MECHANISM DESIGN WITH LIMITED COMMUNICATION:Implications for Decentralization1

Dilip Mookherjee and Masatoshi Tsumagari2

This Version: June 23 2010

Abstract

We develop a theory of mechanism design in a principal-multiagent set-ting with private information, where communication involves costly delay.The need to make production decisions within a time deadline preventsagents from communicating their entire private information to the principal,rendering revelation mechanisms infeasible. Feasible communication proto-cols allow only finite number of possible messages sent in a finite numberof stages. An extension of the ‘Revenue Equivalence’ Theorem is obtained,and used to show that an optimal production allocation can be computed bymaximizing virtual profits of the Principal subject to communication con-straints alone. In this setting delegation of production decisions to agentsstrictly dominates centralized production decisions, and decentralized com-munication protocols dominate centralized ones. The value of decentralizingcontracting decisions depends on the ability of the principal to verify mes-sages exchanged between agents.

KEYWORDS: communication, mechanism design, decentralization, incentives,principal-agent, organizations

1We thank Ingela Alger, Richard Arnott, Hideshi Itoh, Ulf Lilienfeld, Bart Lipman, MichaelManove, Preston McAfee, Alberto Motta, Andy Newman, Roy Radner, Stefan Reichelstein,Ilya Segal, Hideo Suehiro and seminar participants at Boston College, Brown, Northwestern,Hitotsubashi, Keio, Kobe, Nagoya, Padua, Paris, Stanford and Toulouse for useful comments.

2Boston University and Keio University, respectively

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1 Introduction

This paper develops a theory of mechanism design with limitations on the ca-pacity of agents and Principal to communicate, which prevent implementation of‘complete’ contracts or revelation mechanisms. We derive implications for optimalallocation of control rights over different components of the mechanism — con-tracting, communication and production decisions. The main point of departurefrom standard mechanism design theory is incorporation of constraints on the abil-ity of agents and principal to communicate with one another. As is well known,the existing literature on mechanism design focuses almost entirely on revelationmechanisms. In such mechanisms agents simultaneously send a report of their en-tire private information to the Principal, instead of communicating directly withone another. Agents are not delegated authority over (verifiable) production deci-sions: they await instructions from the Principal on what to do (see, e.g., Myerson(1982)). It is not possible to derive a theory of allocation of control rights in thatsetting as a completely centralized mechanism can always replicate the performanceof any other mechanism. Most organizations involve substantial decentralizationof decision-making and direct, interactive communication among agents. The aimof this paper is to explore circumstances where this can be explained by limits oncommunication abilities on otherwise strategic and rational organization members.Limitations on communication can be justified by the fact that reading and writingmessages are time-consuming tasks, and decisions have to be made under some timedeadlines. This is particularly so when agents (e.g., doctors, engineers or financialanalysts) have technical expertise not shared by the principal (e.g., patients orclients). Another example is the richness of information related to specialization ofagents in certain regions or areas — a diplomat assigned to a certain country, a salesmanager assigned to a particular territory, a real estate agent working in a givenneighborhood, all of whom may not be able to communicate detailed knowledgeof their assigned jurisdictions to a central government or corporate headquarters.In such organizations, therefore, even after the exchange of information agentswill still know more about their respective environments than the Principal or anycentral coordination device. It may then make sense to delegate some decisionsto agents. Such ideas can be traced back to Hayek (1945) as well as the ‘messagespace’ literature (Hurwicz (1960, 1972), Mount and Reiter (1974), Segal (2006)).Most of the preceding references as well as the theory of teams developed byMarschak and Radner (1972), abstract from incentive problems by assuming thatall agents in the organization share the same goals as the Principal. In such a con-

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text, communication limitations typically justify decentralized decision-making, asthey permit production decisions to be based on better information. In the pres-ence of incentive considerations where agents pursue self-interested goals, this isno longer so obvious: decentralization may encourage agents to pursue their ownagendas at the same time that it brings to bear better information on decisions.This creates a possible trade-off between the potential ‘control loss’ or ‘abuse ofpower’ against the ‘flexibility’ advantages of delegation.This paper explores this trade-off in the context of a standard team productionmodel with two agents with independent private costs. Each agent produces aone-dimensional real-valued input at a real-valued unit cost which it is privatelyinformed about. The inputs of the two agents have to be coordinated to jointlyproduce a benefit for the Principal. Agents are provided transfers by the Principalto provide them with suitable incentives to report their costs and produce inputs.Contracts are signed at an ex ante stage, and communication occurs at the interimstage according to a selected protocol, following which production decisions aremade.Our formulation of the communication protocol extends and generalizes those inexisting literature, which includes centralized networks (where all agents communi-cate only with one central agent or Principal), public broadcasting networks (whereeach agent sends the same message simultaneously to all others in the organiza-tion), and decentralized networks (where agents communicate directly with oneanother with privately addressed messages). The formulation is based on the as-sumption that only writing or sending messages is time consuming, while readingor listening to messages received is not. Section 9 explains how the results canbe extended when both reading and writing are time-consuming activities. Theprotocol allows for an arbitrary number of rounds of communication with assignedhistory-dependent message spaces of finite size. More rounds of communicationand larger message spaces would involve greater time delays and resulting loss ofperformance, generating a trade-off between extent of information exchanged anddelays in decision-making. We fix the amount of delay and allow the mechanismdesigner to select a communication protocol which is consistent with it, along withcontracts and allocation of decision rights over production. We obtain compar-isons between different assignment of control rights which do not depend on theparticular level of delay chosen.Our model focuses exclusively on the implications of time limitations on communi-cation at the interim stage — after agents have received their private information,and before production decisions are made. In particular, we abstract from limita-

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tions of rationality of agents, or on the complexity of contracts: we assume there areno time limitations at the ex ante stage when contracts are designed, offered andread. We also abstract from limitations on ex post verifiability of performance andmessages on which transfers are conditioned. Under centralized contracting, agentscannot collude by entering into unobserved side contracts or private communica-tion. Proposition 1 shows that under these assumptions, decentralized contractingcannot outperform centralized contracting.We thereafter explore how the mechanism design problem can be posed within theclass of centralized contracts. Propositions 3 and 4 show how it can be simplifiedin a way analogous to methods used for auction design or related problems withquasilinear utilities in the absence of communicational constraints. A version ofthe ‘Revenue Equivalence’ theorem is obtained, enabling optimal transfers to becomputed as a function of production allocations, so that the problem can becast in terms of choice of the latter alone. Under the standard assumption ofmonotone hazard rates for the cost distributions, the problem reduces to selection ofa production allocation rule to maximize ‘virtual’ expected profit of the Principal,subject only to communicational feasibility restrictions. This implies a convenientseparation of the incentive and communicational aspects of the design: the formerare entirely incorporated in the objective function (in the form of incentive rents ofthe agents which cause actual costs to be inflated to virtual costs). Improvementsin communication technology relax the constraints on the problem, leaving theobjective function unchanged. Communicational algorithms to maximize virtualprofit can thus be devised without worrying about incentive compatibility of thechosen communication strategies.This characterization leads to our main result in Section 7. Delegating the produc-tion decision for each agent to that agent itself is strictly superior to any systemwhere they are set production targets by others, provided the production functionsatisfies Inada conditions. This is a consequence of the characterization of optimalproduction allocations. Owing to the finiteness of the communication protocol,agents are better informed about their own environments at the end of the com-munication stage. Moreover, the interests of agents and the principal are perfectlyaligned by the choice of the incentive mechanism: there is no problem of ‘loss ofcontrol’ associated with delegation. Allocating control rights to agents over theirown production levels then allows a more flexible production allocation, withoutattendant problems of control loss.Another implication concerns the design of communication protocols, discussed inSection 8. Owing to the alignment of incentives, greater ‘flexibility’ of decision-

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making and information flows across agents is desirable. Communication protocolsshould thus be designed to maximize the flow of information across agents. Ifonly the writing or sending of messages is time-consuming, we show that atten-tion can be restricted to completely decentralized communication networks whereagents communicate directly with one another, and there is no centralization ofinformation. Moreover, these strictly dominate completely centralized networksin which agents communicate only with the Principal and not with one another.Centralizing the communication network creates bottlenecks that restrict the flowof information.An implication of these results for organization design is the importance of “mixed”mechanisms in the presence of communication constraints. They indicate the valueof mechanisms that combine centralized contracting with decentralized decision-making and communication networks. Such mechanisms have been emphasized asdistinctive features of ‘Japanese’ firms (see, e.g., Aoki (1990)).Section 9 explains how the results can be extended to the context where agentsare limited in their capacity to read as well as to write messages. Section 10 thendiscusses the implications of limited ability of the Principal to verify messagesexchanged between agents. We provide an example where decentralized contract-ing strictly dominates centralized contracting. Finally, Section 11 concludes bydescribing questions left for future research.

2 Relation to Previous Literature

The classical message-space literature (Hurwicz (1960, 1972), Mount and Reiter(1974)) measured communicational complexity by the dimensionality of messagespace sizes in iterative communication protocols, and posed the question of theminimal dimensionality required to implement given resource allocations. Theyabstracted from incentive considerations. More recent literature uses differentmeasures of communicational complexity (e.g., which incorporate both messagespace size and number of rounds of communication), and asks how the minimalcommunicational complexity required to implement given allocations increases inthe presence of incentive problems (Reichelstein and Reiter (1988), Segal (2006),Fadel and Segal (2009), van Zandt (2007)).An alternative approach is to fix a limited communication protocol and ask whatis the constrained optimal allocation mechanism. The Marschak-Radner theoryof teams can be viewed as belonging to this approach, in the absence of incentive

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considerations. Some authors have attempted to extend this to contexts withincentive problems (Green and Laffont (1986, 1987), Melumad, Mookherjee andReichelstein (MMR) (1992, 1997), Laffont and Martimort (1998)).3 This literature,however, typically imposes severe restrictions on communication protocols. Forinstance, MMR assume only one round of communication, and message sets of given(finite) size. They also compare two specific mechanisms: one where contracting,communication and production decisions are all centralized, with another wherethey are all decentralized.This paper extends the MMR approach to accommodate a wider range of bothcommunication protocols and mechanisms. We allow an arbitrary number of com-munication rounds and message space sizes at each round, and arbitrary communi-cation network structures (i.e., who communicates with whom). We do not restrictthe class of mechanisms either. ‘Mixed’ mechanisms are allowed, where differentcomponents of the mechanism: contracting, communication and production deci-sions can be independently centralized or decentralized. This enlarges the relevantclass of mechanisms considerably, with varying notions of ‘decentralization’.For instance, the owner of a firm can contract with all workers, and make produc-tion decisions personally, but may allow agents to directly communicate with oneanother. Or the owner may additionally delegate production decisions to workerson the assembly line or shop floor. Going one step further, the owner could contractwith a single designated agent or ‘manager’, delegating to this manager the author-ity to contract with the other workers, and make (or further delegate) productionand payment decisions. MMR restricted attention to the two polar mechanismswhere all three components — contracting, communication and decision-making —were centralized or decentralized. In this paper we consider intermediate notionsof decentralization as well.It is also important to emphasize that earlier approaches such as MMR (1992)imposed ad hoc restrictions on the size of message spaces, which were not derivedfrom an explicit theory of limited communication.4 This paper grounds the theoryin an explicit model of communication. This ensures a more careful treatment ofthe comparative performance of centralized and decentralized contracting. Thesource of the superior ‘flexibility’ of decentralized contracting in MMR (1992) wasthe ability of an intermediate ‘manager’ to condition subcontracts offered to sub-

3Deneckere and Severinov (2004) develop a theory of mechanism design where truthful mes-sages can be sent costlessly, while non-truthful messages may entail some cost.

4MMR (1997) was based instead on a restriction on the complexity of contracts, measured bythe number of contingencies incorporated — rather than limitations on communication.

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ordinates on information held privately by the manager at the time of contracting.At the same time the ‘manager’ was assumed unable to communicate a similaramount of information to the principal. Proposition 1 shows that this asymme-try cannot arise in a model of limited communication, provided the Principal canverify ex post messages exchanged by agents.5

3 Model

There is a Principal (P ) and two agents 1 and 2. Agent i = 1, 2 produces a one-dimensional nonnegative real valued input qi at cost θiqi, where θi is a real-valuedparameter distributed over an interval Θi ≡ [θi, θi] according to a positive, contin-uously differentiable density function fi and associated c.d.f. Fi. The distributionsatisfies the standard monotone hazard condition that Fi(θi)

fi(θi)is nondecreasing, im-

plying that the ‘virtual cost’ vi(θi) ≡ θi + Fi(θi)fi(θi)

is strictly increasing. θ1 and θ2 areindependently distributed, and these distributions F1, F2 are common knowledgeamong the three players.The inputs of the two agents combine to produce a gross return V (q1, q2) for P ,whose net payoff is V − t1 − t2, where ti denotes a transfer from P to i. Thepayoff of i is ti − θiqi. All are risk-neutral and have autarkic payoffs of 0. We shallassume that V is twice continuously differentiable, strictly concave and satisfies theInada condition ∂V

∂qi−→ ∞ as qi → 0. We shall also assume that the production

function is non-separable: V12 �= 0 for all q1, q2. This creates a need to coordinateproduction assignments between the agents.A mild variation of the assumptions on the production function is needed to incor-porate the case of a procurement auction design problem (with private values). Ifthe production function V equals V if q1 + q2 ≥ 1 and 0 otherwise, with V suffi-ciently large, the desired sum q1 +q2 will equal 1 irrespective of type realizations ofthe agents. Owing to the linearity of costs, the problem then reduces to selectionof one of the two agents to supply 1 unit of the good. While this production func-tion is not strictly concave, or satisfy the Inada condition, there exist productionfunctions arbitrarily close which do satisfy these conditions. Hence most results ofthe paper extend to the auction setting. The only ones that do not are the strictrankings of decentralized and centralized production decisions and communication

5However, it can be justified if there are restrictions on ex post verifiability of messages. Weprovide an example of this in Section 10.

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networks (Propositions 5 and 6 respectively).

4 Contracting and Communication

4.1 Timing

At t = −1, a centralized contracting mechanism is composed of a communicationprotocol (explained further below) and set of contracts offered by P to both agents.In decentralized contracting, P contracts with one of the two agents (i) and designsa bilateral communication network with i. This is followed by i’s design of acontract for j and a bilateral communication protocol between i and j. There isenough time between t = −1 and t = 0 for all agents to read and understand theoffered contracts.At t = 0, each agent i privately observes the realization of θi, and independentlydecides whether to participate or opt out of the mechanism. If either agent opts outthe game ends; otherwise they enter the planning or communication phase whichlasts until t = T . We treat the deadline T as given; it can be chosen subsequentlyby the organization designer to trade off the cost of delayed production with thebenefit of added communication. Our results do not depend on the specific deadlinechosen.At t = T , each agent i = 1, 2 selects production level qi. This does not necessarilymean that i ‘decides’ qi in any meaningful sense. As discussed further below,someone else may set a target for qi — in the form of a message sent to i bythe target-setter during the communication phase — and the incentive schemefor i may effectively force i to meet this target. Messages exchanged during thecommunication phase thus include ‘instructions’ or ‘targets’ that might be set whenproduction decisions are not decentralized to the concerned agents.Finally, after production decisions have been made, payments are made accordingto the contracts signed at the ex ante stage, and verification by the Principal ofmessages exchanged by agents.

4.2 Communication Technology

First we describe the communication technology, i.e., limitations on communicationcapacities of agents within the time available at the interim stage.

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The primitives consist of: (i) a language for communication, represented by a set Lof possible messages that can be communicated by any agent to any other in anygiven round, and (ii) restrictions on communication capacity at each round whichare explained as follows.Communication takes place in a number of successive rounds t = 1, . . . , T . Eachmessage m in L has an integer-valued length l(m) which describes the amount oftime involved in communicating m. The language includes the null message ∅, i.e.,where no message is sent, and the length of a null message equals zero. It is simplestto consider the case where receiving (reading or listening) the message does notentail any time delay: e.g., in verbal exchanges where speaking is time-consumingand the audience listens at the same time that the speaker speaks. This is theimplicit assumption in most of the existing literature on costly communication. Tokeep the exposition simple, we shall focus on this case. The length of a messagecan thus be interpreted as the time required by the sender to send (e.g., write,speak) it. Section 9 below explains how most of the results extend to the casewhere listening or reading messages is also time-consuming.The key restriction is the following:

Assumption 1. For any finite integer n, the set of messages in L with length notexceeding n is finite.

For instance if communication takes place in binary code and it takes one unitof time to communicate one bit of information, at most a finite number (2n) ofmessages can be sent in n units of time. Alternatively if the language consists of afinite alphabet and each letter or word takes a minimum amount of time to say orwrite, at most a finite number of messages can be sent in finite time.Next, define a communication network as follows. Let I denote the set of all agentsin the organization. In our setting, this is {P, 1, 2}, but the following definitionsapply to larger organizations as well. A communication network N specifies foreach i ∈ I a subset J(i) ∈ I \ {i} of agents to whom i can send messages to.Examples of specific networks in our setting are the following. The centralizednetwork NCC is one where J(1) = J(2) = {P}, J(P ) = {1, 2}, i.e., each agentcommunicates only with P . The completely decentralized network NDD has J(1) ={2}, J(2) = {1}, J(P ) = ∅, i.e., the two agents communicate (only) with oneanother. A decentralized network is one where some agent i sends messages directlyto some other agent j (i.e., j ∈ J(i), i, j �= P ).Let mij denote a message sent by i to j. For any given communication network N ,

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the communication capacity of any agent i ∈ I is a message set Ri(N ) ⊆ L#J(i),which contains all message vectors {mij}j∈J(i) with mij ∈ L that i can feasiblysend in any given round of communication.For example, if the network involves separate messages addressed to different re-cipients, and ni is the maximum length of messages that can be sent by i in a singleround, then Ri(N ) consists of {mij}j∈J(i) such that

∑j∈J(i) l(mij) ≤ ni. Here the

aggregate length of all messages sent to all others by i should not exceed the timeavailable for i to write messages. If on the other hand the communication technol-ogy involves public broadcasts of a common message by each agent to all others inthe organization, the network N has J(i) = I \ {i} for all i, and Ri(N ) consistsof {mij}j∈J(i) satisfying the restriction that mij = mik = mi for all j, k �= i withl(mi) ≤ ni.The communication technology is summarized by the language L for communica-tion, and the communication capacity of each agent, represented by the feasiblemessage sets Ri(N ) for each agent i in any given network N .

Assumption 2. For any communication network N , and for every i ∈ I, thefollowing is true: There exists integer n such that

∑j∈J(i) l(mij) < n for any

message vector mi = {mij} ∈ Ri(N ).

Assumption 2 says there is a finite upper bound to the total length of messagessent by any agent in any round. Combined with Assumption 1 it ensures thatevery participant is restricted to finite message sets.

4.3 Communication Protocol

Given a communication technology, the designer can select a communication pro-tocol, which is a specific network N and rules defining message sets for agentsin any given round which may depend on the history of messages exchanged inprevious rounds. Message histories and message sets are defined recursively asfollows. Let mij,t denote a message sent by i to j in round t. Given a his-tory hi,t−1 of messages exchanged (sent and received) by i until round (t − 1),it is updated at round t to include the messages exchanged at round t: hit =(hi,t−1, {mij,t}j∈J(i), {mki,t}{k:i∈J(k)}). And for every i, hi0 = ∅. The message set fori at round t is then a subset of Ri(N ) which depends on hit.Formally, the communication protocol specifies for every agent i: a message setMi(hi,t−1) ⊆ Ri(N ) for every possible history hi,t−1. Hence protocols include rules

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for the order in which different agents speak and/or respond to others’ messages.Let a protocol be denoted by p, and let the set of protocols be denoted by P .

4.4 Communication Plans and Strategies

Given a protocol p ∈ P, a communication plan for agent i specifies for every roundt a message mit(hi,t−1) ∈ Mi(hi,t−1) for every possible history hi,t−1 that couldarise for i in protocol p until round t − 1. The set of communication plans for i inprotocol p is denoted Ci(p). It is evident from the definition of a feasible protocolthat this is a finite set.Given a protocol p ∈ P, a communication strategy for agent i is a mapping ci(θi) ∈Ci(p) from the set [θi, θi] of types of i to the set of communication plans for i.In other words, a communication strategy describes the dynamic plan for sendingmessages of every possible type of the agent. The finiteness of the set of dynamiccommunication plans implies that it is not possible for others in the organizationto infer the exact type of any agent from the messages exchanged. Perforce, non-negligible sets of types will be forced to pool into the same communication plan.

4.5 Centralized Contracts and Production Decisions

Having completed a description of interim communication, we are now in a positionto explain the nature of contracts and production decisions in centralized contract-ing. Recall that at t = −1, P selects a feasible communication protocol p as well ascontracts offered to two agents. The chosen communication protocol may or maynot entail a centralized (NCC) network. Irrespective of the protocol, P is assumedto be able to verify all messages exchanged until round T by all agents. Even if thecommunication network is decentralized, P is able to verify all messages exchangedbetween the agents. This amounts to assuming absence of collusion among agents.This assumption is undoubtedly restrictive, but a necessary first step in the theory(just as in mechanism design theory with unlimited communication) and is madeby existing literature on costly communication.We also assume that at the ex post stage P costlessly observes inputs qi supplied byeach agent, in addition to the message histories. Hence transfers can be conditionedon production decisions and exchanged messages.In what follows, the history of messages exchanged hiT until the last round T byany agent i will be denoted by hi.

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A centralized contract for i is represented by a transfer rule ti(qi, qj, hi), specifyingtransfers from P to i as a function of productions of the two agents, and messagehistory for i. It can be checked that no benefit can be derived by conditioning thepayment to i on message histories of other agents (i.e., hj).At the deadline T each agent i will select a production level qi. This will dependon its type θi and message history hi. The restrictions on communication forceproduction decisions to depend on ‘coarse’ messages about the state of the otheragent: the production of agent i must depend on θj only through hi. Specifically,agent i’s production level is represented by qi(θi, θj) = qi(θi, hi(ci(θi), cj(θj), cP )).They can be fine-tuned to information about i’s own cost θi, which constitutes thepotential ‘flexibility’ advantage of delegating production decisions.Formally, we shall say that the production decision qi(θi, θj) is centralized if it ismeasurable with respect to hP (c1(θ1), c2(θ2), cP ), and partially decentralized if itis measurable with respect to hj(c1(θ1), c2(θ2), cP ), j �= i, P but not with respectto hP (c1(θ1), c2(θ2), cP ). In these cases, production decisions concerning qi can bethought of as being made by P or j respectively.In contrast, the production decision qi is said to be completely decentralized if itis not the case that qi(θi, θj) is measurable with respect to hP (c1(θ1), c2(θ2), cP )or hj(c1(θ1), c2(θ2), cP ), j �= i, P . The production decision of i cannot then bepredicted by P or j at t = T based on the messages they have sent or read.In other words it is not possible that agent i is assigned a production target bysomeone else, combined with an incentive scheme that forces i to abide by thetarget. Instead, i will make the production decision personally, a choice that maybe influenced, though not completely determined, by messages sent by others thatenter as arguments of the incentive scheme.

4.6 Decentralized Contracting

In decentralized contracting, P contracts with the manager (agent i), who subse-quently contracts with the employee (agent j). The communication network is alsohierarchical: the employee communicates with the manager, and the manager withP . Contracts are offered at t = −1: P offers a contract for i and selects an ‘upper-level’ communication protocol pU between herself and the manager. The managerthen offers a subcontract to j, and selects a ‘lower level’ communication protocol pL

between i and j. The set of communication protocols consistent with decentralizedcontracting do not allow any direct communication between the employee j and

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the owner P . Each of them communicates only with the manager i. Moreover, Pdoes not monitor communication between the manager and the employee, neitherdoes the employee monitor communication between the manager and P .Formally, let PN denote the set of feasible communication protocols in networkN . Let U denote the upper-level network defined by J(i) = P, J(P ) = i, L thelower-level network defined by J(i) = j, J(j) = i, and H the resulting hierarchicalnetwork with J(i) = {P, j}, J(j) = J(P ) = {i}. Then P selects an upper-levelcommunication protocol pU ∈ PU . Given pU , agent i is constrained to choose aprotocol for the lower level network from the set

PL(pU) ≡ {pL ∈ PL | (pU , pL) ∈ PH}where (pU , pL) denotes the protocol for H obtained by combining pU and pL.It is evident that any protocol feasible under decentralized contracting is also fea-sible under centralized contracting, while the converse is not true. In centralizedcontracting, P has the option of selecting the same hierarchical communicationprotocol as in decentralized contracting. However, decentralized contracting mustnecessarily involve a hierarchical protocol, whereas centralized contracting is notso constrained.At t = −1, in addition to the selection of pU , P offers a contract to agent i based onobserved production and messages exchanged between P and i. P may be able tomonitor the inputs supplied by each agent separately, as in centralized contracting.P may also be able to monitor the payments made by i to j in the subcontract (e.g.,if i is the manager of a division of a firm owned by P , and j is an employee of thatdivision). This corresponds to observability of ‘cost’ incurred by the manager. Inwhat follows we consider the scenario most favorable to decentralized contracting,where both production assignments and transfers between i and j are contractible.We show below that any allocations attainable under decentralized contracting willbe attainable under centralized contracting as well; a fortiori the same will be trueif contracting options under decentralization are more restricted.A decentralized contract between P and the manager is a transfer rule ti = ti(q1, q2, tj, h

Ui )

for payments made by P to i, as a function of production levels, observed ‘cost’or transfer payment made by i to j, and hU

i which – denotes messages sent (orreceived) by i to (or from) P during the communication phase.6

The subcontract offered by the manager i to the employee j specifies transfers6Note that by the definition of message histories, hU

i = hP , i.e., message histories are the samefor i and P in pU .

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tj from i to j as a function of q1, q2 and messages exchanged on the lower levelnetwork hL

j (= hLi ). Here hL

j and hLi denotes messages sent or received by j and i

respectively among one another.Production decisions qi, qj are made at t = T by i, j respectively, based on their per-sonal information at that point. The manager decides qi(θi, hi) where hi ≡ (hU

i , hLi ),

and the employee decides qj(θj, hj) where hj = hLj . Production decisions may be

centralized or decentralized, as in the centralized contracting regime. The same for-mal definitions of (completely, partially) decentralized and centralized productiondecisions apply here as in the centralized contracting regime.

5 Allocations Attainable under Centralized andDecentralized Contracting

Given a particular set of contracts offered by P in centralized contracting and agiven communication protocol p ∈ P, we have a well-defined Bayesian game ofincomplete information.An allocation attainable under centralized contracting is a state-contingent produc-tion and transfer rule qa(θ1, θ2), ta(θ1, θ2), a = 1, 2 such that there exists a commu-nication protocol p ∈ P, centralized contracts ta(q1, q2, ha), a = 1, 2, and an asso-ciated tuple of communication and production strategies c(.) ≡ {ci(θi), cj(θj), cP }and q(.) ≡ {qi(θi, hi), qj(θj, hj)} which constitutes a Perfect Bayesian Equilibrium(PBE) of the corresponding subgame, such that for any state θ ≡ (θ1, θ2) and anya = 1, 2:

qa(θ) = qa(θa, ha(c(θ))) (1)ta(θ) = ta(q1(θ), q2(θ), ha(c(θ))) (2)

Under decentralized contracting, a different Bayesian game is induced by choiceof a contract for the manager and ‘upper-level’ communication protocol pU by P .Agent i, the manager, must select contracts for agent j, a communication protocolpL ∈ PL(pU), and a communication strategy ci(θi) to be executed by i duringthe communication phase. Production decisions and the strategies of agent j aresimilar to what they are under centralized contracting.An allocation attainable under decentralized contracting is a state-contingent pro-duction and transfer rule qa(θ1, θ2), ta(θ1, θ2), a = 1, 2 such that there exists a con-tract ti(q1, q2, tj, h

Ui ) and communication protocol pU ∈ PU , and a Perfect Bayesian

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Equilibrium (PBE) of the associated subcontracting subgame consisting of a sub-contract offered by i: tj(q1, q2, hj), a communication protocol pL ∈ PL(pU), a tupleof communication strategies c(.) ≡ {ci(θi), cj(θj), cP } and production strategiesq(.) ≡ {qi(θi, hi), qj(θj, hj)}, such that for any state θ ≡ (θi, θj):

qa(θ) = qa(θa, ha(c(θ))), a = 1, 2ti(θ) = ti(q1(θ), q2(θ), tj(θ), hU

i (c(θ))) − tj(θ)tj(θ) = tj(q1(θ), q2(θ), hj(c(θ)))

The following result is a trivial consequence of the fact that decentralized con-tracting involves a restricted set of communication protocols relative to centralizedcontracting.

Proposition 1 Any allocation attainable under decentralized contracting can alsobe attained under centralized contracting.

Proof. Consider an allocation qa(θ), ta(θ), a = i, j attained by decentralizedcontracting with protocols pU , pL at the two layers, transfer rules ti, tj, communi-cation and production strategies c, q1, q2. Recall that the communication protocolp ≡ (pU , pL) is feasible in centralized contracting. Recall also the assumption thatP can verify all messages sent and received by all agents in centralized contract-ing. Therefore hj is verifiable by P in centralized contracting. So P can select theprotocol p, and the following contracts:

tj(q1, q2, hj) = tj(q1, q2, hj)ti(q1, q2, h

Li , hU

i ) = ti(q1, q2, tj(q1, q2, hj), hUi ) − tj(q1, q2, h

Li )

Then from t = 0 the continuation game involving choice of communication andproduction strategies by the agents is the same as under decentralized contracting,so the same strategies constitute a PBE of this game.

The argument of Proposition 1 resembles that underlying the Revelation Principle:under identical communication and contracting constraints, centralized contractscan be designed by the Principal to duplicate any mechanism based on decen-tralized contracts. It confirms the notion that delegation of subcontracting offersno advantages over centralized contracting, once we are careful to incorporate the

15

communication inherent in the act of selecting and offering a subcontract by the‘managing’ agent. In contrast, the superiority of decentralized contracting in MMR(1992) owed to the implicit assumption that agent i can offer a subcontract con-tingent on the true realization θi of its type. In that case the subcontract has tobe offered at the interim stage. This requires agent i to be able to communicatethe realization of a real-valued variable in finite time, which is not feasible in oursetting.However, the argument underlying Proposition 1 utilizes the assumption that Pcan costlessly verify ex post messages exchanged by the agents. The role of thisassumption will be discussed in Section 10.

6 Characterizing Optimal Allocations in Central-ized Contracting

Proposition 1 pertains to allocation of control over contracting, but says nothingabout control over production, or the design of communication. Having designedcontracts, P needs to decide whether to retain control rights over production aswell. For instance, following exchange of messages with the agents, should thePrincipal set production targets? Or should the Principal let the agents decidewhat to produce, under the influence of an incentive mechanism where transfersdepend on outputs and exchanged messages?In order to address this, we need to make some progress in characterizing optimalallocations subject to the communication restrictions.One restriction on output allocations is obvious: the output of any given agentcan depend on information about the type of the other agent only in a coarsemanner, on the basis of exchanged messages. The finiteness of the message spacesimplies that an agent cannot signal its entire private information to others. Thisis represented in the following notion of communication-feasibility.

Definition 1 The output allocation (q1(θ1, θ2), q2(θ1, θ2)) is communication feasi-ble if and only if there exists p ∈ P, c(θ) = (ci(θi), cj(θj), cP ) ∈ C(p) and qi(θi, hi)so that for all (θi, θj):

qi(θi, θj) = qi(θi, hi(c(θi, θj)))

Of course, i’s production qi can depend finely on θi , provided this decision is (com-pletely) decentralized to i. Whereas if the Principal makes production decisions

16

then both q1 and q2 will depend coarsely on θ1, θ2. The key question concernswhich system is in the Principal’s ex ante interest.The first step of the analysis is to note a ‘rectangle property’ that holds for anyfeasible communication protocol. The proof of this and of all subsequent results isprovided in the Appendix.

Lemma 1 [Rectangle property]: Consider any communication protocol p ∈ P.Let Hit(p) denote the set of possible message histories realized by i in this protocoluntil round t. Then for any hit ∈ Hit(p):

{c ∈ C(p) | hit(c) = hit}is a rectangle set in the sense that if hit(ci, c−i) = hit(c

′i, c

′−i) = hit for (ci, c−i) �=

(c′i, c

′−i), then

hit(c′i, c−i) = hit(ci, c

′−i) = hit

Lemma 1 has the following implication. Consider any history hit observed by iuntil t. Then (given knowledge of the communication plan cP of the Principal),the set of possible configurations of communication plans of the two agents thatcould have generated hit can be expressed as the (Cartesian) product of C1(hit)and C2(hit), where Ca(hit) is the smallest subset of Ca(p), a=1,2, that is consistentwith hit. So defining

Θa(hit) ≡ {θa|ca(θa) ∈ Ca(hit)}it follows that the set of types (θ1, θ2) that could have generated the history hit

can be expressed as the Cartesian product of subsets Θ1(hit), Θ2(hit). We note thisformally below.

Lemma 2 Given a communication protocol p ∈ P, a configuration of commu-nication strategies (c1(θ1), c2(θ2), cP ) ∈ C(p), and any history hit that could begenerated by these strategies for i until round t, the set of possible types that couldhave generated this history can be expressed as

{(θ1, θ2) | hit(c(θ1, θ2)) = hit} = Θi(hit) × Θj(hit). (3)

Hence upon observing the history hit, all types θi ∈ Θi(hit) of agent i will updatetheir prior beliefs about θj in the same way, i.e., by conditioning on the event thatθj ∈ Θj(hit).

17

Another implication of the rectangle property will prove useful later.7

Lemma 3 Without loss of generality, the Principal can restrict attention to com-munication protocols p ∈ P with the following properties: (i) #CP (p) = 1, and (ii)for every possible communication plan ca ∈ Ca(p) for agent a ∈ {1, 2}, there existstype θa such that ca = ca(θa).

Part (i) states that the Principal (who has no private information by assumption)commits to a specific communication plan through the design of communicationprotocol. To simplify exposition we shall often suppress cP , the communicationplan of the Principal, since it is interpreted as part of the mechanism and can betaken as given by the agents.Part (ii) states that attention can be restricted to protocols with no off-equilibriumor unused communication plans. This is because unused communication plans canbe deleted, without affecting the set of histories generated by the chosen com-munication strategies. This simplifies the representation of incentive constraintssubsequently: the Principal needs only to deter deviations by any type θi ∈ Θi(hit)following history hit, to mimicking the communication plan chosen by some otherθ′

i ∈ Θi(hit) who has pooled so far with θi. This is noted in the following result, aversion of which appears also in Fadel and Segal (2009, Proposition 3).

Proposition 2 An allocation (ti(θi, θj), qi(θi, θj)) is attainable under centralizedcontracting if and only if:

(i) There exists communication protocol p ∈ P with production strategies qi(θi, hi),and communication strategies ci(θi) ∈ Ci(p), i = 1, 2, such that for all (θi, θj):

qi(θi, θj) = qi(θi, hi(ci(θi), cj(θj)))

(i.e. qi(θi, θj) is communication feasible) and every communication plan isused by some type:

Ci(p) = {ci(θi) | θi ∈ Θi}.

(ii) Defining the set of possible histories

Hit(p) ≡ {hit(c(θ)) | θ ∈ Θ1 × Θ2}7It may be useful to clarify the precise meaning of the statement ‘without loss of generality’

in the following Lemma. It means that given any protocol and communication strategy vector,there exists another protocol and communication strategy vector satisfying the properties listedin the Lemma which generates the same outcomes.

18

that could be generated under p, the following incentive constraint holds forany t ∈ {0, 1, 2, ...T}, any hit ∈ Hit(p) and any θi, θ

′i ∈ Θi(hit):

Eθj[ti(θi, θj) − θiqi(θi, θj) | θj ∈ Θj(hit)]

≥ Eθj[ti(θ

′i, θj) − θiqi(θ

′i, θj) | θj ∈ Θj(hit)]

(iii) Eθj[ti(θi, θj) − θiqi(θi, θj)] ≥ 0 for any θi

As Fadel and Segal (2009) explain, this result indicates a trade-off between com-munication efficiency and incentive constraints. To render feasible any productionallocation that depends more finely on the realized types of agents, communica-tion protocols with more stages or larger message spaces are needed. These tendto impose more incentive constraints: enlarging the number of stages or rangeof message options at any stage is associated with a corresponding enlargementof the number of incentive constraints that the allocation must respect. For in-stance, along an iterative process of communication agents learn something aboutthe types of other agents; this should not distort their subsequent communicationor output decision choices.We now arrive at the main result of this section, a characterization of outputallocations that are attainable under centralized contracting (in combination withsome transfer rules), and the associated maximum ex ante profit for the Principal.It is a generalization of the characterization underlying the Revenue EquivalenceTheorem in auction theory (Myerson (1981)).

Proposition 3 The output allocation qi(θi, θj) is attainable under centralized con-tracting (in combination with some transfer rule) if and only if

(i) qi(θi, θj) is communication feasible.

(ii) If the associated communication protocol and strategies that implement qi(θi, θj)generate the set of message histories Hit for each agent i at date t, thenEθj

[qi(θi, θj) | θj ∈ Θj(hit)] is non-increasing in θi on Θi(hit) for any hit ∈ Hit

and any t.

The maximum expected payoff for P which implements qi(θi, θj) is

E[V (qi(θi, θj), qj(θi, θj)) − vi(θi)qi(θi, θj) − vj(θj)qj(θi, θj)] (4)

19

The proof of Proposition 3 is long and detailed, and is presented in the Appendix.The necessity of conditions (i) and (ii) follows straightforwardly from the neces-sity of the corresponding conditions (i) and (ii) in Proposition 2. The sufficiencyargument presents a number of complications. One has to construct a rule fortransfers that will preserve the dynamic incentive constraints ((ii) in Proposition2) at every date and following every message history. Moreover, the set of typesΘ(hit) pooling into message history hit need not constitute an interval. The mono-tonicity property (ii) for output decisions holds only ‘within’ Θ(hit), which mayspan two distinct intervals. The monotonicity property may therefore not hold fortype ranges lying between the two intervals, which complicates the conventionalargument for construction of transfers that incentivize the output allocation.To overcome these problems, previous versions of this paper imposed strong dis-tributional conditions (such as exponentially distributed types) to ensure that theadditional dynamic incentive constraints do not bind. The construction used in theproof of Proposition 3 however works quite generally, allowing us to dispense withany special distributional conditions (apart from the monotone hazard rate condi-tion standard in this class of models). It is based on a construction of a relatedoutput allocation which coincides with the actual allocation on equilibrium-pathhistories, and is globally non-increasing (with respect to types whose communica-tion strategies could be mimicked on or off the equilibrium path). The constructionis involved as it has to ensure that this property holds at every date and followingevery history. The transfer rule is designed to implement this related output allo-cation subject to incentive and participation constraints at minimum cost, as in theconventional analysis of private value auction problems. The global monotonicity(in combination with the monotone hazard rate property of the cost distribution)ensures the incentive compatibility of the rule, while the fact that the rule agreeswith the actual output allocation on the equilibrium path ensures that it enablesthe Principal to realize her virtual expected profit.Proposition 3 implies the mechanism design problem can be simplified as follows.

Proposition 4 The mechanism design problem can be represented as choice of acommunication protocol, communication strategies and output decision strategiesto maximize the Principal’s ‘virtual’ profit (4) subject to communication feasibilityalone.

In the case of unlimited communication, this reduces to the familiar propertythat an optimal output allocation can be computed on the basis of unconstrainedmaximization of expected virtual profits.

20

The proof of Proposition 4 relies on the argument that the solution to the problemof maximizing expected virtual profit subject to communication feasibility alone(i.e., ignoring the monotonicity constraint (ii) in the statement of Proposition 3)must have the following property. At every date and following every possible mes-sage history, the continuation communication strategy of any agent must maximizethe expected virtual profit (conditional on the information revealed by the messagehistory so far), given the communication and production strategy of others. In par-ticular, it should not be possible for the agent to increase (conditional expected)virtual profit by switching to the continuation communicational strategy used bysome other type with whom the agent has pooled so far. A standard ‘revealedpreference’ argument then implies (given the monotone hazard rate property fordistribution of types) that the monotonicity constraint (ii) in Proposition 3 mustbe satisfied. Hence this constraint cannot bind, and can be dropped from thestatement of the problem.Proposition 4 implies a simple and convenient separation between costs imposed byincentive considerations, and those imposed by communicational constraints. Theformer is represented by the replacement of production costs of the agents by theirincentive-rent-inclusive virtual costs in the objective function of the Principal, inexactly the same way as in a world with costless, unlimited communication. Thecosts imposed by communicational constraints are represented by the restrictionof the feasible set of output allocations, which must now vary more coarsely withthe type realizations of the agents. This can be viewed as the natural extension ofMarschak-Radner characterization of optimal team decision problems to a settingwith incentive problems. In particular, the same computational techniques canbe used to solve these problems both with and without incentive problems: onlythe form of the objective function needs to be modified to replace actual costs byvirtual costs. The ‘desired’ communicational strategies can be rendered incentivecompatible at zero additional cost.Van Zandt (2007) and Fadel and Segal (2009) discuss the question of ‘communica-tion cost of selfishness’, which relates to a different notion of separation betweenincentive and communicational complexity issues. In their context they take anarbitrary social choice function (allocation in our notation) and examine whetherthe communicational complexity of implementing it is increased by the presence ofincentive constraints. If not, communicational and incentive aspects of the prob-lem can be separated in the sense that optimal communication protocols can bedesigned independently of incentive considerations. In our context we fix communi-cational complexity and select an allocation to maximize the Principal’s expected

21

profits (the exact representation of which depends on whether or not incentiveproblems are present). We do not know if there is a connection between the twoseparation properties.

7 Optimality of Decentralized Production Deci-sions

The preceding characterization of the mechanism design problem allows us to provethat an optimal mechanism must completely decentralize production decisions toagents, provided the production function satisfies Inada conditions. The intuitivereason is that with incorporation of agents’ information rents in the objective func-tion, the incentives of agents and the Principal are aligned. Hence delegation ofproduction decisions allows greater ‘flexibility’ with respect to their cost realiza-tions. Since virtual costs vi(θi) are strictly increasing in θi, and agent i alwaysproduces a strictly positive level of output, an optimal mechanism entails outputdecisions which are strictly decreasing in θi, conditional on any given message his-tory. This can be achieved only if the decision over qi is delegated to i, since onlyi knows the exact realization of θi.

Proposition 5 In any optimal mechanism, production decisions must be com-pletely decentralized.

The logic of the argument is as follows. The finiteness of the set of feasible commu-nication plans for every agent implies the existence of non-negligible type intervalsover which communication strategies and message histories are pooled. Conse-quently if someone else in the organization decides the production level for agent i,the production decision for i must be pooled in the same way. Instead if productiondecisions are left to agent i, the production decision can be based on the agent’sknowledge of its own true type. This added ‘flexibility’ will allow an increase inthe Principal’s ‘virtual’ profit (4) while preserving communication feasibility, andthe result then follows upon using Proposition 4.

8 Decentralization of Communication

Proposition 4 also has useful implications for the ranking of different communi-cation protocols. Given any set of communication strategies in a given protocol,

22

in state (θi, θj) agent i learns that θj lies in the set Θj(hi(ci(θi), cj(θj))), whichgenerates an information partition for agent i over agent j’s type.Say that a protocol p1 ∈ P is more informative than another p2 ∈ P if for anyset of communication strategies in the former, there exists a set of communicationstrategies in the latter which yields (at date T ) an information partition to eachagent over the type of the other agent which is more informative in the Blackwellsense in (almost) all states of the world.It then follows that a more informative communication protocol permits a widerchoice of communication feasible output allocations. Hence Proposition 4 impliesthat the Principal prefers more informative protocols. She has no interest in block-ing the flow of communication among agents. This result in turn has interestingimplications for the ranking of centralized and decentralized communication pro-tocols.The following result applied to the case when only the writing of messages is time-consuming but reading is not. This is represented by the following assumption.We use the notation S(mi) to denote the set of non-null messages in a messagevector mi sent by i in any given round.

Assumption 3. Let N ≡ {J(i)}i∈I and N ≡ {J(i)}i∈I be two communicationnetworks, in which mi ≡ {mij ∈ L}j∈J(i) and mi ≡ {mij ∈ L}j∈J(i) are respectivemessage vectors for i, such that S(mi) = S(mi). Then mi ∈ Ri(N ) impliesmi ∈ Ri(N ).

Assumption 3 says that if two message vectors contain the same set of non-nullmessages, feasibility of one implies the other is feasible also, irrespective of whothey happen to be addressed to. For if the same set of messages happen to beaddressed differently, the time and effort required to write the message remainsunchanged. It embodies the assumption that only the writing of messages is time-consuming. If reading messages were also to consume time, this assumption maynot be valid, as the redirection of messages may over-burden some recipients whoend up having more messages to read than they have time for.

Proposition 6 Suppose Assumption 3 holds. Then:

(i) A completely decentralized communication network attains optimal profits forthe Principal in centralized contracting.

(ii) A completely decentralized communication network allows the Principal to

23

earn a strictly higher profit than any completely centralized communicationnetwork.

The main idea underlying this is the following: the formal proof is in the Ap-pendix. For (i), given any communication protocol, we can construct a completelydecentralized protocol which generates at least as high a level of expected profit,as follows. In the decentralized protocol, let the two agents send directly to eachother the entire set of messages that they were sending out to all network par-ticipants in the original protocol. The same messages are being sent, except thatthey are addressed differently: messages previously sent to the Principal are nowsent to other agents instead. Assumption 3 ensures that this is consistent withthe communication capacity of both agents. Each agent is then at least as wellinformed as in the original protocol, since it now directly receives messages pre-viously addressed by other agents to the Principal in addition to those that theyused to get directly from one another.The strict inferiority of a completely centralized protocol, asserted in part (ii)above, follows from the observation that in any such protocol there can be no use-ful transmission of messages at the last round T from the agents to P , since thereare no subsequent rounds for P to send follow-on messages to the agents whichprovides them additional information. Hence in the completely decentralized pro-tocol constructed in the proof of part (i)— wherein agents directly send each otherthe messages they previously sent to the Principal — there will be no messagesexchanged at the last round. The messages that P would be sending at the lastround in the completely centralized protocol would be based on messages sent inearlier rounds by the two agents to P . These would have already been exchangeddirectly by the agents in the constructed decentralized protocol in previous rounds.Hence there is scope for additional information to be exchanged at the last roundin the constructed protocol, which allows the agents to better coordinate theirproduction decisions.

9 Communication with Limited Reading Capac-ity

We now explain we can extend the model and most of the preceding results to thecase where both writing and reading messages are time-consuming activities.The notion of communication capacity of a participant now has to incorporate

24

time required to both read and write messages. The time to read pertains tomessages received from others. So communication feasibility can no longer bedefined independently for different agents. There must be enough time for messagesboth to be written by their senders, and read by their receivers.Formally, given a language L and a network N ≡ {J(i)}i∈I , communication fea-sibility is now a property of a configuration of message sets {Ri(N )}i∈I , one foreach member of the organization. In the case of privately addressed messages, theproperty is the following. Let r(m) denote the time taken by any participant toread a message m ∈ L, and let mi ≡ {mij}j∈J(i) denote a message vector sentby i in any round. Then every message configuration {mi}i∈I ∈ {Ri(N )} mustsatisfy the restriction that

∑j∈J(i) l(mij) +

∑j:i∈J(j) r(mji) ≤ ni for every i ∈ I.

In the case of publicly addressed messages (where each participant i sends a mes-sage mi ∈ L to all others in the organization), the condition required is insteadl(mi) +

∑j �=i r(mj) ≤ ni.

With this re-interpretation of communication feasibility, all our results continueto go through, with the exception of those concerning ranking of centralized anddecentralized communication networks. Part (i) of Proposition 6 may no longerapply, since the Principal may play a useful role in helping process information tobe exchanged by the agents.However, Part (ii) of Proposition 6 will apply to a number of contexts where readingmessages is also time-consuming. For instance, it extends to the case where eachagent is at least as efficient as the Principal in reading messages. The constructeddecentralized protocol involves agents rather than the Principal reading messages,which preserves feasibility of the communication. We conjecture the result extendsto more general settings.

10 Limited Verifiability of Messages

Now consider the implications of limited ability of the principal to verify messagesexchanged between agents. Proposition 1 relied on the assumption of perfect veri-fiability. We provide an example here indicating how this result no longer extendswith limited verifiability.Consider the example of a procurement auction, where V equals some large numberV if q1 + q2 ≥ 1, and 0 otherwise. Suppose additionally that θi is uniformlydistributed on the unit interval. This reduces to a symmetric private value auction.

25

Assume also that P does not have any capacity to communicate with the agents atthe interim stage. At the same time, agents have a large capacity to communicatedirectly with one another. This capacity is large enough that the Principal’s payofffrom decentralized contracting can be approximated closely enough by what couldbe achieved with unlimited communication among agents (i.e., where each agentcan send a real-valued report of his own cost to the other agent at the interimstage).In this setting, decentralized contracting approximately achieves the second-bestoutcome. The outcome is close to what would be achieved with unlimited commu-nication, which is the following ‘cost center’ arrangement (applying MMR (1992)):P offers a contract t1 = K1 + t2

2 to agent 1, which induces the latter to offer asubcontract to agent 2 with a linear price of θ1 for each unit of output delivered by2, upto a maximum of one unit. (With large but finite communication capacity,agent 1 will offer a price which is a suitable discrete approximation of θ1.) This isaccepted by 2 if θ2 < θ1. It is not accepted if θ2 > θ1, whence agent 1 producesthe entire amount herself. The constant K1 can be calibrated to ensure that thesecond-best profit is attained by P .Now consider centralized contracting, where P has no ability to communicate withthe agents at the interim stage, nor verify messages exchanged by the agents.As before, we assume the agents cannot side-contract with one another. If Pselects a centralized communication protocol, the two agents will not be able tocommunicate at all at the interim stage (since P is unable to communicate withthem, and all messages between the agents have to be routed through P ). Thenthe coordination of production necessary to achieve the second-best cannot beattained.On the other hand, if P designs a decentralized communication protocol, the agentscan communicate with one another, and finely coordinate their production deci-sions. However, if P cannot verify the messages exchanged, the transfers can-not be conditioned on messages. They will depend on observed production levelsalone. But these production levels cannot reveal the information necessary to de-sign second-best transfers. For instance, consider two states (θ1, θ2) and (θ1, θ

′2)

where θ1 < θ2 < θ′2. If the second-best could be achieved (or approximated closely

enough), agent 1 must produce 1 and agent 2 will produce nothing in both states.But the second-best transfer to agent 1 must vary between the two states. This isimpossible if P can neither communicate with the agents nor verify messages ex-changed between the two agents. Hence the result of Proposition 1 will be reversedin this setting.

26

The superiority of decentralized contracting here rests on its enabling the design ofcontract offers to agent 2 that are sensitive to agent 1’s information. Centralizedcontracting cannot allow such flexibility, since P can neither communicate with 1,nor later verify messages that might be sent by 1 to 2 at the interim stage. In thissetting, the argument of MMR (1992) is vindicated.Two points deserve to be noted in light of this example. First, it could be arguedthat the shortcoming of centralized contracting is that it does not allow any side-contracting between the agents. It is true that centralization with side-contractingcan always perform at least as well as decentralized contracting. This follows fromthe fact that decentralized contracting is a special case of centralized contractingwith side-contracting, where P does not contract or communicate with one of theagents. In that sense the comparison between centralization and decentralizationof contracting is a trivial consequence of the definition of these two respectivecontracting regimes. The less trivial question concerns the relative performance of‘pure’ centralized contracting sans side-contracts, with decentralized contracting.Second, the superiority of the decentralized regime in the example arises from itsability to condition the subcontract for agent 2 on agent 1’s cost realization. Ifthe subcontract is enforced by a third party, the latter has to be able to verify theconditions of this subcontract, which is tantamount to verifying a real-valued costreport made by agent 1. It is then not clear why the centralized regime cannot relyon such a third party to enforce a similar contract mandated by the Principal andconditioned on a cost report by agent 1. In other words, the centralized regimeturns out to be inferior owing to a limitation on verifiability of messages relativeto the decentralized regime. If instead the two regimes have the same capacityto verify messages, they would permit contracts to be based on similar contingen-cies. It is then likely that centralized contracting will again be able to replicatethe performance of decentralized contracting. The argument for decentralized con-tracting must be based then on an innate comparative advantage with regard tothe verification ‘technology’.

11 Summary and Concluding Comments

This paper developed a theory of mechanism design for a production team in a con-text where agents and Principal have limited capacity to communicate with oneanother. The main finding is that the argument for decentralization of decision-making based on limitations on ability of agents to communicate, extends to set-

27

tings with an incentive problem. As is well-known, in a context of unlimited com-munication and commitment ability of the Principal, attention can be focused onrevelation mechanisms every aspect of which — contracting, production decisionsand communication — are centralized. This flies in the face of pervasiveness ofdelegation of decision-making from owners of firms to managers, or customers toprimary contractors or trading intermediaries. Previous attempts to adapt mech-anism design theory to contexts of limited communication (such as MMR (1992))in order to explain the value of decentralized mechanisms were based on a numberof ad hoc assumptions.The approach followed in this paper avoided imposing ad hoc restrictions on theclass of mechanisms. A partial analogue of the ‘Revelation Principle’ was obtained,under the assumption of perfect ex post verifiability of messages: attention can berestricted to centralized contracting mechanisms. However, under weak conditionson the technology (‘essentiality’ of both agents), production and communicationsystems must be decentralized. This helps clarify the nature of the precise ar-gument for decentralization, i.e., under the stated assumptions that it pertainsto production decisions and communication, rather than contracting rights. Weshowed by an example that decentralized contracting may outperform centralizedcontracting if the principal has limited ability to verify messages, as well as com-municate directly with them.In general, we obtain the following insight concerning relative advantages of cen-tralized and decentralized contracting. Limited verifiability of variables requiredfor P to evaluate the performance of agents in centralized contracting — eithermessages exchanged between agents, or their respective contributions to team out-put — constitutes a drawback relative to decentralized contracting. This has to betraded off against the possible ‘loss of control’ resulting in decentralized contract-ing. Future research needs to explore this trade-off in further detail, going beyondthe simple example described in Section 10.We also hope that the approach of this paper can form a foundation for posing arange of ancillary questions concerning organizational design. Under what condi-tions does the presence of a third party facilitate communication and coordinationamong production agents? This would provide insight into the role of managerswho do not participate in production activities, whose only role is to process infor-mation communicated by production agents and help formulate production plans.In the model presented here, such third party ‘coordinators’ would have no roomfor strategic behavior, owing to the assumption of absence of collusion. If the modelwere extended to accommodate collusion, it would lead to a theory of hierarchies

28

where intermediaries not directly involved in production play a coordinating roleand behave strategically.Another question pertains to the effects of changing communication technologyon organizational design. Comparative statics of such a model with respect toinformation technology could generate predictions that could be tested againstempirical patterns of how these have been changing in recent times (a brief overviewof which is provided in Mookherjee (2006)).

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Appendix: Proofs

Proof of Lemma 1: The proof is by induction. Note that hi0(c) = φ for any c,so it is true at t = 0. Suppose the result is true for all dates up to t − 1, we shallshow it is true at t.

Note thathit(ci, c−i) = hit(c

′i, c

′−i) = hit (5)

implieshiτ (ci, c−i) = hiτ (c

′i, c

′−i) = hiτ (6)

for any τ ∈ {0, 1, .., t − 1}. Since the result is true until t − 1, we also have

hiτ (c′i, c−i) = hiτ (ci, c

′−i) = hiτ (7)

for all τ ≤ t−1. So under any of the configurations of communication plans (ci, c−i),(c′

i, c′−i), (c′

i, c−i) or (ci, c′−i), member i experiences the same message history hi,t−1

until t − 1. Then i has the same message space at t, and (5) implies that i sendsthe same messages to others at t, under either ci or c

′i.

(6) and (7) also imply that under either c−i or c′−i, others send the same messages

to i at all dates until t − 1, following receipt on the (common) messages sent byi until t − 1 under these different configurations. The result now follows from thefact that messages sent by others to i depend on the communication plan of i onlyvia the messages they receive from i. So i must also receive the same messages att under any of these different configurations of communication plans.

Proof of Proposition 2: Necessity follows straightforwardly upon using Lemma3. To prove sufficiency, given p ∈ P, c(θ) ∈ C(p) and qi(θi, hi) which satisfies (i),define ti(qi, hi) by

ti(qi, hi) ≡ E(θi,θj)[ti(θi, θj) | qi(θi, θj) = qi, hi(c(θ)) = hi]

provided{(θi, θj) | qi(θi, θj) = qi, hi(c(θ)) = hi} �= φ,

otherwise set ti(qi, hi) = 0. For hi ∈ Hi(p) ≡ HiT (p), define

Qi(hi) ≡ {qi(θi, hi) | θi ∈ Θi(hi)}

32

andΘi(hi, qi) ≡ {θi ∈ Θi(hi) | qi(θi, hi) = qi}.

Then for (qi, hi) such that qi ∈ Qi(hi),

{(θi, θj) | qi(θi, θj) = qi, hi(c(θ)) = hi} = Θi(hi, qi) × Θj(hi).

It implies

ti(qi, hi) = Eθi[Eθj

[ti(θi, θj) | θj ∈ Θj(hi)] | θi ∈ Θi(hi, qi)].

The incentive constraint (ii) at t = T implies that Eθj[ti(θi, θj) | θj ∈ Θj(hi)]

must be independent of θi on Θi(hi, qi), since Eθj[qi(θi, θj) | θj ∈ Θj(hi)] =

qi(θi, hi) is independent of θi on the same set by definition. Therefore with θi ∈Θi(hi(c(θi, θj)), qi(θi, θj)),

ti(qi(θi, θj), hi(c(θi, θj))) = Eθj[ti(θi, θj) | θj ∈ Θj(hi(c(θi, θj)))].

For any t and any hit ∈ Hit,

Θj(hit) = {Θj(hi(c(θi, θj))) | θj ∈ Θj(hit)}

for any θi ∈ Θi(hit). This implies that for any θi ∈ Θi(hit),

Eθj[ti(qi(θi, θj), hi(c(θi, θj))) | θj ∈ Θj(hit)] = Eθj

[ti(θi, θj) | θj ∈ Θj(hit)].

Now consider the mechanism with protocol p, where agent i is paid ti(qi, hi) onlyif messages sent by i at every date are consistent with the communication planassociated with some type θi; otherwise the agent is paid nothing. In this mecha-nism, suppose that agent i with type θi uses the communication plan ci(θi) untildate t − 1, and hit is realized at t. Then possible deviations by i at date t can berestricted to switching to the communication plan of by some other θ

′i ∈ Θi(hit)

from date t onwards. Using Lemma 2, the resulting expected payoff of i is givenby

Eθj[ti(θ

′i, θj) − θiqi(θ

′i, θj) | θj ∈ Θj(hit)].

Condition (ii) ensures no such deviation is profitable for the agent. And condi-tion (iii) ensures participation of i at t = 0. Hence this mechanism implements(ti(θi, θj), qi(θi, θj)) as a PBE allocation.

33

Proof of Proposition 3: Necessity of conditions (i) and (ii) are straightforward.So we prove sufficiency of these two conditions for an output allocation to beattainable under centralized contracting in combination with a transfer rule, whichgenerates an expected profit of

E[V (qi(θi, θj), qj(θi, θj)) − vi(θi)qi(θi, θj) − vj(θj)qj(θi, θj)]

For the proof, the following notation will be useful.

(i) hit � his if t ≥ s and Θi(hit) × Θj(hit) ⊂ Θi(his) × Θj(his)

(ii)Hit(θi, his) ≡ {hit | hit � his, θi ∈ Θi(hit)} for (θi, his) such that θi ∈ Θi(his).

(iii) hit(θi, θj) ≡ {hit ∈ Hit | (θi, θj) ∈ Θi(hit) × Θj(hit)}.

In (i), the history hit at date t is a successor of history his at date s: it results fromfurther exchange of messages between dates s+1 and t after his has occurred. Theset Hit(θi, his) is the set of all possible histories that type θi could observe at t,following history his observed at date s. And hit(θ) is the history observed by i att in state θ.

The following Lemma will prove useful.

Lemma 4 Choose arbitrary t ∈ {1, .., T}, hi,t−1 ∈ Hi,t−1 and θi ∈ Θi(hi,t−1). Thenfor any hit, h

′it ∈ Hit(θi, hi,t−1),

Θi(hit) = Θi(h′it).

The argument is the following. Messages sent by i at t depend only on θi andhi,t−1. Hence conditional on θi and hi,t−1, the succeeding histories hit, h

′it can differ

only because of differing messages received by i at t, in turn owing to differentrealizations of θj. Hence the set of types θi that observe the history hi,t−1 and sendthe same messages as θi equals both Θi(hit) and Θi(h

′it).

Lemma 4 implies that the set {hit | hit � hi,t−1} of states succeeding hi,t−1 canbe partitioned into a collection of subsets of hit with the property that Θi(hit) isequal among all hit included in the same subset. Let these subsets be numbered

34

from 1 to L(hi,t−1) where L(hi,t−1) is the total number of these subsets, and letHit(k, hi,t−1) denote the subset corresponding to k ∈ {1, .., L(hi,t−1)}. Then

Θi(hit) = Θi(h′it)

(which can also be expressed as Θi(k, hi,t−1)) for any hit, h′it ∈ Hit(k, hi,t−1). More-

over∪hit∈Hit(k,hi,t−1)Θj(hit) = Θj(hi,t−1)

for any k and∪L(hi,t−1)

k=1 Hit(k, hi,t−1) = {hit | hit � hi,t−1}.

We are now in a position to set out the key steps of the proof of sufficiency ofconditions (i) and (ii) of the proposition.

Claim 1

For qi(θi, θj) which satisfies conditions (i) and (ii) in the proposition, there existsqi(θi, hit) defined on [θi, θi] × ∪T

τ=0Hiτ which satisfies the following conditions (a),(b) and (c).

(a) For any hit ∈ ∪Tτ=0Hiτ , qi(θi, hit) is non-increasing in θi on [θi, θi].

(b) For any hit ∈ ∪Tτ=0Hiτ and any θi ∈ Θi(hit),

qi(θi, hit) = Eθj[qi(θi, θj) | θj ∈ Θj(hit)]

(c) For any hi,t−1 ∈ ∪Tτ=0Hiτ and any k ∈ {1, .., L(hi,t−1)},

Σh′it∈Hit(k,hi,t−1)

Pr(Θj(h′it))

Pr(Θj(hi,t−1))qi(θi, h

′it) = qi(θi, hi,t−1)

Claim 1 states that there exists an ‘auxiliary’ output rule qi as a function of typeθi and message history which is globally non-increasing in type (property (a))following any history hit, and nevertheless equals the expected value (conditionalon hit) of output in the allocation rule that is being sought to be implemented(property (b)). Property (ii) in the statement of the Proposition only allows the

35

latter to be non-increasing over the set of types that arrive at that history hit on theequilibrium path. This auxiliary output rule will be used later in the constructionof transfer payments that efficiently incentivize the desired output allocation.

In order to establish Claim 1, the following Lemma is needed.

Lemma 5 For any B ⊂ R+ which may not be connected, let A be an intervalsatisfying B ⊂ A. Suppose that Fi(a) for i = 1, ..., N and G(a) are functionsdefined on A, each of which has the following properties:

• Fi(a) is non-increasing in a on B for any i.

• ΣipiFi(a) = G(a) for any a ∈ B and for some pi so that pi > 0 and Σipi = 1.

• G(a) is non-increasing in a on A.

Then we can construct Fi(a) defined on A for any i so that

• Fi(a) = Fi(a) on a ∈ B for any i.

• ΣipiFi(a) = G(a) for any a ∈ A and for the same pi

• Fi(a) is non-increasing in a on A for any i.

This lemma says that we can construct functions Fi(a) so that the properties offunctions Fi(a) on B are also maintained on the interval A which covers B.

Proof of Lemma 5

If this statement is true for N = 2, we can easily show that this also holds for anyN ≥ 2. Suppose that this is true for N = 2.

ΣNi=1piFi(a) = p1F1(a) + (p2 + ... + pN)F−1(a)

withF−1(a) = Σi�=1

pi

p2 + ... + pN

Fi(a).

Applying this statement for N = 2, we can construct F1(a) and F−1(a) whichkeeps the same property on A as on B. Next using the constructed F−1(a) instead

36

of G(a), we can apply the statement for N = 2 again to construct desirable F2(a)and F−2(a) on A based on F2(a) and F−2(a) which satisfy

p2

p2 + ... + pN

F2(a) + [1 − p2

p2 + ... + pN

]F−2(a) = F−1(a).

on B. We can use this method recursively to construct Fi(a) for all i.

Next let us show that the statement is true for N = 2. For a ∈ A\B, define a(a)and a(a), if they exists, so that

a(a) ≡ sup{a′ ∈ B | a

′< a}

anda(a) ≡ inf{a

′ ∈ B | a′> a}.

It is obvious that at least one of either a(a) or a(a) exists for any a ∈ A\B.

Let’s specify F1(a) and F2(a) so that F1(a) = F1(a) and F2(a) = F2(a) for a ∈ B,and for a ∈ A\B as follows.

(i) For a ∈ A\B so that only a(a) exists,

F1(a) = F1(a(a))

F2(a) =G(a) − p1F1(a(a))

p2

(ii)For a ∈ A\B so that both a(a) and a(a) exist,

F1(a) = min{F1(a(a)),G(a) − p2F2(a(a))

p1}

F2(a) = max{F2(a(a)),G(a) − p1F1(a(a))

p2}

(iii)For a ∈ A\B so that only a(a) exists,

F1(a) =G(a) − p2F2(a(a))

p1

F2(a) = F2(a(a))

37

It is easy to check that Fi(a) is non-increasing in a on A for i = 1, 2 and

p1F1(a) + p2F2(a) = G(a)

for a ∈ A. This completes the proof of the lemma.

Proof of Claim 1:

Choose arbitrary t ∈ {1, ..., T} and hi,t−1 ∈ Hi,t−1, and k ∈ {1, 2, ..., L(hi,t−1)}.Lemma 5 implies that for qi(θi, hi,t−1) which satisfies (a) and (b) for this hi,t−1, wecan construct a function qi(θi, hit) for any hit ∈ Hit(k, hi,t−1) so that (a), (b) and(c) are satisfied. This result is obtained upon applying the Lemma with

B = Θi(k, hi,t−1)

A = [θi, θi]

a = θi

G(θi) = qi(θi, hi,t−1)

Fhit(θi) = qi(θi, hit)

phit=

Pr(Θj(hit))Pr(Θj(hi,t−1))

for any hit ∈ Hit(k, hi,t−1) where each element of the set Hit(k, hi,t−1) correspondsone-to-one to each one of the set {1, ..., N} in Lemma 5. This means that forqi(θi, hi,t−1) which satisfies (a) and (b) for any hi,t−1 ∈ Hi,t−1, we can constructqi(θi, hit) which satisfies (a)-(c) for any hit ∈ Hit.

With hi0 = φ, since qi(θ, hi0) = Eθj[qi(θi, θj)] satisfies (a) and (b), qi(θi, hi1) is

constructed so that (a)-(c) are satisfied for any hi1 ∈ Hi1. Recursively qi(θi, hit)can be constructed for any hit ∈ ∪T

τ=0Hiτ so that (a)-(c) are satisfied.

Claim 2

For qi(θi, hit) constructed in Claim 1,

Eθj[qi(θi, hiT (θi, θj)) | θj ∈ Θj(hit)] = qi(θi, hit)

for any θi ∈ Θi(hit).

38

Proof of Claim 2

From the construction of qi(θi, hit) which satisfies (c), for any θi ∈ Θi(hit),

qi(θi, hit) = Σhit+1∈Hit+1(θi,hit)Pr(Θj(hit+1))Pr(Θj(hit))

qi(θi, hit+1)

= Σhit+1∈Hit+1(θi,hit)Pr(Θj(hit+1))Pr(Θj(hit))

Σhit+2∈Hit+2(θi,hit+1)Pr(Θj(hit+2))Pr(Θj(hit+1))

qi(θi, hit+2)

= Σhit+1∈Hit+1(θi,hit)Σhit+2∈Hit+2(θi,hit+1)Pr(Θj(hit+2))Pr(Θj(hit))

qi(θi, hit+2)

= Σhit+2∈Hit+2(θi,hit)Pr(Θj(hit+2))Pr(Θj(hit))

qi(θi, hit+2) = · · ·

= ΣhiT ∈HiT (θi,hit)Pr(Θj(hiT ))Pr(Θj(hit))

qi(θi, hiT )

The first and second equalities come from (c) and the fact that there exists k sothat Hiτ (θi, hi,τ−1) = Hiτ (k, hi,τ−1) for any θi ∈ Θi(hi,τ−1). The fourth equalitycomes from

Hit+2(θi, hit) = {Hit+2(θi, hit+1) | hit+1 ∈ Hit+1(θi, hit)}.

SinceHiT (θi, hit) = {hiT (θi, θj) | θj ∈ Θj(hit)},

ΣhiT ∈HiT (θi,hit)Pr(Θj(hiT ))Pr(Θj(hit))

qi(θi, hiT )

= ΣhiT ∈{hiT (θi,θj)|θj∈Θj(hit)}Pr(Θj(hiT ))Pr(Θj(hit))

qi(θi, hiT )

= Eθj[qi(θi, hiT (θi, θj)) | θj ∈ Θj(hit)]

This completes the proof of the Claim.

We are now in a position to complete the proof of sufficiency. Based on qi(θi, hit)which was constructed in Claim 1, consider the following mechanism: (ti(θi, θj), qi(θi, θj))with

ti(θi, θj) = θiqi(θi, θj) +∫ θi

θi

qi(θi, hiT (θi, θj))dθi

39

Then for any t, any hit ∈ Hit and any θi, θ′i ∈ Θi(hit),

Eθj[ti(θ

′i, θj) − θiqi(θ

′i, θj) | θj ∈ Θj(hit)]

= (θ′i − θi)Eθj

[qi(θ′i, θj) | θj ∈ Θj(hit)] +

∫ θi

θ′i

Eθj[qi(θi, hiT (θ

′i, θj)) | θj ∈ Θj(hit)]dθi

= (θ′i − θi)qi(θ

′i, hit) +

∫ θi

θ′i

qi(θi, hit)dθi

≤∫ θi

θi

qi(θi, hit)dθi

=∫ θi

θi

Eθj[qi(θi, hiT (θi, θj)) | θj ∈ Θj(hit)]dθi

= Eθj[ti(θi, θj) − θiqi(θi, θj) | θj ∈ Θj(hit)]

The second equality comes from (b) in Claim 1 and the statement of Claim 2 andthe third inequality comes from (a) in Claim 1. The fourth equality comes fromthe statement of Claim 2. This means that the incentive constraint is satisfied atall stages of communication.

At the interim stage with t = 0,

Eθj[ti(θi, θj) − θiqi(θi, θj)]

=∫ θi

θi

Eθj[qi(θi, hiT (θi, θj))]dθi

=∫ θi

θi

Eθj[qi(θi, hiT (θi, θj)) | θj ∈ Θj(hi0)]dθi

=∫ θi

θi

qi(θi, hi0)dθi

=∫ θi

θi

Eθj[qi(θi, θj)]dθi

The third equality comes from the statement of Step 2, and the fourth one is from(b) in Claim 1. This implies that the interim participation constraint is satisfiedand P ’s ex-ante payoff is equal to

E[V (qi(θi, θj), qj(θi, θj)) − vi(θi)qi(θi, θj) − vj(θj)qj(θi, θj)].

40

which is the maximum possible payoff under the constraint that θi is private infor-mation of i. This completes the proof of the proposition.

Proof of Proposition 4:

We show that the solution of the problem without the constraint (ii) in Proposition3 satisfies this constraint. Suppose not. Let (q1(θ1, θ2), q2(θ1, θ2)) be the solutionwithout (ii). Hit, Θi(hit) and Θj(hit) are defined for this communication feasibleproductions. Then there exists t, hit ∈ Hit and θi, θ

′i ∈ Θi(hit) with θi > θ

′i so that

Eθj[qi(θi, θj) | θj ∈ Θj(hit)] > Eθj

[qi(θ′i, θj) | θj ∈ Θj(hit)].

This implies that at least either one of

E[V (qi(θ′i, θj), qj(θ

′i, θj)) − vi(θi)qi(θ

′i, θj) − vj(θj)qj(θ

′i, θj) | θj ∈ Θj(hit)]

> E[V (qi(θi, θj), qj(θi, θj)) − vi(θi)qi(θi, θj) − vj(θj)qj(θi, θj) | θj ∈ Θj(hit)]

or

E[V (qi(θi, θj), qj(θi, θj)) − vi(θ′i)qi(θi, θj) − vj(θj)qj(θi, θj) | θj ∈ Θj(hit)]

> E[V (qi(θ′i, θj), qj(θ

′i, θj)) − vi(θ

′i)qi(θ

′i, θj) − vj(θj)qj(θ

′i, θj) | θj ∈ Θj(hit)]

holds. This means that if at least one type of either θi or θ′i takes other type of

communication plan, P ’s payoff is improved. This is a contradiction.

Proof of Proposition 5:

Suppose that (q∗1(θ), q

∗2(θ)) solves the maximization problem with an associated

communication protocol p∗ ∈ P∗, communication strategies c∗(θ) ∈ C(p∗), and pro-duction strategies q∗

i (θi, hi). Consider any history (hi, hj) ∈ H∗ ≡ {(hi(c∗(θ)), hj(c∗(θ)) |θ ∈ Θi × Θj} that arises with positive probability, i.e., so that the associated setof types Θi(hi) × Θj(hj) has positive measure.

Note that Θi(hi) ⊂ Θi(hj), since all types θi ∈ Θi(hi) of agent i send and receive thesame messages, so any other agent j would not be able to distinguish between them.This implies that q∗

j (θi, θj) does not vary with θi ∈ Θi(hi), for any θj ∈ Θj(hi) andany hi ∈ Hi. Moreover,

E[V (qi, q∗j (θi, θj)) | θj ∈ Θj(hi)]

41

does not vary with θi over Θi(hi). Denote this conditional expectation by V (qi, hi).Evidently q∗

i (θi, hi) = arg max V (qi, hi) − vi(θi)qi. From the properties of V (qi, qj),V (qi, hi) is strictly concave for qi, satisfying the Inada condition. Given the mono-tonicity of vi(θi), q∗

i (θi, hi) is strictly decreasing in θi on Θi(hi). Since hj(c∗(θi, θj))and hP (c∗(θi, θj)) are independent of θi on Θi(hi), q∗(θi, θj) = q∗

i (θi, hi(c∗(θi, θj)) isnot measurable with respect to hj(c∗(θi, θj)) and hP (c∗(θi, θj)).

Proof of Proposition 6 For part (i), take any mechanism involving a communica-tion protocol p and associated communication strategies c(θ) ∈ C(p). Let agent i’scommunication strategy involve sending message mik,1(θi) at round 1, and at roundt the message mik,t(mt−1

Pi , mji,t−1, miP,t−1, mij,t−1; mPi,t−2, mji,t−2, miP,t−2, mij,t−2; ...|θi)to k = j, P if his type happens to be θi. At round 1, there is no scope for P

to communicate anything (mPi,1 = ∅) as P has no private information. Fromround t = 2 onwards, let P ’s communication strategy involve sending messagemPi,t(mPi,t−1, mPj,t−1, miP,t−1, mjP,t−1; mPi,t−2, mPj,t−2, miP,t−2, mjP,t−2; ...).

We now consider the completely decentralized network where only agents i andj communicate with one another, and construct the following communicationstrategies. First, we have miP,t = ∅ for all t. And at round 1, mij,1(θi) =(miP,1(θi), mij,1(θi)), i.e., i sends the vector of messages sent in c(θ) but they arenow all addressed to agent j. Similarly, at round 2, messages are constructed asfollows. At the end of round 1, agent i has received mji,1 ≡ (mjP,1(θj), mji,1(θj))for some θj. In round 2, agent i with message history hi1 ≡ (mji,1, mij,1) sendsj the message mij,2(hi1|θi) = {mik,2(mPi,1 = ∅, mji,1, miP,1, mij,1|θi)}k=P,j. Thisgenerates messages at round 2. At round 3, agent i with message history hi2 ≡(mji,2, mji,1, mij,2, mij,1) sends j the message

mij,3(hi2|θi) = {mik,3(mPi,2(miP,1, mjP,1), mji,2, miP,2, mij,2; mPi,1 = ∅, mji,1, miP,1, mij,1|θi)}k=P,j

and so on in future rounds.

In words, agent i sends to j the set of messages that i would have sent both P

and j in c(θ). Knowing the messages they had respectively sent to P in earlierrounds, and knowing the communication strategy of P in the original mechanism,they can work out what message P would have sent them corresponding to eachhistory. They can therefore reconstruct what messages they would have receivedunder c(θ) in the original protocol, and can calculate the set of messages that they

42

would have responded with in the original protocol, in the subsequent round. Theynow send this entire set of messages to j.

It is evident that the set of non-null messages being sent by each agent is exactlythe same as in the original protocol, in every history. So communication feasibilityis preserved, by virtue of Assumption 3.

Moreover, each agent knows at least as much as in the original protocol in everyhistory: they additionally learn what messages were exchanged by the Principalwith the other agent. Hence the constructed decentralized protocol is at least asinformative as the original one, which establishes part (i).

For part (ii), note that in the last round T there is no value in any agent i sendingany messages to P , as there are no subsequent rounds at which these messagescan be used by P to inform j about the state of agent i. Hence without loss ofgenerality, c(θ) involves mT

iP = ∅. If the original protocol was completely cen-tralized, agents i and j will not communicate anything to each other in roundT in the communication strategies mij(·|θi) in the completely decentralized pro-tocol constructed in proof of part (i) above. Hence there is scope for them toengage in some additional communication in round T . This will enable a strictimprovement in the informativeness of the communication protocol. In turn thiswill enable a strict increase in the expected profit of P . To show this, suppose that{q∗

i (θ)}i=1,2 is the optimal output under the centralized network, p∗ and c∗(θ) arethe associated centralized communication protocol and strategies, and q∗

i (θi, hi) isthe associated output strategies. Inductively we define the following notations:H∗ ≡ {(hi(c∗(θ)), hj(c∗(θ))) | θ ∈ Θ} and H∗

i ≡ {hi(c∗(θ)) | θ ∈ Θ}.

Without loss of generality, focus on the region Θi(hi)×Θj(hi) for arbitrary hi ∈ H∗i .

Suppose that the information of agent i with θi ∈ Θi(hi) is improved from Θj(hi)to information partition {Θ′

j, Θj(hi)\Θ′j} for Θ′

j ⊂ Θj(hi). On the other hand, theinformation of i in other regions and of j is kept constant. We show that thereexists Θ′

j ⊂ Θj(hi) so that P ’s payoff is improved.

Choose arbitrary hi ∈ H∗i . In the optimal solution, q∗

i (θi, hi) satisfies

q∗i (θi, hi) = arg max

qi

E[V (qi, q∗j (θi, θj)) | θj ∈ Θj(hi)] − θiqi

Since Θi(hi) ⊂ Θi(hj) for any hj so that (hi, hj) ∈ H∗, q∗j (θi, θj) is constant on

Θi(hi) for any θj ∈ Θj(hi). Since Θj(hj) ⊂ Θj(hi) for (hi, hj) ∈ H∗ and q∗j (θi, θj) is

43

strictly decreasing in θj on Θj(hj) (from the result of Proposition 5), q∗j (θi, θj) is not

constant on θj ∈ Θj(hi). On the other hand, q∗i (θi, θj) is constant on θj ∈ Θj(hi)

for any θi ∈ Θi(hi). Then Vqi(q∗

i (θi, θj), q∗j (θi, θj)) is not constant on θj ∈ Θj(hi)

for any θi ∈ Θi(hi) and is continuous and strictly increasing in θi ∈ Θi(hi). Hencethere exists θi ∈ Θi(hi) so that both

{θj ∈ Θj(hi) | Vqi(q∗

i (θi, θj), q∗j (θi, θj)) ≥ Eθj

[Vqi(q∗

i (θi, θj), q∗j (θi, θj)) | θj ∈ Θj(hi)]}

and

{θj ∈ Θj(hi) | Vqi(q∗

i (θi, θj), q∗j (θi, θj)) ≤ Eθj

[Vqi(q∗

i (θi, θj), q∗j (θi, θj)) | θj ∈ Θj(hi)]}

have positive measure. Let the former set be denoted by Θ′j. Then

E[Vqi((q∗

i (θi, θj), q∗j (θi, θj)) | θj ∈ Θ

′j] > E[Vqi

((q∗i (θi, θj), q∗

j (θi, θj)) | θj ∈ Θi(hi)\Θ′j]

By the continuity of Vqi(q∗

i (θi, θj), q∗j (θi, θj)), the above inequality also holds in a

neighborhood of θi with positive measure. This implies that

Pr(Θ′j)Eθi∈Θi(hi)[max

qi

E[V (qi, q∗j (θi, θj)) − θiqi | θj ∈ Θ

′j]

+ Pr(Θj(hi)\Θ′j)Eθi∈Θi(hi)[max

qi

E[V (qi, q∗j (θi, θj)) − θiqi | θj ∈ Θj(hi)\Θ

′j]

> E[V (q∗i (θi, θj), q∗

j (θi, θj)) − θiq∗i (θi, θj) | θj ∈ Θj(hi)].

i.e, a strict improvement of P ’s payoff on Θi(hi) × Θj(hi). QED

44