized mechanism design. Accordingly, the principal-

agent contracting game over mechanisms is played as follows:

In stage 1, PL proposes to the agents a mechanism, which is commonly observable.

In stage 2, the agents unilaterally learn their own true types and simultaneously send reports

to PL.

In stage 3, through the pre-o¤ered mechanism, PL assigns contracts to the agents after

learning their reports.

In stage 4, after the agents�participation,17 the contracts are simultaneously executed.

Due to legal customs, technological restriction, or other reasonable constraints, either col-

lective mechanisms or bilateral mechanisms may be available a priori to PL. They are de�ned

below.

De�nition 1 A collective (respectively, bilateral) mechanism is a list of Borel measurable

functions k = (ki : � ! Ki)i2N satisfying for each � 2 �; (k1(�); � � � ;kn(�)) 2 K (respectively,

k = (ki : �i ! Ki)i2N satisfying (k1(�1); � � � ;kn(�n)) 2 K ), where each of its component ki(respectively, ki) speci�es a contract to Ai for each type report pro�le of all agents (respectively, of

single Ai). Let F(�;K) and F(�;K) respectively denote the collection of collective mechanismsand that of bilateral mechanisms.

Collective mechanisms evaluate relative information (all agents� type reports) for specify-

ing every individual agent�s contracts in nature, whereas bilateral mechanisms ignore it and

merely evaluate absolute information (every individual agent�s type reports) for specifying every

individual agent�s contracts.

15Note that this assumption must hold if � is compact, since u is continuous and K is compact.16The setting in this part is similar to Chen and Wu (2015).17 It is permitted that not all the agents eventually participate.

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The well-known revelation principle allows us to restrict attention to Bayesian incentive

compatible direct mechanisms out of general Bayesian mechanisms.18 Thus, we focus on direct

mechanisms in this paper.

Each mechanism o¤ered by PL induces a simultaneous-moved subgame for the agents in

which Bayesian Nash equilibrium (BNE) is adopted as the solution concept.

De�nition 2 A collective mechanism k (respectively, bilateral mechanism k) is Bayesianincentive compatible (BIC) if it induces truthful reporting as the BNE for all the agents,

i.e., for each i 2 N and �i 2 �i,Z��i

vi(k(�); �)��i(d��ij�i) �Z��i

vi(k(�0i; ��i); �)��i(d��ij�i);

(respectively,Z��i

vi(k(�); �)��i(d��ij�i) �Z��i

vi(ki(�0i);k�i(��i); �)��i(d��ij�i); )

for all �0i 2 �i.

Thus, two corresponding PL�s optimization problems address contracting games over Bayesian

mechanisms below.

(P1) collective BIC mechanism design problem:

maxk2F(�;K)

Z�u(k(�); �)�(d�)

s.t. k is BIC :

(P2) bilateral BIC mechanism design problem:

maxk2F(�;K)

Z�u(k(�); �)�(d�)

s.t. k is BIC.

3.2 Contracting Games over Bayesian Menus

The other contracting procedure is the decentralized menu design, in which PL avoids to process

decentralized information or have the agents send messages to specify contracts for the agents.

Instead she can design a (joint) menu, i.e., a subset of the joint contract set, for the agents and

allow them to simultaneously pick the contracts from such a menu.

Accordingly, the principal-agent contracting game over mechanisms unfolds as follows:

At stage 1, PL proposes to the agents a joint menu, which is commonly observable.

18Easy to check the revelation principle also holds for bilateral mechanisms.

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At stage 2, the agents unilaterally learn their own true types and simultaneously select the

contracts from the pre-o¤ered joint menu.

At stage 3, after the agents�participation, the contracts are simultaneously executed.

Each joint contract menu C is a subset of K. We consider a reasonable set of all joint contractmenus

Pf (K) = fC � KjC is a nonempty, closed subsets of Kg.

Each Ai�s strategy is a function eki : �i ! Ki which denotes Ai�s contract selection accordingto his type. Let Fi denote the collection of all arbitrary function eki�s. Write ek = (eki)i2N ,ek(�) = (eki(�i))i2N , and ek�i(��i) = (ekj(�j))j2Nnfig.

The contract selection pro�le under a menu C 2 Pf (K) is ek 2 Fc, whereFc = fekjek(�) 2 C for each � 2 �g.

Each menu C o¤ered by PL actually imposes a constraint on the agents�strategy pro�les and

induces a simultaneous-moved subgame played by the agents with BNE as the solution concept.

All the agents observe all the possible optional contract pro�les in each preo¤ered menu. A

certain contract pro�le within the menu needs to be simultaneously agreed on by the agents as

the feasible outcome, that is, ek(�) 2 C for each �. Thus, the subgames induced by preo¤ered

menus are related to the "generalized games" or "constrained games" introduced by Arrow and

Debreu (1954), and Rosen (1965).

De�nition 3 A contract selection pro�le ek 2 Fc is a BNE under a joint menu C if for each

i 2 N , �i 2 �i,Z��i

vi(ek(�); �)��i(d��ij�i) � Z��i

vi(eki0(�i);ek�i(��i); �)��i(d��ij�i);for all eki0 2 Fi satisfying (eki0(�i);ek�i(��i)) 2 C for some ��i 2 ��i. Such a joint menu C is

called a Bayesian (joint) menu.

PL can deduce that the agents will have the BNE contract selection pro�le in the subgamede�ned by a Bayesian menu and hence has an optimization problem to address this contracting

game, that is, Bayesian menu design problem (P3):

maxC2Pf (K)

Z�maxek2Fc u(ek(�); �)�(d�)

s.t. ek is the BNE under C.In view of tie-breaking, PL may designate or recommend ek in her best interest for the agentswith type pro�le � to follow.

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3.3 Delegation Principle for Bayesian Implementation

When we examine the decentralizability of Bayesian Implementation in the generalized multi-

agency environment, the �rst important observation summarized in Proposition 1 below is a

complete characterization of all bilateral BIC mechanisms via Bayesian menus.

First consider the set-valued mapping : �� Pf (K)� K de�ned by

(�; C) = fk 2 C : k = ek(�), where ek is the BNE under Cg:It is deducible by PL and represents the �-type-pro�le agents�joint BNE response to any menu

o¤er C. A well-de�ned (�; C) implies that there exists at least one Bayesian menu C.

De�nition 4 Given C 2 Pf (K), (�; C) is well-de�ned if (�; C) is nonempty for each �.

Proposition 1 Given a contracting mechanism k 2 F(�;K), the following statements are equi-valent:

(i) k is BIC.

(ii) There exists a joint menu C 2 Pf (K) such that (�; C) is well-de�ned, and k is a Borel-measurable selection from (�; C), that is, k(�) 2 (�; C) for all � 2 �.

Proof. See Appendix.Proposition 1 helps establish the delegation principle for Bayesian implementation presen-

ted in Proposition 2 below. This delegation principle identi�es that Bayesian menu design

is strategically equivalent to bilateral Bayesian mechanism design, which simpli�es collective

Bayesian mechanism design by ignoring relative information evaluation, even in a quite gen-

eral multi-agency situation permitting "full-blown" interdependence, including correlated types,

externalities in contracts, and interdependent valuations.

Proposition 2 (Delegation Principle for Bayesian Implementation).(i) If k

�solves the contracting problem over bilateral BIC mechanisms given by (P2), then

C� =nQi=1cl19f(k�i (�i) : �i 2 �ig solves the contracting problem over Bayesian menus given by

(P3).(ii) If C� solves (P3), then k

�satisfying k

�(�) 2 argmaxek(�)2(�;C�)u(ek(�); �) for each � 2 � solves

(P2).Moreover, the optimal objective values of the two problems are equal.

Proof. See Appendix.

Remark 1 In some cases, the contract set may contain some primitive constraints across the

contracts for di¤erent agents, that is, K �nQi=1Ki in general. For instance, in auction mechanism

19cl denotes the closure.

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design, the sum of the probability assignments for di¤erent agents must not be greater than 1. It

is not technically di¢ cult to �nd this delegation principle also holds in this generalized situation

with C� =nQi=1clf(k�i (�i) : �i 2 �ig\ K instead in part (i) of Proposition 2.

An immediate consequence of our delegation principle is an interesting corollary that the

general menu design problem (P3) can be reduced to a product (joint) menu design problem(P4)

maxC2PPf (K)

Z�maxek2Fc u(ek(�); �)�(d�)

s.t. ek is the BNE under C;where the product menu set is PPf (K) = fC = (C1; � � � ; Cn) � KjCi is a nonempty, closedsubsets of Kig. Note that the closure of the range of each component of a bilateral mechanismki must be contained in the compact set Ki as a component of K. Clearly, there exactly exists anoptimal menu with product structure given the existence of optimal bilateral BIC mechanism.

On the other hand, even if there exists an optimal Bayesian product menu, the �-type-pro�le

agents�joint BNE response to such a menu will still have a Borel-measurable selector which is

an optimal bilateral BIC mechanism with the product structure of the contract constraint set.

Corollary 1 (i) If k� solves the contracting problem over bilateral BIC mechanisms given by

(P2), then C� =nQi=1cl20f(k�i (�i) : �i 2 �ig solves the contracting problem over Bayesian menus

given by (P3). C� is also the solution to (P4).(ii) If C� solves (P3) or (P4), then k

�satisfying k

�(�) 2 argmaxek(�)2(�;C�)u(ek(�); �) for each

� 2 � solves (P2).Moreover, the optimal objective values of the two problems are equal. Therefore, (P4) is alsostrategically equivalent to (P2) and (P3).

Corollary 1 indicates that we can further restrict attention to the optimal Bayesian product

menu design problem out of the optimal Bayesian general menu design problem without any loss

of generality in our analysis. Since no primitive constraint is across the contracts for di¤erent

agents, PL can simply design individual-speci�ed (product) menus and permit di¤erent agents

to separately choose the contracts within the individual-speci�ed menus on their own accord.

4 Availability of Decentralization in Bayesian Implementation

Now we discuss the availability of decentralization in contracting games with Bayesian imple-

mentation in further details. An immediate implication from our delegation principle is that20cl denotes the closure.

11

contracting are decentralizable in the realistic environments where the centralized mechanisms

are ad hoc restricted to be bilateral, even if the impacts of the agents�asymmetric information

are signi�cantly interrelated.

But when will PL have incentive to still rely on decentralization (or bilateral centralization)

if the centralized mechanisms are allowed to be collective? First note that the collection of

bilateral BIC mechanisms is essentially equivalent to a subset of the collection of collective

BIC mechanisms. Because ((ki � �i)i2N ) 2F(�;K) is equivalent to k 2 F(�;K), where �i :� ! �i is the projection function de�ned by �i(�1; � � � ; �n) = �i. Therefore, the optimal

collective mechanism will clearly make PL better o¤ than the optimal bilateral mechanism

and the optimal menu. But this is just a weak dominance result. It does not completely

rule out the possibility of the full equivalence of those relevant contracting procedures. If the

optimal bilateral BIC mechanism (and equivalently optimal Bayesian menu) brings to PL the

same objective value (expected payo¤) as the optimal collective BIC mechanism does, the full

equivalence occurs. PL will hence have incentive for decentralization (or bilateral centralization)

even if collective centralization is possible, considering usually higher "transaction costs" of using

collective centralization.

Our delegation principle suggests that bilateral BIC mechanism can indeed serve as a bridge

from the analytical perspective, even if available mechanisms are not restricted to be bilateral,

to connect centralization and decentralization in contracting games and compare them. Hence,

our subsequent analysis boils down to the comparison between the collective BIC mechanism

design problem (P1) and the bilateral BIC mechanism design problem (P2).There are two trivial situations for this full equivalence. The �rst situation is that the only

optimal joint-based BIC mechanisms are (�-almost everywhere) constant mechanisms. Thus,

the optimal individual-based mechanisms will take the same constants (�-almost everywhere)

and the optimal menu can be a singleton. The second situation is that the multiple agents are

completely separate, independent individuals from PL�s viewpoint. Solving (P1) is equivalentto simultaneously solving n independent sub-problems only with respect to individual agents.

However, it is more desirable to �nd certain economically intuitive conditions on the primit-

ives for this full equivalence. A key idea is that such simpli�cation of the contracting procedure

tends to be o¤set by �ner information structure (common knowledge) in the contracting envir-

onment. We can adopt this idea to identify a class of economically intuitive conditions in which

the full equivalence holds by introducing interim payo¤ equivalence in Bayesian implementation.

4.1 Interim-Payo¤-Equivalent Mechanisms

First we de�ne the quasi-separable environment as an extension of the separable environment

introduced in Chung and Ely (2006). It is applicable to a large class of economic scenarios and

previous studies. In the quasi-separable environment, interdependent valuations and correlated

types are permitted. Each agent can separate his direct utility from his own contract and type

12

and valuation adjustment from all agents�types in a linear form of his payo¤.

De�nition 5 A contracting game is played in a quasi-separable environment if (1) Ki =miQj=1

Kij for each i 2 N and some mi 2 Z+. Its element is ki =miQj=1

kij. Each Kij is compact

and connected, and (2) for each i 2 N ; j 2 f1; � � � ;mig, vi(k; �) �miPj=1

hij(kij ; �i)wij(�) + qi(�)

for some continuous functions hij : Kij � �i ! R, wij : � ! R satisfying either wij(�) is

non-negative for all �, or wij(�) is non-positive for all �, and qi : �! R satisfying qi(�i; �) areintegrable with respect to ��i for each �i. We call hij(kij ; �i) the direct utility from kij and �i,

and call wij(�) the (interdependent) valuation adjustment of kij.

Then, in a quasi-separable environment we can identify a bilateral mechanism interim-payo¤-

equivalent (short for IPE) to any collective mechanism by the Bayesian (interim) belief structure.

De�nition 6 A collective mechanism k 2F(�;K) and a bilateral mechanism k 2 F(�;K) areinterim-payo¤ -equivalent (with respect to Ai) if for each �i 2 �i,Z

��i

vi(ki(�i); �)��i(d��ij�i) =Z��i

vi(ki(�); �)��i(d��ij�i):

Proposition 3 In a quasi-separable environment, for any collective (respectively, collective

BIC) mechanism k with its i-th coordinate ki =miQj=1

kij, there exists a bilateral (respectively,

bilateral BIC) mechanism k interim-payo¤-equivalent (with respect to all agents) to k. The con-

verse is also true, i.e., for any bilateral (respectively, bilateral BIC) mechanism k, there exists

a collective (respectively, collective BIC) mechanism k interim-payo¤-equivalent to k.

Proof. For each i, given ki;�i; j, since hij(�; �i) is continuous, compactness and connectednessof Kij implies that the range of hij(�; �i) is also compact and connected in R and therefore shouldbe a closed interval [aij(�i); bij(�i)],21 and hij(�; �i) is onto from Kij to its range.

Then we de�ne the inverse set-valued function of hij(�; �i) as ��1ij;�i : [aij(�i); bij(�i)] � Kijby

��1ij;�i(x) = fkij 2 Kij : hij(kij ; �i) = xg

By the closed map lemma, hij(�; �i) is a continuous closed map. Thus, ��1ij;�i is a measur-

able set-valued function. Since hij(�; �i) are continuous, ��1ij;�i must be nonempty closed-valued.Kuratowski-Ryll-Nardzewski Selection Theorem implies that ��1ij;�i must admit a Borel-measurable

selector, say 'ij;�i : [aij(�i); bij(�i)]! Kij .21Note that if we only focus on this IPE between bilateral mechanisms and collective mechanisms regardless of

menu, then in fact we do not require the compactness of Kij to prove the result.

13

Now de�ne �i;kij : �i ! R by

�ij;kij (�i) =

R��i

hij(kij(�); �i)wij(�)��i(d��ij�i)R��i

wij(�)��i(d��ij�i)

Obviously, �ij;kij is a Borel-measurable function of �i. Moreover, for all �;

aij(�i) � hij(kij(�); �i) � bij(�i):

Since wij is non-negative or non-positive, �ij;kij (�i) 2 [aij(�i); bij(�i)] for all �i.Therefore, we can de�ne a function kij : �i ! Kij by

kij(�i) = 'ij;�i(�ij;kij (�i)); for each �i 2 �i:

Hence kij is clearly a Borel-measurable function, and k is a well-de�ned bilateral mechanism.

Next by the de�nitions above, for each �i 2 �i,

hij(kij(�i); �i) =

R��i

hij(kij(�); �i)wij(�)��i(d��ij�i)R��i

wij(�)��i(d��ij�i):

Thus, Z��i

hij(kij(�); �i)wij(�)��i(d��ij�i) = hij(kij(�i); �i)

Z��i

wij(�)��i(d��ij�i)

=

Z��i

hij(kij(�i); �i)wij(�)��i(d��ij�i)

In sum, ki is interim-payo¤-equivalent to ki. Moreover, since k is interim-payo¤-equivalent

to k, and k is BIC, k is clearly BIC too. The converse is straightforward. �

Remark 2 In the constructive proof, bilateral mechanism k IPE to any given collective mech-anism k, allowing hij(�; �i) is not one-to-one for each �i. If hij(�; �i) is one-to-one, then ��1ij;�imust be a well-de�ned function. Thus, we can simply de�ne the desirable kij by kij(�i) =

'ij;�i(�ij;kij (�i)), for each �i.

Remark 3 Quasi-separable environment is of signi�cance for the interim payo¤ equivalence.

First, if contract externalities are permitted, it is di¢ cult for each Ai to form a kj (j 6= i)

coupled with ki such that k IPE to k. The situation free of contract externalities raises the

degree of freedom to �nd IPE bilateral mechanisms.Moreover, if quasi-separable forms of the agents�payo¤ functions or compactness and connec-

tedness of the contract sets are violated, interim payo¤ equivalence may fail. Consider a simple

�nite case as follows. N = f1; 2g. K1 = f0; 1g. �1 is a singleton. �2 = fL;Hg. �2 are equallydistributed. Let v1(0; L) = v1(1;H) = 1, and v1(1; L) = v1(0;H) = 0. Then consider k1 such

14

that k1(L) = 0 and k1(H) = 1.R�2v1(k1(�); �)�2(d�2) =

12(v1(k1(L); L) + v1(k1(H);H)) = 1.

But it is unlikely to �nd a (constant) bilateral k1(�1) 2 K1 such thatR�2v1(k1(�1); �)�2(d�2) = 1.

Remark 4 This proposition also implies that all BIC bilateral mechanisms must be IPE to

some BIC collective mechanisms in quasi-separable environments.

Proposition 3 implies that in the quasi-separable environment the collection of bilateral

mechanisms IPE to all collective BIC mechanisms is exactly the collection of all bilateral BIC

mechanisms. In this respect, Proposition 3 completely characterizes collective BIC mechanisms

via interim-payo¤-equivalence with bilateral BIC mechanisms. Thus, the comparison of central-

ization and decentralization boils down to the comparison between the highest payo¤ brought

by collective BIC mechanisms and the highest payo¤ brought by the IPE bilateral mechanisms

in the quasi-separable environment. This hints at a possibility of full equivalence from PL�s

viewpoint.

4.2 Full Equivalence Results

With additional assumptions on PL�s payo¤ function related to the agents� payo¤ functions

and belief structure, Proposition 3 can usher in a few results on the full equivalence between

collective and bilateral BIC mechanism designs and therefore Bayesian menu design due to

our delegation principle. The key for the full equivalence is to test whether the optimal BIC

collective mechanism and at least one of bilateral BIC mechanisms IPE to collective BIC mech-

anisms can bring the same expected payo¤ to PL.22 In other words, we need to test whether

maxk is BIC

R� u(k(�); �)�(d�) = max

k is IPE to BIC k

R� u(k(�); �)�(d�).

For the rest of this paper, we �rst need to assume that for each i 2 N the interim belief

��i(�j�) is derived from the prior �, that is, for any �-integrable functions � : �! R,Z��(�)�(d�) �

Z�i

Z��i

�(�)��i(d��ij�i)�i(d�i):

4.2.1 Linearly Additive Payo¤ Relations

Based on Proposition 3, if additionally PL�s payo¤exhibits a certain linearly additive separability

with each component as a linear transformation of each agent�s payo¤ (given his own type), the

full equivalence can be ensured when the agents have general payo¤ structures without contract

externalities.

Corollary 2 In a quasi-separable environment, if

u(k; �) �nXi=1

[

miXj=1

(aij(�i)hij(kij ; �i)wij(�))] + L(�)

22Yet our subsequent analysis centers on the conditions on the primitives for the overall equivalence.

15

for some continuous functions L : �! R and aij : �i ! R for each i 2 N , j 2 f1; � � � ;mig, hijand wij are given as in the de�nition of quasi-separable environment,

then for any optimal collective mechanism k�, there exists its IPE bilateral mechanism k�

bringing to PL the same expected payo¤. Thus, Bayesian menu design is equivalent to both

bilateral and collective BIC mechanism designs.

Proof. See Appendix.

Remark 5 Linearly additive separability of PL�s payo¤ with the agents�payo¤s can clearly helppreserve the interim-payo¤ equivalence and establish the full equivalence given interdependent

valuations.

Remark 6 Note that aij can be either positive or negative. It is usually involved with partner-ship or social e¢ ciency that a0ijs take positive signs. In contrast, it re�ects con�icts of interests

between PL and the agents that a0ijs take negative signs, especially in the common principal-agent

relationship.

Remark 7 It would be di¢ cult to �nd conditions on the primitives for the exact equivalencebetween collective mechanisms and its IPE bilateral mechanisms if we allow non-separable re-

lation between the agents� payo¤s and the principal�s payo¤. Because ki(�) and k�i(�) may

simultaneously be integrated out with respect to ��i under ��i.

In Corollary 2, we additionally assume that PL�s payo¤ function takes a "separably" additive

form linearly related to the agents� payo¤s. In this pattern PL can predict a bilateral BIC

mechanism interim-payo¤-equivalent to the optimal collective BIC mechanism for her. Such

bilateral mechanism will also be the optimal bilateral BIC mechanism. Then the full equivalence

can be achieved. There are a few examples below in which Corollary 2 is applicable.

Example 4 (Vertical Contracting 1) A manufacturer (PL) wants to sell his products to n retail-ers in segmented markets. She just controls the wholesale prices p = (p1; � � � ; pn) as the contract.pi 2 [0; pi]. Retailer i receives a local baseline demand signal �i. The prediction about actualmarket demand for i is wi(�). Let Pi(wi(�)) be the continuous (estimated) inverse demand based

on market demand wi(�) for i. Then i�s pro�t is Pi(wi(�))wi(�) � piwi(�). It implies interde-pendent valuations.23 The manufacturer�s production cost c 2 R does not rely on the wholesaleprices. Her ex post payo¤ function can hence be her revenue from wholesale:

nPi=1piwi(�).

Example 5 (Vertical Contracting 2) Now the contract for retailer i is (qi; ti), where qi 2 [0; qi]is the quantity that the manufacturer sells to retailer i and ti 2 [0; ti] is the payment from retailer23This may be a remarkable phenomenon in modern practice. It has recently attracted more attention in the

context of vertical contracting or procurement. For instance, Han (2013) in his recent paper addresses suchinterdependent valuation in the analysis asymmetric �rst-price menu auctions in the procurement environment.

16

i to the manufacturer. Each retailer is a price taker. Let Pi(�i)qi denote i�s revenue, where the

market price Pi is a positive continuous function of the private signal about local market demand

�i. Then i�s pro�t is Pi(�i)qi � ti. The manufacturer has the production cost as b(nPi=1qi) for

some b 2 R. So her payo¤ isnPi=1ti � b(

nPi=1qi).

Example 6 (Ex Ante E¢ cient Allocation) Consider a classic resource allocation context. Con-

tracts consist of the assignments of some divisible resources x =nQi=1xi 2

nQi=1[0; xi] and the

monetary transfers (from the agents to the social planner) t =nQi=1ti 2

nQi=1[ti; ti]. Each agent i

has a private evaluation �i about the his assignment. His quasi-linear payo¤ function

vi(x; t; �) = wi(�)hi(xi; �i)� ti:

Then the planner�s ex post payo¤ function (social welfare) isnPi=1[wi(�)hi(xi; �i)], and she con-

siders ex ante e¢ cient allocation.

Example 7 (Teamwork) A headquarter assigns production tasks of a homogenous good to twodownstream branches (indexed by i = 1; 2). Each branch i has an e¢ ciency parameter as

its private type �i 2 �i. The units of good the branch i produces is xi 2 [0; xi]. Contracts

consist of the assignments of productions. Retailer i has a pro�t function wi(�)hi(xi; �i). The

headquarter has a constant management cost c > 0, and then needs to maximize the full pro�tPni=1wi(�)hi(xi; �i)� c.

4.2.2 Non-linearly Additive Payo¤ Relations

In some applications, PL may have a di¤erent payo¤ functional form from the agents. So

PL�s payo¤ may be related with the agents�payo¤s in a non-linearly additive pattern. In the

quasi-separable environment, if PL�s payo¤ accordingly exhibits a certain non-linearly additive

separability with each component as a concave transformation of each agent�s direct utility

(given his own type), the full equivalence can be ensured.

Corollary 3 In a quasi-separable environment, if(i) wij(�) � wij(�i), for each i 2 N , j 2 f1; � � � ;mig, and

(ii) u(k; �) �nPi=1[miPj=1

Gij(hij(kij ; �i); �i)] + L(�) for some continuous functions L : � ! R

and Gij : R��i ! R for each i, j satisfying Gij(�; �i) is a concave transformation for each �i;then for any optimal collective mechanism k�, there exists its IPE bilateral mechanism k

�

bringing to PL the same expected payo¤. Thus, Bayesian menu design is equivalent to both

bilateral and collective BIC mechanism designs.

Proof. See Appendix.

17

There is an example of procurement in which Corollary 3 is applicable.

Example 8 (Procurement 1) A buyer (PL) needs procurement of two imperfectly substitutivegoods respectively from two producers indexed by i = 1; 2. i receives a production cost signal

�i 2 [0; 1]. Contracts consist of the quantities of procurement x =nQi=1xi 2

nQi=1[0; xi], where xi is

the quantity of the good purchased from i, and the monetary transfers t =nQi=1ti 2

nQi=1[0; ti], where

ti is the monetary payment to i. Each producer i0s payo¤ is ti�ci(xi; �i), where ci(xi; �i) = �ix2iis the production cost of xi. The buyer�s payo¤ is

nPi=1lnxi �

nPi=1ti. lnxi is the payo¤ the buyer

can draw from consumption of xi. Here lnxi =lnx2i2 . It is an increasing concave transformation

of x2i .

5 Discussions

5.1 Approximation of Centralization

The value of our �ndings may also lie in approximation of centralization. It means that we

can approximate optimal collective BIC mechanism design by optimal bilateral BIC mechanism

design or optimal Bayesian menu design, even if the full equivalence does not exactly hold. For

instance, although PL�s payo¤ function may have non-separable relation with respective agents�

payo¤ functions, linear approximation (�rst-order Taylor expansion) of PL�s payo¤ function in

contracts can help to this end. Thus, our delegation principle and Proposition 3 may still be

applicable to the linear approximation situation for the full equivalence in quasi-separable envir-

onments. Linear approximation can also measure the di¤erence or approximation error between

centralization and decentralization with a bound. In fact, we will still prefer decentralization if

such a bound is smaller than the practical cost of performing centralized contracting relative to

decentralized contracting.

Now consider a quasilinear quasi-separable environment. �i 2 �i with associated

probability measures � and ��i(�j�i) as we assume in the general setting. The contract availableto Ai consists of assignment xi 2 Xi = [xi; xi] and transfer ti 2 Ti = [ti; ti]. Write x = (xi)i2Nand t = (ti)i2N . Ai�s payo¤ is

vi(xi; ti; �) = hi(xi; �i)wi(�)� ti;

where hi is a continuous real-valued function, and wi is a continuous function. We assume there

exists an x�i 2 Xi for each � and i such that hi(x�i ; �i) = 0. Let 0 denote the n-dimensional

vector with all coordinates equal to 0. PL�s payo¤ is

u(x; t; �) = G(x; �) +

nXi=1

ti;

18

whereG is a continuous real-valued function. A collective mechanism is a list of Borel-measurable

functions (x; t) = ((xi : �! Xi)i2N ; ti : �! Ti)i2N ). A bilateral mechanism is a list of Borel-

measurable functions (x; t) = ((xi : �i ! Xi)i2N ; ti : �i ! Ti)i2N ).

Let U� denote the optimal value of collective BIC mechanism design problem and U�� denote

the optimal value of bilateral BIC mechanism design problem or Bayesian menu design problem.

Lemma 1 In the quasilinear quasi-separable environment, if for each i,j,and �,(i) G(x; �) is continuously second-order di¤erentiable in each xi24 and continuous in �. Gi

denotes the partial derivative with respect to xi, and Gij denotes the ij-th second order derivative

with respect to xi and xj,

(ii) 0 2 Xi and Gi(0; �) exists,(iii) jGij(x; �))j �M for all x, and

(iv) for any optimal collective BIC mechanism (x�; t�), there exists a bilateral BIC mechan-

ism (x�; t�) such that Z

�Gi(0; �)x

�i (�i)�(d�) =

Z�Gi(0; �)x

�i (�)�(d�); (1)

and t�i (�i) =R��i

ti(�)��i(d��ij�i) for each i,then U� � U�� � M(

Pni=1

Pnj=1 �ij), where �ij = max

xi2Xi;xj2Xjjxixj j. Moreover,

��U��U��U��

�� �M(

Pni=1

Pnj=1 �ij)

minf���� infx2X;�2�

G(x;�)+Pni=1 ti

����;����� supx2X;�2�

G(x;�)+Pni=1 ti

�����g; if inf

x2X;�2�G(x; �) and sup

x2X;�2�G(x; �) are �-

nite.

Proof. See Appendix.

Remark 8 Second-order derivatives of G(�; �) represents the degrees of the strategic interde-pendence (either strategic complementarity or strategic substitutivity) in PL�s payo¤ between

individual agents�actions.25 The polar case is that G is linearly additive in all xi�s. It actually

represents strategic independence in PL�s payo¤ between any two individual agents�actions, that

is, any Gij = 0. Lemma 1 as an asymptotic result indicates that menu and bilateral mechanism

designs are approaching collective mechanism design in PL�s viewpoint as those degrees of the

strategic interdependence approach zero. Moreover, the weaker the strategic interdependence is,

the more likely menu and bilateral mechanism designs are close to collective mechanism design

in the PL�s viewpoint.

Remark 9 Lemma 1 also implies that the loss of menu or bilateral mechanism designs may

increase as the number of agents increases, since each �ij must be positive.

24More rigorously, given �, G is continuously second-order di¤erentiable in xi over the interior of Xi, andleft(respectively, right)-continuously-second-order-di¤erentiable at the right (respectively, left) ending point ofXi.25Generally speaking, this includes the "nterdependence" between any individual agent�s actions and his actions

themselves, which is re�ected by second-order derivatives with respect to one xi itself.

19

Based on Lemma 1 we can establish the further approximation result through IPE and our

full equivalence results.

Proposition 4 In the quasilinear quasi-separable environment, if(i) G(x; �) = g((hi(xi; �i))i2N ; �), where g : Rn ��! R is a function which is continuously

second-order di¤erentiable in each of �rst n arguments26 and continuous in �,

(ii) for all i and �i, hi(xci ; �i) = 0 for some xci 2 Xi,

(iii) for all i and �, gi(0; �) = ai(�i)wi(�) for some continuous function ai : �i ! R, and(iv) for each i,j,and �, jgij((hi(xi; �i))i2N ; �)j �M for all x,

then U� � U�� �MR�(Pni=1

Pnj=1 �ij(�i; �j))�(d�), where

�ij(�i; �j) = maxxi2Xi;xj2Xj

jhi(xi; �i)hj(xj ; �j)j:

Moreover,����U� � U��U��

���� � MR�(Pni=1

Pnj=1 �ij(�i; �j))�(d�)

minf���� infx2X;�2�

G(x; �) +Pni=1 ti

���� ;����� supx2X;�2�

G(x; �) +Pni=1 ti

�����g;

if infx2X;�2�

G(x; �) and supx2X;�2�

G(x; �) are �nite.

Proof. We now consider the entire hi(xi; �i) as xi for given i and � in Lemma 1. Thus,

Proposition 3 and hypothesis (iii) implies that for any optimal collective BIC mechanism (x�; t�)

there exists a bilateral BIC mechanism (x�; t�) satisfying hypothesis (iv) in Lemma 1. Therefore,

the results are straightforward. �In Proposition 4 gij denotes the degree of the strategic interdependence between any two

agents�direct utilities from individual actions and types. Thus, if G(x; �) can be expressed as

a composite function of hi(xi; �i)0s;and gi(0; �) can be a linear transformation of wi(�) by a

multiplier ai(�i);menu and bilateral mechanism designs are approaching collective mechanism

design in PL�s viewpoint through IPE as those degrees of the strategic interdependence approach

zero. Moreover, the loss of menu or bilateral mechanism designs may increase as the number of

agents increases. Here are several examples to which Proposition 4 can apply.

Example 9 (Vertical Contracting 3) Consider a vertical contracting case �tting our quasilinearquasi-separable environment. A supplier (PL) needs to sell a certain product to two buyers

indexed by i 2 N . i has a private evaluation over the good �i 2 [0; 1]. The quantity of thegood purchased from i is xi 2 [0; xi]. The monetary payment by i is ti 2 [0; ti]. Each buyeri0s payo¤ is xiwi(�) � ti, where wi(�) is i�s valuation per unit of xi. The supplier�s payo¤ is26More rigorously, given �, g is continuously second-order di¤erentiable in the i-th argument over the interior of

the range of hi(�; �i), and left(respectively, right)-continuously-second-order-di¤erentiable at the right (respect-ively, left) ending point of the range.

20

nPi=1ti � c(x), where c(x) = 1

2(Pni=1 xi)

2 is the cost function of the supplier. ci(x) =Pni=1 xi.

ci(0) = 0. So we can choose ai(�i) � 0. For each i,j, cij(x1; x2) = 1. Then Proposition 4

implies U� � U�� �Pni=1

Pnj=1 �ij, where �ij = max

xi2Xi;xj2Xjjxixj j:

Example 10 (Procurement 2) A buyer (PL) needs procurement of two imperfectly substitutivegoods respectively from n producers indexed by i = 1; 2. i receives a production cost signal

�i > 0. Contracts are the same as in example 8 of Procurement 1. Each producer i0s payo¤

is ti � ci(xi; �i), where ci(xi; �i) = �ix2i is the production cost of xi. The buyer�s payo¤ is

B(x) �nPi=1ti, where B(x) = �e�(x

21+x

22) is the bene�t the buyer can draw from consumption of

x. Let B(x) = G(x21; x22). Gi(x

21; x

22) = e�(x

21+x

22), and Gi(0) = 1. We can choose ai(�i) � 1

�i.

Moreover, Gij(x21; x22) = �e�(x21+x22). Clearly, jGij(x21; x22)j � 1. Then Proposition 4 implies

U� � U�� �Pni=1

Pnj=1 �ij, where �ij = max

xi2Xi;xj2Xjx21x

22:

Example 11 (Resources Allocation) A social planner allocates a homogenous good to two agents(indexed by i = 1; 2). The units of good agent i receives is xi 2 [0; xi]. The monetary transferfrom i to the planner is ti 2 [0; ti]. Each agent i has a private type �i 2 �i and then a payo¤function wi(�)xi�ti. But such good has some negative externality e¤ect C(x) = (

Pni=1 xi)

2. The

planner�s payo¤ is the social surplus G(x; �) =P2i=1wi(�i)xi� (

P2i=1 xi)

2. Given �, Gi(x; �) =

wi(�) � 2(x1 + x2) and Gi(0; �) = wi(�). For each i,j, Gij(x1; x2) = �2. Then Proposition 4implies U� � U�� � 2

Pni=1

Pnj=1 �ij, where �ij = max

xi2Xi;xj2Xjx1x2:

5.2 Primitive Constraints across the Contracts for Di¤erent Agents

The aforementioned results do not address the explicit primitive constraints across the contracts

for di¤erent agents under which K is not directly equal to the product of the agents�contract

sets. But if the IPE bilateral BIC mechanism k still satis�es the primitive constraint, i.e.,

k(�) 2 K for each �, Proposition 3 will still hold. Even as long as the BIC bilateral mechanismIPE to optimal collective mechanism k� still satis�es the primitive constraint, Corollaries 2 and

3 still hold. Here is a simple example to this end.

Example 12 (Procurement 3) One producer procures two input goods separately from two inputsuppliers denoted by i = 1; 2. Contracts are the same as in example 8 of Procurement 1. �i�s

are independently distributed. i�s payo¤ function vi(ti; xi; �i) = ti � �ixi. The producer�s payo¤u(x; t; �) = x�1x

1��2 � t1� t2. x�1x1��2 denotes the Cobb-Douglas (monetary) production function,

where � 2 (0; 1). There is a constraint over x0is : x�1x1��2 � q, where q denotes the capacity

limit. The producer should not purchase the bundle of (x1; x2) beyond the production capacity

constraint. For each i, collective (respectively bilateral) BIC assignment rule for i is xi : �! R(respectively, xi : �i ! R). Suppose optimal collective BIC assignment rule is x�. Then its IPE

21

bilateral assignment rule x� can be de�ned by

x�i (�i) =

Z��i

x�i (�)��i(d��i); i = 1; 2:

Clearly, if for each �;x��

1 (�)x�1��2 (�) � q, then x��1 (�1)x�

1��2 (�2) � q.

Yet in some cases, IPE bilateral mechanisms may not necessarily preserve some primitive

constraints across the contracts for di¤erent agents, especially for those linear combination

inequality constraints, such as the natural requirement on probabilistic assignments in auction

design. It is generally di¢ cult to provide conditions on the primitives for such preservation. Such

preservation may need some requirement on the properties of the optimal collective mechanism

per se. For instance, the symmetric mechanism design with ex ante identical agents may help

to this end, especially in auction contexts.

5.3 Individual Rationality Constraints

The aforementioned analysis also does not include the participation constraints modeled as

the individual rationality conditions. It can be formulated as follows. Suppose that Ai has

the commonly-observable reservation utility ri(�i) 2 R based on his type �i. A collective BIC

mechanism k is also (Bayesian) Individual Rational (IR) if for all i 2 N , � 2 �,Z��i

vi(k(�); �)��i(d��ij�i) � ri(�i): (IRc)

A bilateral BIC mechanism k is also (Bayesian) IR if for all i 2 N , � 2 �,Z��i

vi(k(�); �)��i(d��ij�i) � ri(�i): (IRb)

A BNE contract selection pro�le ek is also (Bayesian) Individual Rational (IR) under a jointex post menu C if for each i 2 N , � 2 �,Z

��i

vi(ek(�); �)��i(d��ij�i) � ri(�i): (IRm)

Such menu C is hence said to be a (joint) Bayesian menu with IR constraints.From the mathematical perspective, IR conditions serves similar to corresponding BIC condi-

tions or BNE condition in the constraints of the multi-agency contracting problems. Apparently,

given �i, the right hand sides of the IR conditions (the interim payo¤s under relevant mechanisms

or contract selections) remain the same, and the left hand sides of the IR conditions (ri(�i)) are

just some constants. Thus, it is not technically di¢ cult to incorporate the individual rationality

conditions in all the aforementioned results.

22

6 Conclusion

Decentralizability of multi-agency contracting with Bayesian implementation is attainable in

several economically interesting situations. When available mechanisms must be bilateral, we

can always switch to decentralization with no loss of generality. Even if available mechanisms

can be collective, Bayesian updated beliefs as common knowledge and interim-payo¤-equivalence

can help provide the possibility to establish the full equivalence between centralization and de-

centralization. One can bene�t from the �ner information structure in Bayesian implementation

relative to ex post (or dominant-strategy) implementation to deal with information asymmetry.

These facts make decentralizability of multi-agency contracting considerable for practical de-

cision makers in real life. Our analysis may have many economic applications, including nonlin-

ear pricing, resource allocation, regulation, insurance, public choice, etc. It could also be more

tractable to establish the decentralizability under more concrete application environments. In

large contracting games, since every agent becomes negligible somehow, the decentralizability is

worth further discussion as well. Our approximation results may also be useful in the empirical

study of mechanism design.

Appendix

Lemma 2 For any C 2 Pf (K) satisfying (�; C) is well-de�ned, (�; C) is a compact-valuedBorel-measurable set valued function from � to C:

Proof. First claim that (�; C) has a closed graph for any C 2 Pf (K) satisfying (�; C) iswell-de�ned. For simplicity, let FC(�) = (�; C). We need to show that GrFC = f(�;ek(�)) 2�� Cjek is the BNE under Cg is closed.

First �x � 2 �. Pick any arbitrary sequence f(�l;ek(�l))gl in Gr'C satisfyingek(�l) 2 FC(�l), and (�l;ek(�l))! (�;ek(�)), as l!1:

Thus it su¢ ces to show that ek(�) 2 FC(�), that is, for each i 2 N ,Z��i

vi(ek(�); �)��i(d��ij�i) � Z��i

vi(eki0(�i);ek�i(��i); �)��i(d��ij�i);for all eki0 2 Fi satisfying (eki0(�i);ek�i(��i)) 2 C for some ��i 2 ��i.

For each i 2 N ,Z��i

vi(ek(�l); �l)��i(d��ij�li) � Z��i

vi(k0i(�

li); k

l�i(�

l�i); �

l)��i(d��ij�li);

for all eki0 2 Fi satisfying (eki0(�i);ek�i(��i)) 2 C for some ��i 2 ��i. Since vi is continuous and

23

is bounded on �, Delbaen�s Lemma (1974)27 implies thatZ��i

vi(ek(�); �)��i(d��ij�i) � Z��i

vi(eki0(�i);ek�i(��i); �)��i(d��ij�i);for all eki0 2 Fi satisfying (eki0(�i);ek�i(��i)) 2 C for some ��i 2 ��i. Therefore, the graph of(�; C) is closed in �� C, i.e., (�; C) is closed-valued.

Moreover, (�; C) is compact-valued, as K is a compact metric space. Note that � and C areboth Borel spaces. Thus, by Theorem 3 in Himmelberg, Parthasarathy and Van Vleck (1976),

(�; C) is Borel-measurable. �

Proof of Proposition 1.(i))(ii). Assume that k 2 F(�;K) is BIC. De�ne

C =

nYi=1

clf(ki(�i) : �i 2 �ig \ K.

First claim that k(�) 2 (�; C) for all � 2 �, that is, for each i 2 N , and each �i 2 �i,Z��i

vi(k(�); �)��i(d��ij�i) �Z��i

vi(ek0i(�i);k�i(��i); �)��i(d��ij�i);for all eki0 2 Fi satisfying (eki0(�i);ek�i(��i)) 2 C for some ��i 2 ��i. Suppose not. Then forsome agent j, some �0j 2 �j , and some ekj 0 2 Fj satisfying ( ekj 0(�0j);k�j(��j)) 2 C for some

��j 2 ��j ,Z��j

vj(kj(�0j);k�j(��j); �

0j ; ��j)��j(d��j j�0j) <

Z��j

vj(ek0j(�0j);k�j(��j); �0j ; ��j)��j(d��j j�0j):(2)

Because of the de�nition of C, any section of C is still closed. Thus, for any ��j , there exists a

sequence of type f�j;lgl in �j such that (kj(�j;l);k�j(��j))! (ek0j(�0j);k�j(��j)) in C, as l!1.Hence, by the continuity of vj and Delbaen�s Lemma (1974), for l large enough, (2) impliesZ��j

vj(kj(�0j);k�j(��j); �

0j ; ��j)��j(d��j j�0j) <

Z��j

vj(kj(�j;l);k�j(��j); �0j ; ��j)��j(d��j j�0j):

This contradicts the fact that k is BIC and proves the claim.

Thus, (�; C) is clearly well-de�ned. By Lemma 1, (�; C) is Borel-measurable. Therefore,k is actually a Borel-measurable selection from (�; C).

(ii))(i). Assume that k(�) 2 (�; C) � C for all � 2 �. For all i 2 N , all �i 2 �i, and all27Another description about this lemma can be found in Page (1987)

24

eki0 2 Fi satisfying (eki0(�i);ek�i(��i)) 2 C for some ��i 2 ��i.Z��i

vi(k(�); �)��i(d��ij�i) �Z��i

vi(eki0(�i);k�i(��i); �)��i(d��ij�i):Since there are some eki0 satisfying eki0(�i) = ki(�0i) for any �0i 2 �i, we haveZ

��i

vi(k(�); �)��i(d��ij�i) �Z��i

vi(ki(�0i);k�i(��i); �)��i(d��ij�i);

for all �0i 2 �i. Thus, k is BIC. �

Remark on Proposition 1.Furthermore, when (�; C) is well-de�ned for some C 2 Pf (K), the Bayesian menu design

problem (P2) can be rewritten in a compact way:

maxC2Pf (K)

Z�

maxek(�)2(�;C)u(ek(�); �)�(d�):The feasible bilateral BIC mechanism set is de�ned as

ICI =�k 2 F(�;K) : k is BIC.

The bilateral BIC mechanism design problem (P10) can also be stated compactly as

maxk2ICI

Z�u(k(�); �)�(d�):

Moreover, the equivalent mechanism set induced by a joint menu C 2 Pf (K) is de�ned by

�(C) =�k 2 F(�;K) : k(�) 2 (�; C) for all � 2 �

: (3)

It denotes the set of all measurable selections from (�; C) in F(�;K) for a given menu C 2Pf (K). Next, the full equivalent mechanism set induced by all joint menus is de�ned by

� =[

C2Pf (K)�(C).

Indeed, Proposition 1 is equivalent to say ICI = �. �

Lemma 3 For each C 2 Pf (K) satisfying (�; C) is well-de�ned, there exists some k 2 �(C)such that

u(k(�); �) = maxek(�)2(�;C)u(ek(�); �);

25

for all � 2 �. Moreover, the function � 7! maxk2(�;C)

u(k; �) is Borel measurable.

Proof. Note that � and K are Borel space. By Lemma 2, for each C 2 Pf (K) satisfying(�; C) is well-de�ned, (�; C) is Borel-measurable and compact-valued. We know u is Borel-measurable and u(�; �) is continuous. Then by Theorem 2 in Himmelberg, Parthasarathy and

Van Vleck (1976), there exists some Borel measurable selector k in �(C) for the set-valued

function(�; C) such that u(k(�); �) = maxek(�)2(�;C)u(ek(�); �) for all � 2 �. Moreover, the function� 7! maxek(�)2(�;C)u(ek(�); �) is also Borel measurable. �

Remark. (Theorem 2 in Himmelberg, Parthasarathy and Van Vleck (1976))

Let S and A be Borel spaces, let F be a Borel measurable compact valued multifunction

from S to A, and let u : GrF ! R be a Borel measurable function such that u(s; �) is anupper-semicontinuous function on F (s) for each s 2 S. Then there exists a Borel measurableselector f : S ! A for F such that

u(s; f(s)) = maxa2F (s)

u(s; a)

for all s 2 S. Moreover, the function v de�ned by v(s) = maxa2F (s)

u(s; a) is Borel measurable.

Proof of Proposition 2.(i). By the proof of Proposition 1, k

�(�) 2 (�; C�), we haveZ

�maxek(�)2(�;C�)u(ek(�); �)�(d�) �

Z�u(k

�(�); �)�(d�):

Thus, for all k 2 ICI , Z�u(k

�(�); �)�(d�) �

Z�u(k(�); �)�(d�):

Then, by Proposition 1, ICI = � =S

C2Pf (K)�(C). Hence, for all k 2

SC2Pf (K)

�(C), we have

Z�u(k

�(�); �)�(d�) �

Z�u(k(�); �)�(d�): (4)

Moreover, by Lemma 3, for each C 2 Pf (K), there exists some k0 2 �(C) such that u(k

0(�); �) =

maxek(�)2(�;C)u(ek(�); �) for all � 2 �. Thus, by (3), for each C 2 Pf (K), we haveZ�u(k

�(�); �)�(d�) �

Z�

maxek(�)2(�;C)u(ek(�); �)�(d�):Therefore,

R� maxk2(�;C�)

u(k; �)�(d�) �R� maxek(�)2(�;C)u(ek(�); �)�(d�) for all C 2 Pf (K). Hence,

C� solves the given contracting game over menus.

26

Clearly,

maxC2Pf (K)

Z�

maxek(�)2(�;C)u(ek(�); �)�(d�) =Z�u(k

�(�); �)�(d�) = max

k2ICI

Z�u(k(�); �)�(d�):

(ii). By Lemma 3, for each C 2 Pf (K), there always exists some k such that

u(k(�); �) = maxek(�)2(�;C)u(ek(�); �)for any � 2 �.

Now consider k� 2 �(C�) satisfying k

�(�) 2 argmaxek(�)2(�;C�)u(ek(�); �) for all � 2 �. For each

C 2 Pf (K), by hypotheses,Z�u(k

�(�); �)�(d�) =

Z�

maxek(�)2(�;C�)u(ek(�); �)�(d�)�

Z�

maxek(�)2(�;C)u(ek(�); �)�(d�) =Z�u(k(�); �)�(d�);

for all k 2 �(C) satisfying u(k(�); �) = maxek(�)2(�;C)u(ek(�); �) for any � 2 �. It implies thatZ�u(k

�(�); �)�(d�) = max

k2S

C2Pf (K)�(C)

Z�u(k(�); �)�(d�): (5)

Also, by Proposition 1,

ICI = � =[

C2Pf (K)�(C): (6)

Hence, by (4) and (5), we have

maxC2Pf (K)

Z�

maxek(�)2(�;C)u(ek(�); �)�(d�) =

Z�

maxek(�)2(�;C�)u(ek(�); �)�(d�)=

Z�u(k

�(�); �)�(d�)

= maxk2

SC2Pf (K)

�(C)

Z�u(k(�); �)�(d�) = max

k2ICI

Z�u(k(�); �)�(d�):

Therefore, k�solves the given contracting game over mechanisms. �

Proof of Corollary 1.Given the optimal collective mechanism k� solving P1, Proposition 3 implies there must be

27

a bilateral mechanism k�interim-payo¤-equivalent to k�. ThenZ

�u(k�(�); �)�(d�)

=

Z�(nXi=1

[

miXj=1

(aij(�i)hij(k�ij(�); �i)wij(�))] + L(�))�(d�)

=

Z�(

nXi=1

[

miXj=1

(aij(�i)hij(k�ij(�); �i)wij(�))]�(d�) +

Z�L(�)�(d�)

=

nXi=1

miXj=1

(

Z�(aij(�i)hij(k

�ij(�); �i)wij(�))�(d�)) +

Z�L(�)�(d�)

=nXi=1

miXj=1

(

Z�i

aij(�i)

Z��i

hij(k�ij(�); �i)wij(�)��i(d��ij�i)�i(d�i)) +

Z�L(�)�(d�)

=nXi=1

miXj=1

(

Z�i

aij(�i)

Z��i

hij(k�ij(�); �i)wij(�)��i(d��ij�i)�i(d�i)) +

Z�L(�)�(d�)

=nXi=1

miXj=1

[

Z�(aij(�i)hij(k

�ij(�); �i)wij(�))�(d�)] +

Z�L(�)�(d�)

=

Z�u(k

�(�); �)�(d�):

Thus, bilateral BIC mechanism k�brings to PL the same expected (ex ante) payo¤ as k�

does. Hence, P1 is strategically equivalent to P10 and therefore P2. �

Proof of Corollary 2.By Proposition 3, we can always �nd a bilateral BIC mechanism k interim-payo¤-equivalent

to k�. Due to hypothesis (ii), we have

hij(k�ij(�i); �i) =

Z��i

hij(k�ij(�); �i)��i(d��ij�i) (7)

Hence,

28

Z�u(k�(�); �)�(d�)

=

Z�(nXi=1

miXj=1

(Gij(hij(k�ij(�); �i); �i) + L(�))�(d�)

=

Z�(nXi=1

miXj=1

(Gij(hij(k�ij(�); �i); �i)�(d�) +

Z�L(�)�(d�)

=nXi=1

miXj=1

(

Z�Gij(hij(k

�ij(�); �i); �i)�(d�)) +

Z�L(�)�(d�)

=nXi=1

miXj=1

(

Z�[

Z��i

Gij(hij(k�ij(�); �i); �i)��i(d��ij�i)]�(d�)) +

Z�L(�)�(d�)

�nXi=1

miXj=1

(

Z�i

Gij(

Z��i

hij(k�ij(�); �i)��i(d��ij�i); �i)�i(d�i)) +

Z�L(�)�(d�)

(By Jensen�s inequality.)

=nXi=1

miXj=1

(

Z�i

Gij(hij(k�ij(�i); �i); �i)�i(d�i)) +

Z�L(�)�(d�) (By (6))

=nXi=1

miXj=1

(

Z�i

[

Z��i

Gij(hij(k�ij(�i); �i); �i)��i(d��ij�i)]�i(d�i)) +

Z�L(�)�(d�)

=

Z�u(k

�(�); �)�(d�):

Since k� is the optimal solution to (P1),R� u(k

�(�); �)�(d�) =R� u(k

�(�); �)�(d�). �

Proof of Lemma 1.By Taylor expansion Theorem, for any � 2 �,

G(x; �) = G(0; �) +

nXi=1

[Gi(0; �)xi] +R2;

where R2 = 12

Pni=1

Pnj=1Gij(x

c�; �)xixj , for some xc� in between x and 0.

Clearly in quasi-linear environment given t, then by hypothesis (iv) the di¤erence between

U� and U��

U� � U��

�Z�G(x�(�); �)�(d�)�

Z�G(x�(�); �)�(d�)

=1

2

Z�fnXi=1

nXj=1

Gij(xc�; �)x�i (�)x

�j (�)�

nXi=1

nXj=1

Gij(xd�; �)x�i (�)x

�j (�)g�(d�);

29

for some xc� between x�(�) and 0 and some xd� between x�(�) and 0 for each �.

Due to the optimality, we must have U� � U��. Thus by hypothesis (iii),

U� � U��

� 1

2

Z�fj

nXi=1

nXj=1

Gij(xc�; �)x�i (�)x

�j (�)�

nXi=1

nXj=1

Gij(xd�; �)x�i (�)x

�j (�)jg�(d�)

� 1

2

Z�fj

nXi=1

nXj=1

Gij(xc�; �)x�i (�)x

�j (�)j+ j

nXi=1

nXj=1

Gij(xd�; �)x�i (�)x

�j (�)jg�(d�)

� 1

2

Z�fnXi=1

nXj=1

jGij(xc�; �)x�i (�)x�j (�)j+nXi=1

nXj=1

jGij(xd�; �)x�i (�)x�j (�)jg�(d�)

� 1

2

Z�fnXi=1

nXj=1

M�ij +

nXi=1

nXj=1

M�ijg�(d�)

= M(

nXi=1

nXj=1

�ij)

Clearly, jU��j � minf���� infx2X;�2�

G(x; �) +Pni=1 ti

���� ;����� supx2X;�2�

G(x; �) +Pni=1 ti

�����g;so we can obtainthe second result. �

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