Auctions with an Informed Seller: Disclosed vs

Secret Reserve Prices

Ben Jarman

October 24, 2007

Faculty of Economics and Business

University of Sydney

email: b.jarman@econ.usyd.edu.au

1

Thesis Title: Essays in Auction Design

Supervisor: Dr Abhijit Sengupta

Auctions are typically modelled as a selling mechanism between multi-

ple potential buyers, each with private preference information, and a seller

whose preferences are known. This significantly influences the predictions

of the theory, since the existence of potential gains from trade is commonly

known, and the seller cannot earn any informational rent. Further, the issue

of interdependence between the preferences of buyers and and the seller is

ignored. This thesis studies auction design under two sided private informa-

tion, and the associated repercussions for the auction design objectives of

profit and efficiency. This paper addresses an aspect of the interdependency

issue, namely how a seller who holds private information about the quality

of a good should conduct the auction.

1

Abstract

In recent work Cai, Riley and Ye (2007) have demonstrated that

a seller who holds private information about the quality of a good

can use a disclosed reserve price to signal this information in an auc-

tion. Given that signalling is costly, (reserves are higher than would

be optimal under complete information) it is reasonable that the seller

would benefit from some policy that can transmit similar information

to bidders without the need for such announcements. We examine the

returns to the seller in an English auction from using different types of

secret reserves. Surprisingly, we find that when the reserve acts as a

take it or leave it offer to the winner in the event of non-clearance, full

disclosure is preferable to later announcement of a reserve regardless of

whether the reserve itself is selected before or after the auction. Sale

occurs less often during the auction for a given reserve price strategy

under secrecy, which increases the incentive of the seller to misrepre-

sent her type when she does eventually announce the reserve price,

making signalling more costly than under immediate disclosure. How-

ever, operating the right of refusal (i.e. never announcing the reserve)

dominates full disclosure whenever the latter more efficient at the in-

terim stage. Also, a seller who will eventually announce a reserve

price benefits by assigning the auction winner exclusive rights to ex

post negotiation, a feature that is uncommon in standard models but

protected by law in real estate auctions in NSW.

2

1 Introduction

1.1 Disclosed vs Secret Reserve Prices

In auction theory the reserve price is typically treated as publicly known.

However, in many market situations bidders do not know the reserve when

they are bidding, and may even be unsure as to whether one exists. Ashen-

felter (1989) comments on this phenomenon in auctions for wine and art;

‘...every item is hammered down and treated as though it were sold. Only

after the auction does the auctioneer reveal whether and at what price the

item may have actually been sold. In short, the auctioneers do not reveal

the reserve price and make it as difficult as they can for bidders to infer it.’

This behaviour is also common in auctions of real estate in New South Wales.

The NSW Office of Fair Trading website reports that1 ‘the seller will nom-

inate a reserve price which is usually not told to the interested buyers.’

Within the standard framework of Independent Private Values (IPV)

models the analysis of auctions is unaffected by the specification of a se-

cret reserve2. In particular, these models ignore the fact that the seller’s

pricing decisions (and announcements of those decisions) can influence buy-

ers’ opinions regarding the quality of a good. In many situations the seller,

as a market expert or as custodian of the object holds private information as1Source listed in references.2The reason is that the seller’s type is known (usually normalized to zero), so bidders

can infer the reserve she will select anyway. In some formats, for example the Second

Price Sealed Bid and English auctions, bidders may not even take this step. They have

a dominant strategy to bid their valuation regardless, provided they do not incur any

participation cost, see Vickrey (1961).

3

to its quality3. To the extent that the seller’s valuation, and thus the reserve

price she will set are increasing in this signal, an announced reserve price

represents an (indirect) announcement of this information. A celebrated

result in auction theory seems applicable here4;

Theorem 1.1 Linkage Principle: If signals are independent and the lowest

type of bidder pays nothing, any two auctions with the same allocation rule

generate the same expected revenue.

A casual application of this result suggests that under the above conditions

the seller should have no preference between announcing her private in-

formation and keeping it secret, provided bidders interpret announcements

correctly so that the ordering of bidders (and thus the allocation rule) is

preserved5. We demonstrate that this intuition is incorrect. The allocation

rule of an auction specifies not only an ordering of players (i.e. an auction

winner), but defines the rules of trade between the buyer and seller, that

is, the reserve price. The problems of reserve price selection and informa-

tion disclosure cannot be separated, because they lead to different incentive

constraints for the seller, and therefore a different allocation rule. When

the seller announces a reserve price, she wants to claim a high value. In

order to credibly signal such information the seller needs to take a costly

action, specifically she must announce a higher reserve than she would un-3This is often referred to as the ‘lemons’ problem, and was first addressed by Akerlof

(1970).4The general argument and several implications for revenue rankings in the various

auction forms were originally presented in in Milgrom and Weber, 1982. For a formal

statement and proof see Krishna (2002).5Indeed this implication was noted by Milgrom and Weber (1982) for the case of ver-

ifiable information. Here we address the case in which there is no impartial third party

who can be employed to verify quality.

4

der complete information. The separating equilibrium that describes reserve

price signalling under these conditions has been characterized by Cai et. al.

(2007) for the case in which the reserve is announced before bidding com-

mences. One feature of their results is the presence of adverse selection. The

lowest type of seller loses nothing; she can implement the same strategy she

would under complete information, because announcing the lowest possible

type is always credible. For all other types of seller, a lower level of trade is

realised than would occur if her type were known.

A secret reserve on the other hand seemingly avoids this issue since it

removes the ability of the seller to influence bidding behaviour during the

auction through announcements of her type. The question is when, if at all,

a secret reserve should be announced. In an English auction the fundamental

function of the reserve price is that it represents a posted price in the event

of non-clearance at auction6. If a price will be posted ex post in the event of

non-clearance, then signalling remains an issue for the seller. An interesting

question is whether the ability to postpone the announcement (and possibly

selection) of a reserve is a useful tool for the seller. We compare the ex ante

expected profits to the seller from the separating equilibrium that involves

full disclosure of a reserve price to those that involve announcement after

the auction.6At least, this is the treatment the reserve usually receives in auction theory. It is this

property that leads the seller to act like a monopolist, setting a price above her private

valuation (or marginal cost), see Bulow and Roberts (1989). This does seem to correspond

to common practice in real estate auctions, though perhaps not in those for wine and art

which are usually run through auction houses (Recall the previous quote from Ashenfelter,

1989). This distinction is important and shall be discussed later.

5

We show that for a given reserve price strategy ; (1) Bids clear the reserve

less often at auction under secrecy, but (2) After accounting for transactions

after the auction, the allocation rule is preserved (i.e. the LP can be ap-

plied) and both auctions generate the same revenue. (1) and (2) together

imply that sale occurs more often through ex post offers under secrecy. This

greatly simplifies the analysis because we can examine the interim effect of

increases in the share of profits from ex post offers on reserve price selection.

The intuition of these results is as follows. For any given reserve price strat-

egy bids are less likely to clear the reserve at auction under secrecy because

they involve a conditional expectation over the seller’s signal given that the

reserve has been cleared. Alternatively, in the separating equilibrium un-

der disclosure, players learn the seller’s information before the auction and

so bid their valuations. Bids under secrecy therefore incorporate bad news

at the margin (i.e. when a valuation would just meet the reserve) that a

bid under disclosure does not. This shifts more importance to the reserve

price announcement under secrecy, increasing the incentives for the seller to

misrepresent her type, and thus makes signalling even more costly in equi-

librium. The Linkage Principle dictates that revenue equivalence between

secret and disclosed reserve auctions must hold if we concentrate on a partic-

ular reserve price rule and presume it constitutes a separating equilibrium.

We can therefore fix the allocation probability in a separating equilibrium

and analyze the effect of such differences on reserve price selection. This

reduces the revenue ranking problem to an interim comparison of reserve

price strategies under secrecy and disclosure. Later announcement leads to

higher reserve prices which is detrimental to profits.

6

These results beg the consideration of one final auction rule. The seller

may benefit by dispensing with ex post offers and simply choosing to ac-

cept or reject the auction price. We shall describe such an auction rule as

the seller’s Right of Refusal. Bidders facing this rule will still condition on

clearing a secret reserve, but the choice of reserve price will be much lower

than when signalling is present; the seller has a dominant strategy to ac-

cept any price above her reservation value. She therefore faces a tradeoff

between allocating more often, and exercising less price setting power. We

show that whenever the equilibrium under full disclosure is incentive effi-

cient (so that the lowest type of seller always allocates the object), then

the Right of Refusal is best. Our results therefore help to explain Ashenfel-

ter’s (1989) observation about the pervasiveness of secret reserves in sales

run by auction houses, in particular why they appear to take the extreme

form of the Right of Refusal. We also demonstrate the importance of the

assignment of winner’s rights in secret reserve auctions that involve ex post

offers. Specifically, the seller benefits by giving the auction winner exclusive

rights to purchase after the auction, despite the fact that ex post she would

achieve the same revenue by posting this offer and selecting a willing buyer

at random. This is of particular interest since it is not a feature of standard

auction models, but is a right that is protected by law in NSW real estate

auctions; ‘the highest bidder is the purchaser, subject to any reserve price.’7.

We proceed with some definitions and a description of the auction games,

followed by an example to demonstrate the intuition of the results. The fol-

lowing section characterizes bids under a secret reserve and expected profits

for a given reserve price strategy. In Section 3 we revisit the model of Cai et.7Again, see the Department of Fair Trading website, address in references.

7

al. (2007) and then develop conditions for a revenue ranking. We then de-

rive equilibrium reserve prices under secrecy and verify that Full Disclosure

is optimal. Section 5 discusses these results, as well as the importance of the

assignment of winner’s rights. We then give conditions such that the Right

of Refusal dominates Full Disclosure and the other Secret Reserve auctions.

Section 6 concludes.

1.2 Preliminaries

Call the set of players I = {0, 1, ..., N}, where Player 0 is the seller, and

I/{0} is the set of bidders8. Player i ∈ I receives a non-negative real valued

signal labelled {S, X1, X2, ..., XN} respectively from the space [θ, θ], where

θ ≥ 0. We shall suppose that signals are statistically independent, and are

distributed (symetrically) according to Fi(.) = F (.) with density fi(.) =

f(.)9.

Bidders’ valuations are symmetric and non-negative, given by Vi = v(Xi, S).

The function v is strictly increasing in both a bidder’s own signal and in that

of the seller, and is separable in these two components. For unknown S, the

expected value of a player when S ≤ s is

w(x, s) ≡ E[v(x, S)|S ≤ s]

The seller’s valuation is v0(S). It is useful to define for a strictly increas-

ing reserve price strategy R(.) and strictly increasing symmetric bidding

function βR(.) the filters mR and mR where

mR(s) := inf{x : v(x, s) ≥ R(s)}8Throughout the number of bidders N is fixed and this is known to all players.9Henceforth realisations of a variable will be written in lower case, uppercase letters

denote the random variable itself.

8

and

mR(s) := inf{x : βR(x) ≥ R(s)}

These represent the lowest bidder types such that a valuation or equilibrium

bid βR(x) respectively will clear the reserve given seller type s, and reserve

price strategy R(.). We shall return to verify the existence and properties

of such points in specific settings later.

1.3 The Game

A single indivisible object is sold by means of an English auction, to be

modelled in the familiar ‘clock’ format10. The auction begins with the price

at 0, then ascends with bidders indicating their participation at each level.

A bidder who is willing to buy the object at any price p is said to be ‘active’

at p. Bidders exit publicly, voluntarily and irrevocably11. As soon as only

one bidder remains, the auction ends and this individual is said to have won

the auction. A reserve price acts as a take it or leave it offer to the winner in

the event that the reserve has not yet been met, otherwise the object is sold

to the winner at the auction price. The problem for the seller is to decide

ex ante when she will select and announce her reserve price. We compare

the outcomes of three particular strategies of timing;

• (1) Full Disclosure (FD): The seller selects and announces a reserve

price upon observing her type and before the auction commences.10This set-up has become standard in the literature, see Milgrom and Weber (1982).11These assumptions are merely for simplicity and have little bearing on our results.

Irrevocability is only a constraint when bidders learn new information during the auction

that makes it worthwhile to re-enter. In the equilibria of the auction games we consider

this is never occurs.

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• (2) Envelope (EN): A reserve is selected by the seller upon observing

her type and before the auction game, but is only announced once the

auction is over. The seller commits to this reserve price, sealing it in

an envelope and giving it to the auctioneer to be revealed later.

• (3) Wait and See (WS): The seller selects and announces a reserve

price after the auction ends.

We shall call the EN and WS games Secret Reserve Auctions, since the

particular value of the reserve is unknown to bidders during the auction.

However, we shall assume that the seller’s choice from the above policies

is known to bidders12. Under Full Disclosure a reserve price is announced

before the auction, and this transmits information to bidders that will influ-

ence their beliefs about the value of the object, and hence their behaviour

in the auction and ex post. In any such signalling game players on the unin-

formed side of the market are wary of the incentive for all types of informed

agent (the seller) to claim favourable information. However, since higher

types of seller (those with a higher reservation value) would set a higher

reserve under complete information, there is scope for signalling to occur in

equilibrium13. Cai et. al. (2007) have verified the existence of a separating

equilibrium under Full Disclosure (FD), and analyzed its properties14. The12This distinction is necessary so that bidders can distinguish between the EN and WS

auctions.13This property is commonly referred to in the literature as ‘single crossing ’. Specifically

the condition says that the indifference curve that relates the cost of announcing a higher

type (here lost sales due to higher reserves) to its benefit (higher valuations and hence

bids) must be decreasing in the seller’s true type. Signalling can therefore be credible in

equilibrium because a higher type of seller loses less by raising the reserve.14They demonstrate that a consequence of signalling is that the reserve price is increas-

ing in the number of bidders, in contrast to the optimal reserve price under complete

10

most important feature of this equilibrium is that reserve prices are higher

than would occur under complete information; this is the costly action the

seller must take to convince bidders of her type. We characterize equilibrium

for Secret Reserve Auctions, and find somewhat surprisingly that regardless

of when the particular value of the reserve is selected (EN or WS), this

policy is never as good as Full Disclosure.

We show that even when bidders do not know the reserve price, they still

incorporate some information about the seller’s signal, since they condition

their bid on clearing the reserve to avoid the winner’s curse. The conse-

quence of this is that expected profits are the same for a given reserve price

strategy in the FD and Secret Reserve Auctions, in accordance with the

Linkage Principle. However, this equivalence belies a shift in the source of

profits to the seller. We demonstrate that bidders are less likely to clear any

given reserve price during the auction under secrecy, since at the margin

(i.e. for seller signal values such that a given bidder’s valuation would just

reach the reserve), her bid under secrecy will incorporate ‘bad news’ that

would not be present under complete information. Thus there is a greater

probability that the reserve will determine the price than under FD for any

given probability of allocation, which increases the incentive for the seller to

misrepresent her type. Secrecy therefore requires the seller to select an even

higher reserve to credibly signal her type, so Full Disclosure is preferable.

We now proceed with an example that demonstrates the flavour of our re-

sults, followed by a more thorough analysis of bidding behaviour under a

secret reserve price.

information which is independent of N .

11

Example 1: A single indivisible object is offered for sale by a seller to N

bidders by way of an English auction. Let these players receive signals S,

X1, X2 etc. that are all distributed identically, independently and uniformly

on [0, 1]. The bidders have symmetric valuations of the form

Vi = Xi + S

the seller’s private value is S, and suppose that she always uses the known

reserve price strategy R(S) = 32S. Assume for simplicity that, announced

or otherwise, the seller simply accepts all prices at or above R(s), otherwise

she refuses sale15. We are interested in comparing the expected profits to

the seller when this reserve is announced to that when its particular value

is secret. Under disclosure bidders have ‘complete’ valuations, that is, they

know that s = 23r, and so in an English auction they can do no better than

to bid their valuation16. Then for a bidder of type x it is a dominant strategy

to bid

b = v(x, s) = x + s

How would the same bidder bid if this reserve were secret? At any particular

price p one can evaluate the expected value conditional on winning the object

at p subject to the reserve price strategy R(S). This is given by

w(x,R−1(p)) = x + E[S|S ≤ 23p]

It is a symmetric Nash equilibrium17 for a bidder with signal x to stay active

until the price p = w(x,R−1(p)). From our distributional assumptions, we15This format implements a reserve-like mechanism through the seller’s Right of Refusal,

which will be of interest later.16This fact is well known, see Vickrey (1961).17We shall return to treat equilibrium bidding under secrecy more rigourously later.

For this particular case, we refer the reader to Milgrom and Weber’s (1982) equilibrium

12

can write

p = x +23.p

2

p =32x

which is of course not a function of s18. It follows that the ranking of bids

will depend on the particular realisation of the seller’s signal. For example,

taking s = 0 bids are higher under secrecy since 32x > x, while for s = 1

valuations exceed bids under secrecy. However, note that the reserve is

cleared more often under disclosure for every s, since this occurs whenever

x ≥ s2 under disclosure, while under secrecy we require x ≥ s. When we take

expectations over all values of s, it is the latter property, the probability of

allocation, that determines the revenue ranking, rather than the expected

value of bids. To see this, compute the expected revenue to the seller when

the object is sold under secrecy. Noting that the secret reserve is cleared

at auction when the second highest bidder’s type x ≥ s19, and calling f2(x)

the density of the second highest bidder type, we can express this expected

revenue as

Π(Secret) =∫ 1

0

∫ x

0

32xf(s)f2(x)dsdx

of the second price sealed bid auction with interdependent values. Our case only differs

from their analysis due to the asymmetry between the bidder and seller strategies and

valuation functions here, so that the winner’s curse information cannot be summarized

solely by monotonicity in types.18Note that in this example the winner’s curse is present for all but the very highest

type of bidder, since clearing the reserve implies s < x, which is a positive probability

event for all but the highest bidder type. This is by no means a general property. In

other cases some (high) bidder types may always clear the reserve and hence encounter

no winner’s curse. These types would bid their unconditional expected value.19i.e. when the second highest bid, the price at which the second last bidder drops out

32x is greater than 3

2s.

13

=∫ 1

0

32x2f2(x)dx

For the disclosure case, we have

Π(Disclosed) = Π(s

2≤ x ≤ s,Disclosed) + Π(x ≥ s,Disclosed)

Concentrating on the second term above, we can perform a similar calcula-

tion to that for secrecy over the set of types x ≥ s;

Π(x ≥ s,Disclosed) =∫ 1

0

∫ x

0(x + s)f(s)f2(x)dsdx

=∫ 1

0[xs +

s2

2]x0f2(x)dx

=∫ 1

0

32x2f2(x)dx

= Π(Secret)

Since Π(x ≥ s,Disclosed) is only a fraction of profits under disclosure, we

have found that disclosure is preferable. The explanation for this is intu-

itive. Bids under secrecy incorporate a conditional expectation over s. If we

integrate over all outcomes in which this expectation is correct, i.e. when

the reserve is cleared under secrecy (x ≥ s), we achieve the expected value

of receiving the second highest valuation over these events. The same ex-

pected profits would be achieved for signals in this region under disclosure,

since bids are equivalent to valuations. So the revenue ranking is determined

by the fact that the good is allocated more often under disclosure. In this

simple example we have ignored the critical issues of post auction offers and

equilibrium reserve price selection20. Regardless, the property that the rev-

enue ranking is determined by the probability of allocation is unaffected by

these complications. We now present a generalization of this result formally.20And indeed, the relationship between the two.

14

Result 1.1 In an English auction with bidder valuations Vi = v(x, s), if a

secret reserve price is higher than would be announced under Full Disclosure

for every s, then Full Disclosure is optimal.

Proof: See Theorem 3.3.1.

Our analysis will indeed show that higher reserve prices are selected under

secrecy, so that Full Disclosure is optimal. In the above example, our sim-

plifying assumption that the reserve acts only as a right of refusal meant

that the reserve is cleared more often under Full Disclosure. As we shall see

in the next section, this property is preserved when the reserve also acts as

a take it or leave it offer to the winner in the event of non-clearance.

2 Bidding Under Secrecy and Expected Profits

2.1 Bidding Under Secrecy

We now present a general treatment of bidding under a secret reserve price.

The analysis here is restricted to strictly increasing reserve price strategies

in which the probability of accepting the post auction take it or leave it offer

is increasing in x and decreasing in s. Recalling the definition of mR(s);

mR(s) := inf{x : v(x, s) ≥ R(s)}

we require mR(s) to exist and be strictly increasing for every s, as similarly

for its inverse; ∃sR(x) := sup{s : V (x, s) ≥ R(s)} that is increasing for

every x. Later in Sections 3 and 4 we shall verify these properties in the

appropriate auctions. We now turn to the analysis of bidding behaviour

in the Secret Reserve Auctions. A symmetric equilibrium of an English

auction is a function β(.) that specifies for every bidder type X a price

15

β(X) at which she will withdraw from the auction. For a given strictly

increasing reserve price strategy of the seller R(.) and associated symmetric

equilibrium bidder response βR(.), we can express the expected profit of

bidder type x from announcing type z when all others announce truthfully

as21;

E[Π(z, x)] =∫ z

θ

∫ R−1(βR(y))

θ[v(x, s)− βR(y)]f(s)fY (y)dsdy

+∫ z

θ

∫ s(x)

R−1(βR(y))[v(x, s)−R(s)]f(s)fY (y)dsdy (1)

Here y and fY (y) represent the realisation and density (respectively) of the

highest type of a bidder’s competitors. In other words, Y is the first order

statistic22 of (N −1) draws from the distribution F (.). The first component

of the above expression represents the expected profits from announcing type

z and winning the auction at a price above the reserve. This involves taking

expected values over the events in which the best competitor’s bid (the price

you will pay if you win at auction) clears the reserve, or S ≤ R−1(βR(y)),

and your bid wins, so that Y ≤ z. This is the ‘bad news’ about the seller’s

signal the winner needs to incorporate in order to avoid the winner’s curse

in this outcome. The second term corresponds to profits from receiving a

take it or leave it offer of the reserve price, which occurs when one wins the

auction, but at a price (highest competitor’s bid) below the reserve. This

event yields surplus to the bidder whenever the reserve is above the auction

price β(y), but below one’s value so that S ≤ s(x), given that R(s) is an

equilibrium announcement of s. Maximizing with respect to z and setting

the first order condition equal to 0 achieves the following equilibrium given21Here we appeal to the Revelation Principle.22Alternativly, the convention in statistics (as opposed to auction theory) is to state

order statistics in increasing order, so Y represents the (N − 1)th order statistic.

16

increasing reserve price strategy R(.)23;

βR(x) = w(x,R−1(βR(x))) +

∫ s(x)R−1(βR(x))[v(x, s)−R(s)]f(s)ds

F (R−1(βR(x)))

The symmetric equilibrium is to bid equal to one’s expected value from

just winning the auction, i.e. at βR(x) = βR(y). The first term of the

expected value is familiar from Example 1, it is the expected value of the

object conditional on winning at a price above the secret reserve. The bidder

adjusts her valuation of the good since beating the reserve implies an upper

bound on the seller’s signal s. The second component represents the option

value of learning the reserve, and hence one’s valuation, from a post auction

offer. Having observed an offer R(s) and learned her valuation v(x, s), the

winner can do no better than to accept the price iff v(x, s) ≥ R(s), or

s ≤ s(x). This term therefore incorporates the expected value of this surplus

when it is positive, weighted by the relative probability of achieving such a

surplus relative to winning at the auction price24.

Since βR(x) is not a function of s, it will typically differ from the true

valuation v(x, s), leading to different bidding behaviour than if the reserve

was disclosed. In particular, bidders will bid too little for high values of s

and too much for low values, as in Example 1. It is relevant to ask whether

bids clear the reserve more or less often here than under Full Disclosure. It

turns out that even though bids can be higher in either auction depending

on the realisation of s, we can show that for a given reserve price strategy

R(s) with associated increasing mR(s), the reserve is cleared at auction more23See the Appendix for the derivation.24The weightings are somewhat implicit because they partially cancel with the denom-

inator in the conditional expectations. We refer the reader to the derivation in the Ap-

pendix for the demonstration and a more thorough explanation.

17

often for every s under disclosure than with the same reserve price strategy

under secrecy, just as in Example 1. Recalling the definition of mR(s);

mR(s) := inf{x : βR(x) ≥ R(s)}

and calling βR(x) the bidders’ symmetric equilibrium response to strictly

increasing secret reserve price strategy R(.), the above statement can be

phrased as follows25;

Lemma 2.1.1 For any increasing R(.),mR(.) and βR(.), if ∃ a bidder value

mR(s) such that βR(mR(s)) = R(s), then mR(s) > mR(s).

Proof:

βR(x) is independent of s and R′(s) > 0, so if mR(s) exists it is unique and

strictly increasing. For any s and its associated mR(s) the reserve price can

be written as R(s(mR)) = v(mR, s(mR)). Now

v(mR, s(mR)) ≥ w(mR, s(mR))

= w(mR, R−1(v(mR, s(mR))))

The weak inequality arises from the event that s(mR) = θ, because v(mR, θ) =

w(mR, θ). In this case the reserve is equally likely to be cleared in secrecy

and disclosure. Since this event occurs with probability zero, we choose

to state the result in the strong form, although it is not necessary for the

result. Note that the strict form of the above inequality is of the form25In the following Lemma we treat the possibility of reserve clearance at any s under

secrecy as an open question. We do not want to preclude the possibility that mR(s) is

defined but mR(s) is not for some values of s, so that allocation will never happen in these

cases under secrecy. This would be the case if for example β(θ) < R(θ) < v(θ, θ). In such

cases it is trivial that allocation occurs more often under FD.

18

p > w(mR, R−1(p)). Substituting mR into the bidding function, we have

βR(mR) =

∫ R−1(βR(mR))θ v(x, s)f(s)ds

F (R−1(βR(mR)))+

∫ s(mR)R−1(βR(mR))[v(mR, s)−R(s)]f(s)ds

F (R−1(βR(mR)))

=

∫ R−1(βR(mR))θ v(x, s)f(s)ds

F (R−1(βR(mR)))+ 0

= w(mR, R−1(βR(mR)))

The second term in βR(mR) disappears because a bidder of type mR gains

zero surplus from receiving an offer at the reserve price; by definition she

would pay exactly her valuation. At this price a bidder with type mR

would have already dropped out under secrecy26. This is of the form p =

w(mR, R−1(p)). In order for the equilibrium response βR(.) to exist, it is

necessary that the function p − w(x,R−1(p)) is strictly decreasing27. But

we have shown that this function is positive at p = v(mR, s(mR)) and 0 at

p = βR(mR). Therefore

v(mR, s(mR)) > βR(mR)

and

R(s(mR)) > βR(mR)

So mR(s) > mR(s) since mR is strictly increasing. Q.E.D.

The only cases that this Lemma does not directly address are those when

mR(s) does not exist for some s and the reserve is never cleared under

secrecy in this region, and when mR(s) is defined but mR(s) is not. In the

former case it is trivial that allocation occurs more often under disclosure,26This is consistent with the fact that a bidder of type mR makes zero profit from

learning the reserve price.27See the derivation of βR(x) in the Appendix.

19

and the latter is impossible because it requires v(θ, s) < R(s), in which case

βR(x) = w(x,R−1(βR(x))), because the ex post offer yields zero surplus, and

we know that w(x,R−1(β(x))) < v(x,R−1(β(x))). If β(x) clears the reserve,

then R−1(β(x)) ∈ [θ, θ], and for some s < R−1(β(x)) we have v(x, s) = R(s)

and the reserve is also cleared under disclosure. It follows that the reserve is

cleared more often at auction for every s under FD for a given reserve price

strategy. The next section demonstrates the significance of the above result

to the revenue ranking.

2.2 Ranking Expected Profits

In order to compare the EN and WS strategies to the FD case, it is helpful

to know how keeping a given reserve price secret affects expected profits.

Despite the reduced probability of sale at auction under secrecy, the ability

to make an offer after the auction ensures that nothing is lost by keeping

the reserve secret, provided the seller can implement the same reserve price

strategy. We demonstrate this result below.

Theorem 2.2.1 For any increasing R(.), mR(.) and mR, secret and dis-

closed reserves yield the same expected profits to the seller.

Proof:

The ex ante expected profit from using the increasing secret reserve price

strategy R(.) and symmetric bidder with equilibrium βR(.) is given below28.

E[Π(Secrecy)] =∫ θ

θ

∫ θ

m[β(x)− v0(s)]f2(x)f(s)dxds

28Henceforth we repress the subscript R where appropriate. Again, for the moment we

assume that m and m are defined for all s.

20

+∫ θ

θ[F2(m)− F1(m)][R(s)− v0(s)]f(s)ds

Again, these terms represent expected profits from sale at the auction price;

when the second highest player type is above m, and at the reserve price;

when that type (and hence the price) is too low for sale at auction, but the

winner’s type is above m so she will accept the ex post offer. Changing the

order of integration in the first term achieves

E[Π(Secrecy)] =∫ θ

θ

∫ R−1(β(x))

θ[β(x)− v0(s)]f(s)f2(x)dsdx

+∫ θ

θ[F2(m)− F1(m)][R(s)− v0(s)]f(s)ds

Substituting in β(x) obtains

E[Π(Secrecy)] =∫ θ

θ

∫ R−1(β(x))

θ[w(x,R−1(β(x)))− v0(s)]f(s)f2(x)dsdx

+∫ θ

θ

∫ R−1(β(x))

θ

∫ s(x)R−1(β(x))[v(x, s)−R(s)]f(s)ds

F (R−1(β(x)))f(s)f2(x)dsdx

+∫ θ

θ[F2(m)− F1(m)][R(s)− v0(s)]f(s)ds

Recalling that w(x,R−1(β(x))) is the expected value conditional on S ≤

R−1(β(x)), and integrating the first two terms over s, we have

E[Π(Secrecy)] =∫ θ

θ

∫ R−1(β(x))

θ[v(x, s)− v0(s)]f(s)f2(x)dsdx

+∫ θ

θ

∫ s(x)

R−1(β(x))[v(x, s)−R(s)]f(s)f2(x)dsdx

+∫ θ

θ[F2(m)− F1(m)][R(s)− v0(s)]f(s)ds

=∫ θ

θ

∫ θ

m[v(x, s)− v0(s)]f2(x)f(s)dxds

+∫ θ

θ[F2(m)− F1(m)][R(s)− v0(s)]f(s)ds

= E[Π(Disclosure)]

21

When we take expectations over values of s such that the bidder’s condi-

tional expectation is correct (i.e. when they are allocated the object), then

we are left with the expected profits that would be made under complete

information in this range. The crucial factor that distinguishes this result

from that in Example 1 is that while it is still true that the reserve is cleared

more often under disclosure, here the seller can still make an ex post offer

when the object fails to sell at auction. Bidder’s then incorporate the value

of receiving this offer into their bid, and are correct in expectation.

In Example 1, disclosure was optimal for a given reserve price rule because

the reserve was cleared more often (by a similar argument to Lemma 2.1.1.),

and our assumption that the seller used the right of refusal meant that

this was equivalent to a higher allocation probability. In the auctions we

consider here, the seller can make an ex post take it or leave it offer to

the auction winner, so that nothing is lost by having a lower probability

of reserve clearance during bidding under secrecy. The allocation rule is

therefore preserved; the object is awarded to the auction winner iff her type

x ≥ m. Given a particular reserve price rule, both auctions allocate in the

same circumstances, so they generate the same expected profits. This result

would also be predicted by the Linkage Principle, which we discussed in

Section 1.129.

Our analysis will demonstrate that in equilibrium Full Disclosure is opti-

mal as a direct consequence of the Linkage Principle and our Lemma 2.1.1.

Given this lemma, the equivalence dictated above means that the share of29It can also be viewed as an application of the Revenue Equivalence Principle of My-

erson (1981).

22

seller profits that arise from sale at the reserve price will be higher under

secrecy for any m and s. This greatly simplifies the analysis by reducing the

revenue ranking to a comparison of interim reserve price setting behaviour.

To this end we now review Cai et. al.’s (2007) treatment of equilibrium

under the benchmark case of Full Disclosure, and develop conditions for a

revenue ranking. In Section 4, we analyze reserve price setting behaviour

in the EN and WS auctions, then apply our conditions to prove that FD is

preferable.

3 Reserve Prices

3.1 Full Disclosure (FD)

Recalling the definition of mR(s) := inf{x : v(x, s) ≥ R(s)}, our task is

to specify a reserve price strategy that results in a separating equilibrium

under FD. Cai et. al. (2007) express this problem as a maximization in

terms of the seller’s announced type s, and the threshold type m subject to

incentive compatibility30, so that m(s) = m(s). The interim expected profit

function for this problem can be written as follows;

U(s, s,m) =∫ θ

m[v(x, s)− v0(s)]f2(x)dx + [F2(m)− F1(m)][v(m, s)− v0(s)]

Here F1(.) and F2(.) again represent the distributions of the first and second

order statistics of bidder types respectively, and v(m, s) = R(s) by definition.

The first component describes the interim expected profit to the seller from

sale at the auction price. This was the focus of our analysis in Example

1. The other component represents profits from the post auction take it or30This follows the methodology of Riley (1979).

23

leave it offer of the reserve price to the winning bidder. Taking the first

order conditions for m and s achieves 31

dU

dm= f1(m)

[v0(s)− v(m, s) +

dv

dx |x=m

(F2(m)− F1(m)f1(m)

)]dU

ds=

dv

ds[1− F1(m)]

Define m∗ to be the be the solution to dUdm = 0 given that s = s. Then m∗

identifies the optimal reserve price under complete information32.

Theorem 3.1 For separable v, there is a unique separating equilibrium un-

der Full Disclosure that is described by the differential equation

dmFD

ds=

dvds [1− F1(mFD)]

f1(mFD)[v(mFD, s)− v0(s)− dv

dx |x=mFD

(F2(mFD)−F1(mFD)

f1(mFD)

)]with initial condition mFD(θ) = m∗

FD(θ), if

d

dx

[v(x, s)− dv(x, s)

dx

(F2(x)− F1(x)f1(x)

)]> 0 (2)

for all s.

Proof: See Cai et. al. Theorem 1 (2007)33.

In this equilibrium, the lowest type of seller chooses the reserve that is

optimal under complete information. It is straightforward to demonstrate

that every other type of seller selects a higher reserve price, and therefore31Here we make use of our assumption that the valuation function v(x, s) is separable

in the private and common value components. Cai et. al. do not require this assumption.32This solution is an analogue of the standard optimal auction reserve, the price that

sets virtual valuations equal to 0, see Myerson (1981).33Cai et. al. show that when signals are independent (2) is satisfied if d2v(x,s)

dx2 < 0 for

all s and the distribution of x is regular.

24

higher threshold m than she would if her type were known. A separating

equilibrium solves34

d

ds[U(s, s,m(s))] =

dU

ds |s=s+

(dU

dm |s=s

)(dm

ds |s=s

)= 0

Now dUds |s=s

> 0 for all s > s, and and dmds > 0 at equilibrium. Therefore

at the equilibrium solution dUdm |s=s

< 0 so under FD seller types S ∈ (θ, θ]

select a higher m than under full information. (2) implies that dUdm |s=s

is

decreasing in m, so it also follows that increasing m further from mFD for

each s reduces the interim expected profit function U(s, s,m(s))|s=s. We

can immediately prove the following theorem.

Theorem 3.1.1 In unique separating equilibria of the English auction with

bidder valuations Vi = v(x, s), if the secret reserve price is higher than

would be announced under Full Disclosure for every s, then Full Disclosure

is optimal35.

Proof:

Recall from Theorem 2.2.1. that the expected profit in a secret reserve

equilibrium is equivalent to that under disclosure for the same reserve price

strategy. Profits in these mechanisms therefore differ only in the sense that

the two auctions may select a different reserve price strategy, and thus a

different m for each s. We have demonstrated that the interim profit function

U(s, s,m(s))|s=s is decreasing in m for m ≥ mFD. Any secret reserve auction

that selects a higher m than under FD for each s therefore reduces the34Again, see Cai et. al.35Again, we prove the case in which mR(s) is defined for every s. For other cases the

proof follows directly by restricting the comparison to seller types where mR(s) is defined,

and noting that FD can do no worse when mR(s) is not defined.

25

integrand of E[Π(Secrecy)] at every s, and results in lower ex ante expected

profits. Q.E.D.

To demonstrate that equilibrium under Full Disclosure (FD) is preferable

to any involving announcement after the auction, it now suffices to verify

that separating equilibria in the Envelope (EN) and Wait and See (WS)

auctions select higher reserve prices than under FD.

4 Equilibrium in the EN and WS Auctions

4.1 Envelope Auction (EN)

In this section we consider the reserve price setting behaviour of a seller who

selects and commits to a reserve price before bidding commences, but seals

it in an envelope (perhaps to be handed to a third party, e.g. an auctioneer)

to be announced after the auction has ended.

The interim expected payoff can be written as follows;

UEN (s, s,m) =∫ θ

m(m)[βEN (x)−v0(s)]f2(x)dx+[F2(m(m))−F1(m)][v(m, s)−v0(s)]

Here βEN (x) is the symmetric equilibrium that describes bidding behaviour

conditional on the equilibrium reserve price strategy that will be selected.

The probability that a bid clears the reserve at auction is summarized by

the filter m, where β(m) = R(s). We shall write this (with a slight abuse

of notation) as m(m), since it is a function of the reserve price rule that is

summarized by m for each s. Following the methodology of Cai et. al, there

exists a unique separating equilibrium given by

dmEN

ds=

[F2(m(mEN ))− F1(mEN )]dvds

f1(mEN )[v(mEN , s)− v0(s)− dv

dx |x=mEN

(F2(m(mEN ))−F1(mEN )

f1(mEN )

)]26

with initial condition m∗EN (θ) equal to the minimum bidder type that would

be selected under complete information provided36

d

dx

[v(x, s)− dv(x, s)

dx

(F2(m(x))− F1(x)f1(x)

)]> 0 (3)

We now show that mFD(s) < mEN (s), which, recalling Theorem 3.1.1 is

sufficient for Full Disclosure to dominate the Envelope auction.

Now m∗EN (θ) > m∗

FD(θ) since both set dUdm = 0 at s = s = θ in their

respective objective functions, and in the envelope auction, we have37

dU

dm |s=s= f1(m)

[v0(s)− v(m, s) +

dv

dx |x=m

(F2(m(m))− F1(m)f1(m)

)]This function only differs from that under FD differ because F2(m) has

been replaced with F2(m(m)). We know that m > m for any m by Lemma

2.1.1, so m∗EN (θ) > m∗

FD(θ). Similarly, dmENds > dmFD

ds because we have

increased the numerator and decreased the denominator by replacing F2(m)

with F2(m(m)). So mEN > mFD for all s, and we can apply the logic of

Theorem 3.1.1; Full Disclosure generates higher expected profit than the

Envelope auction. We briefly withhold a discussion in order to demonstrate

a very similar result for the Wait and See (WS) auction.

4.2 Wait and See Auction (WS)

In the Wait and See auction the seller selects and announces a reserve price

at the auctions end. The seller therefore takes the observed bids of all but

the winner as given, and selects a reserve price with the sole intention of36Refer to the Appendix for the derivation.37We again refer the reader to the Appendix.

27

negotiating with the high bidder38. The interim expected profit is then39

U(s, s,m) =1− F1(m)

1− F1(β−1(p))[v(m, s)− v0(s)]

given that the auction ended at price p. The unique separating equilibrium

is characterized below40

dmWS

ds=

[1− F1(mWS)]dvds

f1(mWS)[v(mWS , s)− v0(s)− dv

dx |mWS

(1−F1(mWS)

f1(mWS)

)]where the initial condition m∗

WS(θ) sets dUdm = 0 at s = s = θ, given that

d

dx

[v(x, s)− dv(x, s)

dx

(1− F1(x)f1(x)

)]> 0 (4)

Assuming that (4) holds, in the WS auction, we know that41

dU

dm |s=s=

f1(m)1− F1(β−1(p))

[v0(s)− v(m, s) +

dv

dx |m

(1− F1(m)f1(m)

)]The only difference between this condition and that for the Full Disclosure

auction is that F2(m) has been replaced by 1. It follows that m∗WS(θ) >

m∗FD(θ). Also, replacing F2(m) with 1 in the dm

ds equation increases the

numerator and decreases the denominator. Thus dmWSds > dmFD

ds , so mWS >

mFD for all s, and by Theorem 3.1.1 Full Disclosure is preferable to the

Wait and See auction.38Of course if the selected reserve price is below the second highest bid she will sell at

the auction price.39The conditional probability in the denominator terms here play no role as the seller

gains no useful information about the high type from observing all others when signals

are independent.40See the Appendix for a full derivation.41Again refer to the Appendix.

28

4.3 On the Increase in Reserve Prices and the Linkage Prin-

ciple

In the EN and WS auctions bidders treat the reserve price as secret during

the auction. For any given reserve price strategy, m > m for all s, so bidders

are less likely to clear the reserve during the auction than if the same reserve

were announced. However, Theorem 2.2.1 (a natural consequence of the

Linkage Principle) demonstrates that for the same reserve price strategy,

this does not affect expected revenue, because sale will occur more often

after the auction to compensate. This means that we can fix a probability

of allocation [1−F1(m)] and analyze the effects of an increase in the share of

profits from ex post negotiation. In the EN and WS games, this shift toward

the bargaining component of the mechanism increases the stakes of selecting

the reserve for the seller, and thus increases her incentive to misrepresent

her type. It therefore requires a higher reserve price to credibly signal the

same information. Conditioning on s = s, we know that reserve prices are

too high under full disclosure, so further increases must reduce expected

profits relative to that auction.

4.4 The Assignment of Winner’s Rights

In the auctions we have studied here, (and in the theory of English auctions

more generally) the reserve acts as a take it or leave it offer to the high

bidder in the event of non-clearance at auction. The fact that winning the

auction assigns one exclusive rights to post auction purchase is neither a

crucial nor commonly noted feature of this set up. This phenomenon is

common however in real world auctions, and is protected by law in real

29

estate auctions in NSW42. It is therefore of interest to note that in our

model, this right is an important feature of equilibrium under secret reserve

prices. To demonstrate, consider a secret reserve price auction in which the

reserve acts as a posted price in the event of non-clearance. Assume that in

this situation the object is won by lottery among the bidders who are willing

to pay the posted price. Given a reserve price strategy R(.), a symmetric

equilibrium is given by43

β(x) = w(x,R−1(β(x)))

Expected profits from winning a posted price lottery after the auction are

independent of one’s announced type during the auction. Bidders therefore

bid the expected value of the object conditioning on clearing the secret

reserve44. This is obviously less than is bid when the winner is assigned

exclusive rights to purchase after the auction, leading to a lower probability

of reserve clearance for a given reserve price rule. Then the reserve price

selected in the EN and WS auctions will be even higher when the seller

uses a posted price after the auction. By an almost identical argument to

Theorem 3.1.1, we then have

Remark 4.4.1 In a secret reserve auction, it is better to assign the auction

winner exclusive rights to the object. That is, it is better to make a take it or

leave it offer to the winner in the event of non-clearance than to use a posted42See Section 1.1.43This can be seen by replacing the second term in the bidder’s objective function (1)

with

Pr[win lottery]×∫ θ

θ

∫ s(x)

R−1(β(y))

[v(x, s)−R(s)]f(s)fY (y)dsdy

which is not a function of z.44This is also identical to the bidding function under a secret reserve price modelled as

the seller’s Right of Refusal, as in Example 1. We return to this in the next section.

30

price. Also, Full Disclosure dominates the EN and WS auctions regardless

of whether the reserve acts a posted price or as a take it or leave it offer in

the event of non-clearance.

The second statement follows from the arguments above, noting that bidding

behaviour is unaffected by the assignment of rights under FD. The reader

may recognize the similarity between bidding behaviour under secrecy with

a posted price and that under the Right of Refusal as in Example 1. While

bidding behaviour is certainly identical in these cases, reserve prices selected

in equilibrium will differ, since under the right of refusal the seller never

needs to make a ‘type’ announcement, whereas a posted price will influence

bidder’s beliefs about the quality of the good, leading to signalling. We now

briefly discuss the possibilities this raises for our revenue ranking.

5 On the Right of Refusal

We have shown that a seller who will eventually announce a reserve price

should do so earlier to signal the same information in a less costly manner.

The possibility remains however that Full Disclosure may be dominated by

a policy of full secrecy. That is, it may be optimal to operate under the

seller’s Right of Refusal (RR) and never reveal this threshold amount to

bidders. There is reason to be optimistic about such a policy, since if the

seller never reveals her information, bidders are never concerned about the

seller’s incentives and costly increases in the reserve price above the complete

information case may be avoided. To the extent that this facilitates a lower

reserve price in equilibrium, allocation may occur more often under secrecy,

rendering Theorem 3.1.1. inapplicable. In fact, it is trivial that in the secret

RR auction the seller has a dominant strategy of R(s) = v0(s), which is

31

certainly lower than under any signalling equilibrium45. The cost of such a

policy is that the seller must sacrifice potential profits by refusing sale when

there is a possibility that the winner would be willing to pay a higher price

if she knew the value of s. The seller’s preference between the FD and secret

RR policies will therefore be determined by the tradeoff between allocating

not enough (costly signalling under FD) and too often (not using monopoly

price setting power under the secret RR).

To investigate this, let βRR(x) and mRR represent the symmetric bidder

equilibrium and threshold value respectively in the secret RR auction. Call

the interim expected profits to the seller in this auction

U(s, mRR(s)) =∫ θ

mRR

[βRR(x)− v0(s)]f2(x)dx

also, recall that in equilibrium under FD we have

U(s, s,mFD(s))|s=s =∫ θ

mFD

[v(x, s)− v0(s)]f2(x)dx

+ [F2(mFD)− F1(mFD)][v(mFD, s)− v0(s)]

We are interested in how these functions change with s. For the RR auction,

we have

d

ds[U(s, mRR(s))] = [βRR(mRR)− v0(s)]f2(mRR)

(dmRR

ds

)+

∫ θ

mRR

v′0(s)f2(x)dx

= 0 + [1− F2(mRR)]v′0(s)

The expression in the square brackets is 0 by the definition of mRR, and the

integral can be evaluated because the seller’s valuation is independent of x.

Performing the same operation for interim profit in the FD auction achieves

45And indeed, is never higher than the complete information optimal reserve.

32

d

ds[U(s, s,mFD(s))|s=s] = [1− F1(mFD)]

dv

ds

+ f1(mFD)[v0(s)− v(mFD, s) +

(dv(x, s)dx

)|x=m

F2(mFD)− F1(mFD)f1(mFD)

]+ [1− F1(mFD)]v′0(s)

=dU

ds |s=s+

( dU

dmFD

)|s=s

(dmFD

ds

)|s=s

+ [1− F1(m)]v′0(s)

= 0 + [1− F1(mFD)]v′0(s)

By a standard envelope theorem argument, infinitesimal changes in the

seller’s type only influence her utility through their direct effect on her pref-

erences in equilibrium. We can therefore write

d

ds[U(s, mRR(s))] >

d

ds[U(s, s,mFD(s))|s=s]

F1(mFD) > F2(mRR) (5)

because v′0(s) > 0, so higher types of seller are more likely to prefer the

RR auction if it allocates more often. Now recall that the interim expected

profit under FD at the lowest seller signal value is equivalent to that which

she would earn under complete information. If valuations are such that the

lowest type of seller always allocates the object, then this expected profit

can be written as

U(s, θ,mFD(s))|s=θ =∫ θ

θ[v(x, θ)− v0(s)]f2(x)dx

By comparison in the RR auction the seller actually earns more than she

would under complete information in this case;

U(θ, mRR(θ)) =∫ θ

mRR(θ)[w(x, v−1

0 (β(x)))− v0(s)]f2(x)dx

>

∫ θ

mRR(θ)[v(x, θ)− v0(s)]f2(x)dx

33

=∫ θ

θ[v(x, θ)− v0(s)]f2(x)dx

where mRR(θ) = θ, because βRR(mRR) > v(mRR, θ) > v0(θ). A seller of

the lowest possible type benefits from secrecy because bidders expect her to

have some interior signal value. We have shown that if the lowest type of

seller always allocates the object under complete information under FD (so

that the probability of allocation is 1 in both FD and RR), then this type

of seller prefers the secret RR auction. Then if (5) holds for all other types,

the seller will always prefer the Right of Refusal auction. It follows that we

can rank expected profits in the FD and RR auctions at the interim stage

(and therefore obviously ex ante) if we can also interim rank them in terms

of efficiency.

This point is worth noting, since under one-sided incomplete information,

there is usually a conflict between efficiency and profit maximization46. Once

we allow both sides of the market private information (and of course the in-

terdependence between valuations), these objectives are realigned47. The

welfare implications of this are obvious, however this also significantly re-

duces the complexity of the design problem for the seller. Under the above

conditions, the auction becomes ‘detail free’ in the sense of the Wilson Doc-

trine. That is, the seller only needs to know her own valuation, then run an

English auction to achieve a superior outcome to that under the extremely

complicated signalling equilibrium of Full Disclosure. Of course, it is pos-

sible that there are still further mechanisms that are more efficient and/or46See Krishna (2002), Chapter 2.47Ausubel and Cramton (1998) show that these objectives also coincide in a Coaseian

environment involving perfect resale after the auction.

34

generate greater revenue than those considered here48. Myerson (1991) de-

velops a result that seems applicable. He shows that if a seller has private

information, any bilateral trading mechanism49 that is interim incentive ef-

ficient allocates with probability 1 when she is the lowest possible type. It

is therefore intuitive that a general result relating global incentive efficiency

to the disclosure problem may be achievable along these lines. This remains

a task for future research.

6 Conclusion

We have investigated the properties of secret reserve auctions in a setting in

which the seller holds valuable information about the quality of the object.

Somewhat surprisingly, we have found that later announcement of a reserve

exacerbates the seller’s incentive problem, so that full disclosure is preferable

to secrecy. The Linkage Principle (in the form of Theorem 2.2.1) played a

crucial role in this ranking, since it allowed us to express the ranking in terms

of interim reserve price selection relative to full disclosure. Conditioning

on truthful revelation of the seller’s type, reserves are too high under full

disclosure. Further increases are therefore detrimental to expected profits.

Our Lemma 2.1.1 proves that any reserve is cleared less often under secrecy,

which increases the importance of ex post sale, and thus the incentives for

the seller to misrepresent her type. Every type of seller must therefore must

select a higher reserve price under secrecy, so full disclosure is preferable.

We have thus shown that Milgrom and Weber’s (1982) result can play a role48For example, we have not considered pooling equilibria, in which a seller announces a

reserve that is constant across her types.49i.e. our model with N = 1.

35

in producing optimal disclosure even when no statistical linkage is present.

However, our conclusions rested heavily on the specification that the reserve

is announced ex post, so that the seller faces the adverse selection problem

eventually. This can be avoided if the seller reserves the Right of Refusal.

If this mechanism allocates at least as often as Full Disclosure, then the RR

auction is preferable. Finally, we have also shown that given that a seller

will announce a reserve after the auction, it is optimal to assign the auction

winner exclusive rights to this offer. This feature is common in practice, but

to our knowledge has not been represented in any other theoretical model.

7 Appendices

7.1 Derivation of βR(x) and Existence

Differentiating (1) with respect to z yields50

dE[π(z, x)]dz

=∫ R−1(β(z))

θ[v(x, s)− β(z)]f(s)fY (z)ds

+∫ s(x)

R−1(β(z))[v(x, s)−R(s)]f(s)fY (z)ds

by Liebniz’ rule. Setting this equation equal to 0 at z = x yields

F (R−1(β(x)))β(x) =∫ R−1(β(x))

θv(x, s)f(s)ds

+∫ s(x)

R−1(β(x))[v(x, s)−R(s)]f(s)ds

or

β(x) = w(x,R−1(β(x))) +

∫ s(x)R−1(β(x))[v(x, s)−R(s)]f(s)ds

F (R−1(β(x)))(6)

50We again suppress the subscript R.

36

This can also be expressed as

β(x) = w(x, R−1(β(x)))+

∫ s(x)R−1(β(x))[v(x, s)−R(s)]f(s)ds

F (s(x))− F (R−1(β(x)))×F (s(x))− F (R−1(β(x)))

F (R−1(β(x)))

the second term is the expected surplus from an ex post offer when it is posi-

tive, weighted by the probability of this event relative to winning at auction.

On the other hand the expected value in the first term w(x,R−1(β(x))) ef-

fectively has a weight of 1. These two weights sum to

F (s(x))F (R−1(β(x)))

> 1

The fact that the equilibrium bid is not a convex combination of expected

values seems to suggest that players are overbidding. In fact, this is not

so; bidders can afford to do this because when the reserve is not cleared at

auction, their bid becomes irrelevant and they have made no commitment,

they can decide later whether or not to buy at a price the seller will decide.

We now turn to the question of existence. At β(x) = p = R(θ), we can

evaluate the right hand side of (5), which gives w(x, θ) + ... = v(x, θ) + ....

Given that s(x) and m(s) exist and are increasing, we know that v(x, θ) ≥

R(θ), so we can say that at p = R(θ), the right hand side of (5) exceeds the

left hand side, so type x is ‘active’ at p = R(θ). Alternatively, at p = R(s(x))

the right hand side is equal to w(x, s(x)) < v(x, s(x)) = R(s(x)). So at

p = R(s(x)) the left hand side of (5) exceeds the right hand side, and type

x would have already dropped out of the auction. Since this expression is

continuous in p, there is a solution to (5) at β(x).

7.2 Derivation of Equilibrium in the EN Auction

Take the first order conditions of UEN (s, s,m(s)) with respect to s and m;

dU

ds= [F2(m(m))− F1(m)]

dv

ds

37

dU

dm= −[β(m(m))− v0(s)]f2(m(m))

dm(m)dm

+ f2(m(m))dm(m)

dm[v(m, s)− v0(s)]

− f1(m)[v(m, s)− v0(s)] + [F2(m(m))− F1(m)]dv

dx |x=m

Following Cai et. al. (Theorem 1), we can verify that the single crossing

condition holds, namely that dds

(dU/dmdU/ds

)> 0, because dU

ds is not a function of

s and dUdm is increasing in s. Recall that a separating equilibrium constitutes

a minimum bidder type m(s) such that dmds > 0, and s = s maximizes

UEN (s, s,m(s)). Setting s = s in the First Order Conditions yields

dU

ds= [F2(m(m))− F1(m)]

dv

ds |s=s

dU

dm= − f1(m)[v(m, s)− v0(s)] + [F2(m(m))− F1(m)]

dv

dx |x=m

= f1(m)[v0(s)− v(m, s) +

dv

dx |x=m

(F2(m(m))− F1(m)f1(m)

)]because β(m(m)) = v(m, s) = R(s). Thus by Theorem 1 of Cai et. al., (3)

is sufficient for the existence of a unique separating equilibrium given by

dmEN

ds= −

dUENds |s=s

dUENdmEN |s=s

=[F2(m(mEN ))− F1(mEN )]dv

ds

f1(mEN )[v(mEN , s)− v0(s)− dv

dx |x=mEN

(F2(m(mEN ))−F1(mEN )

f1(mEN )

)]with initial condition mEN (θ) = m∗

EN (θ).

7.3 Derivation of Equilibrium in the WS Auction

We achieve the following first order conditions;

dU

ds=

1− F1(m)1− F1(β−1(p))

(dv

ds

)dU

dm=

11− F1(β−1(p))

[− f1(m)[v(m, s)− v0(s)] + [1− F1(m)]

dv

dx |x=m

]

38

Again, dUds is independent of s and dU

dm is increasing in m, so the single

crossing condition holds, and at s = s we have

dU

ds=

1− F1(m)1− F1(β−1(p))

(dv

ds |s=s

)dU

dm=

f1(m)1− F1(β−1(p))

[v0(s)− v(m, s) +

dv

dx |m

(1− F1(m)f1(m)

)]Thus by Theorem 1 of Cai et. al., (4) is sufficient for the existence of a

unique separating equilibrium given by

dmWS

ds= −

dUWSds |s=s

dUWSdmWS |s=s

dmWS

ds=

[1− F1(mWS)]dvds

f1(mWS)[v(mWS , s)− v0(s)− dv

dx |x=mWS

(1−F1(mWS)

f1(mWS)

)]with initial condition mWS(θ) = m∗

WS(θ).

8 References

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Ashenfelter, O., “How Auctions Work for Wine and Art”, Journal of Eco-

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Ausubel, L. and Cramton, P., “The Optimality of Being Efficient”, World

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Bulow, J. and Roberts, J., “The Simple Economics of Optimal Auctions”,

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Cai, H., Riley, J. and Ye, L., “Reserve Price Signaling”, Journal of Eco-

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Jullien, B. and Mariotti, T., “Auction and the Informed Seller Problem”,

39

Games and Economic Behaviour 56, 2006.

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