Auction and the Informed Seller Problem

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Auction and the Informed Seller Problem∗ B. Jullien†

T. Mariotti‡

August 2002 revised January 2003

Abstract A seller possessing private information about the quality of a good attempts to sell it through a second-price auction with announced reserve price. The choice of a reserve price transmits information to the buyers. We compare the outcome of a game where the seller runs himself the auction (signaling) and a game where a monopoly broker chooses the trade mechanism (screening). For the former case, we characterize the equilibria of the resulting signaling game and show that they lead to reduced levels of sale compared to the symmetric situation. We compare the unique separating equilibrium with the outcome that would be chosen by the monopoly broker. The ex-ante expected probability of trade may be larger with a monopoly broker, as well as the ex-ante total expected surplus.

∗

We wish to thank Bruno Biais, Bernard Caillaud, David Martimort, Benny Moldovanu and JeanCharles Rochet, as well as seminar participants at Toulouse University, for insighful comments. † University of Toulouse (GREMAQ and IDEI), 21 Allée de Brienne, 31000 Toulouse, France. Email: [email protected]. ‡ University of Toulouse (GREMAQ), 21 Allée de Brienne, 31000 Toulouse, France, LSE and CEPR. E-mail: [email protected]

1. Introduction This paper considers an auction market subject to a lemon problem (Akerlof (1970)) and derives conditions under which a trading process centralized by a monopoly broker can improve on a decentralized market structure. The lemon problem arises because sellers, who design the auctions in a decentralized structure, have access to some non-verifiable information relevant to buyers. This results in an “informed principal” situation (Maskin and Tirole (1992)) which implies sub-optimal trade. The intermediary has no particular expertise except that all trades have to be conducted through his house. He is thus able to design screening mechanisms that induce revelation of the sellers’ information to the buyers, thereby limiting the lemon problem. When the efficiency loss due to monopoly distortion is not too large, the monopoly broker will then generate more trade and larger total surplus than the decentralized structure. Traditional auction theory considers auctions conducted by sellers and attempts to derive the revenue maximizing auction or the efficient auction. Instances where the seller chooses and conducts himself the auction, such as public procurement auctions or treasury bond auctions for example, are not so numerous however. By large, many auctions or auction-like mechanisms in the private sector are conducted by intermediaries. This is the case for instance of IPOs, art items, wine, real estate... The spectacular success of eBay has pointed to the potential of developing auction-like trading mechanisms, when the cost of gathering and processing information is reduced as it is the case for Internet. Notice that one feature of eBay is that it has being successful in solving quality issues for secondary goods, through adequate mechanism design and information feedback (see Dellarocas (2002)). In their seminal work on efficient mechanisms, Myerson and Satterthwaite (1983) addressed from an auction perspective the problem faced by a “broker” who intermediates trades . But since then the impact of the intermediary has not being the object of specific attention in the auction literature. Aside auction theory, there has being however many discussions of the role of intermediation in reducing information related costs. The literature has mostly focused on the expertise of intermediaries. Typically intermediaries have access to some information technology that raises the efficiency of market organization. For instance intermediaries may facilitate the search of a trading partner (Rubinstein and Wollinsky (1987), Yavas (1994)). They may have expertise in certification, screening or monitoring (Diamond (1984), Biglaiser (1993)). In our set-up, the intermediary has no expertise except in conducting auctions and this is precisely the source of the efficiency gains.1 Indeed an informed intermediary may face similar issues than the informed seller as it may not have the proper incentives to reveal the information to the market at the interim stage (Lizzeri (1999), Peyrache 1

Assuming a monopoly intermediary means however that some type of exclusive expertise or specific asset prevents entry in the intermediation market.

1

and Quesada (2002)). Being uninformed, our broker is not constrained by the fact that some choice of trading mechanism may convey a bad signal about the value of the good traded, which is the source of the results. More precisely, we consider in Section 1 a seller who owns an indivisible unit of a good and faces two potential buyers. The seller’s valuation for the good is private information. A buyer’s ex-post valuation for the good includes a common value component that is proportional to the seller’s valuation plus a private value component. In a decentralized market, the seller chooses a reserve price and organizes a second-price sealed bid auction with public reserve price. The game thus amounts to a signaling game, with the reserve price chosen by the seller, followed by the auction.2 This issue of the informed auctioneer has not yet been resolved.3 The corresponding signaling game does not fit standard assumptions. In the first part of the paper we show how to transform the problem and derive general properties of the equilibria of this game. The main result is that any equilibrium with monotone beielfs is characterized by a generalized lemon problem, i.e. the probability of trade is below the symmetric information probability of trade for all realizations of the seller’s information. We also show that any equilibrium satisfies some monotonicity conditions, and that pooling is associated to some form of discontinuity. In Section 3, we characterize the unique separating equilibrium of the signaling game.4 We then turn in Section 4 to the monopoly broker problem. This is a standard screening problem and the optimal mechanism can be interpreted as one in which the broker organizes a second-price sealed bid auction on behalf of the seller, but let the seller announce a reserve price, under the condition that the seller pays a fee contingent on the level of the reserve price. The announced reserve price then reveals the seller’s information to the buyers. In the last section we compare the separating decentralized equilibrium with the broker equilibrium. At the interim stage, it is not possible to rank the two, as the broker sells more for low seller’s valuations but less for high sellers valuation. But, from an ex-ante perspective, 2

Notice that in the last stage of the signaling game (the auction), buyers share the same information on the common value component so that the auction is basically a private value one and revenue equivalence holds. We prove indeed that, provided that the seller’s action is restricted to chosing a selling mechanism and that he is not an active participant of the mechanism once this choice is made (as may occur in a double auction), there is no loss of generality in assuming a second-price sealed bid auction. 3 Milgrom and Weber (1982) show that a seller benefits from revealing his information, but this holds ex-ante for verifiable information, while we analyze the problem at the interim stage for nonverifiable information that is correlated with the sellers opportunity cost of selling. 4 While we put the final touch on this version of the paper, we are informed of a simultaneous related work by Cai, Riley and Ye (2002). They solve for the separating equilibrium of a signaling game in which a seller chooses a mark-up over the buyers’ beliefs. The reserve price is then defined as the sum of the beliefs and the mark-up. This appears to be close to our resolution of the separating equilibrium of the game where the seller chooses directly the reserve price, and their Proposition 1 is closely connected to our Proposition 3.

2

we prove that provided that the uncertainty on the seller’s information is not too large in the sense that its distribution is concentrated enough, the expected probability of sale is larger when trade is intermediated by a monopoly broker. The same holds true for the ex-ante expected surplus. 2. The Model We consider the following common value environment. Let e θ be the valuation of the e e seller, and θi = λθ + (1 − λ)e εi that of buyer i = 1, 2, where e θ, e εi , i = 1, 2, are independently distributed with c.d.f. on [0, 1] given by G and F respectively, and λ ∈ [0, 1]. We assume that both associated densities g and f are strictly positive on [0, 1], and that the usual monotone hazard rate properties are satisfied: µ ¶ µ ¶ d g(θ) d f (ε) < 0, > 0. dθ G(θ) dε 1 − F (ε) It is assumed that e θ, e εi , i = 1, 2 are private information of the seller and the buyers, respectively. In the informed seller game, the seller runs a second-price auction with announced reserve price r. For some beliefs ϕ of the buyers about the expectation of the common value parameter θ, and for some reserve price r, let z(ϕ, r) = (r − λϕ)/(1 − λ). Then, trade occurs if at least one buyer i has a private component e εi above z(ϕ, r). The probability of no sale is thus π(ϕ, r) = F (z(ϕ, r))2 . In what follows we shall use z or π whenever convenient with the convention that π = F (z)2 . One can then check that the utility of the seller can be expressed as: ¶ Z 1 µ 1 − F (ε) − θ dF (ε)2 . U (θ, ϕ, r) = θ + λ(ϕ − θ)(1 − π(ϕ, r)) + (1 − λ) ε− f(ε) z(ϕ,r) The expected gains from trade U(θ, ϕ, r) − θ includes two components. The second component is the expected gain that the seller would obtain under symmetric information (ϕ = θ) for an auction with probability of no sale π(ϕ, r). Under symmetric information, the corresponding reserve price would be λθ + (1 − λ)z(ϕ, r). The first term λ(ϕ − θ)(1 − π) accounts for the fact that a given probability of sale corresponds to higher bids and a higher reserve price if the buyers’ beliefs ϕ on the common value component are biased upward: indeed the bids are raised by λ(ϕ − θ), and so is the reserve price that maintains unchanged the sale probability. Note that U (θ, ϕ, r) depends on the reserve price r only through the probability of not selling the good, π(ϕ, r). Also, by construction, U (θ, ϕ, r) = U (θ, ϕ, λϕ + (1 − b by: λ)z(ϕ, r)). It is therefore convenient to introduce a function U b (θ, ϕ, π) = U (θ, ϕ, λϕ + (1 − λ)(F 2 )−1 (π)). U 3

This measures the expected surplus of the seller as a function of the buyers’ beliefs and the probability of sale. One has then the following helpful lemma. b (θ, ϕ, π) = U b (θ0 , ϕ, π) + (θ − θ 0 )π. Lemma 1 For all (θ, θ0 , ϕ, π), U b: Proof. From the definition of U

¶ µ 1 − F (ε) b (θ, ϕ, π) = πθ + λϕ(1 − π) + (1 − λ) U ε− dF (ε)2 . f (ε) z(π) Z

1

The result is then immediate.

b is that it is strictly increasing As is clear from the definition of U, a key property of U with respect to the buyers’ beliefs ϕ. Moreover: µ ¶ 1 − F (z) b U3 (θ, ϕ, π) = λ (θ − ϕ) − (1 − λ) z − −θ , f (z)

where F 2 (z) = π. As z increases with π,a further important property is that, under the usual monotone hazard rate property for F , the seller’s utility is strictly concave in the probability of not selling the good, π(ϕ, r). 3. Equilibrium Analysis

We start with some basic properties of equilibria of the informed seller game. Next, we investigate the consequences of assuming monotone beliefs. Last, we study in detail the unique separating equilibrium. Finally, we show that the particular signaling game that we analyze, in which signaling occurs through the reserve price, is without loss of generality in a sense to be made precise. 3.1. Preliminaries Some Monotonicity Results. An equilibrium is given by a pair of functions (φ∗ , r∗ ) that satisfies, by Bayes’ rule: h i ∗ ∗ e e φ (r) = E θ|r (θ) = r ,

for all r in the range of r∗ . The equilibrium beliefs of the buyers, when the type of the seller is θ, are then given by ϕ∗ (θ) = φ∗ ◦r∗ (θ). Define z ∗ (θ) = (1−λ)−1 (r∗ (θ)−λϕ∗ (θ)) and π ∗ (θ) = F (z ∗ (θ))2 . Note that if r∗ (θ) = r∗ (θ0 ), then ϕ∗ (θ) = ϕ∗ (θ0 ) and therefore π ∗ (θ) = π ∗ (θ0 ). The incentive compatibility constraints, together with the definition of b , imply that: U b ϕ∗ (θ), π ∗ (θ)) = max U b (θ, ϕ∗ (θ0 ), π ∗ (θ0 )). U(θ) ≡ U(θ, 0 θ

4

(1)

One has then the following lemma. Lemma 2 In any equilibrium, π ∗ is a non-decreasing function of θ. Moreover, U is a continuous function of θ. Proof. Let θ 6= θ0 . Then, by (1) and Lemma 1: Similarly,

b (θ, ϕ∗ (θ0 ), π ∗ (θ0 )) = U(θ0 ) + (θ − θ0 )π ∗ (θ 0 ). U (θ) ≥ U b (θ0 , ϕ∗ (θ), π ∗ (θ)) = U (θ) + (θ0 − θ)π ∗ (θ). U(θ0 ) ≥ U

Combining these two inequalities, one obtains that:

(θ − θ0 )π ∗ (θ) ≥ U (θ) − U (θ 0 ) ≥ (θ − θ0 )π ∗ (θ0 ), which implies the result. It follows in particular that in any equilibrium, the probability of sale is a decreasing function of the seller’s type. Note that the proof implies that U is Lipschitz, hence absolutely continuous on [0, 1]. It is therefore differentiable almost everywhere on [0, 1]. By the envelope theorem applied to (1), incentive compatibility entails that: dU b1 (θ, ϕ∗ (θ), π ∗ (θ)) = π ∗ (θ), (θ) = U dθ

the probability of no sale. Let us introduce the mapping: b (θ, ϕ, π). π b(θ, ϕ) = arg max U π

π b(θ, ϕ) is the optimal probability of no sale for a seller of type θ, given fixed beliefs ϕ. This mapping is well-defined since, under the monotone hazard rate property for F , the b ϕ, π) is strictly concave. Let zb(θ, ϕ) be the corresponding threshold mapping π 7→ U(θ, level for the private value component of the buyers. If θ was symmetric information, then the optimal reserve price would be λθ + (1 − λ)b z (θ, θ), where zb(θ, θ) is the unique solution in ζ to the usual equation: ζ−

1 − F (ζ) =θ f (ζ)

(2)

that defines the optimal reserve price in a second-price auction. With such an allocation, the seller’s utility would be: ¶ Z 1 µ 1 − F (ε) b U (θ, θ, π b(θ, θ)) = θ + (1 − λ) ε− − θ dF (ε)2 . f(ε) zb(θ,θ) 5

Using (2), a direct computation shows that: b dU (θ, θ, π b(θ, θ)) = λ + (1 − λ)b π (θ, θ), dθ

which differs from π b(θ, θ) unless λ = 0, in which case we are back to the private values benchmark. Hence announcing the symmetric information reserve price λθ + (1 − λ)b z (θ, θ) cannot be an equilibrium of the informed seller game. An important corollary of Lemma 2 is that equilibrium reserve prices are monotone functions of the seller’s type. Corollary 1 In any equilibrium, r∗ is a non-decreasing function of θ. Proof. Let θ > θ0 and assume that r∗ (θ) < r∗ (θ0 ). By Lemma 2, π(θ) ≥ π ∗ (θ0 ). Hence, it must be that ϕ∗ (θ) < ϕ∗ (θ0 ), or: h i h i E e θ|r∗ (e θ) = r∗ (θ) < E e θ|r∗ (e θ) = r∗ (θ0 ) .

Two cases must be distinguished. Either there exists θ00 < θ0 such that r∗ (θ00 ) = r∗ (θ), and thus π ∗ (θ00 ) = π ∗ (θ). Since θ00 < θ0 , it follows from Lemma 2 that π ∗ (θ) ≤ π ∗ (θ0 ) and thus π ∗ (θ) = π ∗ (θ0 ). Or, there exists θ 00 > θ such that r∗ (θ00 ) = r∗ (θ0 ), and thus π ∗ (θ00 ) = π ∗ (θ0 ). Since θ00 > θ, it follows from Lemma 2 that π∗ (θ) ≤ π ∗ (θ0 ) and thus b π ∗ (θ) = π ∗ (θ0 ) again. But then, by ϕ∗ (θ0 ) > ϕ∗ (θ) and the strict monotonicity of U with respect to ϕ, b ϕ∗ (θ 0 ), π ∗ (θ0 )) = U b (θ, ϕ∗ (θ0 ), π ∗ (θ)) > U(θ, b ϕ∗ (θ), π ∗ (θ)), U(θ,

which violates the incentive compatibility condition (1).

A second corollary of Lemma 2 is that equilibrium beliefs are monotone functions of the seller’s type. Corollary 2 In any equilibrium, ϕ∗ is a non-decreasing function of θ. Proof. Let r > r0 in the range of r∗ . By Corollary 1, r∗ is non-decreasing. Hence, we have: h i h i ∗ ∗ 0 ∗ e ∗ e 0 e e φ (r) = E θ|r (θ) = r ≥ φ (r ) = E θ|r (θ) = r . It follows immediately that ϕ∗ = φ∗ ◦ r∗ is a non-decreasing function of θ.

It should be noted that Corollary 2 says nothing about out-of-equilibrium beliefs, i.e., on the global shape of φ∗ outside the range of r∗ . The characterization of equilibria 6

is however greatly simplified if we assume, as we later do, that φ∗ is monotone over its entire domain. Bunching. There is no a priori reason to rule out equilibria with bunching, i.e., equilibria in which r∗ is constant on some interval. By Corollary 1, the set of types where there is bunching is composed of an at most countable family of maximal intervals. Bunching creates discontinuity in the allocation and in the reserve price, as the following result shows. Lemma 3 Suppose that (φ∗ , r∗ ) is an equilibrium with bunching, and let [θ, θ] ⊂ (0, 1) be a maximal bunching interval. Then π ∗ , r∗ and ϕ∗ are discontinuous at θ and θ. Proof. Clearly, ϕ∗ is discontinuous at the boundaries of the bunching interval. Suppose b is strictly increasing that π ∗ is continuous at one of the boundaries, say θ. Then, as U in ϕ and continuous, U would be discontinuous at θ, which contradicts Lemma 2. Hence, since π ∗ is non-decreasing, it has an upward jump at θ, just as ϕ∗ . Since r∗ = λϕ∗ + (1 − λ)z ∗ , the result follows. As an immediate corollary, it follows that any equilibrium in which r∗ is continuous must be separating. 3.2. Monotone Beliefs In this subsection, we investigate the consequences of assuming monotone beliefs, i.e., that the mapping φ∗ is non-decreasing on its entire domain. Our first result is a partial converse of Lemma 3. Lemma 4 In any equilibrium with monotone beliefs, π ∗ , r∗ and ϕ∗ are continuous on the interior of the set of separating types. Proof. For ϕ∗ , the result is immediate as ϕ∗ (θ) = θ on S ∗ . Suppose that r∗ is discontinuous at θ in the interior of the set of separating types. Since U is continuous by Lemma 2, a seller of type θ must be indifferent between r∗ (θ− ) and r∗ (θ+ ) and b θ, π ∗ (θ − )) = U b (θ, θ, π ∗ (θ+ )). Since beliefs are monotone, one must have U(θ) = U(θ, each r ∈ (r∗ (θ− ), r∗ (θ+ )) is associated with the same out-of-equilibrium beliefs θ. The strict concavity of the seller’s utility with respect to the probability of no sale π(θ, r) implies then that he could guarantee himself strictly more than U (θ) by choosing r ∈ (r∗ (θ− ), r∗ (θ+ )), a contradiction. Hence, r∗ and thus π ∗ are continuous at θ. Let us introduce the mapping Πφ∗ (r) = F ((r − λφ∗ (r))/(1 − λ))2 . This mapping extends π ∗ by allowing out-of-equilibrium reserve price as its arguments. We know that Πφ∗ is non-decreasing on the range of r∗ , since Πφ∗ (r∗ (θ)) = π ∗ (θ) and both r∗ and 7

π ∗ are non-decreasing. The following result implies that if beliefs are monotone, there is no loss of generality in assuming that Πφ∗ is non-decreasing on its whole domain. Moreover, the belief function can be chosen to be strictly monotone and continuous. Proposition 1 Suppose that (φ∗ , r∗ ) is an equilibrium with monotone beliefs. Then there exists an equilibrium (φ∗∗ , r∗ ) with continuous and strictly monotone beliefs φ∗∗ such that Πφ∗∗ is a non-decreasing function of r. Proof. The proof goes through a series of steps. Step 1: First, we set φ∗∗ = φ∗ over the range of r∗ . Since (φ∗ , r∗ ) is an equilibrium with monotone beliefs, Lemma 4 implies that r∗ is continuous on the interior of the set of separating types. Hence, since φ∗ ◦ r∗ (θ) = θ on the set of separating types, φ∗ and thus φ∗∗ are continuous on the part of the range of r∗ corresponding to the interior of the set of separating types. Now, let θ be a point of discontinuity of r∗ , with r∗ (θ− ) < r∗ (θ+ ). Then, by Lemma 4, θ must be at the boundary of a bunching interval. By Lemma 3, this implies that z ∗ and ϕ∗ are also discontinuous at θ, with π ∗ (θ− ) < π ∗ (θ + ) and ϕ∗ (θ− ) < ϕ∗ (θ+ ). Moreover, since U is continuous by Lemma 2, b (θ, ϕ∗ (θ− ), π ∗ (θ− )) = U(θ, b ϕ∗ (θ+ ), π ∗ (θ + )). U(θ) = U Step 2: We shall now prove that:

b (θ, ϕ∗ (θ− ), π). π ∗ (θ− ) ≥ arg max U π

b (θ, ϕ∗ (θ− ), π∗ (θ+ )) by strict monotonicity of U b with Suppose not. Then, since U (θ) > U respect to ϕ, there exists π 0 ∈ (π ∗ (θ− ), π ∗ (θ+ )) such that: b ϕ∗ (θ− ), π ∗ (θ− )). b (θ, ϕ∗ (θ− ), π 0 ) = U (θ) = U(θ, U

It follows that for all r ∈ (r∗ (θ− ), r∗ (θ+ )), Πφ∗ (r) ∈ / (π ∗ (θ− ), π 0 ). Otherwise, the strict concavity of the seller’s utility with respect to the probability of no sale Πφ∗ (r) and the monotonicity of beliefs would imply that: b ϕ∗ (θ− ), Πφ∗ (r)) > U(θ), b (θ, φ(r), Πφ∗ (r)) > U(θ, U

which violates the incentive compatibility condition. Since Πφ∗ (r∗ (θ− )) = π ∗ (θ − ) < π ∗ (θ+ ) = Πφ∗ (r∗ (θ+ )), it follows that there exists r ∈ (r∗ (θ − ), r∗ (θ+ )) such that Πφ∗ (r− ) < Πφ∗ (r+ ). But then φ∗ (r− ) > φ∗ (r+ ), which violates the monotonicity of φ∗ . Step 3: We shall now construct the belief function φ∗∗ over (r∗ (θ− ), r∗ (θ+ )). For any b (θ, ϕ, π) = U (θ). This equation has ϕ ∈ [ϕ∗ (θ− ), ϕ∗ (θ+ )], consider the equation in π, U 8

a solution in (π ∗ (θ− ), π ∗ (θ+ )) since:

and

b ϕ, π ∗ (θ− )) − U(θ) = U b (θ, ϕ, π ∗ (θ− )) − U(θ, b ϕ∗ (θ− ), π ∗ (θ − )) > 0 U(θ, b ϕ, π∗ (θ+ )) − U(θ, b ϕ∗ (θ + ), π ∗ (θ+ )) < 0, b ϕ, π ∗ (θ+ )) − U(θ) = U(θ, U(θ,

b with respect to ϕ. Moreover, by Step 2, this solution is by strict monotonicity of U b b is unique and continuous with respect to ϕ. Call it Π(ϕ). Note that the mapping Π b b 0 ). Then U b (θ, ϕ, Π(ϕ)) b = Π(ϕ = one-to-one. Indeed, suppose that ϕ > ϕ0 but Π(ϕ) 0 b b b U (θ, ϕ , Π(ϕ)) which contradicts the fact that U is strictly increasing in ϕ. Hence, as b is continuous and Π(ϕ b ∗ (θ− )) = π ∗ (θ− ) < π ∗ (θ + ) = Π(ϕ b ∗ (θ + )), it follows that Π b is Π ∗ ∗ strictly increasing ³ on´[ϕ (θ− ), ϕ (θ+ )]. It follows that the mapping ϕ 7→ rb(ϕ) = λϕ + b is strictly increasing and continuous, with rb(ϕ∗ (θ− )) = r∗ (θ− ) (1 − λ)(F 2 )−1 Π(ϕ)

and rb(ϕ∗ (θ+ )) = r∗ (θ+ ). Define φ∗∗ to be the inverse of rb on (r∗ (θ − ), r∗ (θ+ )). Then b ◦ φ∗∗ . It follows φ∗∗ is strictly increasing and continuous over this interval, and so is Π that: µ µ ¶2 ¶2 r − λφ∗∗ (r) rb ◦ φ∗∗ (r) − λφ∗∗ (r) b ◦ φ∗∗ (r) =F =Π Πφ∗∗ (r) = F 1−λ 1−λ is also strictly increasing on (r∗ (θ − ), r∗ (θ+ )). To conclude the proof, note that the seller is indifferent between all possible r ∈ [r∗ (θ − ), r∗ (θ+ )] given the belief φ∗∗ , since b (θ, φ∗∗ (r), Πφ∗∗ (r)) = U(θ) for all such r. Hence, since (φ∗ , r∗ ) is an equilibrium, U (φ∗∗ , r∗ ) is an equilibrium as well.

Our objective is now to compare the equilibrium probability of sale with the probability of sale that would arise if the seller could credibly commit to reveal the true value of θ to the bidders. The result is that any monotone equilibrium is subject to a generalized lemon problem, namely that the volume of trade is uniformly below the symmetric information profit maximizing volume of trade.

Proposition 2 In any equilibrium with monotone beliefs, the probability of sale is less than the probability of sale that would arise if the true value of θ where publicly revealed to the bidders: π ∗ (θ) ≥ πˆ (θ, θ). Proof. We first have the following lemma. Lemma 5 Suppose that (φ∗ , r∗ ) is an equilibrium with monotone beliefs. Then π b(θ, ϕ∗ (θ+ )) ≤ π b(θ, ϕ∗ (θ− )) ≤ π ∗ (θ − ) for all θ, where the first inequality is strict if θ is at the boundary of a bunching interval. 9

Proof. Suppose that θ is at the boundary of a bunching interval, and that π b(θ, ϕ∗ (θ− )) > b (θ, ϕ∗ (θ− ), π) is strictly increasing in a right neighπ ∗ (θ− ). Then the mapping π 7→ U ∗ borhood of π (θ− ). From Proposition 1, there is no loss of generality in assuming that φ∗ is strictly increasing and continuous, and that Πφ∗ is non-decreasing and continuous. b is strictly increasing Hence, for r larger but close enough to r∗ (θ− ), one has, since U with respect to ϕ: b φ∗ (r), Πφ∗ (r)) > U(θ, b ϕ∗ (θ− ), Πφ∗ (r)) ≥ U(θ, b ϕ∗ (θ − ), π ∗ (θ− )), U(θ,

which violates the incentive compatibility condition for θ, a contradiction. Thus π b(θ, ϕ∗ (θ− )) ≤ π ∗ (θ− ), as claimed. Now suppose that π b(θ, ϕ∗ (θ− )) ≤ π b(θ, ϕ∗ (θ+ )). b3 (θ, ϕ∗ (θ+ ), π It follows that U b(θ, ϕ∗ (θ− ))) ≤ 0, or: µ ¶ 1 − F (b z (θ, ϕ∗ (θ− ))) ∗ ∗ θ − λϕ (θ+ ) + (1 − λ) − zb(θ, ϕ (θ− )) ≥ 0 f (b z (θ, ϕ∗ (θ− ))) which implies, since ϕ∗ (θ+ ) > ϕ∗ (θ− ): µ ¶ 1 − F (b z (θ, ϕ∗ (θ− ))) ∗ ∗ θ − λϕ (θ− ) + (1 − λ) − zb(θ, ϕ (θ− )) > 0, f (b z (θ, ϕ∗ (θ− )))

b3 (θ, ϕ∗ (θ− ), π i.e., U b(θ, ϕ∗ (θ− ))) > 0, which contradicts that π b(θ, ϕ∗ (θ− )) is optimal for (θ, ϕ∗ (θ− )). Suppose now that θ belongs to the interior of a bunching interval, with b (θ, ϕ, π) has increasing fixed belief ϕ∗ (θ) = ϕ. Note that the mapping (π, θ) 7→ U marginal returns. Hence the mapping θ 7→ π b(θ, ϕ) is strictly increasing on the interval of bunching (see Edlin and Shannon (1998, Theorem 1)). Since π b(θ, ϕ) ≤ π ∗ (θ− ) at the right boundary θ of the bunching interval, we have π b(θ, ϕ∗ (θ)) = π b(θ, ϕ) < π ∗ (θ− ) = ∗ π (θ) on the interior of the bunching interval. The proof for the case where θ is an interior point of the set of separating types is similar to the case where θ is at the boundary of a bunching interval. We are now ready prove the main result of this section. If θ is a separating type, this follows immediately from Lemma 5. Suppose now that θ belongs to a bunching θ) in this bunching interval. As the mapping interval. Let e θ be such that e θ = ϕ∗ (e b ϕ, π) has increasing marginal returns (see above), the mapping θ 7→ (π, θ) 7→ U(θ, π b(θ, ϕ) is strictly decreasing in ϕ. It follows that if θ ≥ e θ = ϕ∗ (θ), π b(θ, θ) ≤ π b(θ, ϕ∗ (θ)) and the result follows from Lemma 5. To conclude, one must prove the result for θ ≤ e θ e in the bunching interval. Since the result is trivially true for θ = θ, we need only prove that the mapping θ 7→ zb(θ, θ) is non-decreasing, which follows from the fact that the b (θ, θ, π) has increasing marginal returns. mapping (π, θ) 7→ U 10

3.3. The Separating Equilibrium We now consider the case of a separating equilibrium, i.e., where r∗ is one-to-one. Some caution is needed here, because it is not clear a priori that π ∗ will be one-to-one in a separating equilibrium. We shall follow the following approach. Instead of considering the original game, where the seller chooses a reserve price r, and has utility U(θ, ϕ, r), b (θ, ϕ, π). This we consider the game where the seller chooses π, and has utility U essentially amounts to say that the seller chooses the probability of selling directly. Of course, any separating equilibrium of this auxiliary game corresponds to a separating equilibrium of the original game. The following lemma shows that the converse is true. ¡ ∗ −λId ¢2 Lemma 6 Suppose that (φ∗ , r∗ ) is a separating equilibrium. Then π ∗ = F r 1−λ is strictly increasing. Proof. The proof is very much like that of Corollary 1. We know from Lemma 2 that π ∗ is non-decreasing. Suppose that π∗ (θ) = π∗ (θ0 ) for θ < θ0 . Since (φ∗ , r∗ ) is separating, b with respect to ϕ implies then ϕ∗ (θ0 ) = θ0 > θ = ϕ∗ (θ). The strict monotonicity of U that: b ϕ∗ (θ 0 ), π ∗ (θ0 )) = U b (θ, ϕ∗ (θ0 ), π ∗ (θ)) > U(θ, b ϕ∗ (θ), π ∗ (θ)), U(θ,

which violates the incentive compatibility condition (1).

A key consequence of Lemma 6 is that there is no loss of generality in focusing on the separating equilibria of the auxiliary game. Building on Mailath (1987), we will construct an equilibrium of this game in which π ∗ is one-to-one, and we will show that this equilibrium is differentiable and characterized by an intuitive differential equation. Moreover, we shall show that this equilibrium is unique, and therefore, by Lemma 6, that the informed seller game has a unique separating equilibrium. b . Note first that the mapping U b satisfies the conditions Regularity Conditions for U (1)-(5) of Mailath (1987), with two minor exceptions. First it is twice continuously b2 is non-negative. Third, U b13 is non-negative. differentiable on [0, 1]2 × R. Second, U b3 (θ, θ, π) = 0 has a unique solution in π, π Fourth, the equation U b(θ, θ), which maximizes b b U (θ, θ, π), and at which U33 (θ, θ, π) < 0. The fifth condition is emptily satisfied since b (θ, θ, π) is strictly concave. The only points of departure from the mapping π 7→ U b2 and U b13 can vanish, respectively for Mailath’s (1987) original condition is that U π = 1 and π = 0. (This follows from the fact that the density f is bounded away from zero on [0, 1].) Special care must thus be taken in applying his results in that respect. Mailath’s (1987) Results. We shall now follow the arguments in Mailath (1987). Hereafter, it is assumed that π ∗ is one-to-one and satisfies the incentive compatibility condition (1). 11

Lemma 7 Suppose that π ∗ is continuous at θ ∈ (0, 1) and that π ∗ (θ) 6= π b(θ, θ). Then, b2 (θ, θ, π ∗ (θ)) π ∗ (θ0 ) − π ∗ (θ) U . = − b3 (θ, θ, π ∗ (θ)) θ0 − θ θ →θ U lim 0

(3)

Proof. This is Proposition 2 in Mailath (1987, Appendix). His proof does not use the b2 and U b13 cannot vanish, and thus is valid in the present setting. fact that U Next, we have the following result.

Lemma 8 Suppose that π ∗ is continuous on an open interval I contained in [0, 1], then π ∗ (θ) 6= π b(θ, θ) for all θ ∈ I.

Proof. We prove first that there is no open interval E of [0, 1] such that π ∗ (θ) = π b(θ, θ) 0 0 ∗ 0 b on E. Otherwise, since θ maximizes U (θ, θ , π (θ )) over θ ∈ E, one would have: db π b2 (θ, θ, π b3 (θ, θ, π U b(θ, θ)) + U b(θ, θ)) (θ, θ) = 0 dθ

b2 (θ, θ, π b3 (θ, θ, π b(θ, θ)) = 0 and U b(θ, θ)) 6= 0 as π b(θ, θ) 6= 1 which is impossible since U for θ ∈ E (note that our regularity assumptions imply that the derivative of π b(θ, θ) is well-defined and continuous). The rest of the proof follows Proposition 3 in Mailath (1987, Appendix). From this, it follows that π ∗ can be discontinuous at at most one point θ0 , and that if it was so, it would be strictly decreasing on one of [0, θ0 ) or (θ0 , 1], and strictly increasing on the other (see Mailath (1987, Proposition 4, Appendix)). But this we know is impossible by Lemma 2. It follows that, if a separating equilibrium exists, π ∗ is continuous on (0, 1), and thus differentiable by Lemma 7. To pin down the equilibrium, we need an initial condition for the differential equation (3), which we rewrite as: dπ ∗ (θ) = dθ

λ (1 − π ∗ (θ)) µ ¶. 1 − F (z ∗ (θ)) ∗ (1 − λ) z (θ) − −θ f(z ∗ (θ))

Let us show that, in a separating equilibrium, an initial condition is provided by π ∗ (0) = π b(0, 0). Lemma 9 In any separating equilibrium, π ∗ (0) = π b(0, 0).

Proof. Suppose that π ∗ is incentive compatible, one-to-one, and π∗ (0) 6= π b(0, 0). If ∗ π b(0, 0) = π (θ), for some θ ∈ (0, 1], then the bidders infer θ when the seller chooses 12

π b(0, 0). One has then:

b θ, π U(0, b(0, 0)) = λθ(1 − π b(0, 0)) + (1 − λ)

Z

1

zb(0,0)

µ

¶ 1 − F (ε) ε− dF (ε)2 f (ε)

µ ¶ 1 − F (ε) ε− > (1 − λ) dF (ε)2 f(ε) zb(0,0) ¶ Z 1 µ 1 − F (ε) ε− > (1 − λ) dF (ε)2 f(ε) ∗ (0) z ∗ b = U (0, 0, z (0)) Z

1

which violates incentive compatibility, a contradiction. The same reasoning shows that if π b(0, 0) ∈ / π ∗ ([0, 1]), a seller of type 0 will always gain from deviating to π b(0, 0), ∗ whatever the beliefs of the bidders following this deviation. Hence π (0) = π b(0, 0), as claimed.

From Mailath (1987, Theorem 2) if π ∗ is one-to-one, satisfies incentive compatibility, and satisfies the initial condition π ∗ (0) = π b(0, 0), then it satisfies the differential ∗ equation (3) on (0, 1), and furthermore, π is continuous on [0, 1]. We are thus led to study the initial value problem: dπ(θ)2 = dθ

λ (1 − π(θ))) ¶ 1 − F (z(θ)) (1 − λ) z(θ) − −θ f (z(θ)) 1 − F (z(0)) 0 = z(0) − f(z(0)) 2 π(θ) = F (z(θ)) µ

(4)

and we seek a solution such that dπ (θ) > 0 on (0, 1). An argument parallel to Mailath dθ (1987, Proposition 5) implies that such a solution exists, and is unique. Call it π S . What must be checked is that π S (θ) < 1 for all θ < 1, and that π S (1) = 1. (Mailath b2 > 0.) This is done in the following does not have this problem, since he assume that U lemma. Lemma 10 Let π S be the solution to (4). Then π S (θ) < 1 for all θ < 1, and π S (1) = 1. Proof. Rewrite the differential equation (3) as: ¢¢ d ¡ ¡ ln 1 − F (z S (θ))2 = dθ

−λ µ ¶. 1 − F (z S (θ)) S (1 − λ) z (θ) − −θ f(z S (θ)) 13

Suppose that limθ0 ↑θ z S (θ0 ) = 1 for θ ∈ (0, 1]. Then, from the above expression, one must have: µ ¶ 1 − F (z S (θ0 )) 0 S 0 z (θ ) − lim − θ = 0, f(z S (θ0 )) θ0 ↑θ or θ = 1. This implies the first assertion. Now, suppose that there exists θ ∈ (0, 1) such that z S (θ) = zb(θ, θ), and take the smallest such θ. Then we have z S (θ 0 ) > zb(θ0 , θ0 ) for all θ0 < θ and: ¯ dF (z S (ϑ))2 ¯¯ lim ¯ 0 = +∞ dϑ θ0 ↑θ ϑ=θ

as z S (θ) < 1, a contradiction. Thus for all θ ∈ (0, 1), z S (θ) > zb(θ, θ). Since zb(1, 1) = 1 and z S (θ) ≤ 1 for all θ, it follows that z S (1) = 1. Finally, one must prove that the solution π S to (4) is indeed incentive compatible. The proof of this result makes use of the single crossing condition, which states that: µ µ ¶¶ 1 − F (z) −z θ − λϕ + (1 − λ) b3 (θ, ϕ, π) U f(z) = b2 (θ, ϕ, π) λ (1 − π) U

is strictly monotone with respect to θ, provided that π ∈ (0, 1). (Again, this follows from the assumption that the density f is bounded away from zero on the interval [0, 1].) It should be noted that the single crossing condition holds globally, and not only in equilibrium. Lemma 11 Let π S be the solution to (4). Then π S is incentive compatible.

Proof. The proof follows closely Mailath (1987, Theorem 3). Suppose that π S (θ) ∈ / b (θ, (π S )−1 (ζ), ζ), and let ζ 0 be an optimal choice. Suppose first that arg maxζ∈π∗ ([0,1]) U b2 (θ, θ 0 , ζ 0 ) > 0 by Lemma 10. The first order condition, (π S )−1 (ζ 0 ) = θ0 ∈ (0, 1). Then U together with the differential equation (3) yields: b3 (θ, θ0 , ζ 0 ) b3 (θ 0 , θ0 , ζ 0 ) U U d(π S )−1 0 (ζ ) = = − b2 (θ, θ0 , ζ 0 ) b2 (θ 0 , θ0 , ζ 0 ) dζ U U

which violates the single crossing condition, as ζ 0 ∈ (0, 1). Suppose now that (π S )−1 (ζ 0 ) = 0, so ζ 0 = π b(0, 0). Then θ 6= 0 as π S (0) = π b(0, 0) is an optimal choice for θ = 0 by Lemma 9. The first order condition can be written as: S −1

d(π ) b2 (θ, 0, π U b(0, 0)) dζ

b3 (θ, 0, π (b π (0, 0)) + U b(0, 0)) ≤ 0, 14

b3 (θ, 0, π which simplifies to U b(0, 0)) ≤ 0 given the behavior of π S at zero. This provides a contradiction since: b3 (θ, 0, π U b(0, 0)) = θ > 0

by definition of π b(0, 0). Suppose now that (π S )−1 (ζ 0 ) = 1, so ζ 0 = 1. Then θ 6= 1 as b (θ, 1, 1) = θ, π S (1) = 1 is an optimal choice for θ = 1. The utility of type θ is then U S b (θ, θ, π (θ)) > θ by not deviating, a contradiction. Hence π S whereas he would get U is incentive compatible, as claimed. Taken together, Lemmas 6-11 imply the following result.

Proposition 3 The informed seller game has a unique separating equilibrium, that is characterized by the solution (π S , z S ) of (4). In this equilibrium, the probability of sale is strictly less than the probability of sale that would arise if the seller could credibly commit to reveal the true value of θ to the bidders, unless θ ∈ {0, 1}. A key feature of the separating equilibrium is that it never arises that the good is not put for sale, i.e., that the probability of a sale is zero, unless of course θ = 1. Things are different if the good is sold through a monopoly broker, as we will later see. 3.4. More General Mechanisms So far, we have assumed a particular game form for the informed seller game, namely that the seller designs a second-price auction with announced reserve price. A priori, the seller could use much more general mechanisms. The most general class of mechanisms that the seller could use are Maskin-Tirole (1992) mechanisms, in which the seller would propose a grand mechanism in which he would have to report his type as well as the agents. We shall not consider these kinds of mechanisms here, but rather focus on the situation in which the seller can publicly propose a revelation mechanism in which only the agents can make reports. Formally, a contract is a pair of mappings {xn (ε1 , ε2 ), tn (ε1 , ε2 )}(ε1 ,ε2 )∈[0,1] , where xn refers to the probability that buyer n receives the good, and tn is the expected transfer to the seller, both conditional on the realization of the valuation vector (e ε1 , e ε2 ). We then have the following proposition. Proposition 3 For any equilibrium of the contract offering game, there exists an equilibrium of the game where the seller offers a second-price auction with announced reserve price that gives the seller the same expected revenue.

Proof. By the Revelation Principle, there is no loss of generality in focusing on the case where the seller offers incentive compatible revelation mechanisms to the buyers. The Revenue Equivalence Theorem (Riley and Samuelson (1981)) holds, because given 15

their common beliefs about θ, the buyers’ private information is independently and identically distributed. Hence, for any incentive compatible direct revelation mechanism, there is a second-price auction with announced reserve price that achieves the same revenue for the seller, given the buyers’ common beliefs about θ. Fixing the buyers’ beliefs as in the contract offering game, it then follows that the incentive constraint of the seller is the same in both games. The result follows. One consequence of this result is that, provided the seller is restricted to use auction mechanisms, focusing on second-price auctions is without loss of generality, as far as the utility of the seller is concerned. 4. Trading with a Monopoly Broker Let us now consider the situation where a monopoly broker, uninformed about e θ and e εi , i = 1, 2, designs the selling mechanism, as in Myerson and Satterthwaite (1983). By the revelation principle, there is no loss in generality in focusing on direct revelation mechanisms {π(V ), t(V )}V ∈[0,1]3 for the seller and {π n (V ), tn (V )}V ∈[0,1]3 , n = 1, 2, for the buyers, where the π component refers to the probability that the agent receives the good, and the t component to the expected transfer to the broker, both conditional on the realization of the valuation vector Ve = (e θ, e ε1 , e ε2 ). The interim utility of the seller is: h i U(θ) = E π(Ve )θ − t(Ve ) | e θ=θ , and that of buyer n = 1, 2 is:

h i θn − tn (Ve ) | e εn = εn . Un (εn ) = E π n (Ve )e

The incentive compatibility conditions imply that U and Un , n = 1, 2, are almost everywhere differentiable and their derivatives are given by the usual envelope conditions. The broker’s maximization program is standard: # " 2 2 X X max E π(Ve )e θ+ π n (Ve )e θ n − U (e θ) − Un (e εn ) , n=1

n=1

under the incentive compatibility and rationality constraints for the seller: h i dU (θ) = E π(Ve ) | e θ=θ dθ U(θ) ≥ θ,

16

the incentive compatibility and rationality constraints for buyer n = 1, 2: h i dUn e (εn ) = (1 − λ)E π n (V ) | e εn = εn dεn Un (0) = 0, and the feasibility constraint: π(V ) +

2 X n=1

π n (V ) ≤ 1.

Note that since dU (θ) ≤ 1, the type-dependent participation constraint for the seller dθ can be replaced by U(1) = 1. Standard computations imply that: " # h i e G( θ) E U(e θ) = 1 − E π(Ve ) , e g(θ) and that for each n = 1, 2,

·

¸ 1 − F (e εn ) e π n (V ) . E [Un (e εn )] = (1 − λ)E f(e εn )

The broker’s problem becomes thus to maximize: " à ! µ µ ¶¶# 2 X e G( θ) 1 − F (e ε ) n θ+ E π(Ve ) e + π n (Ve ) λe θ + (1 − λ) e εn − e f (e ε ) n g(θ) n=1 under the feasibility constraint. Thus the good is sold if: ½ ¾ G(e θ) 1 − F (e εn ) εn − >e θ+ max e . n∈{1,2} f(e εn ) (1 − λ)g(e θ) In particular, define θmax < 1 as the solution of θ+

G(θ) = 1. (1 − λ)g(θ)

Then the good is not sold if e θ > θmax . This implies that selling the good through a monopoly broker imposes an inefficiency that is not present if the good is sold directly by the seller, and the equilibrium of the informed seller game is the separating equilibrium described in Proposition 3, namely that high quality goods are not put for sale. Correspondingly, sellers with e θ > θmax would be better off selling the good themselves 17

than through a monopoly broker. For each θ < θ max , define z A (θ) and π A (θ) as the solutions to: z A (θ) −

G(θ) 1 − F (z A (θ)) = θ+ ; A f (z (θ)) (1 − λ)g(θ) π A (θ) = F (z A (θ))2 .

Then, the probability of sale, conditional on e θ = θ, is given by 1 − π A (θ).

Implementation. A first question is whether the optimal allocation selected by the broker can be implemented by a simple mechanism. Denote rA (θ) the reserve price solution of: rA (θ) = λθ + (1 − λ)z A (θ). Then it is clear that if the buyers know the common component θ, a second price sealed auction with reserve price rA (θ) implements the monopoly broker’s allocation. Notice that from the monotone hazard rate property, the reserve price rA (θ) is increasing in θ. Thus if the sellers is induced to choose a public reserve price rA (θ) when his valuation is θ and a second-price sealed bid auction is organized, the choice of the reserve price will reveal the common value component θ, and the allocation will be implemented. This leads to the following simple mechanism. Define B A (r) on rA ([0, 1]) by the relation: b (θ, θ, π A (θ)) − U(θ), B A (rA (θ)) = U

and consider the following mechanism.

1. The broker sets an expected transaction fee conditional on the reserve price B A (r),which for our purpose can take the form of a fixed registration fee to be paid by the seller in order to participate, but could also obtain with an ex-post transaction fee. 2. The seller announces a public reserve price r. 3. The buyers observe r and B A , and are engaged in a second-price sealed bid auction with reserve price r (or any other equivalent auction under private values), in which no fees are paid to the broker. Then it is immediate to verify that incentive compatibility of the broker’s allocation implies that there is a separating equilibrium of the signaling game in which the seller chooses the reserve price rA (θ) and receive the interim expected utility U(θ), the good is sold with probability 1 − π A (θ), and buyer n’s interim expected utility is Un(εn ). 18

Thus the allocation can be implemented with a simple auction mechanism in which the fees charged to the seller are contingent on the reserve price chosen. An important consequence of this result is that it shows that the comparison between the broker’s solution and the informed seller’s solution will not be driven by the fact that the broker has access to a larger set of mechanisms. Indeed the solution is the same if the broker is restricted to use only second-price sealed bid auctions with announced reserve price. Thus the comparison will only be driven that the fact that the broker can tie the seller’s fee to the reserve price and use that as a screening device. 5. Comparison with the Separating Equilibrium The issue addressed her is whether, despite the distortion due to market power, the monopoly broker can generate a more efficient allocation than a decentralized market structure in which the seller chooses freely the selling mechanism and the reserve price. In doing so, we focus on the separating equilibrium of the signaling game. Typically, π A will be differentiable at zero, in contrast with π S . This implies that for low values of e θ, the allocation implemented by the monopoly broker is more efficient than the allocation in the separating equilibrium of the informed seller game. This result depends crucially on the assumption of common values, i.e., λ > 0. If λ = 0, then we are back to the private values case. It is easy to see that all the characterizations obtained so far generalize to this limit case. In that case, one has π S (θ) = π b(θ, θ) < π A (θ) and the monopoly broker reduces the probability of sale in two ways: some goods are not put for sale (if e θ > θmax ), and the reserve price is higher than if the seller sells the good himself. Thus in the private values case, having a monopoly broker is always less efficient than letting the seller sell the good himself, and the revenue to the seller is also less. To allow the comparison we plot in the next graph the two functions πA and πS . π

6

1 πS π b(0, 0)

πA

-

θmax

1

θ

Figure 1: Comparison of π A and πS 19

As we can see the monopoly broker tends to increase the probability of sale for low values but to reduce it for high values. Typically, to raises its own revenue, the broker must prevent the seller to overstate the value θ. This calls for larger distortions when the sellers claims a high value. Thus at the interim stage, no situation dominates unambiguously. We now turn to the ex-ante stage. Expected Probability of Sale. Suppose to simplify notation and computations that both e εi , i = 1, 2, are uniformly distributed over [0, 1]. Our objective is to compare the expected probability of selling the good in the separating equilibrium of the informed seller game, ΠS , with the expected probability of selling the good in the monopoly broker mechanism, ΠA . In the latter case, one has: h i h i A A e e Π = 1 − E π(V ) = 1 − E π (θ) . Using the fact that both e εi , i = 1, 2, are uniformly distributed over [0, 1], one finds that: ½ ¾¶2 Z 1 µ G(θ) 1 A 1 + min 1, θ + Π =1− g(θ) dθ. (1 − λ)g(θ) 0 4 In the informed seller game, the average probability of selling the good is, in the separating equilibrium: Z 1 S Π = 1− z S (θ)2 g(θ) dθ. 0

A key observation is that z S does not depend on the distribution G, as is usual with a separating equilibrium. We shall show that if G is sufficiently concentrated around some θ0 ∈ (0, 1), then ΠA > ΠS . For this, we introduce the following definition. Definition 1 Let ε ∈ (0, 1) and suppose that θ0 ∈ (ε, 1−ε). Then G is ε−concentrated at θ 0 if: G(θ 0 + ε) − G(θ 0 − ε) ≥ 1 − ε. We then have the following proposition. Proposition 5 For any θ0 ∈ (0, 1) there exists ε0 ∈ (0, 1) such that for all ε ∈ (0, ε0 ), if G is ε−concentrated at θ0 , then ΠA > ΠS . Proof. We first have the following lemma.

20

Lemma 12 Suppose that G is ε−concentrated at θ 0 . Then: Z 1 1 1− (1 + θ)2 g(θ) dθ − ΠA ≡ ∆Π ≤ O(ε). 4 0 Proof. Using the expression for ΠA , one has: ! ½ ¾¶2 Z 1 õ 1 G(θ) 2 ∆Π = 1 + min 1, θ + − (1 + θ) g(θ) dθ (1 − λ)g(θ) 0 4 ½ ¾¶ µ ½ ¾ ¶ Z 1 µ 1 G(θ) G(θ) = 2 + θ + min 1, θ + min 1, θ + − θ g(θ) dθ (1 − λ)g(θ) (1 − λ)g(θ) 0 4 ½ ¾ ¶ Z 1µ G(θ) min 1, θ + ≤ − θ g(θ) dθ (1 − λ)g(θ) 0 Z θ0 +ε Z θ0 −ε Z 1 G(θ) g(θ) dθ + dθ + g(θ) dθ ≤ 0 θ0 −ε 1 − λ θ0 +ε 2ε ≤ε+ , 1−λ which implies the result. To complete the proof, suppose that G is ε−concentrated at θ0 . Then, by Lemma 12, Z 1 Z 1 1 2 S A (1 + θ) g(θ) dθ − z S (θ)2 g(θ) dθ + O(ε) . Π −Π ≤ 4 0 0 By Proposition 3, z S (θ) > (1 + θ)/2 for all θ ∈ (0, 1). Hence: S

A

Π −Π ≤

Z

θ0 +ε

1 (1 + θ)2 g(θ) dθ − 4

Z

θ0 +ε

z S (θ)2 g(θ) dθ + O(ε)

θ0 −ε µθ0 −ε ¶ 1 2 S 2 ≤ (1 + θ 0 + ε) − z (θ 0 − ε) (G(θ0 + ε) − G(θ0 − ε)) + O(ε) 4 ¶ µ 1 2 S 2 ≤ (1 + θ 0 + ε) − z (θ 0 − ε) (1 − ε) + O(ε) 4 0, there exists distributions of the seller’s type such that the ex-ante probability of sale is higher with the broker than in the separating equilibrium of the informed seller game. Ex-Ante Efficiency. Suppose again to simplify notation and computations that both e εi , i = 1, 2, are uniformly distributed over [0, 1]. Our objective is to compare the ex-ante expected total surplus in the separating equilibrium of the informed seller game, ΣS , with the expected total surplus in the monopoly broker mechanism, ΣA . In the latter case, one has: " # 2 X ΣA = E π(Ve )e θ+ π n (Ve )e εn . n=1

Using the fact that both e εi , i = 1, 2, are uniformly distributed over [0, 1], one finds that:   Z 1  Z 1  h i A e ¶ ϑ(ϑ − θ) dϑ g(θ) dθ. Σ = E θ + 2(1 − λ) max 0, µ G(θ) 1   0 1+θ+ (1−λ)g(θ)

2

In the informed seller game, the expected surplus is, in the separating equilibrium: Z 1Z 1 h i S e Σ = E θ + 2(1 − λ) ϑ(ϑ − θ) dϑ g(θ) dθ. 0

z S (θ)

We shall show that if G is sufficiently concentrated around some θ0 ∈ (0, 1), then ΣA > ΣS . Proposition 6 For any θ0 ∈ (0, 1) there exists ε0 ∈ (0, 1) such that for all ε ∈ (0, ε0 ), if G is ε−concentrated at θ0 , then ΣA > ΣS . Proof. First, we have the following lemma. Lemma 13 Suppose that G is ε−concentrated at θ 0 . Then: Z 1Z 1 h i e E θ + 2(1 − λ) ϑ(ϑ − θ) dϑ g(θ) dθ − ΣA ≡ ∆Σ ≤ O(ε). 0

1 (1+θ) 2

22

Proof. Using the expression for ΣA , one has:    Z 1 Z 1  Z 1   ¶ ϑ(ϑ − θ) dϑ  g(θ) dθ ϑ(ϑ − θ) dϑ − max 0, µ ∆Σ = 2 (1 − λ) G(θ) 1 1   (1+θ) 1+θ+ 0 2 2 (1−λ)g(θ)   ¶ µ Z 1 Z 1 1+θ+ G(θ) Z 1  2 (1−λ)g(θ) = 2 (1 − λ) min ϑ(ϑ − θ) dϑ, ϑ(ϑ − θ) dϑ g(θ) dθ 1  1 (1+θ)  (1+θ) 0 2 2 µZ θ0 −ε ¶ Z 1 Z θ0 +ε 2 G(θ) = 2 (1 − λ) dθ + g(θ) dθ + C g(θ) dθ 1−λ 0 θ0 −ε θ0 −ε ¢ ¡ ≤ 2 1 + C 2 − λ ε, where:

C = max

θ∈[0,1]

1³ 1+θ+ 2

G(θ) (1−λ)g(θ)

´

=1+

1 2(1−λ)g(1)

by the monotone hazard rate property, which implies the result. Now suppose that G is ε−concentrated at θ0 . Then, by Lemma 13, Z 1 Z zS (θ) S A Σ − Σ ≤ −2 (1 − λ) ϑ(ϑ − θ) dϑ g(θ) dθ + O(ε) . 0

1 (1+θ) 2

By Proposition 3, z S (θ) > (1 + θ)/2 for all θ ∈ (0, 1). Hence: Z θ0 +ε Z zS (θ) S A Σ − Σ ≤ −2 (1 − λ) ϑ(ϑ − θ) dϑ g(θ) dθ + O(ε) ≤ ≤ ≤