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 F0 on [0, s̄], twice differentiable
with a continuous density f0 . Each buyer i has a private signal xi for the object
independently and identically distributed on F ∈ [0, x̄], twice differentiable with
continuous density f . Buyers valuation v : [0, x̄] × [0, s̄] → 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 v0 : [0, s̄] → 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
3.1
Ex ante Analysis
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(xi , ŝ) = E[Vi |S = ŝ, xi = 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)
Z ω̄
+
[v(x, ŝ) − v0 (s)]f(2) (x)dx
(2)
m(ŝ)
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
∂U
and assumption(1), we
full information minimum type. Then according to ∂m
have


0
∗
m (s) = J −1 (V0 (s))


x̄
if V0 (s) < J(s, 0)
if J(s, 0) ≤ V0 (s) < J(s, x̄)
if V0 (s) ≥ J(s, x̄)
(3)
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.
s0 (m) = −
D3 U DR (s, s, m(s))
D2 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 is1
[D1 U DR (s, s, m(s))] = D2 U DR + (D3 U DR m0 (s))
− v00 (s)[1 − F1 (m(s))]
= −v00 (s)[1 − F1 (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
Es [U
DR
Z
D
(s, s, m(s))] = U (0, 0, m(0)) −
s̄
[1 − G(s)][1 − F1 (m(s))]v00 (s)ds (6)
0
3.2
Right of Refusal (RR)
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 ⊂ Rn , we write Di g(x1 , . . . , xi , . . . , xn ) to respectively
the partial derivative of g with respect to its i-th argument evaluated at the point (x1 , . . . , xn ).
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(xi , s0 ) as the expected value of a bidder i give that
the seller signal is less than s0 . 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(xi , s̄) < v0 (s̄) then the
bidders bid their conditional expected value, otherwise (w(xi , s̄) > v0 (s̄)) they bid
their unconditional expected value.
Following Jarman and Sengupta (2012) proposition 1, the bidding function is
as follows;
(
p = w(xi , v0−1 (b))
if w(x, s̄) < v0 (s̄)
βRR (xi ) =
(7)
w(xi , s̄)
if Otherwise
Let m̂(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 m̂(s) as follows;
(
inf{x : βRR (x) ≥ v0 (s)}
if s ≤ v0−1 (βRR (s̄))
m̂(s) =
(8)
s̄
if s > v0−1 (βRR (s̄))
If there is any bid higher than the seller’s value, m̂(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
Z
ω̄
[βRR (x) − v0 (s)]f(2) (x)dx.
(s, m̂(s)) =
(9)
m̂(s)
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
[D1 U RR (s, m̂(s))] = −[βRR (m̂(s)) − v0 (s)]f2 (m̂(s))m̂0 (s)
− v00 (s)[1 − F2 (m̂(ω̄)) + F2 (m̂(ω̄)) − F2 (m̂(s))]
= −v00 (s)[1 − F2 (m̂(s))] < 0
(10)
By the definition of m̂(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, m̂(s)) = U
RR
Z
(0, m̂(0)) −
s
[1 − F2 (m̂(x))]v00 (x)dx
(11)
0
Taking expectation from (11) over s and rearranging the integrals, we have
E[U
RR
(s, m̂(s))] = U
RR
Z
(0, m̂(0)) −
s̄
[1 − G(s)][1 − F2 (m̂(s))]v00 (s)ds
(12)
0
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(s̃, xi ) = E[vi |s = s̃, s ≤ v0−1 (p)]. According to buyers
valuation, only buyers with valuation v(s̃, xi ) ≥ p are willing to buy. Let m̃(s) be
the minimum buyer’s type who is willing to buy given that the seller’s signal is
s̃. Then the minimum expected value of the buyer who is willing to buy becomes
v(s̃, m̃(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, m̃(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, s̃, m̃(s)) = (v(s̃, m̃(s)) − v0 (s))(1 − F1 (m̃))
(13)
F1 (m̃) 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
D1 U pp (s, s, m̃(s)) = (D3 U P P )(m̃0 (s)) − v00 (s)(1 − F1 (m̃))
(14)
By (14) and the Fundamental Theorem of Calculus we can find the ex ante
expected payoffs to the seller which is
pp
Z
pp
E[U (s, s, m̃(s))] = U (0, 0, m̃(s)) −
s̄
(1 − G(s))(1 − F1 (m̃(s))v00 (s)ds
(15)
0
3.4
Example
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 V0 (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, xi ) =
s+xi . 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, s̄] 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))
Z 1
+
xdF(2) (x)
(16)
m
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
∗
m (s) =


0
1
((γ
2
if (γ − 1)s < J(0)
if J(0) ≤ (γ − 1)s < J(1)
if (γ − 1)s ≥ J(1)
− 1)s + 1)

1
(18)
Since γ is greater or equal to zero and s ∈ [0, 1], then m∗ (s) = 12 ((γ − 1)s + 1).
To calculate the minimum buyer type we need to solve the differential equation
from (4) which is as follows
Z m
γ−1
−γ
s(m) = (1 − F(1) (m))
f(1) (x)(1 − F(1) (x)) J(x)dx
(19)
m
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
2 γ−1
1
Z
s(m) = (1 − m )
1
2
(4x2 − 2x)
dx
(1 − x2 )γ
(20)
Substitute the result into the seller’s expected payoff gives us the following
equation
Z 1
DR
U (s, s, m) =
2(s + x − γs)(1 − x)dx + (2m − 2m2 )(m + s − γs)
(21)
m
3.4.2
Right of refusal (RR)
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
(
2γ
x
if x ≤ γ − 21
βRR (x) = 2γ−1 1
(22)
x+ 2
if x ≥ γ − 21
Calculation of the minimum buyer type who enter the auction is much more
straight forward in this case. To solve m̂(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 m̂(s) we can substitute the result into the seller’s expected
payoff which is the following equation,
9
U
RR
Z
1
[βRR (x) − γs](2 − 2x)dx
(s, m̂(s)) =
(23)
m̂
The main difference here is the biding function which can be either conditional
or unconditional. So after calculation of m̂ 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 v0 (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(s̃, xi ) = E[Vi |s = s̃, s ≤ γp ].
According to the buyers expected valuations, only buyers with expected value
v(s̃, xi ) ≥ 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 + m̃(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 s̃ and m.
where
D2 U P P = 1 − F(1) (m)
(24)
˜
D3 U P P = (γs − s̃ − J(m))f
(1) (m)
(25)
1 − F(1) (m)
˜
J(m)
=m−
f(1) (m)
Now the differential equation becomes equal to
s0 (m) =
˜
(γs − s̃ − J(m))f
(1) (m)
1 − F(1) (m)
Solving this differential equation would result to
Z m
γ−1
−γ ˜
s(m) = (1 − F(1) (m))
f(1) (x)(1 − F(1) (x)) J(x)dx
(26)
(27)
m
For this example m(s) = 21 . We can solve the integral in (27) numerically for a
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, m̃(s)) = (s + m̃ − γs)(1 − m̃2 )
3.4.4
(28)
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., v0 (s) = s and the buyers valuations are symmetric and
linear with the following format: v(xi , s) = xi + 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 s0 ,
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 s0 . Let’s focus on the marginal seller who
has a signal less than s0 . 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 s0 . Thus buyers
form an expectation for the seller’s signal between [0, s0 ] which is, for example,
s̄ 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”, s0 ]
are better off by choosing to reveal their type via a reserve price.
Now buyers know a seller with a signal between [s”, s0 ] 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
D1 U DR (s, ŝ, m(ŝ)) = D2 U DR + (D3 U DR m0 (s))
− v00 (s)[1 − F1 (m(s))]
= −v00 (s)[1 − F1 (m(s))] < 0
15
(29)
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
DD1 U DR (s, ŝ, m(ŝ)) = −v000 (s)[1 − F1 (m(s))] + m0 (s)f1 (m(s))v00 (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
Z s
DR
DR
U (s, s, m(s)) = U (0, 0, m(0)) −
[1 − F1 (m(x))]v00 (x)dx
(31)
0
Differentiating the expected payoff to the secret reserve regime (9) with respect
to s gives us
D1 U RR (s, m̂(s)) = −[βRR (m̂(s)) − v0 (s)]f2 (m̂(s))m̂0 (s)
− v00 (s)[1 − F2 (m̂(1)) + F2 (m̂(1)) − F2 (m̂(s))]
= −v00 (s)[1 − F2 (m̂(s))] < 0
(32)
By the definition of m̂(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
DD1 U RR (s, m̂(s)) = −v000 (s)[1 − F2 (m̂(s))] + m̂0 (s)f2 (m̂(s))v00 (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.
Z s
RR
RR
[1 − F2 (m̂(x))]v00 (x)dx
U (s, m̂(s)) = U (0, m̂(0)) −
(34)
0
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 Es [v0 (s)] − v0 (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 v0 (s) −
v0 (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, m̂(0)) − U DR (0, 0, m(0)) >
1
2
(35)
At the interim we have
∀s, U RR (0, m̂(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 xi will buy the
object if and only if v(s, xi ) ≥ p. Then the expected value for the minimum buyer
type who is willing to buy at the posted price is v(s, m̃(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 s̃ and m̃(s) we have
U pp (s, s̃, m̃(s)) = (v(s̃, m̃(s)) − v0 (s))(1 − F1 (m̃))
∂v(s̃, m̃(s))
(1 − F1 (m̃))
∂s̃
∂v(s̃, m̃(s)) 0
D3 U pp (s, s̃, m̃(s)) =
(m (s)(1 − F1 (m̃))
∂ m̃(s)
− f1 (m̃)(v(s̃, m̃(s)) − v0 (s))
= f1 (m)(v0 (s) − J1 (s̃, m)
D2 U pp (s, s̃, m̃(s)) =
(37)
(38)
(39)
The m̃(.) function which characterizes the equilibrium must be the one which
U (s, ŝ, m̃(s)) = maxU (s, ŝ, m̃(ŝ)). Differentiating that with respect to ŝ and
considering the fact that in equilibrium ŝ = s we have
pp
D2 U pp (s, s, m̃(s)) + D3 U pp (s, s, m̃(s))m0 (s) = 0
(40)
pp
D2 U (s,s,m̃(s))
In equilibrium m0 (s) = − D
pp (s,s,m̃(s)) characterizes the unique separating equi3U
librium. The solution gives the minimum buyer type who maximizes the seller’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
Z
1
(2x − 2x2 ) + (2m2 − 2m3 )
m
Z m
s(m) =
(4x2 − 2x)/(1 − x2 )
=
(41)
(42)
0.5
(
2x
βRR (x) =
x+
1
2
18
if
if
x≤
x≥
1
2
1
2
(43)
U
RR
Z
0.5
Z
1
(x + 0.5)(2 − 2x)dx
(2x − s)(2 − 2x)dx +
=
(44)
0.5
m̂(s)
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 s̈ < ṡ chooses not to reveal
her signal. Then buyers believe for the s̈ is the lowest among all other signal. So
unless s̈ 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 m̂(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 m̂(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)
Z ω̄
[v(x, 0) − v0 (0)]f(2) (x)dx
+
(45)
m(0)
U
RR
Z
m(0)
Z
ω̄
[βRR (x) − v0 (0)]f(2) (x)dx +
(0, m̂(0)) =
m̂(0)
[βRR (x) − v0 (0)]f(2) (x)dx.
m(0)
(46)
By definition of βRR (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, m̂(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. According to (34) and (31), U RR (s, m̂(s)) −
U DR (s, s, m(s)) is equal to
Z s
Z s
RR
0
DR
U (0, m̂(0))− [1−F2 (m̂(x))]v0 (x)dx−U (0, 0, m(0))+ [1−F1 (m(x))]v00 (x)dx
0
0
(47)
For this to be positive we have:
U
RR
(0, m̂(0)) − U
DR
Z
(0, 0, m(0)) >
s
[F1 (m(x)) − F2 (m̂)]v00 (x)dx
(48)
0
Since [F1 (m(x)) − F2 (m̂)] < 1 the necessary condition for (48) to satisfy is the
left hand side greater than than v0 (s) − v0 (0). 20
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2