Auctions in Markets: Common Outside Options and … in Markets: Common Outside Options and the Continuation Value E ffect∗ Stephan Lauermann Gabor Virag June 22, 2010 Abstract We

Auctions in Markets: Common Outside Options and the

Continuation Value Effect∗

Stephan Lauermann Gabor Virag

June 22, 2010

Abstract

We study auctions with endogenous outside options determined through actions

taken in the aftermarket. We show that endogenous outside options have important

consequences for auction design. In contrast to the case of exogenous outside options,

auctions with less information revelation may yield higher revenues. Opaque auctions

decrease the information available to losing bidders, which leads to worse decisions in

the aftermarket. This leads to worse outside options, and thus more aggressive bidding

in the original auction. The timing of information revelation is important: it is never

optimal to reveal information after the auction, while it may be to optimal to reveal

information before the auction.

∗University of Michigan, University of Rochester. We would like to thank Tilman Borgers, ThomasJeitschko, Philipp Kircher, Tymofiy Mylovanov, Peter Norman, Marco Ottaviani, Romans Pancs, Lones

Smith, and various audiences for their helpful comments. All remaining errors are ours.

1

1 Introduction

There are many auctions where unsuccessful bidders may purchase a similar good in an ex-

ternal market. Examples include bidding for a consumer item on e-Bay, Treasury auctions,

electricity auctions and timber auctions. For example, a loser of an auction for golf clubs

on e-Bay can buy an identical or very similar item in an auction offered by another seller.

A main characteristic of such auctions is that bidders need to from beliefs about the state

of the outside market (the next auction on e-Bay) in which they participate upon losing the

original auction. These beliefs play two roles: they directly influence the perceived value

of the outside options, and they determine how a bidder behaves in the outside market.

The existing literature has focused on the first role of those beliefs. However, if the losing

bidder is not a price-taker on the outside market, then the second role of the beliefs become

important. For example, on e-Bay the availability and the price of the product in future

auctions are determined endogenously through actions taken in those auctions.

The fact that information learnt in the auction may be valuable in subsequent market

interaction has so far been overlooked in the literature on auction design. We show that by

studying auctions with endogenous outside options, one may obtain important new insights

into auction design. Most importantly, we show that may prefer non-transparent auctions

as releasing information about bids may decrease revenues. To form intuition, imagine

that the losing bidder’s optimal decision in the post-auction market depends on market

conditions like market tightness. If the other bidders or the auctioneer have information

on the state of the market, then valuable information may be learnt in the auction that

helps a losing bidder to make a better decision in the post-auction market. A transparent

auction allows a losing bidder to obtain precise information about the signals of the other

bidders and the auctioneer, which improves his outside option. Consequently, the bidders

bid less in a transparent auction, an outcome we refer to as the continuation value effect.

Our results provide a new explanation why intransparent auctions arise in markets like

e-Bay auctions. On e-Bay, it is difficult to gather information about aggregate market

conditions, such as information about past ending prices of close substitutes. In fact, in

Germany, it was for a long time impossible to search for past auctions on e-Bay, see Sailer

(2008).1

1Sailer also documents that losing buyers change their bid over time (they increase their bids on average),

a stylized fact that is consistent with our model in which losing bidders’ actions depend on information

learned through participation in auctions. Another example for opaque markets is the over-the-counter

market for corporate bonds. This market was traditionally opaque, but in 2002, the market underwent a

2

We introduce a stylized model to study bidding and auction design with endogenous

outside options. The following is a special case of our model. There are two risk neutral

bidders who participate in a second price auction with a post-auction market. Both bidders

have the same known valuation for the good that is sold in the auction. After one of two

possible states of the market is realized, bidders receive private signals that are correlated

with the true state, and submit their bids. The winner obtains the object, while the loser

participates in a decision problem that we interpret as a reduced model of a market. A

prime example is when the losing bidder can make a take it or leave it offer to a seller he

is randomly matched with.2 The payoffs in the decision problem depend on the state of

the world ∈ {} and the action that is taken: payoffs are () in the high state

and () in the low state. If the losing bidder believes that the probability of the high

state is , the maximized continuation payoff is (). The fact that there is a value of

information in the decision is captured formally by being strictly convex, as opposed to

being linear.

The auctioneer can affect the value of the outside option by controlling the amount

of information released in the auction. This has ramifications for optimal auction design

as captured by the continuation value effect. Building on this insight, we show that the

auctioneer may prefer opaque auctions. In particular,

• the auctioneer can benefit from withholding information before the auction;

• the auctioneer unambiguously prefers auctions that reveal as little information aspossible to losing bidders after the auction;

• the revenue from a second-price auction can be larger than the revenue from an

ascending price auction.

To discuss these results, and understand why opaque auctions may or may not enhance

revenues, it is instructive to compare our model with standard models of common value

auctions. The fact that common outside options introduce common value elements has

fundamental change when the Transaction Reporting and Compliance Engine (TRACE) was introduced;

see Bessembinder and Maxwell (2008).2 In Lauermann, Merzyn, and Virag (2010), we consider a fully structural dynamic model of a market in

which a losing bidder can participate in a sequence of auctions. Expected payoffs depend on the bid and on

the unknown aggregate number of buyers and sellers in the market. We show that the current model with

a single auction followed by a decision problem is the precise reduced form for Lauermann, Merzyn, and

Virag (2010), i.e., the bidders’ problem in the auction stage is strategically equivalent for given continuation

values.

3

been noted before3. Our model has features that resemble common value auctions in two

ways. First, the bidders have a common outside option and thus a common valuation in

the original auction. Second, the value of each bidder’s outside option depends on the

likelihood of each state, so each bidder’s utility depends on the other’s signal. However,

the case of endogenous outside options is not fully captured by common value auctions.

Formally, the outside option is endogenous if is strictly convex (as opposed to being

linear). When this is the case, our model is not embedded in the canonical model of

common value auctions as formulated in Milgrom and Weber (1982).4

The main insight into auction design for common value auctions is the optimality of

"transparent" auctions as implied by the linkage principle. The linkage principle implies

that the auctioneer should reveal his information about the object, and also reveal the

bids placed. As a corollary, an ascending auction is preferable to a second price auction.

When information has no value, and thus is linear, the linkage principle applies to

our model and transparent auctions yield higher revenues. The intuition for the linkage

principle is that transparent auctions reduce the problem of winner’s curse which leads to

low bidding. More precisely, revealing extra information alleviates worries that a bidder

only won because all other bidders had low signals and winning is not profitable.

Our bullet points show that the insight of linkage principle may be overturned when

outside options are endogenous. The first two points imply that it matters whether infor-

mation is released before or after the auction. If a piece of information is only revealed

after the auction (like the winning bid), then the linkage principle effect is absent since

the bidders cannot incorporate such an information into their bids. On the other hand,

the continuation value effect is present, since such an information increases outside op-

tions. Therefore, revealing information after the auction lowers revenues. If information

is revealed before the auction or during the auction (like in an English auction), then the

linkage principle operates to provide a positive revenue effect. However, the continuation

value effect is still present, since any information learned before (or during) the auction

can be used to improve the decision after the auction. Since both of the opposing effects

are present, in general one cannot tell whether transparent or opaque auctions yield higher

revenues. We provide three results to assess this trade-off. First, if the continuation value

3See for example Milgrom and Weber (1982), Goeree and Offerman (2002) who argue that auctions with

post-auction markets can be modelled as common value auctions.4 If is a linear function, then the value of the outside option depends only on the state, and not on the

action taken. Then the auction corresponds to a pure common value auction as studied in Milgrom and

Weber (1982).

4

in one state is much larger regardless of the decision taken, then the linkage principle ef-

fect dominates, since avoiding the winner’s curse is key for increasing revenues. Second,

if the continuation value is very sensitive to the decision taken ( is "very convex"), then

the continuation value effect dominates. Third, if bidders have imprecise signals, then the

winner’s curse is less of an issue, and the linkage principle effect is weak. Then revealing

information only serves to reduce revenues via the continuation value effect.

1.1 Literature review

Our main observation is the fact that, if auctions are embedded in markets, an auctioneer

has incentives to manipulate the information flow because this changes the bidders prefer-

ences in the current auction. The literatures on competing mechanisms5 is related to our

paper, since it also studies auctions embedded in a larger market. In fact, this paper and

the companion paper (Lauermann, Merzyn and Virag (2010)) provide an alternative frame-

work to discuss some interesting problems of auction design when the auctioneer competes

with other sellers on the market. The literature on auctions with resale6 is also related,

because it takes post auction market interaction between bidders seriously. However, we

abstract from strategic interaction of the bidders on the stage following the auction, and

instead concentrate on general features of broader market generated endogenous outside

options. This is in stark contrast with the literature on auctions with resale where the

same bidders interact in the resale stage as in the original auction. A more similar setup

is that of Mezzetti et.al. (2004) who consider a setup where a single seller has multiple

objects, and needs to decide how much information to release about the first auction before

the second auction takes place. Although, the setup is similar the key trade-offs turn out

to be quite different in their work, and it is not possible to describe the intuition using the

effects described in our paper.

A recent literature initiated by Duffie et. al. (2007) studies information percolation in a

system where parties disclose all information to each other. Even if the sellers could decide

how much information to reveal the linkage principle suggests that all information would

be released in the standard setup of exogenous continuation values. Therefore, if society

prefers transparent trading mechanism7, it could rest assured that auctioneers have private

5See for example McAfee (1993) and Peters (1997) among other papers.6See for example Hafalir and Krishna (2008), Cheng and Tan (2009) and Garrat and Troger (2007).7 It was suggested by Duffie et al that efficient (and quick) information percolation is important for the

efficiency of markets such as decentralized over-the-counter trading of financial assets.

5

incentives to offer such auctions. However, with endogenous outside options incentives to

hide information by employing non-transparent mechanisms may exist, as our model shows.

In this case the speed at which information is transmitted in the economy is decreased,

and the market may operate in a less transparent way if left alone.

The two distinct roles of information in a dynamic game has been considered by the

corporate finance literature as well. Holmström and Tirole (1993) considers two types

of information in the corporate governance context. Strategic information is valuable to

make better decisions (after the takeover), while speculative information just relates to

the value of the firm as a consequence of past actions. In our paper the same piece of

information takes on both roles to determine equilibrium willingness to pay for the object.

Goldstein and Guembel (2008). They show that uninformed traders have an incentive

to manipulate the informativeness of the stock price, which changes the firm’s optimal

investment decision. In other words, they acknowledge "the presence of a feedback effect

from the financial market to the real value of a firm", which is an instance of the strategic

value of information. While these papers point out the two possible uses of information,

but ours is the first paper to analyze those roles from the point of view of mechanism

design.

2 Model and preliminary analysis

2.1 Setup

The interaction unfolds in three stages. First, the auctioneer and the bidders receive sig-

nals about the state of the world. Second, the auctioneer runs an auction for an indivisible

object. Third, each of the losing bidders chooses an action in a decision problem.

Information. There are two possible states of the world, ∈ {}, and the realizationis not observed by the bidders. The probability of the high state is 0. The state of the

model is interpreted as the unknown aggregate market condition. The bidders receive

private signals that are correlated with the state, these signals are denoted by 1 2 .

For simplicity, we assume that the auctioneer’s signal 0 is perfectly correlated with the

state, that is 0 = 1 if = and 0 = 0 if = . In state the bidders’ signals

are distributed independently according to , so, the bidders’ signals are conditionally

i.i.d.. We assume that admits a continuous density function . With a signal , the

Bayesian posterior probability of the high state is 0 () ((1− 0) () + 0 ()).

6

We assume that the distributions are such that the signal is equal to the posterior =

0 () ((1− 0) () + 0 ()); this assumption is without loss of generality.

Auction. All bidders participate in the seller’s auction where a single indivisible good

is for sale. We analyze bidding in standard auction formats, including the first-price, the

second-price, and the ascending (English) auction. In order to illustrate the main ideas,

we often assume that the seller uses a second price auction without reservation price.

Equilibrium in the second price auction is generally easy to characterize and provides clean

intuition. It is worthwhile to point out that although the valuations do not depend on the

signals, but the bids may, because the signals influence the beliefs about the state, which

influences the option value from losing as it is described below. Moreover, since bidders

generally have different beliefs about their outside option, they submit different bids in the

auction.

Preferences. The winning bidder receives the object and pays price , while the losing

bidders do not make any payments in the auctions studied. The valuation for the object,

, is the same for all bidders and publicly known. The payoff is for the seller and −

for the successful bidder. Finally, we assume that the bidders are risk neutral, that is they

maximize their expected payoffs. Note, that the state affects the value of the outside option

(as described below), but not the value of winning.

Outside Option: Decision under Uncertainty. The outside option consists of a decision

problem under uncertainty. A losing bidder takes some action ∈ = [ ]. Depending

on the state, payoffs are () or (). We impose the following regularity conditions

on these utility functions. First, are twice differentiable. Second, they are both

strictly concave. Let denote the posterior of a losing bidder. The posterior depends on

the signal, the information contained in the fact of having lost with a certain bid, and infor-

mation learned before and after the auction. Let () ∈ [ ] be the optimal decision rule, () = argmax

∈[] ()+(1− ) () where existence of an optimal decision follows from

Weierstrass’s theorem, and uniqueness from the strict concavity of the utility functions. De-

note the maximized payoff, the value function by () = ( ()) + (1− ) ( ()).

It follows from standard arguments from information economics that the value function

is convex. Unless otherwise noted, we assume that is decreasing in beliefs, i.e., the high

state provides lower outside option for the (losing) bidders.

Digression: A structural model of the decision problem

Let us provide a fully structural model that leads to a continuation value of the form we

7

proposed. After the auction is conducted the loser faces a potential seller with probabilities

and with 0 ≤ ≤ ≤ 1, otherwise the continuation value is zero. The seller’scost (reservation value) is distributed according to cdf and in the two states where

both admit continuous, strictly positive densities. To simplify exposition, assume

that their common support is [ ] with ≤ . This assumption ensures that the

buyer buys with probability strictly between 0 and 1. To obtain a perfect correspondence

with our formulation, focus on the case where the losing buyer makes a take it or leave it

offer to the seller he is matched with.

The buyer has a non-trivial action to make, while the optimal action of the seller

contacted is very easy to characterize: he accepts a price offer if and only if it exceeds

his cost, that is if ≥ . Let us now formally describe the optimal decision of the buyer.

Suppose that a buyer whose value of the object is makes a take it or leave it offer .

Then his utility is if his belief attaches probability to the high state is

= ( − )(() + (1− )())

To obtain the notation from the model description above, let () = ( − )() and

() = ( − )(), and thus = () + (1− )(), a linear function in .

Given our assumptions about the distribution functions the optimal price offer has to

satisfy (an interior) first order condition, and thus

( − )(0() + (1− )

0()) = () + (1− )() (1)

To capture the idea that being in the high state is less profitable for the buyer let assume

that for all it holds that () ≤ () implying that in the high state the seller is less

likely to accept. For technical purpose we need the stronger assumption of reverse hazard

rate dominance that requires that0

0. Then we have the following result after a

matter of simple algebra:

Lemma 1 It holds that 2

0, and thus the utility function satisfies the single

crossing condition in ( ). Therefore, the optimal offer satisfies a monotone comparative

statics in .

The above Lemma implies that is increasing in in the sense of monotone com-

parative statics for correspondences. Take a selection () from the optimal price offer

8

correspondence ∗().8 Then one can define the value function as

() = ( − ())((()) + (1− )(()))

Standard arguments using revealed preference argument imply that is a convex function

of . Moreover, it follows immediately from () ≤ () that is decreasing

in . The above setup is then completely nested into our reduced form functional form

assumptions for the continuation value function.

It is in order to indicate how much one can generalize the bilateral bargaining setup in

a way that it can be still represented as a decision problem in the outside market. First, a

straightforward extension that allows both parties to make the offer with positive proba-

bility in the above bilateral game can be easily accommodated. Second, the continuation

problem can be an auction where the market is mostly anonymous and thus meeting the

same seller or buyers in future interaction is not very likely. This is a good description of

many e-Bay auctions where a standard object is auctioned. In general, any anonymous

outside market where rival bidders of the original auction do not interact, and the market

does follow the outcome of the initial auction fits our requirements.9 Lauermann, Merzyn

and Virag (2010) provide a dynamic model of large markets that satisfies this requirement.

They assume that each period buyers and sellers randomly match in a many-to-one match-

ing, and each seller runs a second price auction. The winner wins the object, while the

losers update their beliefs about market conditions, and continue to another auction in the

next period. This model features can be written as our setup, after taking the (equilibrium)

continuation values of the losers as given.

2.2 Preliminary analysis

Under the assumptions that are both strictly concave and continuous, it follows

that the optimal action correspondence is a strictly monotone and continuous function

of . Therefore, one can normalize the action space and the utility functions such that

8Since a monotone function is continuous almost everywhere, this also implies that the optimal offer

correspondence ∗() is a function for almost every .9 If these requirements do not hold, then the other participants of the outside market are influenced by

the outcome of the initial auction when they take their actions. In this case, the analysis would become

more involved as higher order beliefs about each other’s information in the continuation game would be

necessary to specify. We consider our simplifying assumptions as a first approach to study auction design

when outside options are endogenous.

9

for all it holds that () = .10 Then the value function can be written as () =

()+(1− ) () and the first order condition for the condition = argmax∈[01]

()+

(1− ) () becomes

0() + (1− ) 0() = 0 (2)

Using the above first order condition and that is a decreasing function, we obtain that

for all

0) = ()− () ≤ 0 (3)

Normalize (0) = 1 and let (1) = ≤ 1. Let us show that for () ≤ () to hold

for all , it is necessary and sufficient that (1) ≤ (1) Using that is convex, and

(3) imply that

00() = 0()− 0() ≥ 0 (4)

Formulas (2) and (4) imply that 0() ≥ 0 ≥ 0(), which then indeed yields that(1) ≤ (1) =⇒ ∀ ∈ [0 1] () ≤ ().

Let us summarize our discussion in the following Lemma:

Lemma 2 Under the above assumptions, is a decreasing function if and only if (1) ≤(1).

Intuitively, if (1) = is close to (0) = 1 , then in terms of continuation utilities

it is more important to know the state, and to be able to respond optimally, than it is

to be actually in the low state. In this case the value function is lowest at an interior ,

where the state is known very imprecisely. Therefore, for to be decreasing for all one

needs to assume that there is a large enough gap between and 1. We study a case in

an example where this assumption fails, and show that a monotone equilibrium does not

exist, but the non-monotone equilibrium has some intuitive features. However, for most of

the analysis we assume that is monotone decreasing.

10Define utility functions that satisfy () = (()) and () = (()). Then one can

look at the equivalent decision problem where the agents have utility functions and choose an action ∈ [0 1]. The optimal decision is then to choose = , since for all 6= it holds by assumption that

() + (1− )() = ()

(()) + (1− )(()) =

= () + (1− )()

10

Given (2) there exists a function such that 0 = (1 − )() and 0 = −()Normalizing, (0) = 1 this implies that for some ≤ 1 it holds that

() = −Z 1

(1− )()

and

() = 1−Z

0

()

By the above Lemma, to ensure that is monotone decreasing we need to assume that

(1) ≤ (1) or that

≤ 1−Z 1

0

() (5)

3 Analysis of the Second Price Auction

We start with the case in which the seller does not reveal either his signal or the bids after

the auction. Therefore, the losing bidder does not learn anything beyond the mere fact of

having lost. We need to introduce further notation for our analysis below. Let () denote

an agent’s belief conditional on being tied at the top, 1 = 2 = for all 2. One

can calculate this value as

() =0

2()

−2 ()

02()

−2 () + (1− 0)

2()

−2 ()

Let () denote the probability of the high state conditional on losing with a signal , that

is

() =0()(1−−1

())

0()(1−−1 ()) + (1− 0) ()(1−−1

())

We analyze an equilibrium in monotone and symmetric strategies that is described by

a strictly increasing function . Since both bidders’ (continuation) values are affected by

the signal of the other bidder, therefore the bidders are in a second price auction with

interdependent valuations. Moreover, it is easy to see that signals are affiliated, and thus

our model is a special case of the Milgrom and Weber (1982) setup, except for the fact

that beliefs directly influence valuations. In the standard Milgrom and Weber setup (the

case where and are constant), it is well known that in the symmetric equilibrium of

the second price auction, each bidder bids his valuations assuming that he ties at the top

11

spot. We build on this insight, but need to make an adjustment reflecting the new feature

of our model that allows beliefs to play a role in determining continuation values through

the actions they induce in the future. We can capture this new term, by noting that the

relevant continuation value is the one that is assessed conditional on tieing (that is ), but

assuming that the action taken is induced by the belief upon losing (). Formally, then

the bid of a bidder with type can be written as

() = − [()(()) + (1− ()) (()] (6)

The following summarizes our findings:11

Lemma 3 If (5) holds, then is decreasing, and there exists a unique monotone and

symmetric equilibrium. In this equilibrium both bidders use bid function as defined in

(6).

For the rest of the paper, except when it is stated otherwise, we assume that (5) holds,

and concentrate on the monotone equilibrium in the game with no information revelation.

To calculate the expected revenue, let (2)() denote the density function of the second

largest signal of the signals from an ex-ante perspective. Intuitively, the bidder with

such a type will determine the revenue in the second price auction. Formally,

(2)() = 0()(1−())−1 () + (1− 0)()(1−())

−1 ()

holds. Then the ex-ante expected revenue of the seller can be written as

=

Z 1

0

(2)()() =

= −Z 1

0

(2)()[(){ −Z 1

()

(1− )()}+ (1− ()){1−Z ()

0

()}] =

= −1+(1−)Z 1

0

(2)()()+

Z 1

0

(2)()[()

Z 1

()

(1−)()+(1−())Z ()

0

()]

Define =R 10(2)()[()

R 1()

(1 − )() + (1 − ())R ()0

()]. This variable

can be interpreted in a very simple way for our purposes. The higher is the more

11The formal proof is in Appendix 1.

12

important it is to make the right decision. Suppose that the decision problem is trivial,

which corresponds to the case where () = 0 for all . In this case there is no significant

action to be taken, and indeed = 0 holds. In general, the higher is, the more the post

auction decisions matter, and the higher becomes. With this notation in hand, one can

rewrite the expected revenues from not revealing any information as

= − 1 + (1− )

Z 1

0

(2)()()+ (7)

If the seller reveals his signal ex ante, the symmetric equilibrium is simple. Given the

signal 0 ∈ {0 1}, the bidders know the state. The probability of the high state is 0 andthe payoff in the decision problem is (0). Thus, bids are

( 0) = − (0) .

The ex-ante expected revenue can be simply written as

= − (1− 0) (0)− 0 (1) = − 1 + 0 (1− ) . (8)

3.1 Revenue comparison when the state may be revealed

Using the revenue formulas from above, we are ready to analyze how revealing information

in a second price auction affects revenues. Recall that measures the importance of the

post auction decisions, and measures the difference in expected payoffs in the two states

as defined above. The revenue comparison is as follows:

Proposition 4 The sellers’ revenue is higher without revealing his informative signal, if

the parameters of the decision problem are such that

(1− )[0 −Z 1

0

(2)()()]

Proof. The result is immediate after comparing formulas (7) and (8). QED

The two key comparative statics parameters, and affect whether revealing infor-

mation is beneficial for the auctioneer. If is large, then the decision after the auction

influences continuation values to a large degree. In this case revealing information improves

the value of the outside options, because a better decision can be taken if a bidder contin-

13

ues to the outside market. This leads to lower bids in the current auction, and thus lower

revenues. This is the continuation value effect. When the two states provide very different

utilities, that is is low, then the standard winner’s curse phenomenon arises. In partic-

ular, it becomes very important not to overpay when winning in the low state, and thus

bidders reduce their bids to overcome the winner’s curse. This in turn implies that reveal-

ing any extra information helps the auctioneer, by the well known linkage principle effect.

To summarize: the linkage principle effect is strongest (favoring information revelation)

when the two states are fundamentally very different in terms of payoffs, when is much

lower than 1, and the continuation value effect is strongest (favoring hiding information)

when future actions are important, that is when is high. Therefore, the key condition

of Proposition 4 has a simple explanation: hiding information is revenue enhancing if and

only if the continuation value effect is stronger than the linkage principle effect, that is

when (1− ) is large.

It is interesting to revisit our structural model of the decision problem that we provided

in the Setup section. Assume first, that , but () = () for all . In this

case, there is less chance in the high state to continue, but the optimal price is independent

of the state. Therefore, the optimal action is the same regardless of the beliefs about the

state, and thus function is identically zero, and thus = 0 holds. On the other hand, it

is more profitable to be in the high state than in the low state, and thus 1. Therefore,

the condition of Proposition 4 fails, and revealing information is revenue enhancing. This

result can be interpreted as follows: the continuation value effect is absent, since the

optimal decision does not depend on the information learnt, but the linkage principle effect

is present and thus, following Milgrom and Weber, revealing information is beneficial for

the seller. More generally, in this example the parameters do not influence optimal

actions, and thus only contribute to the linkage principle effect. The distribution functions

influence optimal actions, and thus contribute to the continuation value effect.12

We are ready to discuss the strength of the condition in the Proposition above, that

is how likely it is in general that the continuation value effect dominates. A monotone

equilibrium exists if the two states provide different continuation utilities (see condition

(5)), which is exactly the case when the linkage principle effect is strong. Therefore, it

may happen for some functional form specifications (forms of ) that the

linkage principle effect always dominates the continuation value effect, and thus revealing

12Given their ranking based on first order stochastic dominance, these distribution functions also con-

tribute to the linkage principle effect.

14

information is always beneficial. To study this issue, the following Corollary describes

under what conditions on the signal distribution and utility functions can one find values

of such that revealing the state may decrease expected revenues:

Corollary 5 If 0−R 10(2)()())

R 10(), then there exists a high enough value

of such that revealing information decreases the expected revenues. If this condition does

not hold, then for all values of such that there exists a monotone equilibrium in the game

without state revelation, the expected revenue from revealing the state is higher than the

expected revenue from hiding it.

Let us consider an example for each of the two possibilities.

Example 1: Let 0 = 12 = 2 = 2(1 − ) and = 2, and () = 2,

() = 2− 2, and

() = + [log() + (1− )]

and

() = 1−

for some ≥ 0. We assume that ≤ 1 − , ensuring that (1) ≤ (1) and thus

is decreasing. The condition of the Corollary for hiding the state to be revenue enhancing

becomes 1− = 1

6∗02446 ≈ 068, and the monotonicity condition for is simply1− ≤ 1. The two conditions are compatible, and thus if one chooses appropriately, thena monotone equilibrium exists with . In this case takes the role of , that is

this parameter measures the importance of the decision taken.

Example 2: Let 0 = 12 = 2 = 2(1 − ) and = 2, and () = 2,

() = 2− 2, and

() = − (1− )2

and

() = 1− 2

for some ≥ 0. We assume that ≤ 1 − , ensuring that (1) ≤ (1) and thus

is decreasing. The condition of the Corollary for hiding the state to be revenue enhancing

becomes 1− ≈ 12, and the monotonicity condition for is simply

1− ≤ 1. The twoconditions are not compatible, and thus if one chooses such that a monotone equilibrium

exists then it always holds that . In other words the linkage principle effect

dominates the continuation value effect whenever a monotone equilibrium exists.

15

If one considers the case where a monotone equilibrium does not exist in Example 2,

then it can be shown that the continuation value effect can dominate the linkage principle

effect. This is pursued in the example below.

Example 3:

Let us reconsider Example 2 with = 1. There is no monotone equilibrium if 0.

Since the two states are symmetric, so it is natural to concentrate on a signal symmetric

equilibrium where () = (1 − ) for all . In this case one can recalculate the relevant

tieing posterior as

e() = Pr | 1 = 2 = or 2 = 1− ) =

The relevant losing probability is

e() = Pr( | 1 = 2 ∈ ( 1− )) = .

Then in the case of not revealing the state the equilibrium bid function can be written as

() = − [(1− (1− )2) + (1− )1− 2] = − 1 + (1− ) − 1

The bid with information revelation is = − 1 in both states, and thus the revenuecomparison favors not revealing the state. This example shows that when the two states

are similar, then the linkage principle loses its bite, and although a monotone equilibrium

does not exist, it follows immediately that revealing the state decreases revenues.

3.2 The effect of signal precision on revenue comparisons

It is also interesting to consider the effect of state revelation when the signal precision

varies. Consider three situations: in the first, the bidders obtain a very noisy signal almost

surely, that is for all ≥ 1, is very close to 0 with a very high probability; in the second,the bidders learn the state completely; while the third case is an intermediate situation.

It is immediate that in the second situation the state revelation does not change revenues,

because the bidders have a completely precise signal even without information revelation.

In the first situation revealing the state decreases revenues, because in the absence of state

revelation the bidders compete away all their rents (relative to the continuation values),

and their continuation values are also low. In other words: the continuation value effect

16

is strong, while the linkage principle effect is absent. In the third, intermediate situation,

revealing the state may increase or decrease revenues depending on the parameter values.

Let us take the more interesting case where it increases revenues (compared to the no state

revelation benchmark). In this case, the benefit from revealing the state is negative if the

signals are very imprecise, exactly zero when they are fully precise, and it is positive when

signal precision is intermediate. This highlights the interesting non-monotone dependence

of the effect of signal revelation on revenues when the precision of the bidders’ signals

changes. While this dependence is quite complex, it can be fully understood using our

notions of the continuation value effect and contrasting it with the well known linkage

principle effect. At the two polar situations (situations one and three), the linkage principle

effect is absent, since the rents are competed away even without signal revelation. The

linkage principle effect is thus strongest in the second, intermediate case. On the other

hand, the continuation value effect is stronger, the weaker the initial signal precision is.

The reason is that the noisier the signals are, the more it helps with the decision problem

if the state is revealed. The combined effect is then the difference of a non-monotone

(U-shaped) effect and a monotone decreasing effect, opening the possibility for different

patterns of comparative statics in the precision of the bidders’ signals.

3.3 Revenue when the winning bid can be revealed

Let denote the expected revenue when the winning bid is revealed assuming that

is decreasing (that is (5) holds). The following result is established below:

Proposition 6 Revealing the bids ex post decreases revenue relative to not revealing any-

thing, .

Proof. Suppose the seller reveals only the winning bid. In this case the bid function

becomes

() = − [()(()) + (1− ())(())]

Comparing it with the case of no such bid revelation yields

() = − [()(()) + (1− ())(())]

− [()(()) + (1− ())(())] = ()

17

where the inequality follows from the fact that = argmax∈[01]

() + (1− ) (). That

is the bid of a type if the winning bid is revealed is lower than in the benchmark case of

no information revelation. Therefore,

=

Z 1

0

(2)()()

Z 1

0

(2)()() =

So revealing the bids after the auction decreases revenues. QED

A similar argument implies that revealing any information after the auction is harmful

for the auctioneer. This is true for any continuation value functions as long as a

monotone equilibrium exists (for which it is a sufficient condition to have (1) ≤ (1)

as we argued above).

4 Format comparisons: first-price, second-price and English-

auctions

In the literature initiated by Milgrom and Weber (1982), another interpretation of the

linkage principle is that the more an auction format links the payments to the types of the

other agents, the higher the expected revenue is. For example, in a first price auction the

payment of a type conditional on winning, is just the submitted bid, which does not depend

on the types of the other bidders. In contrast, in a second price auction the winner pays

the second highest bid, and thus the expected payment (conditional on winning) depends

positively on the types of the other bidders. Therefore, the linkage principle implies that

(in the standard case without endogenous outside options, that is when = 0) the second

price auction yields a higher expected revenue than the first price auction. Similarly, an

English-auction links payments to others’ bids (types) even more, since now all the types

except for the two highest are revealed by the bid at which those bidders are dropping

out. Therefore, the English-auction links the payment to bids even more, and thus yields

a higher expected revenue than even the second-price auction.

We saw in the previous Section that in our model with endogenous outside options the

linkage principle effect is counteracted by our continuation value effect, when we analyze

whether the auctioneer should reveal his exogenous information about the state of the

world. It is natural to ask whether the same is true when one compares the three standard

auction formats fixing the information policy. To conduct such a comparison, assume

18

that the auctioneer does not reveal any information and runs a first-price, second-price or

ascending auction.

First, let us compare the revenues between the first-price and second-price formats.

Extending the argument from the standard case, one can show that the revenue ranking is

unchanged, that is the second price auction revenue dominates the first price auction. This

can be done by modifying the analysis of Krishna (2008), Section 7, pages 105-108. The

intuition is simple: upon losing, the same information is learned in the two auction formats.

In both auctions, the losers only learn that there was bidder with a higher signal than

theirs. This implies that they take the same actions in the continuation decision problems,

and therefore the presence of endogenous outside options does not change the comparison

between the two formats as compared to the standard case. The formal argument is

provided in Appendix 2.

4.1 Comparing second-price and English auctions when outside options

are endogenous

The most important novelty when one compares second-price and English auctions is that

the English-auction reveals more information than a second-price auction and thus the

optimal decision after the two auctions are different. In particular, the English auction

allows the losers to take better decisions, and thus the continuation value effect favors the

second price auction over the English auction. Since the continuation value and the linkage

principle effects work in opposite directions, one needs to assess whether the second-price

or the English-auction raises higher revenues.

To highlight the key trade-off, we concentrate on a three player version of Example 1

above.13 That is we assume that 0 = 12 = 3 = 2(1− ) and = 2, and

() = + [log() + (1− )]

and

() = 1−

for some ≥ 0. However, the main insights carry through to more general specifications.We obtain the following result:

13To perform the comparison, we need to have at least three bidders, otherwise the two formats are

strategically equivalent.

19

The sellers’ revenue is higher in the second price auction than in the ascending auction

if the parameters of the decision problem are such that 1 ≥ 1− 005.

Proof. See Appendix 3.

In this example for some parameter values (such that a monotone equilibrium exists)

the continuation value effect is stronger than the linkage principle effect, and revealing

information via holding a more open auction decreases revenues. The comparison has the

same qualitative effects as above: or being high helps the continuation value effect

to dominate, since in this case the continuation value effect is stronger than the linkage

principle effect.

5 Discussion

In this Section we consider some extensions to inspect the robustness of our results. First,

we show that the assumption of two states does not drive our results, and is just made for

ease of exposition. Let ∈ [0 1] denote the state of the world now, and let () denote

the continuation value when action is taken in state . Let denote the conditional

distribution of signals in state , and let the density function for the state of the world.

Assuming that for all () is decreasing in implies that there is an equilibrium with

monotone bidding. Let () be the density of state if one ties at the top with signal

that is

() =()2 ()

−2 ()R 1

0()2()

−2 ()

Let () be the density of state if one lost with signal that is

() =()()(1−−1

())R 10()()(1−−1

())

Then the optimal after losing with signal satisfies

() = argmax

Z 1

0

()()

20

Let us focus on comparing the revenues from the second price auction with and without

the revelation of the winning bid. Without such a bid revelation the equilibrium bid is

() = −Z 1

0

()(())

With bid revelation the tieing loser learns that he in fact tied with the winner and takes

an action

∗() = argmax

Z 1

0

()() (9)

Therefore, the equilibrium bid becomes

() = −Z 1

0

()(∗())

By (9) it follows that () (), so the revenue comparison result follows the same way

as in the two-state model.

Second, suppose that the winning bidder takes an action too, and his continuation

utility functions are

, which have similar properties to , the continuation

utility of the losers. We keep the assumption that the winner obtains a utility from the

object and that there are two states. Let us now concentrate on the question whether in the

second-price format state revelation enhances or reduces revenues; the other questions can

be studied similarly. Following similar argument as in the benchmark case, the equilibrium

bid function without state revelation is

+ [()((())− (())) + (1− ())(

(()− (())]

From the bid function, one can calculate the equilibrium expected revenue when no infor-

mation is revealed by the auctioneer.14 When the state is revealed, in the high state all

bidders bid + (1)− (1), and in the low state all bid + (0)− (0). From these

observations it is obvious that if the winners continuation values are not very sensitive to

the state of the world (that is −

is uniformly close to zero and thus is close to

being a constant function), then the revenue comparison is similar to the benchmark case

where the winner’s continuation problem was omitted. Therefore, our results are robust as

long as the winner’s continuation problem is not too sensitive to the state as compared to

14The main steps are similar to the benchmark case and are omitted.

21

the losers’. It is worth pointing out that if the winner’s continuation problem is more im-

portant than the loser’s, then the continuation value effect favors information revelation,

and thus transparent auctions are revenue enhancing. The reason is that if more infor-

mation is available after the auction, then the winner can make a better decision, which

then makes bidders more aggressive since the winning prize has become more valuable. In

more technical terms, both the linkage principle and the continuation value effects favor

information revelation in this case.

6 Appendix

6.1 Appendix 1

Proof of Proposition 1:

Proof: First, we show that the above defined bid function constitutes an ex-post

equilibrium. Symmetry of is immediate, while monotonicity follows from the facts that

are increasing, and that () ≤ () for all . To see this, note that

[()(()) + (1− ())(())] =

0()((())− (())) + 0()()((1− )− (1− )) 0

follows from the observations above. But this is equivalent to 0() 0.Next, we show that if it is known that 1 = 2 = , then winning with () yields the

same utility as losing and acting in the future as if the probability of the high state was

(). Losing yields a continuation utility that is equal to ()(())+(1−())(())

by construction, while winning with bid () yields a utility − (), which is equal to the

continuation utility upon losing.

It also has to be established that if = then winning against a type with bid

() is unprofitable, while if then winning against type with bid () is profitable.15

Let us just inspect the case, the other one is similar. In this case winning, upon

tieing, yields a utility of

− () = ()(()) + (1− ())(())

15Again, we need to use the relevant tieing belief Pr( | 1 = 2 = ) and the relevant action inducing

belief upon losing Pr( | 1 = 2 ≥ ).

22

To calculate the utility from losing, upon tieing, let us introduce the relevant tieing posterior

when one bids () and has type as follows:

( ) =0()()

−2 ()

0()()−2 () + (1− 0) ()()

−2 ()

By the fact that and satisfy the MLRP, it follows that () ( ) One can

similarly define the relevant losing posterior as

( ) =0()(1−−1

())

0()(1−−1 ()) + (1− 0) ()(1−−1

())

Again, the MLRP condition implies that () ( ). Then the utility upon losing (and

tieing) can be written as

( )(( )) + (1− ( ))(( ))

Note, that

( )(( )) + (1− ( ))(( ))

( )(()) + (1− ( ))(()) (10)

To see this, note that () ( ) ( ) and the first order condition (2) implies that

() + (1− )()− (() + (1− )()) =

=

Z

¡ 0() + (1− ) 0()

¢ =

Z

¡ 0() + (1− ) 0()

¢+

+

Z

(− )( 0()− 0()) =Z

(− )( 0()− 0()) 0

because and 0() ≤ 0 ≤ 0(). Then using (10), the utility difference betweenlosing and winning satisfies

4 = ( )(( ))+(1−( ))(( ))−(()(()) + (1− ())(())) =

() + (1− )()− (() + (1− )()) =

= (− )(()− ()) 0

23

where the last inequality follows, because and () (). Therefore, it is

indeed more profitable to lose against a type than to win if one’s type is . This

concludes the proof of global optimality for the bidders’ problem.

Finally, uniqueness of as in (6) follows from the above argument as well, since upon

tieing indifference has to hold in an ex-post equilibrium which yields exactly (6) after taking

it into account that the equilibrium is symmetric and monotone. QED

6.2 Appendix 2

Comparison of first and second price auctions:

To formalize the argument in the main text, let us introduce the notation ( ) =

Pr( | 1 = 2 = ) and ( ) = Pr( | 1 = 2 ) and

( ) = − [( )(( )) + (1− ( ))(( ))]

The interpretation is that a bidder with type values winning at ( ) if the he bid as if

he had type . Again, the bidder conditions his posterior upon tieing, but knows that he

will make the decision in the continuation problem as if he lost in the auction. Note, that

it holds that () = ( ) () = ( ) and thus ( ) is equal to the equilibrium bid

in the second price auction.

Following the notation of Krishna (2008), let ( ) the expected price paid by a

bidder if he is winning when he receives a signal , but he bids as if his signal was . Let

( ) denote this expected price for the first-price, and ( ) for the second-price

auction. Denoting the equilibrium bid functions in the two formats as and , one can

write ( ) = () and

( ) = [() | 1 = 2 ].

In the monotone equilibrium of auction = each bidder maximizesZ

0

( )[() + (1− )()] − [() + (1− )()]( )

Therefore, the first order condition becomes

[() + (1− )()](( )−( )]− [() + (1− )()]1 ( ) = 0

24

where 1 denotes the partial derivative of with respect to its first argument. This

can be rearranged so that

1 ( ) =

() + (1− )()

() + (1− )()( )− () + (1− )()

() + (1− )()( )

Therefore,

1 ( )−

1 ( ) = −() + (1− )()

() + (1− )()( ( )− ( ))

Now define

∆() = ( )− ( )

so

∆0() = 1 ( )−

1 ( ) + 2 ( )−

2 ( )

Combining these results yields

∆0() = − () + (1− )()

() + (1− )()∆() +

2 ( )− 2 ( )

Now, by construction 2 ( ) 0 =

2 ( ). Thus if ∆() ≤ 0, then ∆0() ≥ 0.

Furthermore by assumption (0 0) = (0) = − (1) = (0) = ( ), so

∆(0) = 0. This then implies that for all ≥ 0, it indeed holds that ∆() ≥ 0, which showsthat the second price auction yields a higher expected revenue than the first price auction.

6.3 Appendix 3

Proof of the revenue comparison result:

Proof. In the second price auction the middle bidder’s bid is the revenue, and the bid

function can be written as

= − [b()(b()) + (1− b())(b())]where ee are the relevant tieing and losing posteriors. These beliefs can be written as

b() = Pr( | 1 = 2 = 3) =2

2 + (2− 2)(1− )

25

and b() = Pr( | 1 2 = 3) =2

2 + (2− 2)(1− )

Let us now calculate the revenue in the English auction. Let be lowest of the three

types, and be the medium one. Then the revenue is equal to

( ) = − [b( )(b( )) + (1− b( ))(b( ))]where b( ) = Pr( | 1 = 2 = 3 = ) =

+ (1− )(1− )

and b( ) = Pr( | 1 2 = 3 = ) =

+ (1− )(1− )

To calculate the expected revenues let

() = 12− 422 + 603 − 304

denote the density of the medium type, and let

( | ) = 2() + 2(1− ())(1− )

()2 + (1− ())(2− 2)

be the density of the low type given the medium types. Then the expected revenues from

the two auctions can be written as

=

Z 1

0

()()

and

=

Z 1

0

()

Z

0

( | )( )

After substituting in the relevant functional form assumptions about and using the

above bid functions the following can be established

⇔

1− 005

while a monotone equilibrium exists if 1− ≤ 1. QED

26

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