Auction versus Posted-price and the Informed
Seller Problem
Peyman Khezr
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 the static
context in which there is a fixed number of buyers. We give conditions under which it is optimal (within the class of mechanisms we study) for the
seller to sell via an auction with a secret reserve price. Second, we study a
dynamic model with finite horizon one, in which prospective buyers arrive
over time. We give conditions under which a posted price can result in a
higher payoff than an auction.
Keywords: Auction, Posted-price, Reserve price, Signaling, Informed seller.
JEL Classification: D44, D80, D82.
PREFACE
Thesis title: Selling Mechanisms with an Informed seller and The Australian
Housing Market
Supervisors: Dr Abhijit Sengupta, Dr Andrew Wait
The aim of this thesis 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. 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 thesis, we intend to look at both these classes of environments. The thesis
will also consider two variants on the set of prospective set of buyers. First,
we will consider the standard paradigm—where the set of prospective buyers is
invariant. The second—perhaps more relevant in the context of sales of residential
real estate—that there is an arrival rate of such buyers over time. The arrival rate
will be taken as exogenous, but the mechanism design will have to be alert to how
the set of prospective buyers varies over time.
The thesis will take the following structure
1. Chapter one: Introduction
2. Chapter two: Auction versus posted price with an informed seller
3. Chapter three: Signaling via choice of mechanism
4. Chapter four: Selling mechanisms and the Australian Housing Market
5. Chapter five: Conclusion
This paper is based on the maximizing the seller’s expected revenue from an ex
ante prospective.
1
Introduction
The aim of this paper is to study and compare some commonly observed selling
mechanisms in an environment in which the seller of an item has private information about its quality that is payoff-relevant to prospective buyers but where the
seller is unable to reveal her information to the buyers at no cost.
Economic settings in which the seller of an item has superior payoff-relevant
private information than prospective buyers are common. Indeed, Akerlof’s classic
paper on the “lemons problem” (Akerlof, 1971) that introduced the analysis of
asymmetric information to 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 bottles of wine offered would typically have private information
about the conditions under which the wine was stored. 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.
Cai, Riley, and Ye (2007) observe that, when the seller lacks access to a technology for costless and credible revelation of her information, an announced reserve
price at an auction can act as a credible device to signal the seller’s information.
This is because the marginal cost of a higher reserve price—a lower probability of
sale—is lower for a seller who has a superior signal and, therefore, a higher use
value for the item. They characterise the unique separating equilibrium in such
a setting. We treat their work as the theoretical benchmark for the environment
that is the concern of this paper. However, reserve prices are almost never disclosed in real-world auctions. 1 We compare the benchmark model with the two
most commonly observed selling mechanisms: the mechanism in which the seller
simply posts a price and the mechanism in which the seller conducts an auction
with a secret reserve price and has the right to retain the item. The three mechanisms differ in the degree to which they reveal the seller’s information. An auction
with a disclosed reserve price fully reveals the seller’s signal (since the optimal
reserve price is increasing in the signal); the posted-price mechanism reveals some
information; in an auction with a secret reserve price, the signal is never revealed.
In our study the form of the mechanism—whether a posted price or an auction
with a disclosed or a secret reserve—is chosen by an auction house before the
realisation of the seller’s signal. Since the auction house typically receives a set
proportion of the revenue from a mechanism, we suppose that the mechanism is
1
Several studies have documented this: see, for example, Ashenfelter (1989), Cassidy (1967)
and Hendricks and Porter (1988); this can also be readily confirmed by visiting the websites
of traditional auction houses like Christies and Sotheby’s that explicitly state that the reserve
prices for items are kept secret.
chosen with a view to maximising expected revenue from an ex ante perspective.
The posted price or the reserve price (whether disclosed or secret) in a particular
auction for an item is of course chosen by the seller at the interim stage when
the seller’s private signal is known to the seller. Thus, the particular value of the
posted or reserve price acts as a signal in the corresponding mechanism but the
choice of the mechanism itself does not.
We compare the three mechanisms first in a static model, i.e., when the seller
faces a known finite number of prospective buyers, each of whom have receives
an independent private signal but also cares about the seller’s private signal. We
then go to compare the mechanisms when such prospective buyers arrive randomly
over time. In each of the contexts, we compare the ex ante expected revenue from
the three mechanisms. We show that, although in a static context, posted-price
mechanism generates lower expected revenue than an auction with a disclosed
reserve, in the dynamic context, the posted-price mechanism can generate a higher
expected revenue.
The two papers most closely related to this paper are Cai, Riley, and Ye (2007),
already mentioned, 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. The first two papers are in the static
context. Jarman and Sengupta (2012) characterise the bidding function and 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 of the static model. We then compare the results
with those in the dynamic model.
Another related work is Wang (1993) who compares auction and posted-price
mechanisms in a dynamic context. Wang’s dynamic model is quite different from
the one studied here. More importantly, in his setup, the seller’s valuation is
common knowledge and there is no role for signalling.
2
The static model
A seller, player 0, with an indivisible object to sell, faces a set of N = {1, . . . , n}
potential buyers, n ≥ 1. Each buyer i receives an independent private signal xi
concerning the value of the object to the buyer, drawn from a distribution function
F with support X = [0, ω̄], and continuous density f . Similarly, the seller has a
private signal s. This signal s is drawn from a distribution function G on [0, s̄] with
density function g > 0. The signals are private valuation but the distributions are
common knowledge. All signals are independently drawn. Buyer i’s valuation is
2
given by Vi = ui (xi , s). Suppose there is a function u on R2 such that for all i,
ui (xi , s) = u(xi , s). Thus, we have a symmetric valuation function for buyers who
care about the seller’s signal in the same manner. We also assume that there is
a function vi (s, x) which is the expectation of Vi conditional on the realization of
the seller signal when buyer i’s signal is the highest signal among other bidders.
Because of symmetry vi (s, x) = v(s, x), and v(s, x) is increasing in both s and
x, integrable and non-negative. Seller’s own valuation for the object is V0 (s) and
increasing in her own signal s.
In this paper we consider the version in which a mechanism is chosen before
the realisation of the seller’s signal. There are two selling mechanisms available to
the seller, posted-price and ascending auction. We consider two candidate reserve
price regimes for the ascending auction. The first is to disclose the reserve price at
the beginning of the auction before bidding starts. The second is to never disclose
the reserve but retain the right to accept or reject the highest bid. When the seller
chooses the reserve, she knows her valuation, but when the mechanism itself is
selected, she does not. We first begin by giving a short-hand to each regime. As
we mentioned the two possible regimes for the reserve are as follows:
1. Disclose her reserve price before bidding starts, or the disclosed-reserve regime
(DR)
2. Never disclose the reserve but retain the right to accept or reject the highest
bid, or the right-of-refusal regime (RR).
Now we need to characterize the bidding behavior for each of the following
cases and then calculate the expected payoff to the seller.
2.1
Disclose the reserve (DR)
Under this regime, the reserve price would be announced at the beginning of
the auction and before bidding starts. This announcement reveals seller’s private
information to the bidders. For this regime most of the results in Cai, Riley, and
Ye (2007) are directly applicable to our analysis. This is a signaling game and the
seller can use the reserve price to signal her type to the potential buyers. Let us
suppose ŝ ∈ [0, s̄] is the information which bidders use, for forming their bids plus
their own signals. By a result in Milgrom and Weber (1982), it is a Bayesian Nash
−1
equilibrium for each buyer i to bid v(ŝ, xi ) = E[Vi |S = ŝ, xi = x−i
(1) ]. Where x(1)
is the highest signal among all other buyers. Each buyer’s type, given the belief
that the seller’s type is ŝ, will enter the auction if her expected valuation is greater
than the reserve price. Let m(s) be the minimum buyer type which enters the
auction. m function is a strictly increasing function because higher reserve price
induces less probability for sell and higher minimum bid to enter the auction.
3
2.1.1
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.
The interim expected payoff to the seller with type s can be written as follows;
Z ω̄
DR
[v(x, ŝ) − v0 (s)]f(2) (x)dx + (F(2) (m) − F(1) (m))(v(m, ŝ) − v0 (s))
U (s, ŝ, m) =
m
(2)
where F(1) (m) is the probability that the highest signal is less than m, and F(2) (m)−
F(1) (m) is the probability of the case that only highest signal is greater than m.
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
full information minimum type. Then according to ∂m
and assumption(1), we
have


x
∗
m (s) = J −1 (V0 (s))


x̄
if V0 (s) < J(s, x)
if J(s, x) ≤ V0 (s) < J(s, x̄)
if V0 (s) ≥ J(s, x̄)
(3)
where J −1 is the inverse of J and according to our model x is equal to zero.
By Theorem 1 in Cai, Riley, and Ye (2007), the following differential equation
characterizes the unique separating equilibrium.
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 is
[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
4
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
2.2
Right of Refusal (RR)
Under this regime, the reserve is never disclosed. The seller simply accepts
or rejects the highest bid at the auction. Let s̄ be the maximum seller’s type in
which the bidder’s expected value is greater than the seller’s valuation. In the ex
ante stage, there is no extra information to the bidders for the seller’s signal. The
expected payoff to a bidder who announces type z when all other bidders play β
is
Z
z
Z
s̄(x)
[v(x, s) − β(y)]g(s)f (y)ds dy,
E[Π(z, x)] =
0
(7)
0
where f is the density function of the highest N − 1 draws from F . We can derive
the bidding function from (7), which is the first order condition for maximizing
the expected pay off.
The bidding behavior for this regime is straightforward. At a symmetric equilibrium, each bidder stays active until her expected valuation if their expected
value is higher than the highest expected reserve, otherwise they stay active until
v(x, v0−1 (b)) which is their expected value conditional on winning the object at b
subject to the seller’s best response under negotiation. v0−1 (b) is the expectation
of seller’s signal conditional on winning at price b. Jarman and Sengupta (2012)
show that this bidding function is
(
v(x, v0−1 (b))
if v0 (s̄) > E[v(x, s)]
βRR (x) =
(8)
E[v(x, s)]
if v0 (s̄) ≤ E[v(x, s)]
Let m̂(s) be the minimum type with the expected value higher than the seller’s
valuation. Since seller accepts any offer greater than her valuation, m̂(s) would be
as follows
(
inf{x : βRR (x) ≥ v0 (s)}
if s ≤ v0−1 (βRR (s̄))
(9)
m̂(s) =
s̄
if s > v0−1 (βRR (s̄))
If there is any bid higher than the seller’s value, then m̂(s) shows the lowest
bidder’s signal which her expected value is higher than the seller’s value, and if
there are no bids higher than the seller’s value, then m̂(s) would be equal to the
5
highest possible signal which simply means no one has the expected value higher
than seller’s value.
We can now derive the interim expected pay off to the seller at this regime;
U
RR
Z
ω̄
[βRR (x) − v0 (s)]f(2) (x)dx.
(s, m̂(s)) =
(10)
m̂(s)
Now to achieve the ex ante expected pay off to the seller, we need to find out how
the interim profit changes when s changes. Derivative of (10) 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(s̄))]
= −v00 (s)[1 − F2 (m̂(s̄))] < 0
(11)
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 (11) we can calculate
the seller’s ex ante expected profit
Z s
RR
RR
U (s, m̂(s)) = U (0, m̂(0)) −
[1 − F2 (m̂(x))]v00 (x)dx
(12)
0
Taking expectation from (12) over s and rearranging the integrals, we have
E[U
RR
(s, m̂(s))] = U
RR
Z
(0, m̂(0)) −
s
[1 − G(s)][1 − F2 (m̂(s))]v00 (s)ds
(13)
0
This is the ex ante expected pay off to the seller for the right of refusal regime.
2.3
Posted-Price
Now we consider a setting in which the seller sells her object through posting a
price p and the set of potential buyers is N . Buyers arrive randomly to the seller.
Each buyer accepts the offer if p is less than her expected valuation and declines
to buy otherwise. The first buyer who accepts gets the object. If there are more
than one offer to buy the object, then one of them would get the object randomly.
Since p has to be greater or equal to the seller’s valuation, then 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
6
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 2.1. In equilibrium seller post 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))
(14)
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.
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))
(15)
By (15) and the Fundamental Theorem of Calculus we can find the ex ante
expected payoffs to the seller which is
pp
pp
Z
E[U (s, s, m̃(s))] = U (0, 0, m̃(s)) −
s
(1 − G(s))(1 − F1 (m̃(s))v00 (s)ds
(16)
0
Proposition 2.2. If there is no cost for auctioning then seller’s expected payoff
from disclosed reserve regime is always greater or equal to the posted-price mechanism.
The proof is simple, seller, by setting a reserve price equal to the posted price
never gets less than the posted price mechanism with the same set of buyers.
3
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
7
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.
3.1
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)
(17)
˜
D3 U P P = (γs − s̃ − J(m))f
(1) (m)
(18)
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 ˜
γ−1
s(m) = (1 − F(1) (m))
f(1) (x)(1 − F(1) (x)) J(x)dx
(19)
(20)
m
For this example m(s) = 21 . We can solve the integral in (20) numerically for a
given γ to find the value of s. If we assume n = 2 then the seller’s expected payoff
according to (14) is
U pp (s, s, m̃(s)) = (s + m̃ − γs)(1 − m̃2 )
8
(21)
3.2
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)
(22)
m
and J(.) becomes equal to
J(x) = x −
(1 − F (x))
f (x)
(23)
For our example which is the uniform case, it is equal to J(x) = 2x − 1 which
is strictly increasing in x.
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
(24)
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
(25)
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
9
(4x2 − 2x)
dx
(1 − x2 )γ
(26)
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)
(27)
m
3.3
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
(28)
βRR (x) = 2γ−1 1
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 (9), 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,
U
RR
Z
1
(s, m̂(s)) =
[βRR (x) − γs](2 − 2x2 )dx
(29)
m̂
The only 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
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 .
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
10
Figure 1: Interim payoff γ = 0.33 , and , γ = 1
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
Dynamic model
As it is mostly common in the auction theory analysis, models are assumed to
be static like our previous section. One of the possible extensions of our model
is to assume there are more than one period available to the seller for selling her
object. There is also an arrival rate of the buyers at each period. Suppose buyers
arrive according to a Poisson process with parameter λ at every period and at the
end of each period all buyers who have arrived leave the market. Assumptions for
the signals and valuations are to be the same as previous model.
To characterize a time horizon for our model, we use the fact that it is so unlike
for a seller to keep selling her object infinitely, which means that apart from the
method of sale, she keeps the object in the market for a finite period of time and
after that if she couldn’t sell it, she would keep the object at its value. Thus in
our model we suppose that there are finite periods available for a seller to sell
her object. The two possible mechanisms for the seller are either posted-price or
auction. Once a seller decides to sell via auction or posted price she cannot change
her mind in the middle of the game, so she has to stick to the method of sale.
Since we are focusing on the markets which the buyers are not patient and they
arrive and leave fast, we do not have a discount factor in the model, but one can
have the same analysis with a discount factor. In that case being patient is more
costly for the seller.
There are T periods in the game, at each period there is an arrival of new
11
buyers according to a Poisson process with parameter λ. Buyers leave the market
at the end of each period. Let’s suppose T is exogenously defined and seller knows
T at the beginning of the game as the maximum time she is going to display the
object. Thus the probability that k buyers arriving until period T is given by
λT k −λT
e
Pk (T ) =
k!
Now we are continuing our analysis separately for each mechanism.
4.1
(30)
Dynamic Posted-Price selling
Since each buyer knows that the seller will never post a price less than her
value, then the expected value of the object to each buyer would be v(s̃, ti ) =
E[Vi |s = s̃, s ≤ v0−1 (p)]. However, the minimum buyer type who is willing to buy
is a little more complicated than the static model. Since in this model the number
of bidders is stochastically defined, seller has an expectation about the number of
bidders who are going to arrive until the end of the game. Since T is exogenous,
the expected number of bidders can be easily defined as
∗
N =
∞
X
k.Pk (T )
(31)
k=0
Define m̃D (s) as the expected minimum buyer type who is willing to buy the
object at the posted price.
Proposition 4.1. In equilibrium seller posts a price equal to the expected value of
the minimum buyer type who is willing to buy which is pD (s) = v(s, m̃D (s)).
Proof. See appendix.
Having the posted price we can now calculate the new interim expected payoff
to the seller for the dynamic case
U
ppd
∞
X
λT k −λT
D
k
D
(s, s̃, m̃ (s)) =
(p − v0 (s))(1 − F1 (m̃ ))[
e
]
k!
k=0
D
(32)
According to the expected number of bidders N ∗ , seller calculates the expected
minimum buyer’s type from the separating equilibrium that is the differential
equation in (4).
12
4.2
Dynamic Disclosed Reserve Auction
Suppose seller chooses an auction with a disclosed reserve price. She is facing T
periods and at each period there is an arrival rate like the previous section. At each
period if she holds an auction, she only faces the buyer’s who have been arrived
to the market at that period. Although the expected number of arrivals at every
period is equal, but it may not be optimal for the seller to keep the reserve price
constant. Since buyers don’t know how many periods the seller was in the market
when they have arrived, upon observing the reserve price the make their believes
ŝ for the seller’s signal. Then each buyer i bids v(ŝ, xi ) = E[Vi |S = ŝ, xi = x−i
(1) ].
−i
Where x(1) is the highest signal among all other buyers at each period.
From the seller point of view, calculating the reserve price with respect to the
expected number of buyers at each period is not optimal, because she does not
consider the fact that if the object is unsold at one period there is another chance
in the remaining periods until period T . In this case seller should calculate her
reserve price with respect to the total expected revenue from all periods. At the
first period her expected payoff is
DRD
Ut=1
(s, ŝ, m1 )
Z
ω̄
[v(t, ŝ) − v0 (s)]f(2) (t)dt
=
m1
+ (F(2) (m1 ) − F(1) (m1 ))(v(m1 , ŝ) − v0 (s))
+ F(1) (m1 )(Ut>1 )
(33)
F1 and F2 are the distribution of the first and second order of the expected
number of bidders at each period. From (33) we can observe that the interim
expected payoff is different for every period. The last element in (33) is the probability that no one has a value higher than the reserve price multiplied by the
expected utility of the future periods. This would result to a different reserve
price at every period t. Differentiating (33) with respect to ŝ and m1 results to
Z ω̄
∂v
∂v
DRD
(F(2) (m1 ) − F(1) (m1 )) +
dF2 (x)
D2 Ut=1 (s, ŝ, m1 ) =
∂ŝ
m ∂ŝ
(34)
∂(Ut>1 )
+ F(1) (m1 )
∂ŝ
DRD
D3 Ut=1
(s, ŝ, m1 ) = f1 (m1 )(Ut>1 − J(ŝ, m1 )) + F1 (m1 )
Proposition 4.2. At every period t, differential equation
D U DRD (s,s,m (s))
s0t (mt ) = − D32 UtDRD (s,s,mtt (s))
t
13
∂(Ut>1 )
∂m1
(35)
characterizes the unique equilibrium for that periods, Thus the set of T differential equations characterize the unique equilibrium of the game.
Proof. Appendix
Proposition 4.3. In equilibrium, R is the vector of reserve prices with T elements
and Rt = v(s, mt (s)). M is the vector of minimum buyer type who is willing to
buy at each period.
Proof. Appendix
In an example with two periods and linear values we show how the seller optimizes her payoff by disclosing a reserve price at each period.
4.3
Right of Refusal Auction
In this section we suppose that the seller decides to sell her object via auction
with a secret reserve price and with a right of refusal. Since we have more than
one period in this model, the seller’s optimal decision at each period is to say no
to any offer less than her valuation and accept any bid higher than her value if it
is also higher than the expected revenue from other periods. Suppose there are T
periods available to the seller. Starting from the last period her optimal decision
is to say yes to any offer greater than her value, since there is no more period left.
At period T − 1, the optimal decision is to any highest bid if it is higher than
her value and the expected payoff from the last period, and so on until the first
period. From the static model we know that the seller’s optimal reserve price for
this regime is her value for the object. But now the seller’s reservation value at
each period varies and at the last period it becomes equal to her value like the
static model. At each period t seller’s reservation value is
Rt = M ax(v0 (s), EUT −t )
(36)
Now the minimum buyer type who is willing to buy is the minimum type who
bids higher than Rt . Having the minimum buyer type at each period the seller’s
expected payoff becomes equal to
RRD
Ut=1
(s, m̂(s))
Z
ω̄
[βRR (x) − v0 (s)]f(2) (x)dx + F2 (m̂)(Ut>1 )
=
(37)
m̂(s)
Starting from the last period we can calculate the seller’s expected payoff and
then continue backward until period one to calculate each period’s reservation
value and the expected payoff.
14
5
Example: with constant arrivals
Now in a linear valuation example we are going to calculate the seller’s payoff
for auction and the posted price. For simplicity suppose there are only two periods
and at each period two buyers arrive and leave at the end of that period. There
are two choices of mechanism for the seller, ascending auction and posted-price.
All assumptions for values are like section (3).
Suppose seller decides to post a price at period one and do not change it until
the end of the game. Her interim expected payoff becomes
U ppd (s, s̃1 , m̃1 (s)) = (s̃ + m̃1 (s) − γs)(1 − F1 (m̃1 ))
(38)
F1 is now the first order distribution of all buyers arriving until the last period.
Differentiating the seller’s payoff with respect to s̃ and m would result to
where
∂U P P
= 1 − F(1) (m̃1 )
∂s̃
(39)
∂U P P
˜ m̃1 ))f(1) (m̃1 )
= (γs − s̃ − J(
∂m
(40)
˜ m̃1 ) = m̃1 − 1 − F(1) (m̃1 )
J(
f(1) (m̃1 )
Following the same steps like the static case, we can write the s(m̃1 ) as follows
Z m̃1
γ−1
−γ ˜
s(m̃1 ) = (1 − F(1) (m̃1 ))
f(1) (x)(1 − F(1) (x)) J(x)dx
(41)
m
As we can see, seller now calculates the minimum buyer type with respect to
the total number of buyers who will come during two periods. With the same
argument like the one in Theorem 2 in (Cai, Riley, and Ye, 2007), we can prove
that the posted price in the dynamic model is greater than the one period version
of the game simply because of more buyers available in the game.
Now suppose seller decides to sell via ascending auction with a disclosed reserve
price. Starting from the second period since it is the last period, her expected
payoff is
DRD
Ut=2
(s, ŝ, m2 ) =γs(F(1) (m2 ) − 1) + ŝ(1 − F(1) (m2 ))+
Z 1
m2 (F(2) (m2 ) − F(1) (m2 )) +
xdF(2) (x)
m2
15
(42)
Which is the same as one period model. Thus all other steps to calculate the
minimum buyer type would be straight forward. At the first period her payoff
becomes
DRD
(s, ŝ, m1 ) =γs(F(1) (m1 ) − 1) + m1 (F(2) (m1 ) − F(1) (m1 ))
Ut=1
Z 1
DRD
xdF(2) (x) + F1 (m1 )(Ut=2
)
+ ŝ(1 − F(1) (m1 )) +
(43)
m1
Differentiating (43) with respect to m1 and ŝ we have
DRD
∂Ut=1
DRD
)
= (γs − ŝ − J(m1 ))f1 (m1 ) + f1 (m1 )(Ut=2
∂m1
(44)
DRD
∂Ut=1
= 1 − F1 (m1 ) + F1 (m1 )(1 − F1 (m2 ))
∂ŝ
(45)
Thus
s0 (m1 ) = −
DRD
)
(γs − ŝ − J(m1 ))f1 (m1 ) + f1 (m1 )(Ut=2
1 − F1 (m1 ) + F1 (m1 )(1 − F1 (m2 ))
(46)
The solution to the differential equation in (46) gives us the minimum buyer
DRD
type in period one. Suppose a = Ut=2
and b = F1 (m2 ), then the solution is
Z m1
γ−1
1−γ−b
s(m1 ) = (1−bF(1) (m1 )) b
f(1) (x)(1−bF(1) (x)) b (J(m1 )+a)(x)dx (47)
m
For any given 0 < γ ≤ 1 the solution to the above integral gives us the inverse
function of the minimum buyer type in the first period, which is obviously different
than the one in the second period. This would result to different reserve prices for
each period. But suppose like the posted price mechanism it is not possible for
the seller to change the reserve price at each period. Then she has to calculate
the reserve price for the last period and keep it constant for all other periods. It
is like doing the single period auction in the first period and continuing the same
strategy for the other periods if the object is unsold after each period. Figure 2
shows the payoffs for two different given γ in the interval. The upper curve shows
the payoffs to the seller for an auction for any given signal in the interval. If γ is
big enough, for example equal to one, it is possible that the payoffs to the seller
for an auction become equal or even less than the posted price for some signals.
Because keeping the reserve price constant is not optimal for the seller. In this
example reserve price for the first period is lower than the optimal reserve which
would result to a lower overall payoff to the seller.
16
Figure 2: Two period payoffs(constant reserve)
6
Conclusion
When a seller of an object has private information about its value which is
important to the buyers and there is no costless revelation of this information, then
because of the adverse selection problem, the seller’s revelation of information is
not credible to the buyers. We showed in a one shot static model if the seller’s
signal is smaller than a threshold it is optimal for her not to reveal any information.
However, if the value of her signal is high enough it is optimal to reveal her type
via a reserve price in an auction or a fixed price in a posted price mechanism.
If the setting is dynamic, that is the seller faces more than one period in the
game and at each period there are number of buyers who arrive at the beginning
and leave at the end of the period, then the signaling strategy is different. She
starts with the highest reserve price in the first period and respectively reduces this
reserve until the last period which the reserve price becomes exactly equal to the
static model. In this case since buyers don’t know at what period they’ve arrived
the reserve price does not act as a full revelation of the seller’s signal anymore.
Thus buyers who bid at the first period over estimate the value of the object. As
we get closer to the end of the game their estimate becomes more accurate. If
the seller decides not to reveal any information, then her secret reserve price is
no longer equal to her value. It is equal to the maximum of her value and the
expected payoff from remaining periods. Thus even if the seller decides not to
disclose her reserve price, her reservation value is higher at the beginning of the
game and in the last period it is equal to her value for the object.
In the dynamic set up it is also reasonable to assume seller cannot change the
reserve price for each period, then she has to calculate the minimum buyer who
is willing to buy from all periods and not only from just one period. In this case
since buyers know that the reserve price is fixed, then the reserve price is much
17
more credible way to signal the seller’s information. We should keep in mind that
in the real world there are markets in which buyers arrive and leave so soon such
as real estate market and sometimes it is not possible to hold up an auction at
all periods. Thus by choosing an auction seller may miss some arriving buyers,
while by posting a price and advertising it she may never miss any arriving buyer
to the market. The cost of keeping an object in the market with posted price
is also cheeper than auction, because as long as the seller advertises the object
at period one, she can keep it continuously in the market. In this case the cost
of being patient for the seller is much less than the auction. We can conclude
if the seller expect high number of arriving buyers she may choose an auction,
then after observing her signal she can choose the reserve price regime, but if the
expected number of arriving buyers are so low then she can post a price and be
more patient to attract more buyers. In this paper we restricted our attention to
the case that the duration of each period is small enough, so that the discounting
affect is negligible, but a possible extension is when there is a discount factor and
the buyers are more patient. Another extension could be the case that the signals
are no longer independent, which would result to a much more general model.
7
Appendix
Proof of Proposition 2.1. 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)) =
(48)
(49)
(50)
The m̃(.) function which characterizes the equilibrium must be the one which
U (s, s, m̃(s)) = maxU (s, ŝ, m̃(ŝ)). Differentiating that with respect to ŝ and
pp
18
considering the fact that in equilibrium ŝ = s we have
D2 U pp (s, s, m̃(s)) + D3 U pp (s, s, m̃(s))m0 (s) = 0
(51)
U pp (s,s,m̃(s))
2
characterises the unique separating equiIn equilibrium m0 (s) = − D
D3 U pp (s,s,m̃(s))
librium. The solution gives the minimum buyer type who maximizes the seller’s
expected payoff. Proof of Proposition 4.1. Since seller cannot change the posted price after
the period one starts, she calculates the price which is constant for all periods and
maximizes her expected payoff. The steps are exactly the same as Proposition 2.1
except in this situation seller calculates the minimum buyer type from the expected
number of bidders who will arrive in all periods until the end of the game. Thus
F1 is now the first order statistics of the total number of buyers who arrive until
period T .
Proof of Proposition 4.2. Starting from the last period seller’s expected
payoff is equal to
Z ω̄
DRD
Ut=T (s, ŝ, mT ) =
[v(x, ŝ) − v0 (s)]f(2) (x)dx
(52)
mT
+ (F(2) (mT ) − F(1) (m))(v(mT , ŝ) − v0 (s))
Differentiating that with respect to ŝ and mT would result to
DRD
∂v
∂Ut=T
=
(F(2) (mT ) − F(1) (mT )) +
∂ŝ
∂ŝ
Z
ω̄
mT
∂v
dF2 (x)
∂ŝ
(53)
DRD
∂Ut=T
= (v0 (s) − J(ŝ, mT ))f1 (mT )
(54)
∂mT
Solving the following differential equation gives us the minimum buyer type for
the last period
(v0 (s) − J(ŝ, mT ))f1 (mT )
(55)
(mT )0 (s) = − ∂v
R ω̄ ∂v
(F
(m
)
−
F
(m
))
+
dF
(x)
T
T
2
(2)
(1)
∂ŝ
mT ∂ŝ
Doing the same steps for all other period until period one gives us the set of T
differential equation and the solution to each of them characterizes the equilibrium
minimum buyer type at each period. Proof of Proposition 4.3. Again starting from the last period if there was
full information for each buyer with signal xi the equilibrium bid is v(s, xi ). Then
the minimum buyer who bids has an expected value equal to v(s, mT (s)). The
equilibrium reserve price must be equal to the minimum buyer who bids in the
auction. This argument is the same for all other periods until period one. Thus
the reserve prices are a vector with T elements and each element is the expected
value of the minimum buyer type who bids in the auction at that period. 19
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21