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Advice by an Informed Intermediary: Can You

Trust Your Broker?

Anton Suvorov and Natalia Tsybouleva

¤

April 1, 2005

Abstract

The paper investigates credibility of the intermediary’s advise in a bi-lateral trade model. A seller and a buyer with private and independent valuations exchange a unit of good, their trade being mediated by a bro-ker, who at the pre-bargaining stage observes a coarse signal about the buyer’s valuation. We show how the optimal direct mechanism maximiz-ing the broker’s expected pro…t incorporates the broker’s signal. We prove that if the valuations of the seller and the buyer are drawn from uniform distributions and the intermediary’s signal speci…es whether the buyer’s valuation exceeds a certain threshold, choosing an appropriate system of two-part tari¤s allows the intermediary to secure in a modi…ed double-auction game the same payo¤ as in the optimal direct mechanism. We then further restrict the contractual framework and assume that the inter-mediary gets only a …xed transaction fee, his goal being now to maximize the expected number of transactions. We show that the intermediary’s information can be credibly transmitted via cheap talk. The transmission of information by the broker increases ex ante welfare of the seller and the broker, but has ambiguous impact on the buyer. If the intermediary ob-serves signals about valuations of both participants, the equilibrium where the broker truthfully communicates both signals exist only under some further restrictions on parameters of the model.

1 Introduction

Bilateral bargaining often takes place under asymmetric information. The par-ties may seek better information about the other side trying to improve their bargaining positions. Natural sources of such information are intermediaries that

¤We are grateful to Thomas Mariotti, Jean-Charles Rochet and Jean Tirole for many useful comments and criticisms. All errors are ours.

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in many cases implement the transactions. Indeed, a large fraction of real estate transactions are mediated by realtors, most IPOs are mediated by investment banks and art sales are often mediated by specialized dealers, etc. These inter-mediaries often have superior information about the demand for (or the supply of) the item being sold, and, sometimes, about the private valuation of a particular buyer or seller.

The informed intermediaries may and do a¤ect the outcomes of the bargaining if they reveal their information in the form of advice. In case of art auctions, auction house experts help sellers to set reserve prices and at the same time provide in the pre-auction catalogues a low and a high price estimate for each item, which may signal the seller’s secret reserve price1. In the real estate sector

the Federal Trade Commission survey reports that 20.9% of buyers and 30.5% of sellers use the real estate agent’s advice as the single most in‡uential source of information to determine their …rst price o¤ers and listing prices, respectively. Despite an apparent con‡ict of interests, the intermediaries may even advise both sides in the bargaining.

While the consumers welcome the intermediaries’ advice, they are also con-cerned about the abuse of the private information by the broker. For example, there was a serious debate on whether the …nancial institutions should be al-lowed to act as realtors. In December 2000 the Federal Reserve and Treasury Department issued a joint proposal which would allow …nancial holding compa-nies (FHCs) and …nancial subsidiaries of national banks to engage in the real estate brokerage. This proposal met a large discontent of the realtors, and one of the concerns was the endangered consumer privacy. The National Association of Realtors (NAR) reports that ”81% of Americans are worried that their bank could use their private information to sell real estate services to them”.2 NAR’s report

further argues that ”FHC-operated real estate brokerage operations could have access to seller-client …nancial records and use that private credit information to the detriment of a home-seller...”.

Whether intermediaries’ advice about other party of potential transaction should be welcomed and trusted, who gains and who loses from the dissemination of information and what is the overall e¤ect on the welfare, is not clear so far.

The role of intermediaries in facilitating bargaining is vastly discussed in the literature. Typically, intermediaries have access to some information technol-ogy that helps bringing two parties together and thus raises the e¢ciency of the market organization. For instance, intermediaries may facilitate the search of a trading partner (Rubinstein and Wolinsky (1987), Yavas (1994)) or they may have expertise in certi…cation, screening and monitoring (Biglaiser (1993), Lizzeri (1999), Peyrache and Quesada (2004)). Another strand of the literature on

inter-1The seller’s secret reserve price, by convention, lies below the low estimate (Ashenfelter and Graddy (2002))

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mediation emphasizes the strategic advantage that a party (a seller or a buyer) may gain if it delegates the bargaining to the intermediary. For example, Cai and Cont (2000) analyze how a principal (a seller) should design the delegation contract in order to provide proper incentives for her delegate and gain a strategic advantage against a buyer.

Gehrig (1993) considers a search model in which a continuum of buyers and sellers privately informed about their valuations engage in bilateral trade once they meet. He shows that an intermediary can reduce the trading frictions in two ways: provide more e¢cient matching and, by committing to …xed buy and sell prices, replace the ine¢cient bilateral bargaining.

Rochet and Tirole (2004) show that the platform (an intermediary) can im-prove the e¢ciency of bilateral bargaining if it concludes the contracts with the traders before they learn their valuations: the intermediary can subsidize trans-actions and recoup its surplus through participation fees.

In this paper we also look at how the intermediary can improve e¢ciency of bilateral trade. The mechanism is quite di¤erent, however: the intermediary facilitates trade by transmitting information at the pre-bargaining stage. We look at a situation of bilateral trade under asymmetric information and assume that the intermediary observes an (exogenous) imperfect signal about the buyer’s valuation. Then, the intermediary’s choice of the contract reveals this information to the seller and a¤ects the subsequent bargaining between the traders. We further show that even if the contract is …xed exogenously, the intermediary is able to transmit the information he observes by cheap talk.3

To the best of our knowledge, the literature that considers information trans-mission by the intermediary is scarce. The closest papers that deal with the question are Baron and Holmstrom (1980), and Baron (1982). These papers study the optimal contracts between an issuer and an investment bank which provides advising and distribution services, assuming that the investment banker is better informed about the capital market than the issuer is. The task of the issuer is to design a contract that both induces the banker to use his superior information to the issuer’s advantage and provides incentives to the banker to ex-ert e¤ort required to sell the issue. The optimal contract determines the banker’s compensation as a function of his report about the market demand for the issue and of the proceeds from the sale. Both papers assume one-sided asymmetric information – only the intermediary has private information. However, in many

3Farrell and Gibbons (1989) claim that cheap talk can matter in bargaining. In their model at the pre-bargaining stage parties may announce whether they are ”keen” or ”not keen” to trade. Farrell and Gibbons show that if saying that one is ”keen” makes one’s partner more likely to negotiate, then it is the keenest types (high-value buyers, low-value sellers) who are most willing to say so, and hence cheap talk conveys some meaningful information. The equilibrium of the bargaining game preceded by a cheap talk involves more trade when one of the parties has a keen type, and less trade when both parties have intermediate types, compared to the ex-ante optimal linear equilibrium of the double auction mechanism without cheap talk.

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circumstances, the seller (issuer) has some important characteristics unobserv-able by the intermediary but a¤ecting the sellers reaction to the intermediary’s advice: while a low-cost seller may decrease a price upon learning about unfavor-able demand, a high-cost seller may simply withdraw from the market. In our paper we look at how inability to perfectly predict the seller’s behavior a¤ects incentives of the intermediary to transmit information.

A recent paper by Calzolari and Pavan (2004) studies an interaction of an agent with two consecutive principals and investigates when the …rst principal wants to sell the information she gets from the agent to the second principal. It shows that under some broad circumstances, it is optimal for the …rst principal to transmit no information since she would have to leave more rents to the agent to compensate him for the loss of privacy. The role of the …rst principal in the interaction between the agent and the second principal is akin to the role of an informed intermediary in our model. The main di¤erence, however, is that the principal in Calzolari and Pavan acquires the information from the agent, while our broker observes an exogenous signal about the buyer.

Our paper proceeds as follows. Section 2 starts by formulating a simple bi-lateral trade model: a buyer and a seller, each having private and independent valuations, are contemplating an exchange of one unit of good. The trade is me-diated by a broker, who designs a contract to maximize his expected pro…t. We assume that prior to writing the contract, the broker observes whether the buyer is eager to trade (has valuation above a certain threshold) or is not eager. Then, we show that in both cases the optimal direct mechanism is the same as the one the broker would o¤er were his information public.

In Section 3 we consider a decentralized model in which the intermediary o¤ers two-part tari¤s to the buyer and the seller, consisting of a participation and a per-transaction fees. The traders then bargain via a double auction mechanism, in which trade happens if the spread between the bids exceeds the per-transaction fee set up by the broker. We show that there exists a separating equilibrium in which the broker selects di¤erent contracts for di¤erent signals he observes, and at the double-auction stage the traders play (piecewise) linear strategies as in Chatterjee and Samuelson (1983). The intermediary’s expected pro…t achieves the upper bound derived in Section 2. The optimal contract speci…es a higher transaction fee when the buyer is eager to trade, as well as a (weakly) higher participation fee.

Furthermore, comparing the parties ex ante welfare in the models with in-formed and uninin-formed intermediaries, we conclude that the intermediary and the seller always gain from the more precise information and thus improved e¢-ciency, while the impact on the buyer depends on the nature of the intermediary’s signal. If the signal is more informative about the weak types (i.e. the threshold is low), the gains from improved e¢ciency outweigh the buyer’s losses from her high eagerness being divulged, and vice versa for the signal that separates really strong buyers.

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Section 4 further restricts the intermediary’s communication opportunities: we assume that the contract is …xed and consists of a mere per-transaction fee. In this setting the broker wants to maximize the probability that the transaction happens without caring about the terms of trade. We allow the broker to send costless messages to the seller prior to the double auction game. We show that there exists a fully revealing equilibrium in which the intermediary truthfully communicates his signal to the seller. What stops the broker from understating the buyer’s eagerness to trade (thus encouraging the seller to charge less) is the understanding that some higher valuation sellers who have no chance to trade with a weak buyer will simply refrain from bidding (provided they incur an in…nitesimal cost of submitting a bid).

Finally, in Section 5 we assume that the intermediary is informed about both sides of the market. We show that while for some parameter values the fully revealing equilibrium in which the intermediary reports truthfully to both parties still exists, for others there is no equilibrium with complete communication: more precise information about the recipient’s type allows the broker to …ne-tune his misreporting.

The Appendix contains some preliminary analysis under more general as-sumptions than those adopted in the main part of the paper and some technical statements used in the proofs of our results.

2 The Model

We consider a simple model of bilateral trade in the presence of an intermediary. There is a seller who can produce a unit of good at cost vS, and a buyer who

contemplates buying it and has valuation vB: It is common knowledge that the

agents’ valuations vSand vBare drawn independently from a uniform distribution

on [0; 1], but the valuations themselves are the agents’ private information. The intermediary (or the broker) has no valuation for the good and only implements a transaction between the buyer and the seller. The intermediary also observes some informative signal about at least one party’s valuation and can use this information in the trading mechanism or communicate it to the parties thus a¤ecting their bargaining strategies. All the parties are risk-neutral and their utility in the absence of trade is normalized to 0.

In the absence of information observed by the broker, Myerson and Satterth-waite (1983) have characterized a direct mechanism maximizing the broker’s ex ante pro…t. They have shown that it is optimal for the broker to implement trade if and only if the buyer’s ”virtual valuation” (i.e. his true valuation minus the information rent required to induce truthtelling) exceeds that of the seller, which for the case of uniform distributions on [0; 1] means that vB ¸ vS+ 12; and to

leave no surplus to the worst buyer (vB = 0) and to the worst seller (vS = 1).4 4These conditions uniquely determine the expected transfers from the buyer to the broker

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What distinguishes our model from Myerson and Satterthwaite (1983) is the assumption that the intermediary observes a coarse signal about the buyer’s type: he learns whether the buyer is ”eager” or ”not eager” to trade:5 v

B 2 [w; 1]

or vB 2 [0; w] for some …xed parameter w 2 (0; 1). This signal is exogenous,

no truthtelling incentive constraints are to be satis…ed for the intermediary to get this information. There are several justi…cations that can be given for this assumption. One is that a professional intermediary has more experience than a seller in interpreting the buyer’s observable characteristics or aspects of pre-play behavior, so his knowledge of the buyer’s valuation is more precise. Another interpretation is that (for similar reasons) the intermediary has better information about the demand for the seller’s good, or at least about the distribution of the types of buyers that he can match with the seller.6

The broker can use the informative signal he gets in designing the optimal mechanism. By the revelation principle, the upper bound of the broker’s expected pro…t can be found through the optimal direct truthful mechanism, which spec-i…es the probability of trade and the expected payments from the buyer to the broker and from the broker to the seller as functions of the three parties’ reports, subject to the incentive compatibility and interim participation constraints. The optimal mechanism7 is described in Proposition 1.

Proposition 1 The mechanism maximizing the broker’s ex ante pro…t imple-ments trade if vB ¸ vS+12 when the buyer is eager (vB 2 [w; 1]) and if vB¸ vS+w2

when the buyer is not eager (vB 2 [0; w]). The buyer with valuation vB = 0 or

vB = w gets no ex ante surplus, as well as the seller with valuation vS = 1:

Proof. Along the lines of Myerson and Satterthwaite (1983) one easily shows

and from the broker to the seller.

5We want to avoid confusion with Farrell and Gibbons (1989), who use ”keen” and ”not keen” for the same purpose. In their model (without intermediary) the subsets of ”keen” and ”not keen” types are determined endogenously in such a way that the information on the player’s belonging to one of the subsets could be credibly transmitted in pre-play cheap talk. In our model, the subsets of ”eager” and ”not eager” types are de…ned exogenously.

6This interpretation seems appropriate for many situations. For example, an art dealer may know better the demand for a particular piece of art than an accidental owner. The same is often true for other types of intermediaries, such as real estate agents, recruiting agencies, etc. 7The mechanism is uniquely de…ned by the probability of trade and the expected utility of the buyers and the sellers with valuations vB= 1, vB= w and vS = 0. Transfer functions can be speci…ed in multiple ways, but the expected payment of each type of agent is the same (see Myerson and Satterthwaite (1983)).

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that the broker’s expected pro…t UI = w 1 Z 0 w Z 0 ·µ vB¡ 1¡ FL B(vB) fL B(vB) ¶ ¡ µ vS+ FS(vS) fS(vS) ¶¸ pL(vS; vB)fBL(vB)fS(vS)dvBdvS +(1¡ w) 1 Z 0 1 Z w ·µ vB¡ 1¡ FH B(vB) fH B(vB) ¶ ¡ µ vS+ FS(vS) fS(vS) ¶¸ pH(vS; vB)fBH(vB)fS(vS)dvBdvS ¡wUB(0)¡ (1 ¡ w)UB(w)¡ US(1); where FL

B(vB) = vwB and FBH(vB) = v1¡wB¡w for our case of uniform distributions,

pL(vS; vB) and pH(vS; vB) are the probabilities of trade for the case of a non-eager

and eager buyer. It is optimal to set pL(vS; vB) = ½ 1 if vB ¸ vS+ w2; 0 otherwise ; pH(vS; vB) = ½ 1 if vB ¸ vS+ 12; 0 otherwise and UB(0) = UB(w) = US(1) = 0:

Proposition 1 shows that the broker indeed wants to use his information in the optimal mechanism. In fact, the broker would obtain the same expected pro…t if he could commit ex ante to share his signal about the buyer’s valuation with the seller and then use an optimal direct mechanism to which only the buyer and the seller would report their valuations, the buyer’s eagerness to trade being now common knowledge between the players. Since the seller’s utility is quasi-linear, the broker would not lose from the early disclosure of his information to the seller (with more general utility functions, the early disclosure would be suboptimal since the broker could gain by pooling the seller’s incentive and participation constraints – see Maskin and Tirole (1990 and 1992)). Note that in our setting where the broker’s own announcement can be veri…ed ex post (assuming that the buyer truthfully reveals his type), it is costless for the broker to induce his own truthtelling once the buyer’s truthtelling is secured. Thus, the ability to commit to disclose its information is super‡uous.

The fact that the broker use the informative signal in designing the optimal mechanism is still valid in a more general framework, where distributions of the seller/buyer valuations are not uniform, and broker’s signal is not a truncation of the interval (see Appendix).

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3 Indirect Implementation

3.1 The Double-Auction Mechanism

From this section on we abandon the complete contracting approach: we spec-ify the trading institution exogenously no longer assuming that the broker can commit to an arbitrary mechanism. In particular, we model the bargaining game between the buyer and the seller as a double auction, and assume that the inter-mediary charges the traders two-part tari¤s consisting of …xed fees for participa-tion (°B for the buyer and °S for the seller) and a commission which can depend

only on the fact of trade.8 Since the parties are risk-neutral and care only about

expected payments, we can assume without loss that the intermediary charges a fee ±=2 from each trader if trade happens and nothing otherwise9.

The (modi…ed) double auction game proceeds as follows. The seller submits a bid pS and the buyer submits pB; trade occurs if pB ¸ pS+ ± at price p¤ = pB+p2 S:

Taking into account the intermediary’s commission, the seller gets p¤

S = pB+p2S¡±

and the buyer pays p¤

B= pB+p2S+±:

There are many equilibria in the standard double auction game (see Leininger et al. (1989)), and even more if one allows the parties to engage in cheap talk prior to submitting the bids (see Farrell and Gibbons (1989) and Mathews and Postlewaite (1989)). These observations obviously remain valid for our modi…ed game. In this paper, however, we shall only consider an (appropriately modi…ed) equilibrium in linear strategies, obtained initially by Chatterjee and Samuelson (1983). We do not allow for a direct pre-play communication between the parties as in Farrell and Gibbons (1989) or Mathews and Postlewaite (1989).

The following lemma10 presents a linear equilibrium of a modi…ed double

auc-tion game under more general assumpauc-tions (we shall need it for further analysis): the buyer’s and the seller’s valuations are independent and distributed uniformly

8More generally, the commission could depend on the trading price. Moreover, in reality (e.g. for real estate or art brokers) the intermediary’s fee is usually a certain percentage of the trading price. In our setting, however, an appropriately chosen …xed fee allows the broker to obtain the same pro…t as he would in the optimal complete mechanism, while a fee which would depend linearly (non-trivially) on the price can be shown to be an inferior instrument.

9We assume that the broker’s commission is divided equally between the parties, but it can be easily shown that the divivsion of the transaction fee between the buyer and the seller is irrelevant, i.e. the market is not two-sided (see Rochet and Tirole (2004) for an overview of two-sided markets).

10To prove the Lemma, one should rede…ne the buyer’s and the seller’s valuations to be ~

vB= vB¡±2 and ~vS = vS+±2 and apply the Chatterjee and Samuelson’s equilibrium strategies to these modi…ed valuations.

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on [vB; vB] and [vS; vS] respectively. Denote ~ pS(vS) : = 2 3vS+ 1 4vB+ 1 12vS¡ ± 4; (1) ~ pB(vB) : = 2 3vB+ 1 12vB+ 1 4vS+ ± 4: (2)

Lemma 2 in the appendix claims that unless all types of both parties are sure to trade were they playing strategies de…ned in (1)-(2), there exists an equilibrium in piecewise linear strategies: those types of seller that are not sure to trade use (1), while those that are sure to trade submit the maximum bid that makes trade happen for sure (similarly for the buyer).

We shall call the equilibrium described in Lemma 2 Chatterjee-Samuelson equilibrium. The parties trade in this equilibrium if and only if vB ¸ vS+14vB¡ 1

4vS+34±: When all types of both parties are sure to trade, so that neither claim

in Lemma 2 can be invoked, the linear equilibrium breaks down. If this is the case, and, additionally, the distributions of the buyer’s and the seller’s types are symmetric with respect to each other (i.e. vS ¡ vS = vB¡ vB), we shall assume

that all types trade at price p¤ = vS+vB

2 : Somewhat loosely, we shall call this

Chatterjee-Samuelson equilibrium when the linear one does not exist.

3.2 Optimal Two-Part Tari¤s

We assume the following timing: First, the broker speci…es a set T of two-part tari¤s (°B; °S; ±). Then, the traders observe their private valuations and the broker learns whether vB 2 [0; w] or vB 2 [w; 1]: The broker publicly announces

a two-part tari¤ from the set T . Next, the parties choose whether they want to participate in the double auction and if they choose to participate, they pay the participation fees and submit their bids. Finally, if the bids are submitted and the spread exceeds ±; the broker implements a transaction.

We shall assume that traders have an in…nitesimal cost of submitting a bid. We do not here model it explicitly, but assume that if the trader is indi¤erent between submitting a bid or not (which can happen in equilibrium only if the trader perceives the probability to perform a pro…table transaction to be 0), she abstains from bidding.11

As the following Proposition shows, the intermediary can achieve the highest pro…t characterized in Proposition 1 through an appropriately chosen system of two-part tari¤s.

Proposition 2 There exists a perfect Bayesian equilibrium, in which the broker gets the same expected pro…t as in the optimal direct mechanism.

11This behavior would be rational even if there were a small probability that the broker makes mistakes in determining the buyer’s eagerness – it su¢ces to require that the probability of mistakes be su¢ciently small (for a given cost of submitting a bid).

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² When the buyer is eager (vB 2 [w; 1]), the optimal tari¤ is ±E = 13; °ES = 0; °E B = ½ 1 3(w¡ 1 2)2 if w > 1 2; 0 if w · 1 2

and the Chatterjee-Samuelson equilibrium is played by the traders.

² When the buyer is not eager (vB 2 [0; w]), the optimal tari¤ is ±NE = w3;

°NE

S = 0; °NEB = 0: The Chatterjee-Samuelson equilibrium strategies are

played by the buyer and by the types of seller which have a positive probability of trade in equilibrium (i.e. by vS 2 [0; v¤S]) and pS(vS) = 1 for vS 2 [v¤S; 1];

where v¤

S = max(34(w¡ ±); 0):

Proof. Assume that the broker sets T = f(°E

B; °ES; ±E); (°N EB ; °NES ; ±N E)g;

where the two tari¤s are as speci…ed in the Proposition. We need to show that the broker’s choice of tari¤ once he observes the signal is incentive compatible and that these tari¤s lead to the same expected pro…t as the optimal direct mechanism. Then the choice of T is an optimal one.

A) Assume that the buyer is eager, vB2 [w; 1] and that w · 12. If the broker

chooses (°E

B; °ES; ±E); the seller infers that the buyer is eager (and it is common

knowledge between the traders), and trade happens i¤ vB ¸ vS+12 (EDF triangle

on Figure 1). Thus, the broker’s expected return is ER = 1 24(1¡w):

Assume that the intermediary deviates and chooses (°N E

B ; °N ES ; ± N E

): The seller then infers that the buyer is not eager (and this is again common knowledge between the traders), and trade happens i¤ vS · w2 (ABCD rectangle on Figure

1): the sellers strategy is to bid pS(vS) = 23vS+w6 if he believes there is a positive

probability of trade and to abstain from bidding otherwise. The seller expects the buyer to bid ^pB(vB) = 23vB + w6 and thus no trade to happen i¤ vS > w2:

The buyer’s best response to this strategy is to bid ~pB(vB) = 5w6 all traders

who submit bids trade with certainty. The broker’s expected return in case of deviation is gER = w2

6 which can be easily checked to be lower than 1

24(1¡w) for

any w 2 [0;1 2]:

B) Assume that the buyer is eager, vB2 [w; 1] and that w > 12. If the broker

chooses (°E B; °ES; ±

E

); the seller infers that the buyer is eager (and it is common knowledge between the traders), and trade happens i¤ vB ¸ vS+ 12. Thus, the

broker’s expected return is ER = °E

B+ °ES + ±EPrfTradeg = 13(w¡ 12)2+ w6:

If the broker deviates and chooses (°N E

B ; °N ES ; ±N E); the seller infers that the

buyer is not eager, trade happens i¤ vS · w2 (the same reasoning as in the previous

case) resulting in the broker’s expected return gER = w62 which can be shown to be lower than ER for any w:

C) Assume that the buyer is not eager, vB 2 [0; w]; and that w · 12. If

the intermediary chooses (°N E

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v

S

v

B 2 1

A B

C

D

E

2 w

w

F

1

1

0

v

S

v

B 2 1

A B

C

D

E

2 w

w

F

1

1

0

Figure 1:

eager, and trade happens i¤ vB ¸ vS+ w2. Thus, the broker’s expected return is

ER = w242: If he deviates and elects (°E B; °ES; ±

E

), there is no trade and gER = 0: D) Assume that the buyer is not eager, vB 2 [0; w]; and that w > 12. If the

intermediary chooses (°NE

B ; °NES ; ±N E); the seller infers that the buyer is not eager,

and trade happens i¤ vB ¸ vS+ w2. Thus, the intermediary’s expected return is

ER = w2

24: If he deviates and elects (° E B; °ES; ±

E), there is trade i¤ v

B ¸ vS + 12

and vB ¸ v¤B; where vB¤ is the type of buyer for whom the expected gain from

participation (see Lemma 3 in the Appendix) is just su¢cient to compensate her for the fee °E

B: UB(v¤B) = 1 2(v ¤ B¡ 1 2) 2¡1 3(w¡ 1 2) 2 = 0: Thus g ER = 1 18w(w¡ 1 2) 2+ 1 3w(1¡ r 2 3)(w¡ 1 2) 3;

which can be shown to be smaller than ER for all w 2 [1 2; 1]:

The seller’s expected return can be checked to be the same as in the optimal direct mechanism.

3.3 Welfare

It is natural that the intermediary gains from learning a signal about the buyer’s valuation. Looking at how the optimal mechanism takes this information into

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account one should expect that the seller also gains: the intermediary charges a lower fee when the buyer is not eager and trade happens more often. As for the buyer, the impact of the loss of privacy on her welfare is ambiguous: on the one hand, when the intermediary learns that the buyer is not eager to trade, he charges a lower transaction fee than he would in the absence of the knowledge on the buyer’s predisposition to trade. On the other hand, the buyer pays a participation fee if she is discovered to be strong, the fee she does not pay if the intermediary is completely agnostic about her willingness to trade. As Proposition 3 shows, which e¤ect dominates is determined by the quality of information: if the signal that the buyer is ”eager” is relatively weak (i.e. w is not too high), the …rst e¤ect dominates and the buyer bene…ts from the …ner information together with the other players; if the signal that the buyer is ”eager” is relatively strong (w is high enough), the second e¤ect dominates and the buyer loses.

Proposition 3 ² The observability of the buyer’s predisposition increases the probability of trade; the ex ante welfare of the seller and of the broker is higher than in the benchmark no-signal case.

² There exists ¹w 2 ¡1 2; 1

¢

such that the buyer gains from her predisposition being discovered and communicated if w < ¹w and loses if w > ¹w.

Proof. Using Lemma 3 one can check that under the optimal two-part tari¤ the ex ante expected utility of the buyer, the seller and the broker in the absence of information about the buyer’s eagerness to trade is the following:

UB0 = US0 = 3 2 27(1¡ ±) 3 = 1 48; U0 I = 32(1¡ ±)2± 25 = 1 24:

When the buyer is discovered to be not eager to trade the expected utilities are: UNE B = 32(w¡ ±)3 27w = w2 48; USNE = 3 2(w¡ ±)3 27w = w2 48 UINE = 3 2(w¡ ±)2± 25w = w2 24: If the buyer is eager and w < 1

2 UBE = USE = 3 2(1¡ ±)3 27(1¡ w) = 1 48(1¡ w); UIE = 3 2(1¡ ±)2± 25(1¡ w) = 1 24(1¡ w):

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If the buyer is eager and w ¸ 1 2 UE B = 1 1¡ w( 3 32(1¡ ±) 3¡ 1 18(w¡ 1 4¡ 3 4±) 3 ¡ 1 18(1¡ w) 3¡ 3 32(1¡ ±) 2(w¡ 1 4 ¡ 3 4±)¡ 1 3(w¡ 1 2) 2) = w 12 ¡ 1 3(w¡ 1 2) 2; UE S = 1 6(1¡ w) 2 +1 4(w ¡ 1 2); UIE = w 6:

Thus, when the intermediary observes the buyer’s eagerness to trade, the ex ante expected utilities for the case w < 1

2 are ¹ UB = ¹US = 1 48 + 1 48w 3; ¹ UI = 1 24+ 1 24w 3

and for the case w ¸ 1

2 they are ¹ UB = w(1¡ w) 12 + 1 48w 3 ¡ 1 3(w¡ 1 2) 2; ¹ US = 1 24 ¡ w 8 + w2 4 ¡ 7w3 48 ; ¹ UI = w(1¡ w) 6 + w3 24:

It is easy to check that ¹US > US0 and ¹UI > UI0 for all w and ¹UB > UB0 for

w < ¹w; while ¹UB < UB0 for w > ¹w; where ¹w is a root of 125w¡ 5 12w 2+1 48w 3¡ 5 48 = 0; ¹ w¼ 0:61:

4 Cheap Talk Communication

We now assume that the tari¤ is …xed before the broker gets the signal about the buyer’s eagerness to trade. Moreover, for simplicity we assume that there are no participation charges and the broker gets only the transaction fee ± divided equally between the traders. However, the broker gets his signal before the parties submit their bids, and, before the bids are chosen, he can send a message m 2 M about the buyer’s eagerness to trade to the seller. The message has no intrinsic cost – it is cheap talk. For such a message to have (at least some) credibility it must be the case that the interests of those who send it and who attend to it be not too far apart. In our setting the interests of the intermediary and the

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seller have an important common element – both want feasible trade to happen. However, the seller faces a trade-o¤ between the probability of exchange and the pro…t she expects from it, whereas the intermediary wants simply to maximize the volume of trade. Thus, it is not a priori clear whether the intermediary is able to communicate the information he observes to the seller.

In contrast to the previous section here we assume that the intermediary’s message is secret, so the buyer’s strategy depends on the message(s) that is expected to be sent in equilibrium and does not change if the broker deviates from his equilibrium announcement strategy. In the case of public messages results are similar – see note (13).

We also keep assuming that the traders that have no chance to trade do not submit bids (i.e. participation has an in…nitesimal cost). As we shall see, it is the abstention of the discouraged types of seller from trade that disciplines the broker.

4.1 Fully revealing equilibria

We shall call an equilibriumfully revealing if the intermediary can credibly com-municate all his information to the seller, i.e. the seller’s beliefs about the buyer’s valuation induced by the intermediary’s message coincide with the intermediary’s own beliefs. We now want to explore the existence of fully revealing equilibria.12

Note that since the intermediary’s information is binary, a binary message space is su¢cient to fully transmit the intermediary’s information. Since we are inter-ested in the existence of fully revealing equilibria, we shall assume without loss that there are just two messages, M = f”eager”,”not eager”g.

Proposition 4 There exists a fully revealing equilibrium in which the interme-diary truthfully reports the buyer’s eagerness to trade to the seller.

Proof. We only need to check that the intermediary has incentives to tell the truth if the parties expect that he will do so.

If the buyer is not eager to trade, the intermediary can only lose from lying – this would induce the seller to increase the price from ~pnot eager

S (vS) = 23vS+14w¡±4

to ~peager

S (vS) = 23vS+ 14 ¡4±; which would only reduce the probability of trade.

If the buyer is eager to trade, lying induces the seller to reduce the price from ~

peagerS (vS) = 23vS+14¡±4 to ~pnot eagerS (vS) = 23vS+144± when vS · vS¤ = 34(w¡±);

but stop bidding when vS > vS¤ , i.e. when there is no chance to trade with the

non-eager buyer.

12There are many equilibria in the game. For example, in this game, as in any cheap talk game, there always exists a babbling equilibrium in which the sender transmits a message uncorrelated with his signal and the receiver gives it no credibility. Thus, a babbling equilibrium at the …rst stage followed by some equilibrium of the double auction (with the intermediary) without communication is an equilibrium of the whole game.

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loss

gain

v

S

v

B

)

(

4 3

w

δ

δ

4 3 4 1

+

w

δ

4 3 8 1 8 3

w

+

A B

C

D

E

Figure 2:

Under truthtelling trade happens when vB ¸ vS + 14 + 34± (recall also that

vB ¸ w since the buyer is eager to trade). When the intermediary lies, trade

happens when vB ¸ vS + 38w ¡ 18 + 34± and vS · 34(w ¡ ±): When vB ¸ w;

the second condition implies the …rst, so the gain from lying is represented by the area of ABC triangle, and the loss by the area of CDE (see Figure 2). It is straightforward to check that the loss exceeds the gain for any value of w 2 (0; 1). When it is common knowledge that the buyer is not eager to trade, the sellers with low valuation (vS < vS¤) charge a lower price than in the equilibrium where

the buyer is eager to trade. Sellers with higher valuations (vS > vS¤) do not trade

in equilibrium and, so, they lose nothing if they abstain from bidding (abstention would be a dominant strategy for them if the participation cost were modelled explicitly). Since the intermediary is interested in maximizing the volume of trade, it might seem that he is tempted to deviate from the equilibrium behavior and always report that the buyer is not eager to trade because such a message makes the seller reduce the price when vS < vS¤. However, as Proposition 4 shows,

the potential gain from such a deviation is more than o¤set by the loss coming from non-participation of the sellers with vS > vS¤:13

13If the messages are public, a fully revealing equilibrium also exists, where the traders play the same equilibrium strategies as in the equilibrium with private messages described in Lemma 4.

Once again, the broker can be only tempted to understate the buyer’s eagerness. Assume that he does so. The di¤erence now is that the buyer can detect the broker’s deviation and thus he plays a best response to the seller’s strategy ~pnot eager

S (vS) = 23vS+14w¡±4; which can be easily shown to be pB= 34w + 14±: The probability of trade is the same as after a deviation

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4.2 An alternative signal structure

To check the robustness of the fully revealing communication, we assume an alternative information structure and establish the following

Proposition 5 Assume that the intermediary observes to which element [i n;

i+1 n ]

of a uniform n¡element partition of the unit interval belongs the buyer’s type. Then there exists a fully revealing equilibrium in which the intermediary truthfully reports the buyer’s eagerness to trade to the seller.

Proof. Similar to Proposition 4.

This result can be pushed even further: in the limit, when the broker knows with certainty the buyer’s type, there exists a fully revealing equilibrium in which the broker reveals truthfully the buyer’s valuation, the buyer bids pB = 34vB+14±

and the seller bids pS = 34(vB¡ ±) if vS < 34(vB¡ ±) and any bid above 34(vB¡ ±)

otherwise.

The result is in sharp contrast with the standard intuition of cheap talk com-munication models à la Crawford-Sobel, where the sender can communicate its information to the receiver only with some noise, the amount of noise being in-creasing as the interests of the parties diverge. In our model there is an additional complication: the sender (i.e. the broker) is uncertain about the receiver’s (i.e. the seller’s) reaction to his announcement since it does not observe the seller’s type. Despite an apparent con‡ict of interests (the seller is concerned with both the probability of trade and the expected surplus in case of trade, whereas the broker cares only about the probability of trade), the broker can communicate arbitrarily …ne information to the seller. The reason is that the broker’s inabil-ity to predict the seller’s reaction prevents him from understating the buyer’s valuation. As Section 5 shows, when the broker becomes better informed about the seller’s own valuation and thus its ability to predict the seller’s reaction is improved, no fully revealing equilibrium can exist for some values of parameters.

4.3 Welfare

The welfare implications of the observability of the buyer’s predisposition to trade are similar to those obtained in Proposition 3 for the case of two-part tari¤s. The seller and the intermediary gain from a …ner information structure, while the impact on the buyer depends on the quality of the signal observed by the intermediary.

Proposition 6 ² If w · ±, the observability of the buyer’s eagerness to trade has no e¤ect.

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² If ± < w, the observability of the buyer’s predisposition increases the prob-ability of trade and the ex ante welfare of the seller and of the broker is higher than in the benchmark no-signal case.

² If ± < w < 7+p13 18 +

11¡p13

18 ±; the buyer gains from her predisposition being

discovered and communicated. ² If 7+p13

18 + 11¡ p

13

18 ± < w; the buyer loses from her predisposition being

dis-covered and communicated.

Proof. Tedious calculation using Lemma 3, similar to the proof of Proposition 3.

5 Two-sided advise

We have seen that when the intermediary gets a signal about the demand for the seller’s good, it can credibly transmit this information to the seller and thus increase the e¢ciency of trade. We now explore what happens when the inter-mediary is able to observe signals about the valuations of both participants. Is it still able to share this information with the parties? In contrast to Lemma 4, we shall see that the answer depends on the parameters of the problem.

The intermediary now observes two signals: as before, it learns whether the buyer is eager to trade (vB 2 [w; 1]) or not (vB 2 [0; w]); the second signal reveals

whether the seller is eager to trade (vS 2 [0; 1 ¡ w]) or not (vS 2 [1 ¡ w; 1]).

In particular, we assume for simplicity that the signals are symmetric and thus have equal informativeness (across agents, not across states!). The intermediary’s reports to the parties are assumed to be secret.

Proposition 7 A fully revealing equilibrium with Chatterjee-Samuelson bidding strategies14 exists if and only if w · 3p2¡2

4 + 3(2¡p2) 4 ± or w ¸ 5 8 + 3 8±:

Proof. To prove the ”if” part, we, as in Lemma 4, only need to check that the intermediary wants to say the truth if it expects the parties to believe its messages. It is clear that the intermediary never wants to tell that a party is eager to trade if this is not true: it would only decrease the probability of trade (follows immediately from the bidding strategies (1)-(2)).

Let us verify that the intermediary will tell the truth if both parties are eager to trade. When w ¸ 5

8 + 3

8±, both parties would be sure to trade were they to

use the linear strategies (1)–(2). Thus, they trade with probability 1 at price 1 2;

and the intermediary has no incentive to lie.

14As we mentioned above, in the case when all the types of both parties are sure to trade if they were playing the linear strategies (1)–(2), we call Chatterjee-Samuelson equilibrium a symmetric one with the unique price 1

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gain

v

S

v

B

)

(

4 3

w

δ

δ

4 3 4 1

+

w

δ

4 3 8 1 8 3

w

+

A B

C

D

loss

E

1-w

F

gain

v

S

v

B

)

(

4 3

w

δ

δ

4 3 4 1

+

w

δ

4 3 8 1 8 3

w

+

A B

C

D

loss

E

1-w

F

Figure 3: Assume that 1 4+ 3 4±· w · 4 7+ 3

7±: Then, the situation is represented on Figure

3. The gain from lying to the seller about the buyer’s type15 is the area of ABC

triangle, while the loss is the area of CDEF trapezoid (after a false report to the seller and a truthful report to the buyer trade occurs when vS · v¤S = 34(w¡±) – all

the types of the seller that submit a bid are sure to trade). It can be checked that the loss exceeds the gain if and only if w · 3p2¡2

4 +

3(2¡p2)

4 ±: When this inequality

is not satis…ed, truthtelling cannot be induced if the parties play the CS-bidding strategies at the double auction stage, so the fully revealing equilibrium does not exist.

When 4

7+37± · w · 58+38±; there is no cost of lying and a positive gain: since

S = 34(w¡ ±) > 1 ¡ w; lying would result in trade with probability one were the

parties believing the intermediary’s reports. A fully revealing equilibrium does not exist for this range of w. In contrast, along similar lines it is easy to verify that lying is not bene…cial if w < 1

4 + 3 4±:

Analogous consideration shows that the intermediary never wants to lie when only one trader is eager to trade.

As the proof of Proposition 7 shows, truthtelling constraints are hardest to satisfy when the intermediary learns that both types are eager to trade. If w ¸

5 8 +

3

8± though, the parties are sure to trade when it is common knowledge that

both are eager, so the intermediary has no interest in falsifying this information.

15It can be checked that the gain from lying to both traders is (weakly) lower than the gain from lying to just one party. Since the parties are symmetric, it su¢ces to check that the intermediary does not want to lie to the seller.

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When w · 3p2¡2

4 +

3(2¡p2)

4 ±; i.e. the remaining uncertainty about the trader’s

types once they are known to be ”eager” is still high, falsi…cation of the report can be su¢ciently costly to the intermediary if the types who have no chance to trade with non-eager partner refuse to trade. In the intermediate case, truthtelling cannot be induced.

Corollary 1 Assume that the intermediary observes the type of both parties but gives advise to only one of them (say, the seller). Then, truthtelling can be induced only if w · 3p2¡2 4 + 3(2¡p2) 4 ± or w ¸ 5 8 + 3 8±:

6 Conclusions

Our analysis shows that the intermediary can credibly transmit the information he possesses about one of the bargaining parties to the other one. An important element of our analysis is two-sided uncertainty: the intermediary does not know the willingness to trade of the seller (the advisee) and thus cannot predict her reaction to a report about the willingness to trade of the buyer. Although the intermediary is tempted to understate the buyer’s willingness to trade and thus encourage the seller to reduce price, she prefers not to do so since the withdrawal of the pessimistic sellers from the bargaining process after such a report outweighs the gains from price reduction by those sellers that keep trading.

We would like to extend our analysis to address the following questions. Firstly, assume that the sellers can either apply to an intermediary to be matched with a buyer or go directly to the search market (as in Gehrig (1993)), and the only di¤erence is that the intermediary can give them some useful information about the buyer (like in our model). Then, what types of sellers will choose to trade through the intermediary? If the buyers also have choice, what buyers will choose to be matched through the intermediary?

It is also important to investigate the intermediary’s incentives to collect in-formation. In our analysis we have assumed that the intermediary always gets an exogenous signal, but in some circumstances it may be more natural to presume that the intermediary has to exert e¤ort to get a signal. Another interesting extension is allowing for competition between the intermediaries. We leave it to future research.

7 Appendix

7.1 The optimal direct mechanism: a general case

We shall generalize here the results of Proposition 1. We abandon the uni-form distributions assumption and suppose now that the seller’s valuation is

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distributed on [0; 1] with density fS(vS)¸ 0 and the buyer’s valuation with

den-sity fE

B(vB)¸ 0 on [0; 1] if she is eager to trade and fBN(vB)¸ 0 otherwise. Denote

fB(vB) = ¸fBN(vB) + (1¡ ¸)fBE(vB) the expected density of the buyer’s

valua-tion. We assume that the eager buyers are indeed more willing to trade: fBE(vB)

fN B(vB)

is weakly increasing on [0; 1].

The intermediary knows whether the buyer is eager to trade. The traders do not know this: the seller believes that the buyer is not eager with probability ¸ and eager with probability 1 ¡ ¸: The buyer has the same prior beliefs as the seller, but updates them after learning her private valuation: she believes that she is not eager to trade with probability ¸B(vB) =

¸fN B(vB)

fB(vB) : In particular, she

can deduce perfectly the distribution from which her valuation has been drawn if one of the densities is zero, as in the case that we considered in Proposition 1 where the distributions were complementary truncations of the uniform one. Proposition 8 Assume that the broker can commit to reveal truthfully the buyer’s eagerness after both traders report their valuations. Then, the optimal direct mechanism maximizing the broker’s ex ante pro…t implements trade if the buyer’s virtual valuation vB ¡

1¡Fi B(vB)

fi

B(vB) ; where i = E if the buyer is eager and i = N;

exceeds the seller’s virtual valuation vS+FfSS(v(vSS)): The buyer with valuation vB = 0

as well as the seller with valuation vS = 1 gets no ex ante surplus.

Proof. A direct mechanism consists of the probability of trade p(r; vS; vB);

the transfers from the buyer to the broker xB(r; vS; vB) and from the broker to

the seller xS(r; vS; vB); that all are functions of the reports r 2 fE; Ng of the

intermediary, vS and vB of the seller and the buyer.

Let ¹ pS(vS) = ¸ 1 Z 0 p(N; vS; tB)fBN(vB)dtB+ (1¡ ¸) 1 Z 0 p(E; vS; tB)fBE(vB)dtB

be the expected probability of trade conditional on the seller’s valuation vS and

¹ pB(vB) = ¸B(vB) 1 Z 0 p(N; tS; vB)fS(tS)dtS+ (1¡ ¸B(vB)) 1 Z 0 p(E; tS; vB)fS(tS)dtS

be the expected probability of trade conditional on the buyer’s valuation vB:

The expected transfers ¹xS(vS) and ¹xB(vB) are de…ned similarly, as well as the

expected utilities US(vS) and UB(vB) for the traders and UI(N) and UI(E) for

the intermediary. We shall also use ¹pB(r; vB) for the expected probability of trade

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As in Myerson-Satterthwaite (1983) one easily shows that incentive compati-bility constraints for the traders imply

US(vS) = US(1) + 1 Z vS ¹ pS(tS)dtS (3) and UB(vB) = UB(0) + vB Z 0 ¹ pB(tB)dtB: (4)

We now use these formulae for the expected payo¤s to express the expected gains from trade:

¸ 1 Z 0 1 Z 0 (vB¡ vS)p(N; vS; vB)fS(vS)fBN(vB)dvSdvB +(1¡ ¸) 1 Z 0 1 Z 0 (vB¡ vS)p(E; vS; vB)fS(vS)fBN(vB)dvSdvB = 1 Z 0 US(vS)fS(vS)dvS+ ¸ 1 Z 0 UB(vB)fBN(vB)dvB +(1¡ ¸) 1 Z 0

UB(vB)fBE(vB)dvB+ ¸UI(N) + (1¡ ¸)UI(E)

= US(1) + UB(0) + ¸UI(N) + (1¡ ¸)UI(E)

+ 1 Z 0 1 Z vS ¹ pS(tS)fS(vS)dtSdvS+ 1 Z 0 vB Z 0 ¹ pB(tB)fB(vB)dtBdvB

= US(1) + UB(0) + ¸UI(N) + (1¡ ¸)UI(E) + 1 Z 0 1 Z vS ¹ pS(tS)fS(vS)dtSdvS + 1 Z 0 vB Z 0 ¸B(vB)¹pB(N; tB)fB(vB)dtBdvB+ 1 Z 0 vB Z 0 (1¡ ¸B(vB))¹pB(E; tB)fB(vB)dtBdvB

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= US(1) + UB(0) + ¸UI(N) + (1¡ ¸)UI(E) + 1 Z 0 1 Z vS ¹ pS(tS)fS(vS)dtSdvS +¸ 1 Z 0 vB Z 0 ¹ pB(N; tB)fBN(vB)dtBdvB+ (1¡ ¸) 1 Z 0 vB Z 0 ¹ pB(E; tB)fBE(vB)dtBdvB

= US(1) + UB(0) + ¸UI(N) + (1¡ ¸)UI(E) + 1 Z 0 ¹ pS(vS)FS(vS)dvS +¸ 1 Z 0 ¹ pB(N; tB)(1¡ FBN(vB))dvB+ (1¡ ¸) 1 Z 0 ¹ pB(E; tB)(1¡ FBE(vB))dvB

= US(1) + UB(0) + ¸UI(N ) + (1¡ ¸)UI(E)

+¸ 1 Z 0 1 Z 0 p(N; vS; vB)FS(vS)fBN(vB)dvSdvB +(1¡ ¸) 1 Z 0 1 Z 0 p(E; vS; vB)FS(vS)fBE(vB)dvSdv +¸ 1 Z 0 1 Z 0 p(N; vS; vB)(1¡ FBN(vB))fS(vS)dvSdvB +(1¡ ¸) 1 Z 0 1 Z 0 p(E; vS; vB)(1¡ FBE(vB))fS(vS)dvSdvB:

It follows that for the intermediary it is optimal to set p(r; vS; vB) = 1 if and

only if vB¡ 1¡ Fr B(vB) fr B(vB) ¸ vS + FS(vS) fS(vS)

and to set UB(0) = US(1) = 0. Transfers that support this mechanism can be

found as in Myerson and Satterthwaite (1983). Note again that we abstracted from the intermediary’s incentive constraints at this stage, assuming that he can commit to reveal his signal truthfully.

The Proposition shows that the intermediary can do no better than simply to reveal his signal to the traders and then to implement the optimal direct mechanism characterized by Myerson and Satterthwaite (1983).

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7.2 Cheap talk communication: the general case

(prelim-inary)

Let us consider the possibility of communication by the broker via cheap talk in the general case. Assume as before that the intermediary can get two signals about valuation of the buyer vB 2 [0; 1].

With probability 1 ¡ w the signal is ”eager”, and vB is drawn from the

dis-tribution with cdf FE

B(vB) and density fBE(vB), with probability w the signal is ”

not eager” and vB is drawn from FBN(vB) (density fBN(vB)). Monotone likelihood

ratio property is satis…ed for the two distribution functions: fBE(vB)

fN

B(vB) is increasing.

The seller’s valuation vS is distributed on [0; 1] according to cdf FS(vS);and

that the cost of participation in bargaining for the seller equals c ¸ 0: As in the previous section the tari¤ is …xed before the broker gets the signal about the buyer’s valuation: the broker gets fee ± if transaction happens.

We shall assume that at the bargaining stage the seller makes take-it-or-leave-it o¤er to the buyer after learning the message of the intermediary; the buyer accepts the o¤er if his valuation exceeds the price.16 Suppose that the seller

with valuation vS believes that valuation of the buyer is drawn from FBI(vB);

i2 fL; Hg: Then she o¤ers to sell the object at price pi(v

S) such that: pi(vS) = arg max p (1¡ F i B(p))(p¡ ± ¡ vS) that gives pi(v

S) that is determined from the following equation:

pi(vS) = vS+ ± + 1¡ Fi B(pi(vS)) fi B(pi(vS)) (5) To ensure that the seller’s objective function is concave it su¢ces to assume that fi

B(vB) is increasing, i.e. FBi(vB) is convex. The seller chooses to participate in

the bargaining i¤ his expected surplus from trade is greater than vS :

c· (1 ¡ FBi(pi(vS))(pi(vS)¡ ± ¡ vS)

Using (5), the seller with a given vS bargains only if cost c is su¢ciently small:

c· (1¡ F i B(pi(vS))2 fi B(pi(vS)) : (6)

Since the seller’s expected gain from trade is non-increasing in vS, the seller

will choose to submit an o¤er if and only if vS 2 [0; vSi]; where vSi is determined

from (6) holding as equality:

(1¡ Fi

B(pi(vS))2

fi

B(pi(vS))

= c: (7)

16We could consider a more general setup where at the bargaining stage the seller and the buyer make take-it-or-live-it o¤ers to each other with probabilities µ and 1 ¡ µ correspondingly. The point is that the buyer’s o¤er is not a¤ected by the message sent by the intermediary.

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It is easy to see that vN

S · vSE: if the buyer is eager, the seller could imitate the

strategy she would employ were the buyer not eager and make at least the same pro…t since 1 ¡FE

B(p)¸ 1 ¡ FBN(p) (…rst order stochastic dominance follows from

the MLRP).

In fact, in this general setup the intermediary may have incentives to misreport his signal in both directions: claiming that the buyer is ”not eager” when in fact she is, the broker increases the probability of trade due to the reduction in the o¤er price; claiming that the buyer is ”eager” when she is not, he increases the number of sellers participating in the bargaining. The following lemma characterizes conditions su¢cient to induce truthtelling in each case.

Lemma 1 Assume that the seller believes the intermediary’s messages. If d dp[ 1¡ Fi B(p) fi B(p) ] < 1; (8) fS is (weakly) decreasing and fBi is (weakly) increasing, then:

² Gains from reporting the true signal increase with cost of participation c when the buyer is ”eager”; moreover, there exists cL2 (0; 1) such that the

intermediary weakly prefers to report the truth when the buyer is eager if c > cL:

² Gains from reporting the true signal decrease with cost of participation c in the case when the buyer is ”not eager”; moreover, there exists cH 2 (0; 1)

such that the intermediary weakly prefers to report the truth when the buyer is not eager if c < cH:

Proof. Condition (8) puts some regularity constraints on distribution func-tions Fi

B(vB) to get ”natural” behavior of sellers: it ensures that the optimal price

chosen by the seller after each report by the intermediary is an increasing function of the seller’s valuation.17 All distributions with monotone increasing densities

and some monotone decreasing ones satisfy (8), examples include uniform and exponential distributions.

Now let us compute the probability of trade for di¤erent messages of the intermediary.

Suppose that the buyer is ”eager”, intermediary reports ”eager” and seller trust him. Then probability of trade Ptruth equals:

Prtruth("eager") = vE S Z 0 A Z pE(v S) dFBE(vB)dFS(vS) = vE S Z 0 (1¡ FBE(pE(vS))dFS(vS) (9)

17To see that (8) implies the monotonicity of the optimal price one just applies the implicit function theorem to (5) which determines the optimal price.

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Suppose that the buyer is ”eager”, but intermediary misleads the seller and reports ”not eager”. Then the price pN(v

S) posted by seller will be lower18, but

fewer sellers will participate in the bargaining (vN

S · vES). The probability of trade Plie is Prlie("eager") = vN S Z 0 A Z pN(vS) dFBE(vB)dF S(vS) = vN S Z 0 (1¡ FE B(pN(vS))dFS(vS) (10)

Let us examine how the probabilities of trade will change with the cost c. First we’ll compute dvi

S

dc : Di¤erentiating (7) with respect to c;we …nd:

dvi S dc =¡[ (fi B)2 2(1¡ Fi B)(fBi)2+ (fBi)0(1¡ FBi)2 ]¢ 1 dpi=dv S : Di¤erentiating (5) with respect to vS; we get

dpi dvS = (f i B)2 2(fi B)2+ (fBi)0(1¡ FBi) : These two equations give:

dvi S dc =¡ 1 1¡ Fi B(pi(vSi)) : (11) Then d dc[Pr

truth("eager"¡ Prlie("eager")]

= d dc[ vE S Z 0 (1¡ FBE(pE(vS))dFS(vS)¡ vN S Z 0 (1¡ FBE(pN(vS))dFS(vS)] = [1¡ FE B(pE(vES))]fS(vES)¢ dvE S dc ¡ [1 ¡ F E B(pN(vNS))]fS(vSN)¢ dvN S dc = [1¡ FE B(pE(vES))]fS(vES)¢ (¡ 1 1¡ FE B(pE(vSE)) ) ¡[1 ¡ FE B(pN(vSN))]fS(vNS)¢ (¡ 1 1¡ FN B(pN(vSN)) ) = 1 f(vN S) [1¡ F E B(pN(vNS)) 1¡ FN B(pN(vSN)) ¡ fS(v E S) fS(vSN) ]: 18Indeed, since fE B fN B is increasing, then 1¡FE B(p) fE B(p) ¸ 1¡FN B(p) fN B(p) and p ¡ 1¡FE B(p) fE B(p) · p ¡ 1¡FN B(p) fN B(p) for

every p: From this and (5) it follows that pN(v

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It is easy to see that it is su¢cient to require f to be a decreasing function to guarantee that d

dc[Pr

truth("eager")¡Prlie("eager")] > 0: Under these assumptions

we can claim that there exists a level of cost cL 2 [0; 1]; such that if c ¸ cL the

intermediary tells the truth and if c < cL; then he lies. Note that when c = 0;

there is no cost of lying so cL> 0: On the other hand there exist such a cost c < 1

that no sellers participate if the buyer is not eager (vN

S = 0) but some participate

if the buyer is eager (vE

S > 0). Hence cL< 1 which …nishes the proof of the …rst

part of the Lemma.

In the same way one can show that d dc[Pr

truth("not eager")

¡Prlie("not eager")] < 0 to prove the second part of the lemma.

We want now to formulate conditions that would ensure that the intermediary tell the truth in both cases. More precisely, we are looking for conditions su¢cient for the critical levels of the cost parameter from Lemma 1 to satisfy cL< cH.

Assumption 1 We say that the distributions F (v) and G(v) satisfy the (w; ")-separation condition if there exists w such that F (w) < " and G(w) > 1 ¡ ".

Our preliminary conjecture is that under condition 1 (or similar) the range of cost parameters in which there exists a separating equilibrium is non-empty. Conjecture 2 Assume that fS is (weakly) decreasing and fBi are (weakly)

in-creasing and satisfy (8). Then, if FE

B(vB) and FBN(vB) satisfy the (w; ")-separation

condition for w large enough and " small enough, the critical levels of the cost parameter from Lemma 1 satisfy cL < cH and for c 2 (cL; cH) there exists a

separating equilibrium in which the intermediary communicates to the seller the true signal about the buyer’s demand.

7.3 Some technical results

Lemma 2 (i) pS(vS) = ~pS(vS); pB(vB) = ~pB(vB) is an equilibrium if ~pS(vS) >

~

pB(vB)¡ ± and ~pS(vS) > ~pB(vB)¡ ± (i.e. no type of either party is sure to

trade under these strategies).

(ii) pS(vS) = max(~pS(vS); ~pB(vB)¡ ±) and pB(vB) = min(~pB(vB); ~pS(vS) + ±)

is an equilibrium if ~pS(vS) · ~pB(vB) ¡ ± or ~pS(vS) · ~pB(vB)¡ ±; but

~

pS(vS) > ~pB(vB)¡ ± (i.e. some types of at least one party are sure to trade,

but not all types of both parties are sure to trade).

Lemma 3 Assume that the traders play (piecewise) linear strategies described in Lemma 2 and the broker charges the per-transaction fee ± and no participation fees. It is common knowledge that vS 2 [0; 1] and vB 2 [vB; vB]: Then, the parties’

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(i) For the case [vB; vB] = [0; 1]; UB(vB) = ½ 1 2(vB¡ 3 4±¡ 1 4)2 if vB > 1 4 + 3 4±; 0 if vB · 14 + 34±; US(vS) = ½ 1 2( 3 4(1¡ ±) ¡ vS) 2 if v S < 34(1¡ ±); 0 if vS ¸ 34(1¡ ±); UI = 32(1¡ ±)2± 25 :

(ii) For the case [vB; vB] = [w; 1]; w > 14+ 34±;

UB(vB) = 8 < : 2 3(vB¡ 3 4±¡ 1 4) 2¡ 1 6(vB¡ w) 2 if v B > 14 +34±; +1 3(vB¡ 3 4±¡ 1 4)( 3 4± + 1 4 ¡ w) 0 if vB · 14 + 34±; US(vS) = 8 < : (1 2w + 1 4 ¡ 3 4±¡ vS) if vS < w¡ 1 4 ¡ 3 4±; 1 2(1¡w)( 3 4(1¡ ±) ¡ vS) 2 if w ¡ 1 4 ¡ 3 4± · vS < 3 4(1¡ ±); 0 if vS ¸ 34(1¡ ±); UI = ( 1 2w + 1 4¡ 3 4±)±;

(iii) For the case [vB; vB] = [w; 1]; w· 14 +34±;

UB(vB) = ½ 1 2(vB¡ 3 4±¡ 1 4) 2 if v B > 14 +34±; 0 if vB · 14 +34±; US(vS) = ½ 1 2( 3 4(1¡ ±) ¡ vS)2 1 1¡w if vS < 3 4(1¡ ±); 0 if vS ¸ 34(1¡ ±); UI = 32(1¡ ±)2± 25(1¡ w) :

(iv) For the case [vB; vB] = [0; w]; w > 14 + 34±;

UB(vB) = ½ 1 2(vB¡ 3 4±¡ 1 4w) 2 if v B > 14w + 34±; 0 if vB · 14w + 34±; US(vS) = ½ 1 2w( 3 4(w¡ ±) ¡ vS)2 if vS < 3 4(w¡ ±); 0 if vS ¸ 34(w¡ ±); UI = 32(w¡ ±)2± 25w :

(v) For the case [vB; vB] = [0; w]; w· 14 +34±;

UB(vB) = US(vS) = UI = 0:

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8 References

Ashenfelter, O. and K. Graddy (2002) ”Auctions and the Price of Art”, Discussion paper ISSN 1471-0498, University of Oxford.

Baron, D. and B. Holmstrom (1980) ”The Investment Banking Contract For New Issues Under Asymmetric Information: Delegation and the Incentive Prob-lem”,Journal of Finance 35(5): 1115-1138.

Baron, D. (1982) ”A Model of the Demand for Investment Banking Advising and Distribution Services for New Issues”,Journal of Finance 37(4): 955-976.

Cai, H. and W. Cont (2000) ”Agency Problems and Commitment in Delegated Bargaining”, mimeo, UCLA.

Calzolari G. and A. Pavan (2004) ”On the Optimality of Privacy in Sequential Contracting”, mimeo.

Chatterjee K. and W. Samuelson (1983), ”Bargaining under Incomplete In-formation”, Operations Research,31, pp. 835-51.

Farrell, J. and R. Gibbons (1989) ”Cheap Talk Can Matter in Bargaining”, Journal of Economic Theory 48: 221-237.

Jullien, B. and T. Mariotti (2003) ”Auction and Informed Seller Problem”, mimeo, IDEI.

Katzman B. and M. Rhodes-Kropf (2002) ”The Consequence of Information Revealed in Auctions”, Columbia Business School, Economics and Finance Work-ing Paper.

Leininger W., P. Linhart, and R. Radner (1989) ”Equilibria of the sealed-bid mechanism for bargaining with incomplete information”, Journal of Economic Theory,48: 63-106.

Lizzeri A. (1999) ”Information Revelation and Certi…cation Intermediaries”, Rand Journal of Economics,30(2): 214-231.

Maskin, E. and J. Tirole (1990) ”The Principal-Agent Relationship with an Informed Principal: The Case of Private Values”, Econometrica,58: 379-440.

Mathews S. and A. Postlewaite (1989) ”Pre-play Communicaion in Two-Person Sealed-Bid Double Auctions”,Journal of Economic Theory, 48: 238-263. Miceli, T., K. Pancak and C. Sirmans (2000) ”Restructuring Agency Relation-ships in the Real Estate Brokerage Industry: An Economic Analysis”,Journal of Real Estate Research 20: 31-47.

Myerson, R. and M. Satterthwaite (1983) ”E¢cient Mechanisms for Bilateral Trading”,Journal of Economic Theory, 29: 265-281.

The National Association of Realtors (2001) ”The Consequences of Mixing Banking and Commerce”.

Rochet J.C. and J. Tirole (2004) ”Two-Sided Markets: An Overview”, mimeo, IDEI.

Seidmann, D. J. (1990) ”E¤ective Cheap Talk with Con‡icting Interests”, Journal of Economic Theory, 50: 445-458.

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Yavas, A. (1994) ”Middlemen in Bilateral Search Markets”, Journal of Labor Economics,12, 406-429.

References

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