Accept or reject? An organizational perspective

International Journal of Industrial Organization xxx (2014) xxx–xxx

INDOR-02167; No of Pages 9

Contents lists available at ScienceDirect

International Journal of Industrial Organization

j ourna l homepage: www.e lsev ie r .com/ locate / i j i o

Accept or reject? An organizational perspective☆

Umberto Garfagnini a, Marco Ottaviani b,⁎, Peter Norman Sørensen c

a ITAM School of Business, Instituto Tecnológico Autónomo de Mexico, Río Hondo No.1, Col. Progreso Tizapán, Mexico, D.F. 01080, Mexicob Department of Economics and IGIER, Bocconi University, Via Roberto Sarfatti 25, 20136 Milano, Italyc Department of Economics, University of Copenhagen, Oster Farimagsgade 5, DK-1353 Copenhagen, Denmark

☆ We thank the editor, Tommaso Valletti, and two anoncomments and suggestions. Financial support from the#295835 (Ottaviani and Sørensen) and the Asociaci(Garfagnini) is gratefully acknowledged.⁎ Corresponding author.

E-mail addresses: [email protected] (U. [email protected] (M. Ottaviani), peter.sorense

http://dx.doi.org/10.1016/j.ijindorg.2014.03.0040167-7187/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Garfagnini, U., et al.j.ijindorg.2014.03.004

a b s t r a c t
a r t i c l e i n f o
Available online xxxx

JEL classification:D82D83

Keywords:InformationCheap talkDelegationVoting

This paper compares the relative performance of different organizational structures for the decision of acceptingor rejecting a project of uncertain quality. When the principal is uninformed and relies on the advice of an in-formed and biased agent, cheap-talk communication is persuasive and it is equivalent to delegation of authority,provided that the agent's bias is small. When the principal has access to additional private information,cheap-talk communication dominates both (conditional) delegation and more democratic organizationalarrangements such as voting with unanimous consensus.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Mortgage applications are channeled through loan officers who aretypically rewarded upon successful completion of the loan. Final accep-tance or rejection depends on the judgement of a bank's underwritingdepartment. Similarly, headquarters decide whether or not to financeprojects based on proposals by division managers who typically standto gain from the approval. And before deciding whether to accept orreject a paper, the journal editor consults an expert referee who, beinga specialist, is known to be biased.

In these situations, an expert principal decides whether to acceptor reject a project in consultation with a biased expert agent whopossesses some additional information relevant for the decision. It isnatural to wonder how the principal should organize the collection ofinformation.

This article compares the performance of different organizationalstructures for aggregating information. Our point of departure isthe pioneering work by Sah and Stiglitz (1986) on how the organiza-tion of experts affects the quality of decision making. While Sahand Stiglitz do not consider how the experts' reporting changesdepending on the organization structure, we follow Holmström

ymous referees for their helpfulERC through advanced grant

ón Mexicana de Cultura A.C.

fagnini),[email protected] (P.N. Sørensen).

, Accept or reject? An organiza

(1977) and Crawford and Sobel (1982) by focusing on the incentivesfor strategic manipulation by fully strategic players with misalignedpreferences.1 Given our applications to project approval, we focus ona simple model with two actions, two states, and continuous noisysignals. This model is largely based on Li et al. (2001) and Li andSuen (2004).2

The principal and the agent observe continuously distributed,conditionally independent, signals about the state of the world. Afterobserving her signal, the principal can decide the organization ofcommunication. That is, the principal can ask the agent to report(i.e., cheap talk communication), or can delegate authority to the agent(i.e., conditional delegation), or can use more democratic arrangementslike voting. Wewill consider two cases, one where the principal's signalis uninformative (Section 2), the other where the two signals areidentically distributed (Section 3).

When the principal is uninformed, cheap talk communication is“persuasive” if either (i) the bias is small or (ii) theuninformedprincipalfavors acceptance based on prior information, given that we assumethat the agent's signal has an unbounded likelihood ratio. In bothscenarios, the agent is able to influence the principal's decision bysending a recommendation which pools some relatively unfavorablesignals with enough favorable signals to make acceptance optimal.

1 See also the recent contributions by Alonso and Matouschek (2008), Alonso et al.(2008) and Rantakari (2008).

2 Another branch of the literature has focused on the impact of the organizational struc-ture on the quality of decisionmakingwhen experts aremotivated by professional, ratherthan partisan, concerns; see Ottaviani and Sørensen (2001).

tional perspective, Int. J. Ind. Organ. (2014), http://dx.doi.org/10.1016/

http://dx.doi.org/10.1016/j.ijindorg.2014.03.004

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2 U. Garfagnini et al. / International Journal of Industrial Organization xxx (2014) xxx–xxx

An increase in the agent's bias reduces the quality of communicationand increases the probability of accepting a bad project (i.e., Type-Ierror).

When the receiver/principal is uninformed, in our model theinformative equilibrium with cheap talk communication results in thesame outcome as delegation of authority. This result contrasts withresults obtained in a model with continuous action and symmetricloss function by Ottaviani (2000) and Dessein (2002), based onCrawford and Sobel (1982), where delegation dominates communica-tion for small biases.3 This difference stems from the fact that inCrawford and Sobel's (1982) model the agent's bias leads only to aloss of information whereas the equilibrium action is unbiased. Withonly two actions, there is instead persuasion in equilibrium because arestricted (binary) action space increases the informativeness of cheaptalk communication by reducing the set of the sender's incentiveconstraints. 4

When the principal is partially informed, cheap talk communicationdominates conditional delegation of authority to the agent (that is,delegation after the principal has received her information). Withconditional delegation, an expert principal retains authority eitherif she received a very low signal, which makes unilateral rejectionoptimal without the need to consult the agent, or else if she re-ceived a very high signal, which makes unilateral acceptanceoptimal. After receiving authority, the agent infers that the principal'ssignal belongs to an intermediate interval and he best respondsby recommending acceptance when his own signal exceeds acutoff. The principal could instead allow the agent to make anaccept/reject recommendation while retaining authority over thefinal decision. With this restricted cheap talk communication, theagent once again adopts a cutoff strategy which leads the principalto follow the recommendation only when her own signal fallsin an intermediate interval.5 Given that the principal correctlyanticipates the agent's equilibrium strategy under conditionaldelegation, she is indifferent between letting the agent make thedecision or following the agent's recommendation. Therefore,allowing for richer communication protocols improves the qualityof communication which makes cheap talk strictly better thanconditional delegation.6

Cheap talk communication also dominates voting with unanimousconsensus. This occurs because conditional delegation is superior tovoting. The intuition is that delegation makes the agent with authoritybehave more conservatively when deciding to accept. A vote to acceptleads to acceptance if, and only if, the principal also votes to accept.Therefore, the agent can afford to behave in a less cautious mannerwhen voting.

It is natural to wonder how the optimal mechanism withouttransfers would look like in our setting with an expert principal. Weprovide a partial characterization of such mechanism in Section 3.5.When the principal can commit to the way in which she uses theinformation transmitted by the agent, a new trade-off arises. Thistrade-off involves a suppression of information on the principal's side

3 Ottaviani (2000) also shows that when there is uncertainty about the agent's bias inCrawford and Sobel's (1982) model cheap talk dominates delegation, an observationwhich has been investigated further by Rush et al. (2009). Recently, Agastya et al.(2013) have shown that the comparison between cheap talk communication and delega-tion becomes ambiguous even for small biases in the presence of residual uncertainty, thatis, when the agent only observes a dimension of the state of theworld.We instead assumethat the agent's bias is known and that the agent observes a partially informative signalabout the state of the world.

4 As a result, the message space can only contain up to two messages.5 When the principal receives a very high signal, she accepts regardless of the agent's

recommendation. Similarly, she always rejects following a low signal.6 Existence of an informative equilibrium for any size of the bias follows from the fact

that the likelihood ratios are unbounded with our signal structure. If the likelihood ratioswere bounded, unlike in our baselinemodel, the feasibility of communicationwould clear-ly depend on the size of the bias as in standard cheap talk models. Our result would thenhold for small biases.

Please cite this article as: Garfagnini, U., et al., Accept or reject? An organizaj.ijindorg.2014.03.004

in order to reduce the amount of strategic manipulation performedby the agent. By committing to suppress some of her information,the principal induces the agent to adopt a higher standard whenrecommending acceptance, which might increase the quality of deci-sion making.

This article contributes to a growing literature on cheap talk commu-nicationwith a privately informed receiver. In an early analysis of cheaptalk gameswith an informed receiver, Seidmann (1990) shows throughexamples that private information may help communication. Chen(2009) shows that correlation between the sender and the receiver'sinformation generates non-monotone equilibria in the model ofCrawford and Sobel (1982), while Moreno de Barreda (2013) showsthat the quality of communication decreases in the precision of theprincipal's signal about the state of the world in the uniform-quadraticmodel. Ishida and Shimizu (2013) investigate the impossibility ofinformative communication in a binary action-binary state model, asin our framework, in which the sender and receiver observe discretesignals. Unlike those papers, we further investigate the organization ofcommunication.

In a related paper, Harris and Raviv (2005) consider a uniform-quadratic model in which both the sender and the receiver possessindependent pieces of information about different dimensions ofthe state of the world which enter additively in the utility func-tions. When the receiver has no information, their model collapsesto Crawford and Sobel's and delegation dominates communicationas in Dessein.7 Chakraborty and Yilmaz (2013) consider a settingsimilar to Harris and Raviv's with a binary action space and threeplayers: shareholders, who are uninformed, a board and manage-ment who both possess different pieces of information. Whenshareholders can choose the board's bias, the optimal board ren-ders the allocation of authority irrelevant provided that there isconsensus between board andmanagement. In contrast, we consid-er environments in which the sender and the receiver possessindependent pieces of information about the same dimension ofuncertainty.8

2. Uninformed principal dealing with expert agent

A principal, P, needs to make a decision d ∈ {Reject, Accept} re-garding a project of unknown quality. Before making this decision,the principal consults an agent, A. For example, headquarters decidewhether or not to finance a project after consulting an expert divi-sion manager.

There is an unknown state of the world θ ∈ {B, G} which is relevantfor the decision. If the principal rejects, both players receive a payoffwhich is normalized to 0. If the state of the world is B (or bad), accep-tance generates a payoff equal to −RB b 0 for each player. If the stateof the world is G (or good), acceptance generates a payoff equal toRG + bi with RG N 0 for player i∈ {P, A}. We assume that bP = 0 b bA ≡ b.Thus, the principal is unbiased whereas the agent is biased towardacceptance. We denote by p = Prob(θ = G) the common prior aboutthe good state.

Information. The agent receives a signal sA∈ S= [0, 1] about the stateof the world which is distributed according to a continuous densityfunction fθ(⋅) with distribution function Fθ(⋅), θ = B, G. The signal

7 McGee (2013) generalizes the model of Harris and Raviv (2005) by considering thecase in which the marginal contribution of the sender's information to the optimal actiondepends on the principal's information.

8 Marino (2007) studies the trade-off between delegation and veto power in a gamewith a continuous state space. The agent proposes to the principal the size of an invest-ment based on observation of the state of theworld. He shows that veto powermay dom-inate delegation evenwhen the agent's bias is small by inducing the agent to avoid certainlow quality projects. Mylovanov (2008) showsmore generally how veto-based delegationcan be employed by a principal to implement an optimal outcome by carefully selecting adefault decision in case of a veto.




9 This observation follows from the fact that K(s) b l(s) by MLRP.10 Chakraborty andHargaugh (2010) investigate persuasion through cheap talk commu-nication in environments in which information is multidimensional and the agent's pref-erences are state-independent (i.e., b = +∞ in our model).

3U. Garfagnini et al. / International Journal of Industrial Organization xxx (2014) xxx–xxx

structure satisfies the monotone likelihood ratio property (MLRP), thatis, f G sð Þ

f B sð Þ is increasing in s.

We introduce some notation whichwill become useful in the analy-sis. Define l sð Þ ¼ f G sð Þ

f B sð Þ , L sð Þ ¼ 1−FG sð Þ1− FB sð Þ , K sð Þ ¼ FG sð Þ

FB sð Þ , and finally L s1; s2ð Þ ¼FG s2ð Þ−FG s1ð ÞFB s2ð Þ−FB s1ð Þ, for s2 N s1. MLRP then implies the following properties:

1. K(s) b 1 b L(s), for any s ∈ (0, 1);2. K(s) and L(s) are increasing functions of s;3. L(s1, s2) is increasing in both s1 and s2, for s1 b s2;4. For any s1 b s2 b s3, L(s1, s2) b l(s2) b L(s2, s3).

For reasons of tractability, we will often employ a linear signalstructure with fG(s) = 2s and fB(s) = 2(1 − s). This signal specifica-tion corresponds to a binary signal with continuously varying preci-sion and unbounded informativeness at the boundaries. This linearsignal structure guarantees existence and uniqueness of equilibria andsimplifies some of the proofs, but it is unnecessary formany of our char-acterization results.

Organizational structures. The principal can commit ex ante to anorganizational structure. With an uninformed principal, two possibleorganizational structures are feasible. First, the principal could ask theagent to report his signal which the principal uses to make a decision,that is, cheap talk communication. Second, the principal could allow theagent to make the decision that maximizes his own utility based onhis information, that is, delegation.

The principal can makemistakes in her decision to accept or reject aproject. Borrowing from the terminology of statistical hypothesis test-ing, we say that the principal makes a Type-I error (or, false acceptance)when she accepts a bad project, while she makes a Type-II error(or, false rejection) when she rejects a good project.

2.1. Delegation

We start the analysis by considering the simpler case in which theprincipal delegated authority to the agent. The agent uses his signal toupdate the posterior belief about the state to

Prob GjsAð Þ ¼ pl sAð Þpl sAð Þ þ 1−pð Þ : ð1Þ

Given this updated belief, the agent accepts if, and only if, his likeli-hood ratio satisfies l sAð Þ≥ 1−p

pRB

RGþb≡ T bð Þ. MLRP implies that there exists aunique cutoff, tA∗(b), such that the agent accepts if, and only if, his signalexceeds the cutoff. If the principal could instead directly observethe agent's signal, she would accept if, and only if, sA ≥ tP

∗ ≡ tA∗(0). As

tA∗(b) b tP

∗, the agent has higher incentives to accept even after less favor-able signals because of his bias. For example, t�A bð Þ≡ T bð Þ

1þT bð Þ in the contextof the linear signal structure.

2.2. Cheap talk communication

Wenext consider cheap talk communication. The timing of the gameis as follows:

1. Nature draws the state of the world, θ.2. The agent observes a signal sA.3. The agent sends a costless, unverifiable, recommendation m to the

principal.4. Based on the recommendation received, the principal makes

a decision d ∈ {Reject, Accept}. When indifferent, the principalaccepts.

5. The state of the world is revealed and payoffs are realized.

We show in the Appendix A that for any informative equilibriumin which the agent sends more than two messages we can constructanother informative equilibrium in which the agent sends only two


messages and which is payoff-equivalent. So, it is without loss of gener-ality to restrict attention to the message spaceM = {Reject, Accept}.

Suppose that the agent recommends to accept if, and only if, hissignal sA ≥ tA

∗ (b). Given such a strategy, a negative recommendationleads the principal to update her posterior belief downward, that is,Prob(G|sA b tA

∗ (b)) b p. By MLRP, there exists a unique cutoff t�P suchthat the principal accepts if, and only if, t�A bð Þ≥t�P . The principal adoptsa less permissive rule compared to the rule shewould adopt if the signalwere observable, that is, t�Pb t�P .

9 This is because the agent's recommen-dation pools together very bad signals which induces the principal tobehave more conservatively.

When the agent recommends acceptance, the principal accepts if,and only if, t�A bð Þ≥t�P , where t�P Nt

�P . The principal employs a more per-

missive decision rule because a positive recommendation increasesthe principal's posterior belief to Prob(G|sA ≥ tA

∗(b)) N p. With a linearsignal structure, we can solve for the cutoffs directly which gives

t�P ≡min 2T 0ð Þ1þ T 0ð Þ;1

� �and t�P ¼ max T 0ð Þ−1

T 0ð Þ þ 1;0

� �.

Proposition 1 characterizes equilibrium behavior. We say that anequilibrium is informative if d(Accept) ≠ d(Reject), that is, when thefinal decision depends on the recommendation of the agent, and thatan informative equilibrium is unique if all informative equilibria arepayoff-equivalent.

Proposition 1. Suppose thatp b RBRGþRB

, that is, the principal rejects if she hasto decide only based on prior information. Then, there exists a number bN0such that:

1. If the agent's bias isb≤b, the unique informative equilibrium in the com-munication stage is the one in which the agent recommends acceptanceif, and only if, his signal sA ≥ tA

∗(b). The principal always follows theagent's recommendation.

2. If the agent's bias is bNb, then all communication equilibria are uninfor-mative and the principal always rejects, regardless of the agent'srecommendation.

Suppose instead that p≥ RBRGþRB

. Then, the unique informative equilibriumin the communication stage is the one in which the agent recommendsacceptance if, and only if, his signal sA≥ tA

∗(b). The principal always followsthe agent's recommendation.

When the prior belief of a good state is low, cheap talk communica-tion is persuasive in the sense that the principal accepts whenever theagent recommends acceptance, provided that the agent's bias is nottoo large.10 The principal suffers a bias because the biased agent recom-mends acceptance for a region of signal realizations for which theprincipal would have wanted to reject if she had observed those exactsignal realizations. Essentially, in the eyes of the principal, the agentpools some relatively unfavorable signals (for which the principalwould have liked to reject) with some relatively favorable signals thatmake acceptance optimal for the principal. When the conflict of interestis not too high, for the principal the gain associated to the reductionin the probability of a Type-II error (i.e., false rejection) outweighsthe loss associated with the increase in the Type-I error (i.e., falseacceptance).

When the prior belief is high enough that the principal accepts giventhe prior, communication is always informative. This is intuitivebecause the principal would reject only if the signal were sufficientlylow, thus she has no reason to disregard the agent's recommendationto reject given that the agent employs a more demanding standard forrejection.




Fig. 1. Optimal bias when p b RBRGþRB

.


2.3. Cheap talk versus delegation

Provided that the agent's bias lies below the critical level b, the out-come of delegation (i.e., giving authority to the informed agent) isequivalent to the outcome of the informative cheap talk equilibrium.11

So, in this setting when the bias is small cheap talk is exactly equiv-alent to delegation. Note the contrast with Dessein's (2002) resultthat with continuous action and symmetric loss functions (as inCrawford and Sobel, 1982), for small bias the receiver prefers delega-tion to cheap talk.

When the bias is large, the loss outweighs the gain, so the prin-cipal always wants to reject even when the agent wants to accept.The principal prefers cheap talk (which is uninformative) to delega-tion. This result is in line with Dessein—and it is fully general—whenthe cheap talk equilibrium is uninformative, delegation is a badidea.

We summarize the discussion in Proposition 2.

Proposition 2. Cheap talk communication is equivalent to delegationwhenever communication is informative, and it is strictly better otherwise.

2.4. Effect of bias

The next result contains some comparative statics on the quality ofdecision making.

Proposition 3. The following comparative statics holds true:

1. When communication is informative, the probability of a Type-I error(i.e., false acceptance) increases in the bias, b.

2. An increase in the prior belief, p, increases the interval of bias parametersfor which communication is informative, that is, ∂b=∂pN0 for any pb RB

RGþRB.

An increase in the bias leads the agent to recommend acceptancemore often because he has more to lose from rejection. This increasesthe probability of false acceptance whenever communication isinformative. A more optimistic prior belief necessarily increases theprincipal's concern of committing a Type-II error. This implies that theprincipal is more willing to listen to the agent even when the agent ismore biased.

Commitment by the agent. It is important to remark that a binary ac-tion space allows for the persuasiveness of cheap talk communicationby inducing the principal to make a decision which benefits the agent.This contrasts with cheap talk models based on Crawford and Sobel(1982) where the equilibrium action is always unbiased.

In this simple framework, we can thus think about how the agentmight increase his expected utility if he had commitment power. Weconsider the case in which an agentwith bias b N 0 can credibly committo a different bias b̂ ex ante, that is, before observing his own signal.12

Suppose that pb RBRGþRB

, then an agent with bias b N 0 would optimallychoose a bias b∗ to solve

maxb̂≥0

p 1−FG t�A b̂� �� h i

RG þ bð Þ− 1−pð Þ 1−FB t�A b̂� �� h i

RB

n o1

b̂≤b� ��

ð2Þ

Ignoring the constraint imposed by the indicator function, the objec-tive function is quasi-concave and the first-order condition immediatelygives that b∗ must solve l(tA∗(b∗)) = T(b), or b∗ = b. Therefore, for anagent with a bias b≤b, the optimal bias is exactly b. This is because asmaller bias reduces the ex ante probability of acceptancewhile a higher

11 Clearly there is also always an uninformative cheap talk equilibrium, but it makessense to disregard it as not particularly natural.12 Kamenica and Gentzkow (2011) and Kolotilin (2013) investigate optimal persuasionby an agent in a more general framework than ours. More recently, Li et al. (2013) studyhow a principal, like a government, can persuade a heterogeneous audience.


bias increases it but at signal realizations forwhich the agentwould pre-fer the project to be rejected ex post. Agents with a bias greater than bwould instead benefit from the ability to commit ex ante to a biasequal to b. Their bias induces the principal to reject with probability 1,while persuasion occurs when b ¼ b and the principal accepts the pro-ject with (ex ante) positive probability. Those agents strictly benefitfrom informative cheap talk communication compared to no communi-cation. Fig. 1 graphs the optimal bias b∗ as a function of the agent's bias b.

When p≥ RBRGþRB

, equilibrium communication is always informative sothe agent has no incentive to commit to a bias different than his own.The next Proposition 4 summarizes our discussion so far.

Proposition 4. Suppose that an agent with bias b N 0 can commit ex ante

to any nonnegative bias. If p b RBRGþRB

, then the optimal bias isb� ¼ min b; bn o

.

If p≥ RBRGþRB

, then b∗ = b.

The logic behind our persuasion result is reminiscent of the trade-offfaced by a seller in providing information to potential buyers as in Lewisand Sappington (1994). Similar to a seller who wants to maximize theprobability of purchase by an uninformed buyer, our agent wants tomaximize the probability of acceptance by the principal when he findsacceptance optimal.13

3. Expert principal dealing with expert agent

This section considers the more general case in which the principaland the agent observe conditionally independent signals which areidentically distributed. We denote the principal's signal by sP. The com-munication stage now serves as a way for the players to aggregate theirinformation and potentially improve the quality of the final decision.

In the following sections, we consider several organizational struc-tures like cheap talk communication, conditional delegation, and voting,and compare their relative performance.

3.1. Binary cheap talk

Suppose for now that the agent's message space is restricted toM={Reject,Accept}. Consider once again the case inwhich the agent usesa cutoff strategy in the communication stage, that is, he recommends ac-ceptance if, and only if, he receives a signal sA ≥ tA

C(b). Then, following a

13 Johnson and Myatt (2006) generalize the economics behind Lewis and Sappington(1994). More recently, Hoffmann et al. (2013) investigate selective disclosure of informa-tion to buyers with limited attention.





recommendation of acceptance, the principal can update her ownposterior belief to

Prob GjsP ; sA≥tCA bð Þ� �

¼pl sPð ÞL tCA bð Þ

� �pl sPð ÞL tCA bð Þ� þ 1−pð Þ : ð3Þ

The principal accepts following a recommendation of acceptance ifand only if

l sPð ÞL tCA bð Þ� �

≥T 0ð Þ: ð4Þ

Similarly, if the agent recommends rejection, the principal accepts ifand only if

l sPð ÞK tCA bð Þ� �

≥T 0ð Þ: ð5Þ

Fix the strategy of the agent. As long as the likelihood ratio isunbounded, there exist unique solutions to conditions (4) and (5)with equality which we denote by t tCA bð Þ�

and t tCA bð Þ� , respectively.

For the linear signal structure, those functions are given by t tCA bð Þ� ¼T 0ð Þ

T 0ð ÞþL tCA bð Þð Þ and t tCA bð Þ� ¼ T 0ð ÞT 0ð ÞþK tCA bð Þð Þ.

MLRP implies that if sP≥t tCA bð Þ� , the principal accepts regardless of

the recommendation received because her own signal is already verystrong. If sP∈ t tCA bð Þ�

; t tCA bð Þ� , then she accepts only after a positive

recommendation. Finally, if sPbt tCA bð Þ� , she always rejects regardless

of the recommendation because her signal is very low.The principal's decision aggregates her own signal with the informa-

tion shared by the agent. As L(s) N 1 for any s N 0, the principal is using alower cutoff, after a positive recommendation, compared to what shewould use if she had access only to her own information. The principalincreases the acceptance cutoff (K(s) b 1) after a negative recommenda-tion because higher signals are required to counterbalance the negativeevidence provided by the agent.

The next result summarizes the properties of these cutoff functions.

Lemma 5. The threshold functions t sð Þ and t sð Þ satisfy the followingproperties:

1. t sð Þ b t sð Þ, for any s ∈ [0, 1];2. dt sð Þ

ds b 0 and dt sð Þds b 0, for any s ∈ (0, 1);

Proof. The first property follows from the fact that L(s) N 1 N K(s) forany s ∈ [0, 1]. The second property follows from the fact that both L(s)and K(s) are increasing in s by MLRP. □

Given the principal's best-response function, the agent's recommen-dation matters only when the principal's signal lies in the intervalt tCA bð Þ�

; t tCA bð Þ� . Thus, the agent recommends acceptance if and

only if

UA t tCA bð Þ� �

≤ sP b t tCA bð Þ� ��sA

�≥0:

�ð6Þ

The next proposition characterizes equilibrium behavior.

Proposition 6. Suppose that there exists a number tAC(b)∈ (0, 1) such that

the following condition holds:

l tCA bð Þ� �

L t tCA bð Þ� �

; t tCA bð Þ� ��

¼ T bð Þ: ð7Þ

Then, there exists an equilibrium in which the agent recommendsacceptance in the communication stage if, and only if, his signal sA≥ tA

C(b).14

14 As usual, there always exists an equilibriumwhich is completely uninformative, how-ever we disregard that equilibrium as uninteresting.


With a linear signal structure, there always exists a unique interiorequilibrium for any bias b N 0. An increase in the bias increases the agent'sstrategic manipulation of his information, that is, tA

C(b) decreases.

It is important to point out that the characterization of communica-tion equilibria provided by Proposition 6 holds for any signal structurethat satisfies MLRP. It also holds when each player holds informationof different quality. For example, we could modify the signal structureby assuming that player i observes an informative signal with probabil-ity αi and the signal is pure noise with probability 1 − αi. However,existence and uniqueness of equilibrium is not guaranteed with anarbitrary signal structure.

Provided that the likelihood ratio is unbounded, the equilibrium ofthe communication stage is informative for any size of the bias. Theprincipal uses his own signal as well as the information shared by theagent. The equilibrium also exhibits manipulation because the biasedagent recommends acceptance more often than he would if he wereunbiased. The principal responds to this manipulation by acceptingless often than she would with a less biased agent. Intuitively, the prin-cipal compensates for the agent's bias by reducing the agent's influenceover the final decision. An increase in the bias further exacerbates thismanipulation which leads to a larger loss of information.

In equilibrium, the agent bases his recommendation on the event inwhich his recommendation influences the final decision. This occurswhen the principal's information is somewhat inconclusive. However,the agent's equilibrium cutoff determines endogenously the set of theprincipal's signal realizations which provide insufficient evidence for aunilateral decision to be optimal. If the principal were to decide basedonly on her own information, she would reject if l(sP) b T(0) or sP b tP

∗.With communication, the principal finds unilateral rejection optimalonly if l(sP)L(tAC(b)) b T(0) or sPbt tCA bð Þ�

. Since L(s) N 1, then t�P NttCA bð Þ�

. Thus, the principal uses the agent's recommendation for anysignal realization between t tCA bð Þ�

and tP∗ , while she would have

rejected given her private information alone.

3.2. Conditional delegation

After observing her own signal, the principal might delegate author-ity to the agent, what we call conditional delegation.

With an expert principal, delegating authority is informativebecause the agent infers that the principal's signal must be somewhatinconclusive. Intuitively, the principal is unwilling to delegate whenher signal is very optimistic, and her information provides enough evi-dence for acceptance, or else the signal is very pessimistic and rejectionis already optimal without the agent's information. Delegation can onlyoccur for an intermediate set of the principal's signals.15 While delega-tion induces an optimal use of the agent's information, it leads to aloss of information at signal realizations for which the principal isunwilling to delegate.

Proposition 7. (Li and Suen (2004)) For a given bias b N 0, an equilibriumof the delegation game (if it exists) is given by a triple (tAD(b), t1D(b), t2D(b))which solves the following equations:

l tDA bð Þ� �

L tD1 bð Þ; tD2 bð Þ� �

¼ T bð Þ; ð8Þ

l tD1 bð Þ� �

L tDA bð Þ� �

¼ T 0ð Þ; ð9Þ

l tD2 bð Þ� �

K tDA bð Þ� �

¼ T 0ð Þ: ð10Þ

15 The principal could also delegatewhen her signal exceeds a cutoff and retain authorityotherwise. Li and Suen (2004) show that this delegation mechanism is dominated by del-egation only for intermediate signals. Therefore, we focus on the latter type of delegation.




18 Note that the equilibrium of the simultaneous voting game coincideswith the equilib-rium of the game inwhich agents vote sequentially because agents behave as if theywerepivotal in both games.19 Existence is trivial. Uniqueness follows from the observation that l sð Þ

L sð Þ ¼ s1þs is increasing


Given the agent's reporting cutoff t AD(b), the principal selects thecutoff t 2D(b) conditional on the event in which the agent received asignal below t A

D(b) (condition (10)). This implies that the principalwould prefer acceptance for any signal sP ≥ t 2

D(b) when the agentreceived a signal below tA

D(b). Thus, the principal prefers to retainauthority because the agent would otherwise reject. Similarly, theprincipal chooses the cutoff t 1

D(b) based on the event in whichthe agent received a signal above t A

D(b) (condition (9)). In this case,she retains authority for signals sP b t1

D(b) to prevent the agent fromaccepting.

In equilibrium, the agent anticipates that the principal will delegateauthority when her signal lies in the interval [t1D(b), t2D(b)], and retainsauthority otherwise. Thus, the equilibrium reporting cutoff tAD(b) is theone at which the agent is indifferent between recommending accep-tance or rejection (condition (8)).

A linear signal structure guarantees existence and uniqueness ofequilibrium under conditional delegation.16

Cheap talk vs. conditional delegation. We now compare the relativeperformance of cheap talk communication with conditional delegation.

Proposition 8. Cheap talk communication with a binary message space isequivalent to conditional delegation.

Proposition 8 shows that the principal is indifferent betweencheap talk communication with a binary message space and condi-tional delegation. This is because the principal knows, in equilibrium,how the agent will condition his decision on his own information,thus communication becomes unnecessary. This result may be gen-eralized to the case in which the principal and the agent have accessto information of different quality so that the agent has again higherexpertise than the principal, provided that a unique equilibriumexists.

Note again the contrast with Dessein's (2002) result that for smallbiases delegation is strictly preferred to cheap talk. This is the caseeven if a small bias in Dessein's model allows for a higher quality ofcommunication than in this setup in which the agent's message spacehas been restricted to only two messages. Therefore, cheap talk willdominate conditional delegation if there exists an equilibrium inwhich more than two messages are sent.

3.3. Voting

A commonly studied decision procedure with multiple informedagents is voting. So, suppose that both players can express their opinionin favor or against the project, after observing their own signal, througha votingmechanism.17 In particular, suppose that acceptance of the pro-ject requires a unanimous consensus between the two players. Eventhough there is no explicit communication between the players beforevoting takes place, the voting mechanism itself allows for informationsharing.

This organizational structure has been studied by Li et al. (2001).They show that an interior equilibrium (tPV(b), tAV(b)) must simulta-neously satisfy the following two conditions

l tVP bð Þ� �

L tVA bð Þ� �

¼ T 0ð Þ; ð11Þ

l tVA bð Þ� �

L tVP bð Þ� �

¼ T bð Þ: ð12Þ

Then, player i votes for acceptance if, and only if, si≥ tiV(b). Intuitive-

ly, player i's vote is pivotal when the other player votes for acceptance

16 We refer to the proof of Proposition 6 for a formal argument.17 We do not allow the players to abstain.


which, given the equilibrium strategies, occurs with a likelihood ratioequal to L(t− i

V (b)).18 A linear signal structure once again guaranteesthat such an equilibrium exists and is unique.19

A natural question is whether and under what conditions the princi-pal might prefer voting to cheap talk communication (or, equivalently,conditional delegation).

Proposition 9. Cheap talk vs. voting. The principal strictly prefers cheaptalk communication to voting with unanimous consensus.

Proposition 9 shows that voting is always strictly dominated bycheap talk. The principal utilizes the information of the agent morewith cheap talk. Intuitively, the agent considers his vote with votingand his recommendation with cheap talk as pivotal in both organiza-tional structures. However, a decision is pivotal under cheap talk forsignal realizations of the principal for which the optimal decision ismore uncertain.20 Therefore, the agent manipulates his informationless in order to reduce the probability that the principal will make awrong decision.

3.4. Cheap talk with more than two messages

To show that cheap talk communication can dominate conditionaldelegation, it is enough to focus on equilibria with three messages. So,suppose that the agent's message space is M = {m1, m2, m3} and thathe follows the communication strategy21

m sAð Þ ¼m1m2m3

ififif

sAbtA1 ;

t A1≤sAbtA2 ;

sA≥tA2 :

8><>: ð13Þ

Given this strategy, the principal accepts aftermessagem1 orm3 ifsP≥t tA1

� and sP≥t tA2

� , respectively. If she instead receivesmessagem2,

she knows that the agent's signal is between t1A and t2

A. Thus, she acceptsif, and only if, l(sP)L(t 1A, t 2A) ≥ T(0) or sP≥ t̂ tA1 ; t

A2

� . By MLRP, t t A2

� b t̂

t A1 ; tA2

� bt tA2

� . The same pivotal argument employed in the first part

of this section implies that the agent's cutoff types must solve

l tA1� �

L t̂ tA1 ; tA2

� �; t tA1� ��

¼ T bð Þ; ð14Þ

l tA2� �

L t tA2� �

; t̂ tA1 ; tA2

� �� ¼ T bð Þ: ð15Þ

As L t tA2�

; t̂ tA1 ; tA2

� � bl t̂ tA1 ; t

A2

� � bL t̂ tA1 ; t

A2

� ; t tA1� �

by MLRP, it fol-lows that t1A b t2

A. This also shows that whenever such cutoffs exist andare interior, they are part of an informative equilibriumwith threemes-sages. In general, existence clearly hinges on the size of the bias whichputs an upper bound onhowmanymessages can be sent in equilibrium.With a linear signal structure, the unboundedness of the likelihood

in s, so that the best-response functions can only intersect once.20 Recall that the principal follows the agent recommendation only at an intermediaterange of signal realizations.21 For the sake of expositional clarity, we omit the dependence of the cutoffs from theagent's bias b.





ratios implies that an equilibrium with three messages always exists.22

We summarize the analysis so far in the following proposition.

Proposition 10. Cheap talk communication dominates conditionaldelegation.

Themain difference between Proposition 10 andDessein's result liesin the combination of a restricted action space (which is binary here)and information on the receiver's side. Both assumptions are quitenatural in many relevant applications, like grant approvals, etc. Withan expert receiver, the act of delegating authority serves the dualpurpose of allowing for a better use of the agent's information (as inDessein) and communicating information to the agent (which is miss-ing in Dessein). When the principal is uninformed, the second effect isirrelevant but a restricted action space prevents the agent from finetuning his information to the optimal action. This makes cheap talkcommunication and delegation equivalent for the principal. With anexpert principal, conditional delegation also puts an upper bound onhow much information can be inferred from delegating authority tothe agent. Therefore, cheap talk communication allows for finerpartitioning of the agent's information with the additional benefit forthe principal of using her information optimally.23

3.5. Optimal delegation with a binary message space

In this section, we consider the more general case of optimaldelegation. While the full characterization of the optimal mechanismis beyond the scope of this paper, we discuss how the principal mightimprove her payoff through commitment in the context of our leadingexample with a linear signal structure.

Equilibrium of the cheap talk game incorporates the sequentialrationality notion that the principal best-responds with thresholds ttCA bð Þ�

and t tCA bð Þ� to the agent's reporting cutoff tAC(b). We now illus-

trate the potential gain to the principal from committing ex ante (i.e.,before observing her own signal) to different thresholds tbt whichneed not be sequentially rational. More precisely, the principal commitsto the following decision rule

d sP ;mjt ; t� ¼ AcceptReject

if m ¼ Accept & sP≥t ; or m ¼ Reject & sP≥t;otherwise:

�

ð17Þ

The agent takes thresholds tb t as given. With signal sA, the agentoptimally recommends acceptance if and only if

UA t≤sPbtjsA�

≥0; ð18Þ

22 The same construction can be extended to an arbitrary number N of messages. To seethis, let {t 0A=0, t 1A,…, t n − 1

A , t nA =1} denote a partition of [0, 1] such that the agent sends

messagemj if sA ∈ [t j − 1A , t jA). Then, we can define thresholds t̂ t Aj−1; t

Aj

� �n on−1

j¼1such that

the principal accepts after receiving messagemj if, and only if, sP≥ t̂ t Aj−1; tAj

� �. By MLRP, t̂

t Aj−1; tAj

� �N t̂ t Aj ; t

Ajþ1

� �, for any j = 1,…, n − 1. Finally, each cutoff t jA must solve

l tAj� �

L t̂ tAj ; tAjþ1

� �; t̂ tAj−1; t

Aj

� �� ¼ T bð Þ: ð16Þ

Eq. (16) follows again froma pivotal argument. Given the agent's equilibrium strategy, send-ing messagemj + 1 is better than sending messagemj when the principal's signal lies in the

interval t̂ t Aj ; tAjþ1

� �; t̂ t Aj−1; t

Aj

� �h �provided that sA≥ t j

A. This is becausemj+1 induces accep-

tance whilemj leads to rejection. Similarly, the agent optimally sends messagemj when hissignal lies slightly below t j

A. At t jA, the agent must be indifferent between the two messages.23 Recently, Kolotilin et al. (2013) have shown that the principal can benefit from theability to commit ex ante (i.e., before communication) to a restricted action space ex post(i.e., after communication) throughan increase in thequality of communication comparedto the baseline model of Crawford and Sobel (1982). Those authors start from an unre-stricted action spacewhile we consider environments inwhich only two actions are feasi-ble. Thus, it is unclear what the effect of a larger action space would be in our frameworkand if a binary action space would necessarily be optimal.


a slight modification of Eq. (6). Modifying Eq. (7), the agent recom-mends acceptance when sA ≥ tA

O(b) which solves

l tOA bð Þ� �

L t ; t� ¼ T bð Þ: ð19Þ

The linear signal structure implies that

tOA bð Þ ¼ T bð ÞL t ; t� þ T bð Þ ¼

2−t−t�

T bð Þt þ t� þ 2−t−t

� T bð Þ : ð20Þ

Taking into account the agent's best response, the principal's utilityas a function of thresholds tbt is

UP t ; t� ¼ pRG Pr sA≥tOA bð Þ; sP≥t jG

� �þ Pr sAbt

OA bð Þ; sP≥t

��G� �h i

− 1−pð ÞRB Pr sA≥tOA bð Þ; sP≥t jB� �

þ Pr sAbtOA bð Þ; sP≥t

��B� �h i:

ð21Þ

For the purpose of maximization of UP over thresholds tb t, the scaleof UP is irrelevant, so we normalize pRG = 1 which then implies that(1 − p)RB = T(0) N T(b). To simplify the analysis, we further assumethat T(0) = 1. We can then write the principal's expected utility for anarbitrary pair of thresholds t ; t

� as

UP t ; t� ¼ 2t−t2−t2− t2−t2

� �2−t−t� 2−t−t

� T bð Þ2 þ t þ t

� t þ t� þ 2−t−t

� T bð Þ �2

:

ð22Þ

Proposition 11 shows that the principal benefits from commitmentpower.

Proposition 11. Commitment power increases the principal's expectedutility compared to the equilibrium with no commitment.

Furthermore, unconditional delegation, that is, t ; t� ¼ 0;1ð Þ, is never

optimal.

As the principal can always commit to the equilibrium thresholds, itis natural that she finds commitment power beneficial.

Proposition 11 also shows that the principal can never benefit fromdelegating authority unconditionally to the agent, as in Dessein(2002). Suppose that the principal committed to unconditional delega-tion. This implies that the principal behaves as if she were uninformedbecause she will ignore her signal. Thus, only one signal is used in equi-librium. Now fix t ¼ 0 and consider a small reduction in t from t ¼ 1. Byreducing t the principal instead uses her own informationwhen the sig-nal she receives is very informative which increases her expected utilitygiven that more information is used. However, the overall effect of sucha reduction in t depends on its effect on the agent's best-response.Inspection of Eq. (20) shows that a reduction in t always increases theagent's cutoff tAO(b) which is also beneficial for the principal as theagent uses a higher standard for acceptance. Therefore, conditionaldelegation is always preferable.

This discussion highlights two simple but important observations.First, the principal is necessarily better off by exploiting some of herinformation. Second, commitment to thresholds 0btbtb1 can be used toaffect the agent's incentives and, consequently, his strategicmanipulation.

The next Proposition 12 refines the previous discussion by showingthat the principal has at least a local incentive to commit to lowerthresholds compared to the equilibrium ones.

Proposition 12. At the equilibrium thresholds, the principal has a (local)

incentive to commit to lower thresholds, that is, ∂UP t tCA bð Þð Þ;t tCA bð Þð Þð Þ∂t b0 and

∂UP t tCA bð Þð Þ;t tCA bð Þð Þð Þ∂t b0:

To further illustrate these ideas, we consider a simple example.




24 Withmore than twomessages, we say that an equilibrium is informative if there existat least two messages m1 andm2 such that d(m1) ≠ d(m2).25 When sA = tA

∗(b), the agent is indifferent between acceptance and rejection but thisevent happens with zero probability because the signal distribution is continuous.


Example. With a linear signal structure, the equilibriumwhen T(0)=1yields the following best-response functions for the principal

t tCA bð Þ� �

¼ 1−tCA bð Þ2

; t tCA bð Þ� �

¼ 2−tCA bð Þ2

:

The equilibrium condition (7) becomes the quadratic

tCA bð Þ ¼1þ 2tCA bð Þ

� �T bð Þ

3−2tCA bð Þ� þ 1þ 2tCA bð Þ� T bð Þ ; ð23Þ

solved by

tCA bð Þ ¼ 14−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ 1−T bð Þð Þ 5−9T bð Þð Þ

p−2

4 1−T bð Þð Þ : ð24Þ

For instance, when T(b) = 5/9, the solution is t AC(b) = 1/4 with tt CA bð Þ� ¼ 3=8 and t tCA bð Þ� ¼ 7=8. For T(b) = 5/9, the optimal commit-

ment solution is roughly t�; t�Þ ¼ 29=100;3=4ð Þ�with tA

O(b) = 0.339.The commitment solution has lower thresholds than the no commit-ment case, that is, t�b t tCA bð Þ�

and t�b t tCA bð Þ� , which leads to a higher

reporting cutoff tAO(b) N tAC(b). The principal distorts her own thresholds

by committing to follow the agent's recommendation more often afterbad signals and to unilaterally accept more often after good signals.These distortions are optimal because they induce the agent to adopta higher standard when recommending acceptance.

As illustrated in the example, commitment power generates a newtrade-off for the principal between the loss of information due to thesuppression of her own information and the loss of information due tothe agent's strategic manipulation of his own information. This is differ-ent from the standard trade-off between loss of information and loss ofcontrol, as in Dessein (2002), and stems from the fact that the principalis also informed. The example that we analyzed shows that suppressionof information may be inevitable to reduce the agent's equilibriumamount of information manipulation.

4. Conclusion

This paper compares the performance of several organizationalarrangements for aggregating information. We revisit the classictrade-off between centralization and delegation of authority in a frame-work with a binary action space and continuous noisy signals.We showthat delegation is equivalent to cheap talk communication when theprincipal is uninformed and communication is informative. This resultcontrasts with that of Dessein (2002). This is because a restricted actionspace improves the informativeness of cheap talk communication andcan reduce (eliminate, in our case) the information loss experiencedby the principal in standard delegation models based on Crawford andSobel (1982). When the principal is also an expert, we show thatconditional delegation is strictly suboptimal.

While we focused attention on cases in which the principal is onlylimited to commit to a choice among given organizational structures,we sketched a partial characterization of the optimal mechanism withcommitment (but still without transfers) in our setup with an expertprincipal. When the principal can commit to the way in which sheuses the information received by the agent, a new trade-off arisesbetween suppression of information on the principal's side and infor-mation loss due to strategic manipulation on the agent's side. The fullstudy of this trade-off would complement the existing literature ondelegation versus centralization and it is left for future work.


Appendix A. Proofs

Proof of Proposition 1. We first show that for any informative equilib-rium in which the agent sends more than two messages we canconstruct another informative equilibrium in which the agent sendsonly twomessages and which is payoff-equivalent.24 Consider an infor-mative equilibrium inwhich the agent chooses amongmessagesm1,…,mn, with n≥ 2. Given that the final decision is binary, messagemk leadsto acceptancewith probability q(mk). Supposewithout loss of generalitythat q(mk) ≤ q(mk + 1) for k = 1, …, n − 1 with at least one strict in-equality, so that messages are ordered. From previous observations,we know that the agent would like the project to be accepted if, andonly if, sA ≥ t A

∗ (b). When sA N t A∗(b), the agent would be indifferent

among any message that induces the highest probability of acceptance.Similarly, he would be indifferent among any message that induces thelowest probability of acceptance when sA b tA

∗(b).25 Partitioning the setof messages according to their probability of acceptance, we canconstruct another equilibrium in which the agent uses the first andthe last elements of the partition asmessages. These equilibria are clear-ly payoff-equivalent. Thus, we can restrict attention to equilibria with atmost twomessages and, without loss of generality,M={Reject, Accept}.

Suppose that there exists t̂A∈ 0;1ð Þ such that the agent is indifferentbetween inducing acceptance or rejection at signal t̂A. MLRP implies thatthe agent is strictly better off recommending rejection for any signalbelow t̂A, and acceptance otherwise. Thus, in any informative equilibri-um the agent uses a cutoff strategy. Furthermore, MLRP also impliesthat there can only be one signal at which the agent is indifferentwhich corresponds to tA

∗(b).Suppose that p b RB

RGþRBand fix a bias b N 0. Suppose also that the prin-

cipal thinks that the agent recommends acceptance if, and only if, sA ≥tA∗(b). Note that p b RB

RGþRBimplies that 0 b t�P b 1 ¼ t�P. The principal always

rejects after a negative recommendation from the agent which isinterpreted as the event in which sA b tA

∗(b). Conditional on this event,it is indeed optimal for the agent to recommend rejection. So, supposethat sA ≥ t A

∗ (b). Note that an increase in the agent's bias reduces hisown threshold for acceptance, that is, dt�A bð Þ

db b 0. Then, monotonicity and

continuity of t A∗(b) in b imply that there exists bN0 such that t�A bð ÞNt�Pif, and only if, b∈ 0; b

h �. For the linear signal structure, b≡ RG

pRGþ 1−pð ÞRBRB−p RGþRBð ÞN0.

This implies that as long as the agent's bias is below b, the conjecturedcommunication strategy is incentive compatible and the equilibriumis informative, that is, the principal follows the agent's recommenda-

tion. If instead bNb , the previous communication strategy is stillincentive compatible. However, the principal always rejects regardlessof the recommendation. Thus, the equilibrium is uninformative.

Finally, suppose that p≥ RBRGþRB

. The principal is so optimistic aboutthe project that she always accepts after a positive recommendation(i.e., t�P ¼ 0). Thus, the agent is better off recommending acceptanceexactly when sA ≥ tA

∗(b). So, suppose that sA b t A∗ (b). As t�A bð Þbt�P≤1,

the principal always rejects after a negative recommendation. Rejectionis also preferred by the agent. Thus, the conjectured strategy is incentivecompatible and the equilibrium is informative.

Proof of Proposition 3. The probability of a Type-I error is given byPI = (1 − p)Prob(sA ≥ tA

∗ (b)|B) = (1 − p)(1 − FB(tA∗(b))) whenevercommunication is informative. Clearly, an increase in the bias reducesthe cutoff tA∗(b) which necessarily increases the probability of falseacceptance.

Next, differentiating b with respect to the prior belief p gives ∂b∂p ¼

2RBR2G

RB−p RBþRGð Þ½ �2 N 0.





Proof of Proposition 6. The inequality (6) is equivalent to

Prob GjsA; t t CA bð Þ� �

≤sPbt tCA bð Þ� ��

¼pl sAð ÞL t t CA bð Þ

� �; t t CA bð Þ� ��

pl sAð ÞL t t CA bð Þ� ; t t CA bð Þ� � þ 1−pð Þ ≥

RB

RG þ RB þ b:

ð25Þ

The agent recommends acceptance if, and only if, l sAð ÞLt t CA bð Þ�

; t t CA bð Þ� � ≥T bð Þ. Thus, if there exists t AC(b) such that l t CA bð Þ� L

t t CA bð Þ� ; t t CA bð Þ� � ¼ T bð Þ , the agent is indeed (weakly) better off

recommending acceptance if, and only if, sA ≥ tAC(b) by MLRP.

With a linear signal structure, the equilibrium characterization im-plies that the agent’s cutoff, whenever interior, must solve the followingequation

t ¼ 2− t tð Þ þ t tð Þ� �Tðb 2− t tð Þ þ t tð Þ� �T bð Þ þ ðt tð Þ þ t tð Þ ≡ RHS tð Þ: ð26Þ

It is immediate to show that an equilibrium always exists becauseRHS(0), RHS(1) ∈ (0, 1) and dRHS tð Þ

dt N 0, and all equilibria are interiorfor any b N 0. Uniqueness follows from the fact that l tð ÞL t tð Þ; t tð Þ� is increasing in t. Finally, let M tð Þ≡ t tð Þ þ t tð Þ . As an increase inthe bias reduces T(b), it reduces the reporting cutoff tAC(b) because∂RHS tð Þ∂T bð Þ ¼ 2M tð Þ 2−M tð Þð Þ

2T bð Þþ 1−T bð Þð ÞM tð Þ½ �2 is positive by M(t) b 2 for any t ∈ [0, 1].

Proof of Proposition 8. The principal's expected utility in the informa-tive cheap talk equilibrium is given by

UCTP ¼ pRG 1−FG tCA bð Þ

� �h i1−FG tC1 bð Þ

� �h iþ FG tCA bð Þ

� �1−FG tC2 bð Þ

� �h in o

− 1−pð ÞRB

n1−FB tCA bð Þ

� �h i1−FB tC1 bð Þ

� �h i

þ FB tCA bð Þ� �

1−FB tC2 bð Þ� �h io

;

ð27Þ

where t C1 bð Þ ¼ t t CA bð Þ� �

and t C2 bð Þ ¼ t t CA bð Þ� �

. The principal commits a

Type-I error in the bad state and a Type-II error in the good statewhen either she receives a positive recommendation and her signal is

above t tCA bð Þ� �

or if she receives a negative recommendation and her

signal exceeds t tCA bð Þ� �

.

The principal's expected utility from conditional delegation is givenby

UDelP ¼ pRG 1−FG tD2 bð Þ

� �þ FG tD2 bð Þ

� �−FG tD1 bð Þ

� �h i1−FG tDA bð Þ

� �h in o

− 1−pð ÞRB 1−FB tD2 bð Þ� �

þ FB tD2 bð Þ� �

−FB tD1 bð Þ� �h i

1−FB tDA bð Þ� �h in o

:

ð28Þ

A direct comparison of the equilibrium conditions under bothorganizational arrangements together with the requirement that theequilibrium is unique implies that t AD(b) = tA

C(b), t1D(b) = t1C(b) and

t2D(b)= t2

C(b). Thus, a straightforward calculation shows thatUPCT=UP

Del.

Proof of Proposition 9. We can renormalize the principal and agent'spayoffs in terms of costs of making the wrong decision. While RB is thecost of false acceptance, we can think of RG as the cost of false rejectionrather than the payoff from correct acceptance.With this reformulation,the principal's maximization problem can be reinterpreted in termsof minimizing the cost of making a wrong decision and we can thenapply Proposition 6 in Li and Suen (2004) to show that conditionaldelegation, and thus cheap talk, dominates voting.

Proof of Propositions 11 and 12. We investigate how UP varies alongany anti-diagonal defined by M ¼ t þ t , where M ∈ [0, 2]. On such a


line, the free parameter t varies betweenmax{0,M− 1} andM2. Substitut-

ing for M into Eq. (22) gives

UP t ;M−tð Þ ¼ 2 M−tð Þ−t2− M−tð Þ2−M 2−Mð Þ M−2tð Þ 2−Mð ÞT bð Þ2 þMM þ 2−Mð ÞT bð Þ½ �2 ;

ð29Þ

which is a concave quadratic function of our free parameter t . Differen-tiating with respect to t gives

dUP t ;M−tð Þdt

¼ −4t þ 2 M−1ð Þ þ 2M 2−Mð Þ 2−Mð ÞT bð Þ2 þMM þ 2−Mð ÞT bð Þ½ �2 : ð30Þ

At t ¼ 0 and M = 1, the derivative is positive which shows thatthe principal is better off increasing t away from 0. Thus, unconditionaldelegation cannot be optimal.

Finally, evaluating the derivative at the equilibrium thresholdst tCA bð Þ�

; t tCA bð Þ� � gives

∂UP t tCA bð Þ� �

; t tCA bð Þ� ��

∂t ¼ 8T bð Þ3−2tCA bð Þ þ 1þ 2tCA bð Þ

� �T bð Þ

h i2 2tCA bð Þ−12

;

ð31Þ

which is negative because tCA bð Þb 12 from Eq. (24). Similarly, it can be

shown that ∂UP t tCA bð Þð Þ;t tCA bð Þð Þð Þ∂t b0. This shows that the principal can increase

her expected utility compared to the no commitment case by locallydecreasing both thresholds.

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