Efficient Bargaining Through a Broker

Transcription

1 Effiient Bargaining Through a Broker Xingtan Zhang The Wharton Shool of the Uniersity of Pennsylania Deember 31, 2016 Job Market Paper Abstrat A partially informed broker, ating as an intermediary between a buyer and seller, an improe welfare by alleiating ineffiienies aused by asymmetri information. I proide a suffiient ondition on the broker s information to ahiee an effiient outome, in ontrast to the Myerson and Satterthwaite (1983) theorem, whih states that diret bilateral bargaining is generally ineffiient. In the broker s optimal mehanism, while a more informed broker extrats a higher surplus, trade is still more effiient; onsequently, the buyer and seller an be better off trading with a more informed broker. (JEL Code: D82, G23, L12) The Wharton Shool of the Uniersity of Pennsylania, 3620 Loust Walk, Philadelphia, PA xingtan@wharton.upenn.edu. I am indebted to Eduardo Azeedo, Vinent Glode, George Mailath, and Kent Smetters for their inaluable adie and guidane. For helpful omments, I thank Mike Abito, Will Cong, Ulrih Doraszelski, Hanming Fang, Alex Frankel, Itay Goldstein, Joseph Harrington, Yunan Li, Steen Matthews, Rihard MLean, Wojieh Olszewski, Christian Opp, Andrew Postlewaite, Andy Wu, Hongjun Yan, Hanzhe Zhang, and seminar partiipants at Wharton AEW, Wharton BEPP, and Penn Miro Lunh. Any remaining errors are mine. 1

2 1 Introdution Asymmetri information an lead to market failure. In partiular, the seminal work of Myerson and Satterthwaite (1983) shows that regardless of the bargaining game between a buyer and a seller, there is always a positie probability of ineffiient outomes under ery mild onditions. Howeer, real world negotiations often our through third-party brokers. Examples inlude realtors in real estate transations, inestment banks in mergers and aquisitions, brokers in bond markets, et. Moreoer, brokers are likely to be informed about their market. A realtor, experiened in the real estate industry, is well informed about the intrinsi alue of a property, whih is likely orrelated with the seller s and buyer s priate aluation. A finanial intermediary is informed about the supply and demand of finanial assets, and therefore knows more about the reseration alue of the seller and buyer. In this paper, I study a problem in whih a broker ats as an intermediary in the bargaining between a buyer and a seller in the presene of priate information. I assume the broker is informed, in the sense that he obseres imperfet signals about the buyer s and seller s aluations. My main result establishes that bargaining through an informed broker an improe welfare by alleiating ineffiienies aused by asymmetri information. I first examine the possibility of effiient trade. If the broker is uninformed, the problem redues to Myerson and Satterthwaite (1983): Effiient trade is impossible. At the other extreme, if the broker is perfetly informed, an effiient outome an be ahieed. 1 I haraterize a suffiient ondition on what information the broker must hae for effiient trade to be implemented. Importantly, full effiieny an be ahieed with a broker who has imperfet information. The seond part of this paper onerns optimal mehanisms when the broker s information is haraterized by a partition struture. 2 Bargaining through a more informed broker an ahiee higher expeted gains from trade. If the broker hooses a mehanism, a more informed broker 1 In this ase, the broker an buy from the seller at her ost and resell to the buyer at his aluation. Trade is effiient, and the broker reeies all of the surplus. 2 We say that the broker s information is haraterized by a partition struture, if his information partitions the types spae into seeral subinterals. 2

3 an extrat more surplus, but the buyer and seller an also reeie ex ante higher information rents beause trade is more effiient. As a result, bargaining through an informed broker an Pareto dominate bargaining through an uninformed broker. The general intuition that an informed broker an improe effiieny is as follows. The presene of priate information implies that the buyer and seller may misreport their aluations. An uninformed mehanism designer needs to proide proper inenties for them to truthfully report. In Myerson and Satterthwaite (1983), the mehanism designer annot distinguish a buyer who has a aluation and reports ˆ from another buyer who has a aluation but also reports ˆ. In ontrast, onsider bargaining through an informed broker. Sine the broker s information depends on whether the buyer s aluation is or, he an treat the same reports from different types of buyers differently. The broker s orrelated information an help detet misreporting behaiors, therefore making it easier to indue truthful reelation. The idea of using orrelated information in mehanism design was introdued by Crémer and MLean (1985). In an aution enironment, they show that a seller an extrat all of the surplus under ertain onditions on the orrelated information struture aross buyers aluations. 3 MAfee and Reny (1992) extend the result to the ontinuous type ase, and show that a mehanism designer an hae more power in a orrelated information setting. There are two main differenes in my paper. First, I show that an effiient outome, rather than higher effiieny, an be ahieed by a broker who has imperfet information. In my paper, I keep the assumption that the buyer s aluation and the seller s are independent. The key ingredient to ahiee an effiient outome is the orrelated struture between the broker s information and the negotiators aluations. 4 Seond, in ontrast to MAfee and Reny (1992), I show that full effiieny an be ahieed in a broader set of information strutures. For instane, if the broker s information is haraterized by a partition struture, whih fails to satisfy the MAfee and Reny s ondition, the broker an use ross 3 Crémer and MLean (1988) onsider a general setting with orrelated information. 4 MAfee and Reny (1992) show that an effiient mehanism an exist when the buyer s priate aluation is orrelated with the seller s. But the mehanism requires a budget balaner. In the disrete type ase, Gresik (1991) shows a similar result that an effiient outome an exist when the buyer s priate aluation is orrelated with the seller s. Sine I study the ontinuous type ase, I ompare my result to MAfee and Reny s. 3

4 subsidization to failitate effiient trade as I will show in Setion 2. Brokers often play multiple roles in failitating the exhange of a good. For instane, in the sale of a house, the realtor helps in the searh proess by mathing a seller and a buyer; at the same time, the realtor also intermediates negotiations between the two parties. A large literature has demonstrated that brokers an alleiate searh fritions, therefore improing welfare (e.g., Rubinstein and Wolinsky (1987)). My paper does not onsider the searh hannel, and fouses instead on how an informed broker an help in the bargaining. When the bargaining is mediated by a broker, one might expet trade effiieny to be redued, beause a broker s ommission omes from the gains from trade and the broker s inentie may not be aligned with the negotiators. 5 Despite these downsides, I show that inoling an informed broker an improe trade effiieny when diret bilateral bargaining annot reah an effiient outome. 6 Relatedly, Glode and Opp (2016) study the effiieny of trade in a one-sided inomplete information setting in whih an uninformed seller deals with a priately informed buyer. They show that introduing informed brokers in a sequential trading game an expand the set of distributions of the buyer s and seller s priate aluations under whih effiient trade is possible. In subsequent work, Glode, Opp, and Zhang (2016) extend the analysis by showing that generially, if effiient trade an be implemented by an inentie-ompatible mehanism in diret bilateral trading, it an also be ahieed in a sequential trading game with a suffiiently long hain of heterogeneously informed brokers. The major differene between my paper and their papers is that in showing that an effiient outome an be ahieed, I onsider all mehanisms, rather than a speifi game in whih the asset owner makes a take-it-or-leae-it offer. 7 Therefore, my paper differs from their 5 Bazerman et al. (1992) is the first paper that distinguishes between neutral third parties and non-neutral third parties. A ritial issue in the ase of non-neutral third parties is that brokers at to maximize their own personal gain. Consequently, the inentie struture may affet the information transmitted in the negotiation proess. Similarly, Yaas, Thomas, and Sirmans (2001) find that realtors may be harmful in negotiations. On the other hand, a neutral third party for example, a mediator tries to help the parties to see the negotiation from a more rational perspetie (Neale and Bazerman, 1983). That is, the mediator an help the parties make deisions that maximize their own utility. 6 Yaas, Thomas, and Sirmans (2001) examine an experiment in whih the broker has imperfet information. Howeer, in their setting, there is no ineffiieny aused by asymmetri information in diret bargaining beause the buyer s aluation is assumed to always be greater than the seller s ost. My paper offers a rational framework and points out a positie effet of bargaining through a broker from the perspetie of effiieny whih is likely to be a first-order onern, espeially in a market in whih bargaining fritions aused by asymmetri information are substantial. 7 In Glode and Opp (2016) and Glode, Opp, and Zhang (2016), one onsequene of take-it-or-leae-it offer is that 4

5 work in three key ways. First, in their setting the broker may end up with the asset, but in my setting he optimally neer retains the asset. So my paper an address the appliations suh as how, without the possibility of holding the asset, a realtor improes trade effiieny between a buyer and a seller or how inestment banks failitate mergers and aquisitions. Seond, the buyer knows the seller s ost in their bargaining enironment, but in my setting he does not. I onsider a two-sided asymmetri information setting and show that an informed broker an help failitate trade. 8 If I restrit my analysis to their enironment (Setion 6.1), I show that effiient trade an be ahieed for a larger set of distributions of the buyer s and seller s priate aluations. Third, they study partition information strutures. In my setting, an effiient outome an be ahieed with more general information strutures. The rest of the paper proeeds as follows. In Setion 2, I proide a simple example to demonstrate how an informed broker an failitate effiient trade. In Setion 3, I present a model with an informed broker. In Setion 4, I onsider the possibility of effiient trade, and present a ondition on the broker s information suh that effiient trade is implementable. In Setion 5, I study the optimal mehanisms. Setion 6 disusses two alternatie assumptions, relationship to the literature, and empirial impliations. Setion 7 onludes. 2 Illustratie Example In this setion, I proide a simple example to understand the main results in the paper. A seller owns an objet that a buyer is interested in. The seller has a ost of supplying the objet, while the buyer alues it at. Both the ost and the alue are priate information, and are drawn independently from the uniform distribution oer the [0, 1] interal. The effiient alloation of the objet is for the parties to trade wheneer >. The total expeted gains from trade in the effiient ineffiienies are partially drien by market power, whih is absent in my setting. 8 In the working paper ersion, Glode and Opp (2015) show that a similar result holds with two-sided inomplete information. But they fous on the sequential trading game in whih the asset owner has market power (i.e. the asset owner makes a take-it-or-leae-it offer). 5

6 alloation is 1/6. 9 A seminal result in the bargaining literature is the Myerson and Satterthwaite (MS) theorem. The MS theorem shows that there are no budget balaned bargaining mehanisms that implement effiient trade. That is, there are always ineffiienies, regardless of the partiular bargaining game that is played between the parties. For example, onsider the bargaining game in whih the buyer makes a take-it-or-leae-it offer to the seller. The buyer would neer offer his true alue, beause if he does, he will hae to pay wheneer there is a transation, being left with a surplus of 0. If, instead, the buyer shades his bid offering /2 for example he reeies a positie expeted surplus. This means that the buyer always bids less than his true alue, so that trade does not happen, in some ases when >. A key assumption in MS s analysis is budget balane: that the mehanism does not lose money on aerage. There are mehanisms that lose money but guarantee that trade is effiient. For example, onsider the Vikrey-Clarke-Groes mehanism (VCG). The buyer and seller announe and. There is trade if >. In that ase, the buyer pays, and the seller reeies. This mehanism is effiient, beause there is trade wheneer >. Moreoer, both the buyer and the seller hae inenties to delare their true aluations, beause they annot affet the payments that they make or reeie. Howeer, this mehanism is not budget balaned. The mehanism designer loses the prie differene wheneer there is trade, so that the aerage defiit of the mehanism equals the total gains from trade, 1/6. 10 The main ontribution of this paper is to show that an informed broker an failitate effiient trade while running a surplus. To see this, suppose the transation is mediated by a broker. The broker knows whih of the following three subinterals belongs to: [0, 1/3), [1/3, 2/3), [2/3, 1]. Similarly, he knows whih of the three subinterals the seller s aluation belongs to. As shown 9 If the objet is alloated effiiently, the gains from trade are if >, and 0 otherwise. Thus, the expeted surplus is ( )1 > d d = 1 6, where 1 > is the indiator funtion and is equal to 1 if >, and 0 otherwise. 10 Williams (1999) proides a nie proof of MS theorem by showing that the VCG mehanism always runs a defiit, and any effiient mehanism loses at least as muh money as VCG. 6

7 in Figure 1, the broker knows whih of the nine regions the aluations pair belongs to. I show that effiient trade an be sustained by the following mehanism. The broker first announes whih region is true, then the buyer and seller play one of the following three mehanisms: fixed prie mehanism, VCG, or no trade. In the top left region, that is, the seller has a aluation lower than 1/3 and the buyer has a aluation higher than 2/3, the broker buys from the seller at a prie of 1/3 and resells it to the buyer at a prie of 2/3. Trade always ours and the broker earns a profit of 1/3. In the bottom left region, the VCG mehanism is used; in this ase, effiient trade an be ahieed and the broker proides a subsidy. The required subsidy is 1/18, beause the problem is a saled ersion of the original MS problem. 11 Similarly, in the other two diagonal regions, effiient trade is implemented and the broker proides a subsidy of 1/18 in eah region. In the other two regions aboe the diagonal line, effiient trade is implemented at a fixed prie of 1/3 and 2/3, respetiely. Lastly, in all of the other three regions, no trade is effiient. Oerall, an effiient outome is ahieed, and the broker s ex ante expeted profit is positie. 12 This example shows that an informed broker an use ross subsidization to implement effiient trade. In order to get the full effiieny result, the broker must be suffiiently informed. Returning to the example, suppose the broker only knows that the buyer s and seller s aluations belong to either [0, 1/2) or [1/2, 1]. In this ase, the broker annot earn profits in the top left region to subsidize trade in the diagonal regions, so effiient trade annot be ahieed. 13 When the broker s information is gien by a partition struture, Proposition 3 proides a neessary and suffiient ondition on the broker s information suh that an effiient outome an be ahieed. Moreoer, if the broker s information diides the interal into finer partitions, effiient trade is easier to sustain (Corollary 1). 11 The required subsidy to ahiee an effiient outome is ( )1 > 3 3 d d = = 1 18, where 1 > is the indiator funtion and is equal to 1 if >, and 0 otherwise. 12 The broker s ex ante expeted profit is ( ) = In the top left region, the buyer s aluation is in [1/2, 1] and the seller s aluation is in [0, 1/2). The broker an implement effiient trade by setting a prie of 1 2, but he earns a zero profit. 7

8 buyer s aluation seller s ost Figure 1: This figure illustrates that effiient trade an be ahieed with an informed broker, when both the buyer s and seller s priate aluations are drawn independently from the uniform distribution oer the [0, 1] interal, and the broker knows whih of the following three subinterals and belong to: [0, 1/3), [1/3, 2/3), [2/3, 1]. I show that the broker may use profits from the top left region to subsidize trade in the diagonal regions to implement effiient trade. A key assumption underlying the ross subsidization mehanism is that the broker an ommit to subsidize trade in the diagonal regions. If he does not hae ommitment power, then he will not partiipate in the game after he learns that signals are in the diagonal regions. Howeer, as I will show in Proposition 1, wheneer effiient trade an be implemented by ross subsidization, I an onstrut another mehanism that implements effiient trade without requiring the broker s ommitment. The intuition of the onstrution is as follows. Consider a buyer whose aluation belongs to [2/3, 1]. In the ross subsidization mehanism, with probability 2/3, he gets the asset for sure and pays a fixed prie of 2/3; with probability 1/3, he gets the asset when his aluation is greater than the seller s ost, and he pays the seller s ost if there is a trade. In the top left region, any fixed prie below 2/3 an indue the buyer to truthfully report. This indiates that the inentie onstraint is slak. In the top right region, the seller reeies some subsidy from the broker. I an modify the buyer s payments in the top left and top right regions to ahiee effiient 8

9 trade without losing inentie ompatibility. Consider a new mehanism, in whih the buyer and seller simultaneously report their aluations to the broker. 14 In the new mehanism, the buyer pays a lower prie in the top left region and a higher prie in the top right region, so that his expeted payment is unhanged. Sine the buyer reports his aluation without onditioning on the region, he would truthfully report, as in the ross subsidization mehanism. Sine he pays a higher prie in the top right region, the subsidy needed from the broker in the top right region an be redued. By properly adjusting the payments, effiient trade an be ahieed without the broker s losing money in some regions. I explore the issues raised by the example in greater detail. I also study the ase in whih the broker s information is not gien by a partition struture. Proposition 1 shows that it is suffiient to study the ase in whih the broker has ommitment power to derie the ondition to sustain effiient trade. When I restrit to a broker who has ommitment power, the ondition is related to the least expensie way to indue the buyer and seller to partiipate in an effiient mehanism (Proposition 2). So the ondition an be read as the total gains from trade being greater than the minimal information rents left to the buyer and seller. If the broker is informed, information rents are stritly less than that with an uninformed broker. It is easier, therefore, to implement effiient trade with an informed broker. 3 Model 3.1 Enironment Consider a trading problem in whih there are one seller (she), one buyer (he), and one broker. The seller holds an indiisible objet that is aluable for both her and the buyer. The broker does not intend to own the objet in the trade, as the broker does not enjoy onsumption from the objet 14 We an interpret the ross subsidization mehanism as a two-stage game, in whih the buyer and seller play the orresponding mehanism onditional on the broker s announing region. In ontrast, sine I fous on Bayesian implementation, I onstrut the new mehanism in a way that the buyer and seller report their aluations without knowing whih region is true. 9

10 diretly. 15 Let and denote the alue of the objet to the buyer and the seller, respetiely. I assume that these two aluations are independent random ariables. The buyer and seller know his/her own aluation with ertainty, but other agents know the aluation only probabilistially. The buyer s aluation,, is represented by a probability density funtion f, whih is positie in the range [, ]. The orresponding umulatie density funtion is F. The seller s ost,, is represented by a probability density funtion g, whih is positie in the range [, ]. The orresponding umulatie density funtion is G. I assume that 0, 0. For the result to be interesting, I also assume that [, ] and [, ] hae a non-empty intersetion, in whih ase diret bilateral bargaining is neessarily ineffiient, as in Myerson and Satterthwaite (1983). Suppose the transation is mediated by a broker. The broker does not know the buyer s or seller s exat priate aluations, but he may hae some information about them. For example, a realtor, experiened in the real estate industry, is informed about the intrinsi alue of a property, whih is likely orrelated with the seller s priate aluation. A finanial intermediary is informed about the supply (demand) of some finanial asset, and therefore an partially assess the reseration alue of a seller (buyer). Hene, I assume that the broker reeies informatie signals about eah agent s aluation. Prior to trading, suppose the broker priately reeies a signal b B of the buyer s aluation and a signal s S of the seller s aluation. I assume that (, b) is independent of (, s). The broker s signals an be expressed in terms of onditional umulatie distribution funtions, F I (b ) and G I (s ). Let f I (b ) and g I (s ) denote the assoiated onditional probability density funtions. I do not impose assumptions on the ardinality of B and S. The broker s information struture is denoted as I. The distributions (F and G) and the onditional distributions (F I and G I ) are ommon knowledge. Finally, all indiiduals are risk neutral. Both the buyer and the seller hae additiely separable utility for money and the objet. The broker only deries utility from money. 15 Een if I allow the possibility that the broker may retain the objet, the broker would not get it in the ase of either looking for mehanisms to ahiee full effiieny or mehanisms to optimize the broker s (seller s) profit. 10

11 3.2 Diret mehanism The seller, the buyer, and the broker are going to partiipate in some bargaining game. Rather than expliitly modeling the proess, I will study a diret mehanism in whih the probability of trade and payment shedules are determined as a funtion of the agents reported aluations and reported signals. By the Reelation Priniple (see, e.g., Myerson (1979) and Myerson (1981)), the restrition to the diret mehanism is without loss of generality 16 : Any outome assoiated with an equilibrium of some bargaining game will also be an equilibrium outome of some reelation mehanism in whih the buyer and seller truthfully report their aluations and the broker truthfully reports signals. Let p(,, b, s) be the probability that the objet is transferred from the seller to the buyer, if and are the reported aluations of the buyer and the seller, and b and s are the broker s reported signals. I will refer to the probability of trade, p(,, b, s), as an alloation rule. I assume no outsider (other than the buyer, the seller, and the broker) an proide funds; thus, there are only two payment shedules: the payment from the buyer to the broker and from the broker to the seller, denoted as t B (,, b, s) and t S (,, b, s), respetiely. The broker s payoff is t 0 (,, b, s) = t B (,, b, s) t S (,, b, s). Thus, a diret mehanism an be represented as (p(,, b, s), t B (,, b, s), t S (,, b, s)). I define t B (, ˆ) := p B (, ˆ) := S S B B t B (ˆ,, b, s) df I (b ) dg I (s ) dg(), (1) p(ˆ,, b, s) df I (b ) dg I (s ) dg(), (2) U B (, ˆ) := p B (, ˆ) t B (, ˆ), (3) U B () := U B (, ). (4) 16 While the restrition to the diret mehanism is without loss of generality, the restrition on the outome to be probability of trade and payment shedules is not. For example, I annot oer the senario in whih agents are bargaining through time and agents hae different disounting fators. Howeer, if agents hae same disounting fators, the restrition to the outome funtions is also without loss of generality. See Mailath and Postlewaite (1990) for further disussion. 11

12 In other words, if the buyer s aluation is and he reports ˆ, t B (, ˆ) is the expeted payment from the buyer to the broker, p B (, ˆ) is the expeted probability of trade, U B (, ˆ) is the buyer s expeted utility, and U B () is the buyer s expeted payoff. Similarly, for the seller, I define t S (, ĉ) := p S (, ĉ) := t S (, ĉ, b, s) df I (b ) dg I (s ) df (), (5) S B S B p(, ĉ, b, s) df I (b ) dg I (s ) df (), (6) U S (, ĉ) := t S (, ĉ) p S (, ĉ), (7) U S () := U S (, ). (8) If the seller s aluation is and she reports ĉ, t S (, ĉ) is the expeted payment she reeies, p S (, ĉ) is the expeted probability of trade, U S (, ĉ) is the seller s expeted utility, and U S () is the seller s expeted payoff. After the broker reeies a signal of b for the buyer s aluation, the broker updates the distribution of the buyer s aluation. I denote the umulatie distribution as F0 I ( b). After reeiing s, the broker updates the distribution of the seller s aluation. I denote the umulatie distribution as G I 0( s). Thus, if the broker reports (ˆb, ŝ) after reeiing signals of (b, s), his expeted payoff is U 0 (b, s, ˆb, ŝ) := (t B (,, ˆb, ŝ) t S (,, ˆb, ŝ)) dg I 0( s)df I 0 ( b). Define U 0 (b, s) := U 0 (b, s, ˆb, ŝ) to be the broker s expeted profit after reeiing signals of (b, s). For a mehanism to indue honest reporting, it must be inentie-ompatible (IC): = arg max U B (, ˆ), (9) ˆ = arg max U S (, ĉ), (10) ĉ (b, s) = arg max U 0 (b, s, ˆb, ŝ). (11) (ˆb,ŝ) 12

13 I also require that eah indiidual wants to be inluded in the bargaining game. There are two possible indiidual rationality onstraints that I an impose. The stronger ondition requires that eah indiidual earns at least a nonnegatie profit after any realization of aluations and signals. This is an ex post indiidually rational (EXPIR) ondition: t B (,, b, s) t S (,, b, s), if p(,, b, s) > 0. The weaker ondition requires that eah indiidual has a non-negatie expeted gain after eah indiidual learns the priate aluation or signals. This is an interim indiidually rational (INTIR) ondition: U B () 0, U S () 0, U 0 (b, s) 0. I write that an alloation rule p is implementable if there exists payment shedules (t B, t S ) suh that (p, t B, t S ) is inentie-ompatible and interim indiidually rational. The broker s ex ante expeted payoff is gien by the buyer s expeted payment minus the seller s expeted payment: U 0 := (t B (,, b, s) t S (,, b, s)) dg I (s )df I (b )dg()df (). B S If U 0 0, I say that the mehanism is non subsidized. I write that an alloation rule p is implementable under broker ommitment if there exist payment shedules (t B, t S ) suh that (p, t B, t S ) is non subsidized and gies the buyer and seller proper inenties to truthfully report and a nonnegatie interim expeted payoff. In this ase, if the broker has ommitment power, then the alloation rule an be implemented. Ex post effiient trade requires 1 if >, p(,, b, s) = 0 if <, 13

14 i.e., trade should our as long as the buyer s reported aluation is greater than the seller s reported aluation, regardless of the broker s signals. 4 Implementability of Effiient Trade 4.1 Equialene My primary fous is the ase in whih ex post effiient trade is implementable. I first proide a useful equialene result. Proposition 1. If an alloation rule is implementable under broker ommitment, then it is implementable. Proof. See appendix. The proof of Proposition 1 is onstrutie. I proide a sketh here and the details an be found in the Appendix. Suppose (p(,, b, s), t B (,, b, s), t S (,, b, s)) is the mehanism that requires the broker s ommitment. I an onstrut another mehanism that implements the same alloation rule. Let t B(,, b, s) = E,s t B (,, b, s) + E,b t S (,, b, s) E,,b,s t S (,, b, s), t S(,, b, s) = E,s t B (,, b, s) + E,b t S (,, b, s) E,,b,s t B (,, b, s), where E x represents the expetation by integrating oer x. Consider the following new mehanism: (p(,, b, s), t B (,, b, s), t S (,, b, s)). The intuition of the onstrution is as follows. Consider the enironment in Figure 1 in Setion 2. Roughly speaking, in the new mehanism, the buyer whose aluation belongs to [2/3, 1] pays a lower prie in the top left region and a higher prie in the top right region. The buyer s expeted payment is unhanged, so he would truthfully report, as in the ross subsidization mehanism. 14

15 Formally in the proof, the buyer s objetie funtion under t B (,, b, s) is exatly same as his objetie funtion under t B (,, b, s). In addition, the broker s payoff is a nonnegatie onstant under the new mehanism. Thus, the new mehanism an implement the same alloation rule without the broker s ommitment. The definition of implementability imposes more onditions than that of implementability under broker ommitment. Proposition 1 establishes an important equialene result: To study the ondition for implementability, it is suffiient to study the ondition for implementability under broker ommitment. 4.2 Implementability In this subsetion, I proide a general suffiient ondition for ex post effiient trade to be implementable. Consider the effiient alloation rule 1 if >, p(,, b, s) = 0 if <. Thus, a buyer who reports ˆ expets the probability of trade to be G(ˆ), i.e., p B (, ˆ) = G(ˆ). Similarly, a seller who reports ĉ expets the probability of trade to be 1 F (ĉ), i.e., p S (, ĉ) = 1 F (ĉ). I define minimal information rent to the buyer as { } RB I : = inf Ũ B (, ) df () t B (,,, ) s.t. ŨB(, ˆ) = G(ˆ) t B (ˆ,, b, s) df I (b ) dg I (s ) dg(), Ũ B (, ) ŨB(, ˆ),, ˆ, Ũ B (, ) 0,. S B Reall from equation (3) that the first onstraint represents the utility of a buyer who has true aluation and reports ˆ when the payment is gien by t B. Thus, the minimal information rent 15

16 to the buyer, RB I, is the minimal ex ante expeted utility he obtains from any ex post effiient mehanisms that gie him proper inentie to truthfully report and at least a nonnegatie expeted payoff. Note that I take infimum instead of minimum oer all possible payment shedules, beause the minimum alue may not neessarily be ahieed. I say that the infimum problem of R I B is ahieable if there exists t B that gies the buyer ex ante expeted utility exatly equal to R I B. Similarly, I an define minimal information rent to the seller as R I S : = { inf t S (,,, ) s.t. ŨS(, ĉ) = } Ũ S (, ) dg() S B Ũ S (, ) ŨS(, ĉ),, ĉ, Ũ S (, ) 0,. t S (, ĉ, b, s) df I (b ) dg I (s ) df () (1 F (ĉ)), The minimal information rent to the seller, RS I, is the minimal ex ante expeted utility that she obtains from any ex post effiient mehanisms that gie her proper inentie to truthfully report and at least a nonnegatie expeted payoff. I say that the infimum problem of R I S is ahieable if there exists t S that gies the seller ex ante expeted utility exatly equal to RS I. Minimal information rents to the buyer and seller only depend on the fundamentals of the problem (i.e., the distributions of aluations and signals). In partiular, it does not require any knowledge of payments in the bargaining game that they will play. I use notation with a tilde, t i, in the ontext of minimal information rents. I use notation without the tilde, t i, in the ontext of a diret mehanism. Denote the total surplus from an ex post effiient mehanism by W := ( )1 > dg() df (). I now state the main result. 16

17 Proposition 2. A suffiient ondition for ex post effiient trade to be implementable is R I B + R I S < W. A neessary ondition is R I B + R I S W. Furthermore, if the infimum problems of RB I and RI S are ahieable, then a neessary and suffiient ondition for ex post effiient trade to be implementable is R I B + R I S W. Proof. See appendix. Proposition 2 proides a ondition under whih ex post effiient trade exists with an informed broker. If the distributions of the buyer s and the seller s aluations are fixed, RB I and RI S only depend on the distributions of the broker s signals. So the onditions in Proposition 2 impose the ondition on the distributions of the broker s signals. In diret bilateral bargaining, Myerson and Satterthwaite (1983) proe a strong negatie result: that ineffiienies always our with a positie probability in any equilibrium under any possible bargaining game. Proposition 2 says that introduing an informed broker leads to a signifiantly different result: An equilibrium with the effiient outome exists in some bargaining game. I do not laim that the effiient outome will always be obtained. For example, as I disuss in Setion 5, the mehanism that maximizes the broker s ex ante expeted utility is not neessarily ex post effiient. Een if the effiient outome an arise as an equilibrium in a bargaining game, I do not laim that it is the only equilibrium. The result should be interpreted as follows: Under the suffiient ondition there exists a bargaining game with the effiient outome suh that the strong negatie result in Myerson and Satterthwaite (1983) breaks. Bargaining fritions in Myerson and Satterthwaite (1983) are naturally nested in my model 17

18 with an uninformed broker. Minimal information rents with an uninformed broker are exatly gien by the information rents the buyer and seller earn in the VCG mehanism. I show that with an informed broker, the minimal information rents are stritly redued. The hannel that bargaining through an informed broker an resole the ineffiienies is solely through the broker s information. The intuition that an informed broker an improe effiieny is as follows. The presene of priate information implies that the buyer (seller) may misreport their aluations. In Myerson and Satterthwaite (1983), the mehanism designer annot distinguish a buyer who has a aluation and reports ˆ from another buyer who has a aluation but also reports ˆ. Put differently, any buyer who reports ˆ will be treated identially. In ontrast, onsider bargaining through an informed broker. Sine the broker s information depends on whether the buyer s aluation is or, the mehanism an treat the same reports from different types of buyers differently. The broker s orrelated information an help to detet misreporting behaiors, and therefore make it easier to indue truthful reelation. It is worth noting that the suffiient ondition requires strit inequality, while the neessary ondition inoles weak inequality. This is beause the infimum problems of R I B and RI S may not be ahieable. The last part of the proposition says that if I know the infima are ahieable, then I hae the neessary and suffiient ondition. The proposition suggests, therefore, that the ondition with the strit inequality annot be further extended. Lastly, reall that the definition of implementability only requires INTIR instead of EXPIR. Thus, if the suffiient ondition in Proposition 2 holds, I an onstrut an effiient mehanism but it may not satisfy EXPIR. If we further require EXPIR in implementability, the ondition on the broker s information would be more restrited. In the extreme ase, if the broker s signals are perfet i.e., the broker knows the buyer s and the seller s aluations perfetly then an effiient mehanism that is EXPIR exists: The broker buys from the seller at a ost equal to her aluation, and resells to the buyer at a prie equal to his aluation, wheneer trade is effiient. Under this mehanism, the buyer and seller would hoose to partiipate and earn a rent of zero. So, the set of 18

19 onditions for the existene of an EXPIR effiient mehanism is non-empty. To apply the proposition, onsider the illustratie example in Setion 2. I will show in Lemma 2 that R I B = RI S = Total gains from trade W = 1 6. Sine W RI B RI S = 1 54 > 0, an effiient mehanism exists. In Setion 4.4, I study the ase in whih the broker s information has a partition struture. In Setion 4.5, I study the information struture that an fully extrat surplus, as in MAfee and Reny (1992). First, howeer, I present some simple fats about minimal information rents. 4.3 Some fats about minimal information rents To determine whether ex post effiient trade is implementable, it is ruial to know the alues of the minimal information rents to the seller and the buyer. Therefore, I present some simple fats about minimal information rents. Denote as the null information struture in whih the broker is uninformed, i.e., signals are not informatie. Let R B and R S denote the minimal information rents to the buyer and the seller that are assoiated with the null information struture. Lemma 1. The minimal information rents assoiated with the null information struture are RB = RS = G(t)(1 F (t))dt, G(t)(1 F (t))dt. The total gain from trade an be expressed as W = G(t)(1 F (t))dt. For any information struture I, 0 R I B R B W and 0 R I S R S W. 19

20 If the broker has perfet information, then it is lear that the minimal information rent would be zero. The lemma states that for any information struture, the minimal information rent is bounded by the two extreme ases: the perfet information and the null information. In next setion, if the broker s information is gien by partition of an interal, I show that the minimal information rents with an informed broker are stritly less than that with an uninformed broker. As an appliation, it is straightforward to see that as long as (, ) and (, ) hae a nonempty intersetion, RB + RS > W. This is the result in Myerson and Satterthwaite (1983) i.e., if the broker is uninformed, ex post effiient trade an not be implemented without a subsidy. 4.4 Partition struture In this subsetion, I study a partiular type of information struture that the broker may hae: a partition of an interal. Glode and Opp (2016) and Glode, Opp, and Zhang (2016) study this information struture. Suppose the broker knows whih of the following interals belongs to: [ 0, 1 ), [ 1, 2 ),, [ n 1, n ], and whih of the following interals belongs to: [ 0, 1 ], ( 1, 2 ],, ( m 1, m ], where = 0 < 1 < < n =, = 0 < 1 < < m =, and m, n 1. The set of all possible signals an be expressed as B = {1, 2,, n} and S = {1, 2,, m}. For instane, if [ 0, 1 ), then the broker s signal is b = 1. If m = n = 1, then the broker is uninformed. Denote the broker s information struture by I. The following lemma haraterizes the minimal information rents to the buyer and the seller. 20

21 Lemma 2. If the broker s information is gien by a partional struture, then RB I = RS I = n k k=1 k 1 k k m k=1 k 1 k 1 G(t)dtdF (), 1 F (t)dtdg(). In addition, as long as the broker has some information, the minimal information rents will be redued: 1. RB I < R B if and only if n > RS I < R S if and only if m > 1. Proof. See appendix. From Lemma 2, I an alulate the minimal information rents for any partition struture. In addition, if the broker s signal about the buyer s aluation is informatie, the minimal buyer s information rent is stritly less than the minimal information rent assoiated with the null information struture. Suppose the aluations of both the buyer and the seller are drawn from the uniform distribution oer [0, 1] interal. If the broker is uninformed, I find RB = R S = 1. From the lemma, I an 6 alulate the minimal information rents to the buyer and the seller in the illustratie example in Setion 2: R I B = RI S = It is lear that RI B < R B and RI S < R S. Reall that Proposition 2 proides a ondition for the existene of an effiient mehanism. I then obtain a losed-form ondition under whih the broker s information is gien by a partition struture. I summarize the result in the following proposition. Proposition 3. If the broker s information is gien by a partition struture, I, then the neessary 21

22 and suffiient ondition for ex post effiient trade to be implementable is b s n m 0 1 > f() g() b=1 s=1 b 1 s 1 ( F ( b) F () G() G( ) s 1) d d (12) f() g() Proof. See appendix. It is intuitie to interpret the expression in (12). First, F ( b) F () f() aluation term is similar to the irtual 1 F (). The differene omes from the fat that the broker an update the f() distribution of and infer the buyer s aluation from the umulatie distribution funtion ˆF gien by ˆF () = F () F ( b 1) F ( b ) F ( b 1, one the broker knows whih interal belongs to. The assoiated ) probability density funtion is ˆf() = f() F ( b ) F ( b 1. Now the irtual aluation term beomes ) 1 ˆF () ˆf() = F ( b) F (). (13) f() Similarly, + G() G( s 1) g() is the irtual ost onditional on the broker knowing belongs to ( s 1, s ]. So the right-hand side of (12) is the expeted alue, wheneer we want the two parties to trade, of simultaneously buying from the seller and reselling to the buyer. It must be nonnegatie so that the broker is willing to intermediate the bargaining. If m = n = 1, i.e., the broker is uninformed, (12) beomes 0 1 > f() g() ( 1 F () G() ) d d, f() g() whih is exatly the ondition in Myerson and Satterthwaite (1983). Reall that I = {[ 0, 1 ),, [ n 1, n ]} {[ 0, 1 ],, ( m 1, m ]}. The broker s information partitions the types spae into mn regions. Suppose I 1 and I 2 are two 22

23 information strutures. It is natural to say that I 1 is more informatie than I 2 if and only if for any region in I 2, there exists weakly finer regions in I 1, and there exists at least one region suh that I 1 has a stritly finer partition than I 2. It is straightforward to show that a more informatie struture leads to a smaller alue of minimal information rent. I summarize the result in the following lemma. Lemma 3. If I 1 is more informatie than I 2, then R I 1 B RI 2 B and RI 1 S RI 2 S. The equality an hold if and only if I 1 does not reate a finer partition on the aluation of the buyer (seller) than I 2. Diretly applying Proposition 2, I hae the following orollary. Corollary 1. If there exists an ex post effiient mehanism in whih the broker s information is I 2, and I 1 is more informatie than I 2, then there also exists an ex post effiient mehanism in whih the broker s information is I 1. This result says that the ondition is less onstraining when the broker s signals are more informatie. Proposition 4. For any F and G, there exists a partition information struture I suh that the effiient alloation rule an be implemented when the broker s information is I. Proof. See appendix. Proposition 4 says that for any gien distributions, there always exists a fine enough partition suh that an effiient outome an be ahieed with an informed broker. I now proide some examples of when the broker s information is gien by a partition struture. Assume U[0, 1], U[0, 1], and subinterals are equally distributed. Using Lemma 2, I an alulate the minimal information rents: R I B = 3n 1 12n 2, RI S = 3m 1 12m 2. 23

24 From the formula, it is easy to see that RB I goes to 0 at a rate of 1/n. Total gains from trade W = 1/6. Suppose that m = n = 1, whih is the ase in Myerson and Satterthwaite (1983) with an uninformed broker. Then R I B +RI S = 1/3 > W. So ex post effiient trade is not implementable. Now onsider m = 1 and n > 1. Then R I B + RI S W = 3n 1 12n 2 > 0. Ex post effiient trade is not implementable, but the subsidy needed from the broker to reah full effiieny is approahing to 0 as n. Suppose m = n = 2, that is, the broker knows whih of [0, 1/2) or [1/2, 1] the buyer s and seller s aluations fall into. Sine W (RB I + RI S ) = 1/24 < 0, there is no ex post effiient mehanism. Next, I gie an example in whih ex post effiient trade exists when the broker is informed. Suppose that m = n = 3. Then W (RB I + RI S ) = 1/54. Furthermore, by Corollary 1, if the broker s information is finer than m = n = 3, ex post effiient trade is implementable. If m = n = 4, W (RB I + RI S ) = 5/96. Thus, ex post effiient trade is implementable. 4.5 Condition for full surplus extration In this subsetion, I show that minimal information rents are losely related to full surplus extration, as in Crémer and MLean (1988) and MAfee and Reny (1992). Sine the type spaes in my setting are ontinuous, I assume that the signal sets B and S are ompat subsets of R in order to apply the theorem in MAfee and Reny (1992). Without loss of generality, let B = [, ] and S = [, ]. I also assume that both of the joint umulatie distributions, F I (, ) and G I (, ), hae a ontinuous probability density funtion. MAfee and Reny (1992) gie a neessary and suffiient ondition for (almost) full rent extration, whih I first desribe. I then show that if their (almost) full rent extration ondition holds, then minimal information rents are zero. Let (B) denote the set of probability measures on B. Reall that f I ( ) is the onditional probability density funtion. I say that f I ( ) satisfies the MAfee and Reny ondition if, for eery 0 and eery µ (B), µ({ 0 }) 1 implies that f I ( 0 ) f I ( )µ(d). Similarly, I say that 24

25 g I ( ) satisfies the MAfee and Reny ondition if, for eery s 0 and eery µ (S), µ({s 0 }) 1 implies that g I ( s 0 ) gi ( s)µ(ds). I say that the broker s information struture satisfies the MAfee and Reny ondition if both f I and g I satisfy the (almost) full rent extration ondition. Proposition 5. If the broker s information struture satisfies the MAfee and Reny ondition, then R I B = RI S = 0 and ex post effiient trade is implementable.17 Proof. See appendix. To understand why R I B = 0, I first explain the MAfee and Reny ondition. It is easier to explain its disrete ounterpart, in whih the ondition beomes the Crémer and MLean (1988) ondition. Basially, the Crémer and MLean ondition says that the etor of onditional probabilities orresponding to any possible alue is not in the onex hull of the etors of onditional probabilities orresponding to other possible types. This ondition on the information struture is a spanning ondition. Roughly speaking, Crémer and MLean (1988) show that we an design a side bet with the following properties: It gies the buyer a large negatie payoff if he does not report truthfully; it gies the truth-telling buyer the exat amount of payoff suh that he is willing to partiipate. The spanning ondition is exatly what I need to inert a system of linear equations to design the desired side bet. Similar to MAfee and Reny (1992), I onsider first a truth-telling mehanism in the bargaining stage (seond stage) supplementing a pre -mehanism (first stage). In the seond stage, the buyer pays the seller s reported aluation and the seller reeies the buyer s reported aluation, if the buyer s reported aluation is greater than the seller s. Honesty is a dominant strategy for both the buyer and the seller. I then supplement the following first-stage mehanism. The broker offers a set of partiipation fee shedules {z B n ( )} for the buyer. Prior to the bargaining, the buyer hooses one of the partiipation fee shedules z B n ( ), whih requires the buyer to pay z B n (b) if the broker 17 Sine W RS, I obtain the following results by diretly applying Proposition 2: If the broker is uninformed about the seller s aluation and the signals about the buyer s aluation, f I ( ), satisfies the MAfee and Reny ondition, then RB I = 0 and ex post effiient trade is implementable. 25

26 reeies a signal of b. Sine the broker s information struture satisfies the MAfee and Reny ondition, I an design the partiipation fee shedules suh that the buyer earns a rent of no greater than ɛ, ɛ > 0. Finally, let s onsider the buyer s payment shedule, t B (,, b, s) = zn B (b) + 1 >, where zn B (b) is the side bet and 1 > is the seond-stage payment. Sine the side bet depends on the broker s signal and not on the seller s reported type, the payment shedule indues the buyer to truthfully report. In addition, t B (,, b, s) yields an information rent of no greater than ɛ to the buyer. Consequently, the minimal buyer s information rent RB I = 0. Similarly, the minimal seller s information rent R I S = 0. Applying Proposition 2, effiient trade is implementable. Furthermore, sine the MAfee and Reny ondition holds generially (MAfee and Reny, 1992), I onlude that effiient trade an be sustained with a generi informed broker. Howeer, there is a aeat. If a signal ontains ery little information, a large penalty in payment will be required beause of the side bet. In my model, a risk-neutral player does not are about a large penalty, as long as the expeted profit is nonnegatie. Howeer, in reality, agents may be budget onstrained, and in many ases a large penalty is not feasible. If I restrit an exogenous bound on ex post payments, then effiient trade an be implemented only if the broker s signals ontain suffiient information. Example 1. Suppose that, U[0, 1] and F I (b ) = ( ) b θ for b [0, ] and G I ( s) = ( s θ ) for s [0, ], where θ 0 is a parameter. For any θ > 0, the broker s information struture satisfies the MAfee and Reny ondition. If θ = 0, the information struture is ompletely uninformatie. Now onsider θ > 0; we an show that RB I = RI S = 0. Consider t B (,, b, s) = θ+1 b, then θ Ũ B (, ˆ) = ˆ θ+1 ˆbθ ( ) b θ 1 1 db = 0. Consider t 0 θ S (,, b, s) = θ+1(1 )s, then ŨS(, ĉ) = θ (1 ĉ) + 0 θ+1 (1 ĉ)sθ ( s θ 1 1 θ ) db = 0. Thus, RI B = RI S = 0. I now onstrut an ex post effiient mehanism. Consider payment shedules t B (,, b, s) = θ+1 θ+1 b + θ (1 )s 1 and t θ 6 S(,, b, s) = θ+1 θ θ+1 b + (1 )s 1. We an erify that the buyer θ 3 and the seller would truthfully report and earn a profit of zero. The broker s payoff is 1 after any 6 realization. 26