Signaling in an English Auction: Ex ante versus Interim Analysis Peyman Khezr School of Economics University of Sydney and Abhijit Sengupta School of Economics University of Sydney Abstract This paper studies and compares some commonly observed selling mechanisms for a seller of an item who has private information that is payoffrelevant to prospective buyers but where the seller is unable to credibly reveal her information to the buyers at no cost. We first study an English auction format with two different reserve price regimes, and a posted-price mechanism to compare the signaling games from the ex ante point of view. We show in this environment posted price is dominated by an ascending auction with a disclosed reserve price. In the second part of the paper we introduce a designed problem with an informed seller, where the seller observes her signal before she chooses the mechanism. We restrict our attention to an English auction with two reserve price regimes. Our analysis show at any equilibria all types of the sellers chooses the same mechanism after observing their signal. Thus signaling through a reserve price regime is not possible in this environment. Keywords: Auction, Secret reserve, Reserve price, Signaling, Informed seller. JEL Classification: D44, D80, D82.
1 Introduction The aim of this paper is to study selling mechanisms and environments that are relevant to a wide variety of settings but especially salient in the context of real estate markets. One key feature of the environment to be studied is that an indivisible object is offered for sale by a seller who has private information about its quality and prospective buyers care about the seller s information. Examples of economic settings in which there is uncertainty about the quality of an object for sale and where the seller has superior private information than the prospective buyers are legion. Indeed, Akerlof s classic paper on the lemons problem (Akerlof, 1971) that introduced the problem of asymmetric information in economics was concerned with precisely such a setting. The owner of an object often has information about attributes that affect the quality and the desirability of the object from her experience of owning and using it. Thus, the seller of a lot of wine offered at an auction would typically have private information of the conditions under which the wine was stored. And, of course, the owner of a house would typically have a detailed knowledge of the conditions of the house that has obvious bearing on the valuations of prospective buyers. Technically, in the context of auctions, the environment described is a special case of interdependent valuations introduced in (Milgrom and Weber, 1982), in which the valuations of the bidders may depend on the information of other agents. However, the particular special case in which the interdependence is only through the seller s information provides structure that can be exploited and can be especially relevant in many settings such as auction of wines or residential property. A key mechanism to be investigated is one where the object is sold in an ascending auction with no declared reserve price but the seller reserves the right to accept the highest bid or reject it and retain the object. Arguably, this is the most commonly seen auction format in traditional auctions (as opposed to online auctions) and ubiquitous in auctions of art, antiques and wine. Yet this mechanism has not received much attention we are aware of only one study of this mechanism: (Jarman and Sengupta, 2012). The second mechanism is an ascending auction with a disclosed reserve. Finally, for comparison, we will also consider the posted-price mechanism, where the seller simply post a uniform price for the object. The choice of a mechanism to sell an object typically either rests with an institution such as an auction house or with the seller of the object. For art, antiques and wine, established auction houses have, over centuries, designed the rules of the mechanism: for example, an ascending auction with a disclosed reserve or an undisclosed reserve where the seller has the right of refusal. Since an auction house typically gets a fixed share of the revenue generated at the auction, presumably,
the rules are designed with a view to maximizing a seller s expected revenue from an ex ante perspective that is, before the seller learns her information. Of course, at the time a particular seller sets a reserve (whether disclosed or undisclosed), she does so knowing her information. Thus, in an auction designed by auction houses with a disclosed reserve price, the particular value of the reserve acts as a signal but the mechanism itself does not. In contrast, the owner of a house who chooses a mechanism to sell the house does so at the interim stage, after she learns her information. Therefore, this is a design problem with an informed principal. In this paper, we intend to look at both these classes of environments. In the first part of the paper, where the seller observes her private signal after she chooses the reserve price regime, and consequently maximises her ex ante revenue, the main finding is that some types prefer to signal through a reserve price or a posted price, while the others prefer not to disclose any information. Specifically, if the seller observes a signal higher than a threshold, she would be better of by disclosing her information. In the second part of the paper, where the seller observes her signal before she decides to chose a mechanism, we show when there are only two available mechanism here, two type of reserve price regimes the seller of an special type cannot signal her type with the choice of regime, so in the equilibria all types of the sellers choose the same mechanism at the interim stage. Furthermore, the seller s choice of the regime could be different at the ex ante and the interim stages. Thus the time she observers her signal is an important factor for the choice of mechanism(regime). One of the very first studies with a similar model to ours, is Milgrom and Weber (1982). Although the model is more general with affiliation of the buyers signals, but some of their results are useful for our study. They show, when the information is verifiable, the seller s best strategy is to commit to information revelation, which known as Linkage Principle. The main focus of our model is for the case in which the seller s information is not verifiable to the buyers at zero cost. In this case then the seller s best interest could be not to reveal any information. In their environment, the auction price for the English auction cannot be less than the second price and the first price auction (Theorem 11 and 15 Milgrom and Weber (1982)). This result is applicable to our model and it is one of the reasons that we restricted our attention on the open ascending (English) auction throughout this paper. However, according to the assumptions of our model, English auction is strategically equivalent to the second price auction. Then our main concern is to show what level of information revelation is the seller s best interest. The two papers most closely related to this paper are Cai, Riley, and Ye (2007), that studies an auction with a disclosed reserve price and Jarman and Sengupta (2012) that studies an auction with a secret reserve price in an environment similar to this paper. Jarman and Sengupta (2012) characterise the bidding function and 2
the seller s expected revenue in the secret-reserve regime and demonstrate that the seller s ex ante expected revenue can be higher than that in the unique signalling equilibrium characterised in Cai, Riley, and Ye (2007). We discuss and present their results for our analysis. The main difference of our work to Jarman and Sengupta (2012) is the analysis at the interim stage, where the seller observes her signal before she decides to choose a regime. This could change the seller s optimal choice. 2 The Model A seller who has an indivisible object to sell, faces a set of N = {1,..., n} potential buyers n > 1. The seller observes a signal s which is not observable by the buyers, drawn from a known distribution F 0 on [0, s], twice differentiable with a continuous density f 0. Each buyer i has a private signal x i for the object independently and identically distributed on F [0, x], twice differentiable with continuous density f. Buyers valuation v : [0, x] [0, s] R + are symmetric, continuous and increasing functions of their individual signals as well as the seller s signal. Consequently, in this environment, buyers not only care about their own signals but also care about the seller s signal in the same manner. Seller s own value function for the object v 0 : [0, s] R + is a continuous and increasing function of her own signal. There are two selling mechanisms available to the seller, posted-price and ascending auction. Throughout this paper, we consider two candidate reserve price regimes for the open ascending auction. The first is to disclose the reserve price at the beginning of the auction before bidding starts, or the disclosed-reserve regime (DR). The second is to never disclose the reserve but retain the right to accept or reject the highest bid, or the right-of-refusal regime (RR). The posted-price (PP) mechanism is another choice of the seller, where the seller post a uniform price for the object, and prospective buyers who come randomly to the seller, have the choice to accept or reject this offer. The first buyer who accepts the price would own the object. 3 Ex ante Analysis 3.1 Disclose the reserve (DR) Under this regime, the reserve price would be announced at the beginning of the auction and before bidding starts. The stages of the game are as follows. First, the seller observes her private signal. Then she publicly announces a reserve 3
price r(s), as an increasing function of her private signal. After this point the mechanism is an open ascending auction with a reserve price r(s). For this regime most of the results in Cai, Riley, and Ye (2007) are directly applicable to our analysis. They show the single crossing condition holds, so signaling is possible in this situation. Thus the seller can use the reserve price to signal her type to the potential buyers. First, there is a critical assumption for their result as follows; Assumption 1. As in (Cai, Riley, and Ye, 2007), we assume that for any s, J(s, x) = v(s, x) v(s, x) F (2) (x) F (1) (x) x f (1) (x) (1) is strictly increasing in x. This assumption is the generalisaion of the assumption in Myerson (1981), in the context of independent private valuation, that virtual valuation is strictly increasing in x. Starting from the bidding function, if the seller discloses her reserve price before the bidding starts, then according to Milgrom and Weber (1982), it is a Bayesian Nash equilibrium for each buyer i to bid v(x i, ŝ) = E[V i S = ŝ, x i = x i (1)], which is her expected value, given the belief that the seller s type is ŝ, and her own signal is the highest among other bidders. Having the bidding function, each bidder enters the auction if her expected valuation is greater than the reserve price. In this situation the reserve price is a potential signal given the fact that a higher reserve increases the probability of no sale Cai, Riley, and Ye (2007). Define m(s) as the minimum buyer type who enters the auction given the belief that the seller s signal is ŝ, then r = v(m(s), s) is the reserve price which is equal to the expected value of the lowest buyer type who enters the auction. Having the reserve price and the bidding function, a seller with signal s who reports type ŝ would have the interim expected payoffs equal to [ ( ) ( ) ] ( U DR (s, ŝ, m(ŝ)) = F (2) m(ŝ) F(1) m(ŝ) v(m(ŝ), ŝ) v0 (s) ) + ω m(ŝ) [v(x, ŝ) v 0 (s)]f (2) (x)dx (2) where F (1) (m(s)) is the probability that the highest signal is less than m(s), which means that nobody has an expected value higher than the reserve price. F (2) (m(s)) F (1) (m(s)) is the probability of only one(highest) bidder with an expected value higher than the reserve price. 4
If there was full information then s was directly observable, so it becomes ŝ = s. Then seller would choose m to maximize U DR (s, s, m). Let m (s) be the optimal full information minimum type. Then according to U and assumption(1), we m have 0 if V 0 (s) < J(s, 0) m (s) = J 1 (V 0 (s)) if J(s, 0) V 0 (s) < J(s, x) x if V 0 (s) J(s, x) where J 1 is the inverse of J. By Theorem 1 in Cai, Riley, and Ye (2007), the following differential equation characterizes the unique separating equilibrium of the signaling game. (3) s (m) = D 3U DR (s, s, m(s)) D 2 U DR (s, s, m(s)) (4) To find the ex ante expected profit to the seller, we first need to find the changes in U DR when s changes, which is 1 [D 1 U DR (s, s, m(s))] = D 2 U DR + (D 3 U DR m (s)) v 0(s)[1 F 1 (m(s))] = v 0(s)[1 F 1 (m(s))] < 0 (5) By the envelope theorem the first line of (5) becomes zero. By Fundamental Theorem of Calculus after getting expectation and rearranging the integrals, the ex ante expected profit to the seller in this regime becomes E s [U DR (s, s, m(s))] = U D (0, 0, m(0)) 3.2 Right of Refusal (RR) s 0 [1 G(s)][1 F 1 (m(s))]v 0(s)ds (6) Under this regime, the game has two stages. First, bidders bid in an ascending auction with no declared reserve price. Second, after the bidding finishes, the seller observes the highest bid and simply accept or reject it. If she accept, the highest bidder wins the object and pays her bid, otherwise the seller retains the object. Jarman and Sengupta (2012) study the bidding behavior under this regime. It 1 Given a function g : A R, where A R n, we write D i g(x 1,..., x i,..., x n ) to respectively the partial derivative of g with respect to its i-th argument evaluated at the point (x 1,..., x n ). 5
is straight forward to say, given the auction price p at the end of the ascending auction, the seller s optimal decision is to accept p if and only if it is greater or equal to her value. Define w(x i, s 0 ) as the expected value of a bidder i give that the seller signal is less than s 0. Given the other players play β R R the optimal strategy for player i is to stay active until her the price equal to her expected value conditional on being accepted by the seller. If w(x i, s) < v 0 ( s) then the bidders bid their conditional expected value, otherwise (w(x i, s) > v 0 ( s)) they bid their unconditional expected value. Following Jarman and Sengupta (2012) proposition 1, the bidding function is as follows; { p = w(x i, v0 1 (b)) if w(x, s) < v 0 ( s) β RR (x i ) = (7) w(x i, s) if Otherwise Let ˆm(s) be the minimum type with the expected value higher than the seller s valuation. Since the seller accepts any offer greater than her valuation, we can define ˆm(s) as follows; { inf{x : β RR (x) v 0 (s)} if s v0 1 (β RR ( s)) ˆm(s) = (8) s if s > v0 1 (β RR ( s)) If there is any bid higher than the seller s value, ˆm(s) shows the lowest bidder s signal with an expected value higher than the seller s value, otherwise the highest bid would not be accepted by the seller, simply because it is lower than her value. We can now derive the interim expected payoff to the seller at this regime; U RR (s, ˆm(s)) = ω ˆm(s) [β RR (x) v 0 (s)]f (2) (x)dx. (9) Now to achieve the ex ante expected payoff to the seller, we need to find out how the interim profit changes when s changes. Derivative of (9) with respect to s is [D 1 U RR (s, ˆm(s))] = [β RR ( ˆm(s)) v 0 (s)]f 2 ( ˆm(s)) ˆm (s) v 0(s)[1 F 2 ( ˆm( ω)) + F 2 ( ˆm( ω)) F 2 ( ˆm(s))] = v 0(s)[1 F 2 ( ˆm(s))] < 0 (10) By the definition of ˆm(s) the first line becomes zero. Thus the higher the seller s signal would result in the lower expected profit to the seller. According to the Fundamental Theorem of Calculus and using the result in (10) we can calculate the seller s ex ante expected profit 6
U RR (s, ˆm(s)) = U RR (0, ˆm(0)) s 0 [1 F 2 ( ˆm(x))]v 0(x)dx (11) Taking expectation from (11) over s and rearranging the integrals, we have E[U RR (s, ˆm(s))] = U RR (0, ˆm(0)) s 0 [1 G(s)][1 F 2 ( ˆm(s))]v 0(s)ds (12) This is the ex ante expected payoff to the seller for the right of refusal regime. 3.3 Posted-Price In this section, we consider a setting in which the seller decides to choose a posted price p for selling her object to N potential buyers. Buyers arrive randomly to the seller. Each buyer accepts the offer if p is less than her expected valuation, otherwise she declines to buy. The first buyer who accepts p could own the object. Since p has to be greater or equal to the seller s valuation, we have s v0 1 (p). Thus each buyer s expected value for the object with respect to the realization of the seller s signal is v( s, x i ) = E[v i s = s, s v0 1 (p)]. According to buyers valuation, only buyers with valuation v( s, x i ) p are willing to buy. Let m(s) be the minimum buyer s type who is willing to buy given that the seller s signal is s. Then the minimum expected value of the buyer who is willing to buy becomes v( s, m(s)). Proposition 3.1. In equilibrium seller posts a price equal to the expected value of the minimum buyer who is willing to buy which is p(s) = v(s, m(s)). Proof. See appendix. We should mention that assumption (1) needs to be hold here as well, and since this is also a signaling game the differential equation in (4) characterizes the unique separating equilibrium of this game. Having the equilibrium posted price, we can calculate the interim expected payoff to the seller. U pp (s, s, m(s)) = (v( s, m(s)) v 0 (s))(1 F 1 ( m)) (13) F 1 ( m) is the probability that all buyers expected valuations are less than p, and if at least one buyer has an expected value higher than p, she will buy the object at the posted price. 7
Differentiating this expected payoff with respect to s would result to D 1 U pp (s, s, m(s)) = (D 3 U P P )( m (s)) v 0(s)(1 F 1 ( m)) (14) By (14) and the Fundamental Theorem of Calculus we can find the ex ante expected payoffs to the seller which is E[U pp (s, s, m(s))] = U pp (0, 0, m(s)) 3.4 Example s 0 (1 G(s))(1 F 1 ( m(s))v 0(s)ds (15) As an example, we suppose that the valuations of the seller and the prospective buyers are linear functions of the signals. The seller s valuation is a linear function of her signal V 0 (s) = γs for γ > 0. Buyers are symmetric and their valuations are also a simple linear function of their own signal and the seller s signal: v(s, x i ) = s+x i. Suppose all signals are independent and distributed uniformly on [0, 1]. Now we can calculate and compare the seller s payoffs from each mechanism described above. 3.4.1 Disclose the Reserve (DR) In this regime, seller discloses her reserve price at the beginning of the auction upon observing her signal. As we mentioned before, this announcement reveals seller s private information to the bidders and ŝ [0, s] is the common information which bidders use for forming their bids, plus their own signals. According to our example we can rewrite the seller s expected payoff as follows U DR (s, ŝ, m) =γs(f (1) (m) 1) + ŝ(1 F (1) (m)) + m(f (2) (m) F (1) (m)) + 1 m xdf (2) (x) (16) and J(.) becomes equal to J(x) = x (1 F (x)) f(x) (17) For our example which is the uniform case, it is equal to J(x) = 2x 1 which is strictly increasing in x. 8
If we assume m (s) is the optimal reserve price for the case of complete information, then equation (6) becomes 0 if (γ 1)s < J(0) m 1 (s) = ((γ 1)s + 1) if J(0) (γ 1)s < J(1) 2 1 if (γ 1)s J(1) Since γ is greater or equal to zero and s [0, 1], then m (s) = 1 ((γ 1)s + 1). 2 To calculate the minimum buyer type we need to solve the differential equation from (4) which is as follows s(m) = (1 F (1) (m)) γ 1 [ m m ] f (1) (x)(1 F (1) (x)) γ J(x)dx According to (Cai, Riley, and Ye, 2007) for every 0 < γ 1 this is a solution for the separating equilibrium. For a given γ we can solve this differential equation and use the result to calculate seller s expected payoff. When N = 2 it is s(m) = (1 m 2 ) γ 1 1 1 2 (18) (19) (4x 2 2x) dx (20) (1 x 2 ) γ Substitute the result into the seller s expected payoff gives us the following equation U DR (s, s, m) = 1 3.4.2 Right of refusal (RR) m 2(s + x γs)(1 x)dx + (2m 2m 2 )(m + s γs) (21) For the right of refusal regime, there is no extra information available to the bidders, because there is no announced reserve price. According to the bidding function for this regime we have β RR (x) = { 2γ 2γ 1 if x γ 1 2 x + 1 2 if x γ 1 2 (22) Calculation of the minimum buyer type who enter the auction is much more straight forward in this case. To solve ˆm(s) numerically we can use (8), start with a given s in the interval and calculate the minimum buyers type for a given γ. After calculation of the ˆm(s) we can substitute the result into the seller s expected payoff which is the following equation, 9
U RR (s, ˆm(s)) = 1 ˆm [β RR (x) γs](2 2x)dx (23) The main difference here is the biding function which can be either conditional or unconditional. So after calculation of ˆm for a given γ we need to find the related biding function and then substitute it to the seller s expected payoff equation. There could be a case in which both of the bidding functions are relevant so the expected payoff is going to be two different integrals. 3.4.3 Posted-price Since seller s valuation is equal to v 0 (s) = γs, she is willing to sell if and only if γs p. From the buyers point of view, after seller announces the posted-price their expected value for the object would become v( s, x i ) = E[V i s = s, s p γ ]. According to the buyers expected valuations, only buyers with expected value v( s, x i ) p are willing to buy. From the previous section we know that in equilibrium seller post a price equal to the expected value of the minimum type buyer who is willing to buy, that is s + m(s). To calculate the minimum buyer s type we need to calculate the differential equation in (4) by differentiating the seller s payoff for the posted-price with respect to s and m. D 2 U P P = 1 F (1) (m) (24) where D 3 U P P = (γs s J(m))f (1) (m) (25) J(m) = m 1 F (1)(m) f (1) (m) Now the differential equation becomes equal to s (m) = (γs s J(m))f (1) (m) 1 F (1) (m) Solving this differential equation would result to [ m ] s(m) = (1 F (1) (m)) γ 1 f (1) (x)(1 F (1) (x)) γ J(x)dx m (26) (27) For this example m(s) = 1. We can solve the integral in (27) numerically for a 2 given γ to find the value of s. If we assume n = 2 then the seller s expected payoff according to (13) is 10
U pp (s, s, m(s)) = (s + m γs)(1 m 2 ) (28) 3.4.4 Payoff comparison In this section we are going to compare expected payoffs to the seller for each of the mechanisms above. In the signaling games it is not possible to find an analytic solution for s(m) in general. So we first fix any γ, then start with the smallest m in the interval and solve for s(m). After finding a numerical solution for s(m) we can find the expected payoff to the seller. Following graph shows the interim expected payoffs for each mechanism when γ = 0.33, γ = 1. Figure 1: Interim payoff γ = 0.33, and, γ = 1 As we can see when γ = 0.33, the right of refusal(top curve)dominates other two mechanisms, but when γ increases to one, then it is possible that disclosing the reserve price and posted-price dominate the right of refusal if the seller s signal is higher than 0.6. However, in the static model, auction with disclosed reserve price always dominates the posted price. 4 Interim Analysis Throughout the first part of the paper, we assumed the seller observes her signal after she chooses the selling mechanism, thus the choice of mechanism itself could not reveal any information to the buyers. Now we change our assumptions by assuming that the seller observes her signal at the first stage before choosing any mechanism to sell her option. In this section we no longer consider the posted price mechanism as an option to the seller, because in (??) we prove it is dominated by 11
the auction with a disclosed reserve price. From now on we restrict our attention into two extreme case of information revelation in an English auction. 4.1 Mechanisms and the Reserve Price Regimes We consider a seller who is willing to sell her object through an English (ascending) auction with two different reserve price regimes. The steps are as follows: First, the seller observes her signal, then she chooses between two reserve price regimes. One is to disclose the reserve price at the beginning of the auction and before the bidding starts, the other is to keep the reserve price secret forever and reveal no extra information to the buyers. If she decides to disclose the reserve price, she has to commit to it, meaning that she is accepting any highest bid which is higher than the disclosed reserve price. Otherwise, if she chooses not to disclose the reserve price, then it is her choice to accept or reject the highest bid at the end of the auction. After she chooses the reserve price regime and publicly announce it, the bidding starts. If the reserve price is disclosed then the mechanism is an open ascending auction with a reserve price. If the reserve price is kept secret then the mechanism is an ascending auction with the right of refusal to the seller at the end of the bidding, That is, after the bidding finishes the seller decides to accept the highest bidder s bid or reject it. 5 A Motivating Example As an example we consider a case in which the seller s valuation for the object is equal to her signal, i.e., v 0 (s) = s and the buyers valuations are symmetric and linear with the following format: v(x i, s) = x i + s. We also assume all signals are distributed uniformly from [0, 1]. Figure 1 shows the interim expected payoffs to both regimes according to equations (2) and (9) when there are only two buyers. 5.1 Interim Equilibrium Analysis In this section we are going to analyse the seller s behavior at the interim level. A seller after observing her signal must choose between two reserve price regimes we explained before. According to Figure 1 the seller knows what would be the payoff for any given signal, but in our model since she has to choose the reserve price regime after observing her signal, the choice of regime itself reveals information to the buyers, that is, the chosen regime must have a higher interim expected profit to the seller. This information could affect the bidding behavior 12
Figure 2: Interim payoffs FD vs RR of the buyers. We continue with the argument which would result to equilibria. At the beginning of the game, suppose a seller observes a signal less than s, then she knows her expected payoff is higher if she chooses not to disclose her reserve price. But buyers also know if a seller chooses to keep her reserve price secret, then her signal must be less than s. Let s focus on the marginal seller who has a signal less than s. She knows at that signal her interim expected payoff is higher if she chooses the secret reserve. But since she is going to choose the reserve price regime after observing her signal, buyers will figure out any seller who chooses a secret reserve regime must have a signal less than s. Thus buyers form an expectation for the seller s signal between [0, s ] which is, for example, s and strictly less than the seller s actual signal. The bidding function changes according to the new expectation for the seller s signal and the expected payoffs shifts to the left bottom (Figure 2). Thus all sellers with signals between [s, s ] are better off by choosing to reveal their type via a reserve price. Now buyers know a seller with a signal between [s, s ] would also choose a disclosed reserve price because of higher expected payoffs. Thus if a seller with a signal slightly lower than s wants to choose a secret reserve, buyers would bid according to an expectation for her signal between [0, s ], which would result to 13
even lower bids and the expected payoff would shift further to the left bottom. Continuing this argument we can conclude all sellers types are better off by choosing to disclose their reserve price at the beginning of the game except the one with type s = 0 which is indifferent between both regimes. Figure 3: Interim payoffs FD vs RR Proposition 5.1. With the linear valuations and uniform signals on [0, 1], all types of sellers with positive signals would choose the disclosed reserve price to sell their object when they face only two buyers. Proof. See appendix. The result in proposition (5.1) gives an advantage to the regime in which the seller discloses her information, while in the real world, we observe contrary situation, where the seller with a private information tries to keep it secret if the costless access to the information is not possible. There are several assumptions we made to simplify the calculations while they may not be true in the real world. First of all we are going to relax the assumption of two bidders and increase the number of bidders to observe the effects on the previous results. Figure 3 shows the interim payoffs to the seller when the number of bidders are respectively increases. According to Figure 3, proposition (5.1) holds as long as there is an intersection between the expected payoffs for two regimes. When the number of bidders 14
Figure 4: Interim payoffs FD vs RR increases to 10 the interim payoffs to the secret reserve regime dominates the disclosed reserve price regime for all types of the sellers in the entire interval. In this situation they are better off by keeping the reserve price secret and since all types chooses the secret reserve, buyers expectation for the seller s signal would not be affected after the choice of the regime. Observation 1. If one regime dominates the other for all signals, then at interim level the choice of mechanism does not reveal any new information. Proposition 5.2. With the linear valuations and signals with uniform distribution on [0, 1], all types of the sellers in the interval would have a higher payoff by not revealing the reserve price when the number of bidders is big enough, that is, more than 10 bidders. Proof. See appendix 6 Disclosed versus Secret Reserve Price To generalize the previous results we need to investigate how the interim expected payoffs for both regimes changes when the seller s signal changes. By differentiating (2) with respect to s and using the Envelope Theorem we have D 1 U DR (s, ŝ, m(ŝ)) = D 2 U DR + (D 3 U DR m (s)) v 0(s)[1 F 1 (m(s))] = v 0(s)[1 F 1 (m(s))] < 0 (29) 15
Thus the expected payoffs for this regime is strictly decreasing in the seller s signal. If we differentiate (29) another time with respect to s, we have DD 1 U DR (s, ŝ, m(ŝ)) = v 0(s)[1 F 1 (m(s))] + m (s)f 1 (m(s))v 0(s) (30) Using Fundamental Theorem of Calculus and the result in (29), we can represent the seller s expected payoffs in another useful way, that is U DR (s, s, m(s)) = U DR (0, 0, m(0)) s 0 [1 F 1 (m(x))]v 0(x)dx (31) Differentiating the expected payoff to the secret reserve regime (9) with respect to s gives us D 1 U RR (s, ˆm(s)) = [β RR ( ˆm(s)) v 0 (s)]f 2 ( ˆm(s)) ˆm (s) v 0(s)[1 F 2 ( ˆm(1)) + F 2 ( ˆm(1)) F 2 ( ˆm(s))] = v 0(s)[1 F 2 ( ˆm(s))] < 0 (32) By the definition of ˆm(s), the first term becomes equal to zero. Thus the expected payoff for the secret reserve price regime is also strictly decreasing in the seller s signal. The second differentiation would also result to DD 1 U RR (s, ˆm(s)) = v 0(s)[1 F 2 ( ˆm(s))] + ˆm (s)f 2 ( ˆm(s))v 0(s) (33) Furthermore, we can apply the Fundamental Theorem of Calculus on the result in (32) to find another useful way to represent the seller s expected payoffs to the secret reserve price regime. U RR (s, ˆm(s)) = U RR (0, ˆm(0)) s 0 [1 F 2 ( ˆm(x))]v 0(x)dx (34) Proposition 6.1. A seller with a signal equal to zero has a higher expected payoff from the secret reserve price regime than the disclosed reserve price regime. Proof. See Appendix Here for the sake of proposition we assume buyers are not aware of the fact that seller has two options for the reserve price regime. Thus they do not update their information via the choice of reserve price regime. The result here is helpful for the equilibrium argument. Proposition 6.2. If the highest seller s type in the interval has a higher expected payoff for the disclosed reserve price than the secret reserve price, in equilibrium, all the sellers types are better of by choosing the disclosed reserve regime. 16
Proof. See Appendix Proposition 6.3. In equilibria an informed seller chooses a secret reserve price if and only if the expected payoffs to this regime is higher than the disclosed reserve price regime, for all seller s signals as well as the highest type in the interval. Proof. See Appendix for a comprehensive proof. The intuition behind the proposition (6.3) comes from proposition (6.1). Since the lowest type has a higher expected payoff in the secret reserve price regime then if any other types wants to choose the secret reserve price, the expectations of the bidders for those types are less than their actual type or the buyers believe that is a bad seller. Thus they are better off by revealing their true types via a disclosed reserve price. 7 Ex ante versus Interim In this section we are going to compare the seller s optimal choice at the ex ante and the interim stages. Jarman and Sengupta (2012) show the sufficient condition for a seller at the ex ante stage to choose the secret reserve price regime is that the expected payoff of the seller with the lowest signal under the secret reserve price exceeds that under disclosed reserve price by at least E s [v 0 (s)] v 0 (0). Proposition 7.1. For the seller to prefers the secret reserve price after observing her signal, it is sufficient that the expected payoffs to the lowest signal under the secret reserve price exceeds that under disclosed reserve price by at least v 0 (s) v 0 (0). Proof. Appendix Suppose all the assumptions for the seller and the buyers are like the one in (5). Then it is possible that the seller s optimal decision at the ex ante stage is to choose the secret reserve price regime, while at the interim stage she chooses to disclose her reserve price for all types. At the ex ante we have U RR (0, ˆm(0)) U DR (0, 0, m(0)) > 1 2 (35) At the interim we have s, U RR (0, ˆm(0)) U DR (0, 0, m(0)) > s (36) Since the highest s in the interval is 1, then there exists some sellers who satisfy (35) but not (36) 17
8 Appendix Proof of Proposition 3.1. In equilibrium buyer with signal x i will buy the object if and only if v(s, x i ) p. Then the expected value for the minimum buyer type who is willing to buy at the posted price is v(s, m(s)). Thus the equilibrium posted price must be equal to the expected value of the minimum buyer type who is willing to buy the object at that posted price. We need to calculate the minimum buyer type which maximizes the seller s payoff. Differentiate seller s interim payoff with respect to s and m(s) we have U pp (s, s, m(s)) = (v( s, m(s)) v 0 (s))(1 F 1 ( m)) (37) D 2 U pp v( s, m(s)) (s, s, m(s)) = (1 F 1 ( m)) (38) s D 3 U pp v( s, m(s)) (s, s, m(s)) = (m (s)(1 F 1 ( m)) m(s) (39) f 1 ( m)(v( s, m(s)) v 0 (s)) = f 1 (m)(v 0 (s) J 1 ( s, m) The m(.) function which characterizes the equilibrium must be the one which U pp (s, ŝ, m(s)) = maxu(s, ŝ, m(ŝ)). Differentiating that with respect to ŝ and considering the fact that in equilibrium ŝ = s we have D 2 U pp (s, s, m(s)) + D 3 U pp (s, s, m(s))m (s) = 0 (40) In equilibrium m (s) = D 2U pp (s,s, m(s)) D 3 characterizes the unique separating equilibrium. The solution gives the minimum buyer type who maximizes the seller s U pp (s,s, m(s)) expected payoff. Proof of Proposition 5.1. When all signals distributed uniformly on [0,1], we can use equations (2) and (9) to find an expression for the expected payoffs. U DR = 1 s(m) = (2x 2x 2 ) + (2m 2 2m 3 ) (41) m m β RR (x) = 0.5 (4x 2 2x)/(1 x 2 ) (42) { 2x if x 1 2 x + 1 2 if x 1 2 18 (43)
U RR = 0.5 ˆm(s) (2x s)(2 2x)dx + 1 0.5 (x + 0.5)(2 2x)dx (44) It is easy to check that both U DR and U RR are decreasing when the seller s signal increases. In fact we show this is true in general in (29) and (32). Since at s = 0, U RR is greater than U DR and at s = 1, U DR > U RR then by continuity there must be a signal ṡ in which U DR = U RR. There is no point for the signals higher than ṡ to choose the secret reserve regime. Suppose all other signals less than ṡ also choose the disclosed reserve price. We need to show it is not possible to deviate from this strategy for a given signal. Suppose s < ṡ chooses not to reveal her signal. Then buyers believe for the s is the lowest among all other signal. So unless s is not the lowest signal she would be worth of by not revealing her reserve price. Proof of Proposition 5.2. First we check the case with n = 10. It is easy to show U RR > U DR for every s in the interval including the highest, that is, s = 1. Now when n increases we know m(s) and the disclosed reserve price increases(theorem 2 in Cai, Riley, and Ye (2007)), while ˆm(s) is 0.5 and does not change. The bidding function for both regimes increases in the same manner x. Thus for n > 10, U RR U DR cannot be lower than n = 10, which is positive. Proof of Proposition 6.1. When s = 0 then by definition ˆm(0) = 0 but m(0) is positive. According to (2) and (9) we have: [ ( ) ( ) ] ( U DR (0, 0, m(0)) = F (2) m(0) F(1) m(0) v(m(0), 0) v0 (0) ) + ω m(0) [v(x, 0) v 0 (0)]f (2) (x)dx (45) U RR (0, ˆm(0)) = m(0) ˆm(0) [β RR (x) v 0 (0)]f (2) (x)dx + ω m(0) [β RR (x) v 0 (0)]f (2) (x)dx. (46) By definition of β RR (x), the second term in (46) is higher than the second term in (45). The first term in (46) is also higher than the first term in (45). Thus U RR (0, ˆm(0)) > U DR (0, 0, m(0)). 19
Proof of Proposition 6.2. According to (29) and (32) we know the seller s payoffs for both regimes are decreasing when her signal increases. From (6.1) we also know the lowest signal has a higher expected payoff by secret reserve regime. Then by continuity there must be one intersection between both payoffs. Then the argument is the same as the one in (5.1. Proof of Proposition 6.3. First part: Suppose there are some types with a higher payoff from revealing the reserve price, then the highest signal has to be one of them. By (6.2) we know the best equilibrium strategy is to disclose the reserve price for all types. Thus it has to be the case that all types has the higher payoff with a secret reserve. Second part: Suppose all types of the seller has higher payoffs for the secret reserve price. Then there is no profitable deviation from this strategy for all types in the interval. Because if they deviate and choose to disclose their reserve price they would definitely end up with a lower payoff. Proof of Proposition 7.1. U DR (s, s, m(s)) is equal to According to (34) and (31), U RR (s, ˆm(s)) U RR (0, ˆm(0)) s For this to be positive we have: 0 [1 F 2 ( ˆm(x))]v 0(x)dx U DR (0, 0, m(0))+ U RR (0, ˆm(0)) U DR (0, 0, m(0)) > s 0 s 0 [1 F 1 (m(x))]v 0(x)dx (47) [F 1 (m(x)) F 2 ( ˆm)]v 0(x)dx (48) Since [F 1 (m(x)) F 2 ( ˆm)] < 1 the necessary condition for (48) to satisfy is the left hand side greater than than v 0 (s) v 0 (0). 20
References Akerlof, G. A. (1971): The Market for Lemons: Qualitative Uncertainty and the Market Mechanism, Quarterly Journal of Economics, 84(3), 488 500. Ashenfelter, O. C. (1989): How Auctions Work for Wine and Art, Journal of Economic Perspectives, 3(3), 23 36. Bulow, J., and P. Klemperer (1996): Auctions Versus Negotiations, The American Economic Review, 86(1), 180 194. Cai, H., J. G. Riley, and L. Ye (2007): Reserve Price Signaling, Journal of Economic Theory, 135(1), 253 268. Cassidy, R. (1967): Auctions and Auctioneering. University of California Press, Berkeley. Hendricks, K., and R. H. Porter (1988): An Empirical Study of an Auction with Asymmetric Information, American Economic Review, 78(5), 865 883. Jarman, B., and A. Sengupta (2012): Auctions with an Informed Seller: Disclosed vs Secret Reserve Prices, mimeo. Jullien, B., and T. Mariotti (2006): Auction and the Informed Seller Problem, Games and Economic Behaviour, 56(2), 225 258. Kremer, I., and A. Skrzypacz (2004): Auction Selection by an Informed Seller, mimeo. Krishna, V. (2002): Auction Theory. Academic Press, San Diego. Maskin, E., and J. Tirole (1990): The principal-agent relationship with an informed principal: The case of private values, Econometrica, 58(2), 379 409. (1992): The principal-agent relationship with an informed principal: Common values, Econometrica, 60(1), 1 42. Milgrom, P., and R. Weber (1982): A Theory of Auctions and Competitive Bidding, Econometrica, 50(5), 1089 1122. Myerson, R. B. (1981): Optimal Auction Design, Mathematics of Operations Research, 6, 58 73. (1983): Mechanism design by an informed pricipal, Econometrica, 51(6), 1767 1797.
Peters, M., and S. Severinov (1997): Competition among Sellers Who Offer Auctions Instead of Prices, Journal of Economic Theory, 75, 141 179. Riley, J. G. (1979): Informational Equilibrium, Econometrica, 47(2), 331 359. Vulcano, G., G. van Ryzin, and C. Maglaras (2002): Optimal Dynamic Auctions for Revenue Management, Management Science, 48(11), 1388 1407. Wang, R. (1993): Auctions versus Posted-Price Selling, The American Economic Review, 83(4), 838 851. (1998): Auctions versus Posted-Price Selling: The Case of Correlated Private Valuations, The Canadian Journal of Economics, 31(2), 395 410. 2