Q ED

Queen’s Economics Department Working Paper No. 975

Signally by Jump in Private Value

Ruqu Wang Alan Gunderson Queen’s University Industry Canada

Department of Economics Queen’s University 94 University Avenue Kingston, Ontario, Canada K7L 3N6

10-1998

Signalling by Jump Bidding in Private Value

Auctions

Ruqu Wang Alan Gunderson

Department of Economics Industry Canada

Queens University Queen St West

Kingston ON Ottawa ON

KL N KA H

wangrqedeconqueensuca gundersonalanicgcca

Octob er

Abstract

This pap er examines how a bidder can b enet from jump bidding

by using the jump bid as a signal of a high valuation which causes other

bidders to drop out of the earlier than they would otherwise

The information contained in a jump bid must b e sucienttoinduce

a discrete change in the bidding b ehaviour of the other bidders In

an auction for a single item a jump bid signals b oth the identityand

the high valuation of a bidder The existence of a b enecial jump bid

equilibrium requires a gap in the distribution of the jump bidder and

her identitymust b e concealed Concealing the identity of the bidders

p ermits the jump bidder to signal more information through the jump

bid and thus she can b enet more from it In an auction for multiple

items the jump bid signals a high valuation by the jump bidder This

causes a discrete change in the bidding b ehaviour of the other bidder

since it causes this bidder to reduce her demand In b oth a oneob ject

and multipleob ject auctions a seller may exp ect less revenue in a jump

bid equilibrium than a nonjump bid equilibrium

We wish to thank Dan Bernhardt Roger Ware PatrickFrancois Ted Neave and Mak

Arvin for useful comments and suggestions We of course remain solely resp onsible for

all errors Wangs research is supp orted by SSHRC of Canada The views and opinions

expressed in this work are those of the authors and do not necessarily reect those of

Industry Canada or the Canadian federal government Keywords Auction Jump Bidding

JEL Classication D

Intro duction

The eort to design an ecient auction mechanism for the distribution of

sp ectrum rights in the United States and other countries has caused eco

nomic theorists to weigh the b enets and costs asso ciated with several al

ternative auction formats Of particular imp ortance was the choice of how

much information to release to the auction participants during the course of

the bidding If bidders valuations are aliated it is desirable for the seller

to reveal as much information as p ossible so as to assist sensible bidding

and reduce the winners curse Milgrom and Web er However if there

is a relatively small numb er of bidders full information release may increase

the p ossibility of bidder collusion or predatory bidding To discourage such

behaviour the seller maycho ose to conceal the identities of the bidders

The auctions for sp ectrum rights in the United States have undergone

format changes reecting the ab ove concerns For example in the nation

wide narrowband PCS auction the Federal Communications Commission

FCC decided to conceal the identities of the bidders After each round

of bidding the high bid on each licence was p osted but not the name of

the rm that made the bidonly the bidders condential bidder numbers

were used In subsequent auctions however all information regarding bidder

identities was revealed to the auction participants

The choice of the amount of information to release seems to havehada

dramatic impact on bidding b ehaviour For instance in the nationwide nar

rowband PCS auction there were a surprising numb er of jump bids that

is bids greater than the minimum amount necessary to raise the standing

high bid Cramton recounts that of the new high bids in the auc

tion were jump bids Jump bidding is in stark contrast to most

theoretical formulations of auctions Traditionally mo dels presume bidders

bid the minimum incrementabove the previous high bid to avoid the risk of

bidding more than is necessary to win the auction Cramton rationalizes

jump bidding as follows

The basic idea is that the jump bid conveys information ab out a

bidders valuations It is a message of strength conveying that

the bidder has a high value for the particular license Moreover

it conveys this message in a credible way Jump bidding has a

costit exp oses the bidder to the p ossibilityofleaving money on

the table It is precisely this cost that makes the communication

credible A bidder with a lowvalue would not nd it in its

See Chakravorti et al Cramton McAfee and McMillan

b est interest to make a large jump bid The gain increasing the

chance of winning the license would not exceed the cost the risk

of overbidding

Bidding b ehaviour in the nationwide narrowband auction was also likely

inuenced by the presence of asymmetric bidders For example of the

bidders Cramton describ es a few bidders as likely having high values

b ecause of their large market share prior pro duct development or other

advantages

In this pap er weusetwo dierentformulations a oneob ject and a

multipleob ject mo del to explore a bidders rationale for jump bidding In

b oth cases a bidder jump bids to signal that she has a high valuation which

causes a discrete change in the bidding b ehaviour of the other bidders In

order for a jump bid to b e eective it must b e credible and convey sucient

information to b e protable to the jump bidder In the oneob ject mo del

the jump bid signals b oth the identity of the bidder as well as the fact that

she has a high valuation The existence of a jump bid equilibrium requires

certain restrictions on the distribution of the jump bidder In particular a

gap must b e present in her distribution and it must b e sto chastically dom

inated by her rivals distributions over lowvaluations Furthermore the

seller must conceal the identities of the bidders With some information

concealed the jump bid reveals more information and thus the jump bidder

can b enet more from it

In the multipleob ject mo del the conditions required for a jump bid

equilibrium are quite dierent although the motivation is the same as in

the oneob ject mo del In this case the existence of a jump bid equilibriu m

requires the jump bidder to demand fewer items than her rival and her

distribution must b e strictly concave By jump bidding a bidder induces

her rival to strategically reduce her demand for the ob jects she would like

to win so that she pays lower prices for the ob jects she do es in fact win

In the oneob ject case there are twotyp es of bidders that draw their

valuations from a dierent distribution This reects the fact that some bid

ders are likely to value the ob ject more than other bidders p erhaps b ecause

of dierences in the numb er and typ e of other ob jects that they presently

own For example rms that own cellular licences in regions contiguous to

areas where PCS licences are b eing auctioned may place high values on the

PCS licences b ecause they could provide them with ro om to expand their

mobile telephone network

In the multipleob ject case there are also twotyp es of bidders that

draw their valuations from a dierent distribution However they also have

dierent demands for the ob jects up for bid Certain bidders maywishto

win all the ob jects up for bid while other bidders may desire only a few

A couple of recent pap ers have also explored the issue of jump bidding

Daniel and Hirschleifer examine a mo del where the bidders bid in a

sequence and it is costly to submit a bid or bid revision They fo cus on an

equilibrium in which the rst bidder either passes or reveals her valuation

by jumping immediately to a bid whose level is an increasing function of

her valuation Each successive bidder quits if her valuation is less than that

revealed by an earlier bidder or she jump bids to a substantially higher bid

that reveals sucient information to force all preceding bidders to quit In

contrast to traditional bidding mo dels large jumps o ccur at every stage of

the bidding even in the limit as bid costs go to zero Furthermore bidders

with lowvaluations will sometime p ostp one making a bid in order to learn

more ab out their rivals

Avery examines the role of jump bidding in a mo del of common value

He solves for equilibria of sequential bid auctions when jump bidding strate

gies maybeemployed to intimidate ones opp onents In these equilibria

jump bids serve as correlating devices that select asymmetric equilibria to

b e played sequentially Bidders b enet from jump bidding compared to the

symmetric equilibri um of a sealed bid secondprice auction

One of our results conrms that of Daniel and Hirschleifer in the sense

that when bidders are mo delled as drawing their valuations from a contin

uous probability distribution and bidding costs are very small bidders do

not b enet from making jump bids and the seller do es not suer a loss in

revenue Daniel and Hirschleifer show that jump bidding can b enet a bid

der only when there are p ositive bidding costs In a mo del with no bidding

costs we show that jump bidding can b enet a bidder and reduce the sellers

revenue when there is a discontinuity in buyers distributions or if bidders

dier in the numb er of ob jects they would like to win

The pap er is structured as follows In section weshow that if all

bidders are mo delled as drawing their valuations for a single ob ject from

acontinuous probability distribution then b enecial jump bid equilibria

cannot exist This result motivates our development of a new theoretical

formulation in section where bidders are assumed to b e asymmetric In

section we extend our framework to include an auction for multiple ob jects

Section provides a few concluding remarks The pro ofs of all results are

relegated to a technical app endix

Preliminaries

In this section we dene the dierent strategies that bidders may use when

bidding Using these strategies we show that the traditional theoretical

formulation of a oneob ject auction with bidders having continuous distri

butions fails to explain the phenomenon of jump bidding

Standard predicts that in an English op enoutcry auction

where bidders p ossess indep endent private values each bidder continues to

submit bids in the smallest allowable incrementuntil her valution is reached

at whichpoint she drops out of the auction Given this bidding b ehaviour

the winner of the auction is the bidder with the highest valuation and she

pays an amount equal to the second highest valuation This outcome de

p ends critically on the absence of bidding costs With no bidding costs a

bidder always has an incentive to cast another bid although she mayknow

with certainty that a rival has a higher valuation and therefore will win the

ob ject In this pap er although wedonotintro duce bidding costs explicitly

we assume that if a bidder exp ects a nonp ositivepayo from continuing to

bid she immediately drops out of the auction This assumption is motivated

by the notion that it takes time and energy for a bidder to continue in an

auction and it is likely that she will cho ose to drop out if she realizes that

she cannot win with p ositive probability

For simplicitywe mo del an auction as a two stage pro cess where in the

rst stage bidders may make jump bids In the second stage we mo del the

auction as a continuous starting with the highest bid in the

rst stage

If a bidder makes a jump bid in the rst stage then she is said to b e using

a Jump Bidding Strategy JBS These come in twoavors a Separating

Jump Bidding Strategy and a Pooling Jump Bidding Strategy

Denition A Separating Jump Bidding Strategy SJBS consists



of a function B v where B v such that if the bidders valuation is

equal to v then she jump bids to B v in the rst stage A Po oling Jump

Bidding Strategy PJBS consists of a pair a bsuch that if the bidders

valuation is greater than or equal to b then she jump bids to a in the rst

stage Converselyifhervaluation is less than b then she do es not jump bid

in the rst stage

In this pap er we only consider strategies in the form of Denition

and assume that only one bidder intends to use it This may app ear to b e



Daniel and Hirschleifer refer to this as the ratchet solution

a serious restriction on a bidders choice of actions however wewillshow

subsequently that the jump bid equilibrium in the strategy space outlined

ab ove is a separating equilibrium so that even if rms are p ermitted to jump

bid later in the auction they will cho ose not to do so as no information will b e

contained in these jumps Therefore our restriction on jump bids o ccurring

only in the rst stage will b e shown to b e inconsequential for the equilibriu m

wederive It is imp ortant to note however that other equilibria may exist

in strategies with jump bids later in the auction but due to the complexity

of the analysis we do not attempt to characterize them

Using the strategies dened ab ove we next show that no benecial jump

bid equilibria exist when the bidders are mo delled as drawing their valua

tions from a continuous probability distribution By the term b enecial

we are referring to equilibria where a bidder exp ects to receive a greater

exp ected payo from jump bidding than by not jump bidding We include

this analysis to show that conventional auction mo dels without bidding costs

cannot explain the phenomena of jump bidding

Consider n bidders n who wish to buy a single ob ject that is up

for bid The bidders all have indep endentprivate values which are drawn

from the continuous distribution F v For simplicitywe set the sellers

valuation for the ob ject to zero so that she is willing to accept any p ositive

bid

endent private values drawn Prop osition Given n bidders with indep

from a continuous distribution a bidder cannot achieve a strictly greater

expectedpayo by jump bidding than by not jump bidding

Since with indep endentprivate values a jump bid cannot change a bid

ders probability of winningthe ob ject will always go to the bidder with

the highest valuationa bidder can b enet from jump bidding only if she

exp ects to paylessby jump bidding than by not jump bidding If a bidder

do es not jump bid and wins she exp ects to pay the next largest valuation

conditional on her own b eing the largest However a bidder cannot credibly

jump bid to an amount less than this b ecause such a bid would b e mimicked

by another bidder with a slightly smaller valuation than her own Therefore

a separating jump bid equilibrium do es not exist

A similar argument can b e used to show that a p o oling equilibrium do es

not exist The exp ected gain from signalling information through the jump

bid is exactly oset by the exp ected cost of bidding more than necessary to

win the ob ject



The results easily extend to the case of bidders having dierent distribution s eachof

whichiscontinuous

These results are rearmed by Daniel and Hirschleifer They show

that if there are bidding costs then an equilibrium exists where a bidder

receives a strictly greater payo from making a jump bid However as the

bidding costs approach zero the payo from jump bidding approaches zero

Since most of the results in auction theory are derived from mo dels with

continuous distributions these mo dels cannot explain the phenomena of

jump bidding without intro ducing bidding costs

Given the anlaysis ab ove it is apparent that in order to explain the

phenomena of jump bidding and determine its impact on seller revenue we

need to develop an alternative theoretical framework With this in mind we

now turn to section

The OneOb ject Case

The Mo del

There is one seller who wishes to sell a single nondivisable ob ject We

assume that the sellers valuation for the ob ject is zero which implies she is

willing to accept any strictly p ositive bid The seller faces n bidders

one of them we refer to as the special bidder and the others wecall ordinary

bidders The sp ecial bidder draws her valuation from the distribution Gw

with a supp ort over two nonoverlapping intervals v vandv v where

Each ordinary bidder draws her valuation indep endently from the

continuously dierentiable distribution F v with the supp ort v v

The twointerval supp ort for the sp ecial bidders distribution can b e

motivated in a varietyofways For example one can supp ose that ordinary

bidders maybeaware that the sp ecial bidder owns an ob ject that is similar to

the ob ject up for bid but do not know whether the ob jects are complements

or substitutes If they are complements then it is likely that the sp ecial

v Conversely if they are substitutes bidder has a large valuation w v

then the sp ecial bidder is more likely to havealowvaluation w v v

An example of this p ossibility is a rms valuation for a particular sp ec

trum license If the sp ecial bidder already owns a sp ectrum licence in a

similar frequency band and geographic area as the license up for bid she

mayvalue an additional license quite highly if this license complements her

existing one by bringing much needed capacity to her business On the other

hand the sp ecial bidders existing license may provide sucient sp ectrum

capacity for her needs In this case the license up for bid is a substitute

for the one the sp ecial bidder already owns hence she may place a small

value on acquiring an additional license Continuing with this example the

sp ecial bidder may already own a cellular license in the same geographic

area as a PCS license At the time of the MTA broadband auction in the

US it was unclear whether cellular and PCS technology were complements

or substitutes

Another reason b ehind a nonoverlapping twointerval supp ort arises

from the uncertainty that ordinary bidders mayhave concerning the sp ecial

bidders previous RD eorts If the sp ecial bidder has successfully devel

op ed some valuable new technology it is likely that she will place a high

value on the ob ject Conversely if the sp ecial bidders RD eorts have

b een unsuccessful then she will likely havealowvaluation

Equilibrium

In this section we show using this simple mo del that jump bidding can

b enet a sp ecial bidder with a high valuation if bidder identities are con

cealed

v Supp ose she bids Consider a sp ecial bidder with a valuation w v

in the op ening round where w and this bid signals that her valuation

is greater than or equal tov Observing this all ordinary bidders

with valuations less than or equal tov drop out of the auctiontheir

probability of winning the auction is equal to zero If the largest ordinary

bidder valuation is greater thanv then the sp ecial bidder up on winning

the auction exp ects to pay the largest ordinary bidder valuation Therefore

if a sp ecial bidder with valuation w jump bids her exp ected payo is given

by

Z

w

n n

F v w w dF

v

If the sp ecial bidder cho oses not to jump bid her exp ected payo de

p ends on the amount of information bidders are given ab out their rivals

First consider the scenario of complete information release where bidders

are informed as to the identities of the other bidders as well as the number

that drop out as the bidding pro ceedes If the largest ordinary bidder valu

ation is less than or equal tov or is greater than or equal tov then the

sp ecial bidder exp ects to pay the largest ordinary bidder valuation condi

tional on her own b eing the largest If the largest ordinary bidder valuation

lies in the interval v v then all ordinary bidders will drop out when the



See Chakravorti et al



In the case of radio sp ectrum the development of a new digital compression technique

could dramatically increase the value of a sp ectrum license

bidding reachesv since at this p ointtheymust infer that the sp ecial bid

ders valuation is greater than or equal tov In this instance the sp ecial

bidder pays an amountv in return for the ob ject Given this reasoning the

sp ecial bidders exp ected payo under full information release is given by

Z Z

v w

n n n n

w dF w dF F v F v w v

v v

Next consider the exp ected payo to the sp ecial bidder when bidder

identities are concealed but bidders are still able to observe when other

bidders drop out of the auction We refer to this case as that of partial

information release In this instance the sp ecial bidder up on winning the

auction is most likely to pay an amount equal to the largest ordinary bidder

valuation conditional on her own b eing the largest However she can pay

less than this amountif al l the ordinary bidders havevaluations within the

interval v v Given this event when the bidding reaches the p oint

v every ordinary bidder must infer that the sp ecial bidders valuation is

greater than or equal tov and therefore they will all drop out Given

this reasoning the sp ecial bidders exp ected prot under partial information

release is given by

Z Z

v w

n n

v dF F v w dF

v v

Finally consider the case where bidders do not know the identities of

their rivals or whether other bidders have dropp ed out of the auction In

this case of no information release a bidder do es not knowatany particular

momenthowmany rivals she is currently comp eting with This implies that

the sp ecial bidder if she wins the auction exp ects to payanamount equal to

the largest ordinary bidder valuation conditional on her own valuation b eing

the largest among all the bidders Therefore the sp ecial bidders exp ected

prot under no information release is given by

Z

w

n

w dF

v

Since the sp ecial bidders exp ected payo when jump bidding is strictly

decreasing in the amount of the jump bid there exists a unique jump bid for

each information structure where she is indierentbetween jump bidding

and not jump bidding Setting the sp ecial bidders exp ected prot from

jump bidding equal to that from not jump bidding and solving for the jump

bid where she is indierentbetween these two strategies gives

R

v

n

v dF

v

f

v

n

F v

R R

v v

n n

F d v dF F v

v v

p

v

n

F v

R

v

n

F d

v

n

v

n

F v

where f refers to full information release p for partial information release

f p

and n for no information release Note that all three expressions

n

and do not dep end on the precise valuation of the sp ecial bidderit only

matters that it is in the interval v v The reason for this is that the

payos from jump bidding and not jump bidding only dier in the event

that the largest ordinary bidders valuation lies b elowv Hence only

the probabilitybelowv matters

f p n

The relationship among the three expressions and is charac

terized in the following prop osition

Prop osition The maximum jump bid a special bidder with a valuation in

the interval v v is wil ling to make in the case of no information release

is greater than or equal to that in the case of partial information release

which in turn is greater than or equal to that in the case of ful l information

n p f

release That is

This result arises solely from the dierences in the sp ecial bidders ex

p ected payos under the various information structures when she do es not

jump bid These dierences in turn are caused by what the largest valu

ation ordinary bidder with a valuation in the interval v v is able to

learn during the course of the bidding In the case of no information release

each ordinary bidder participates in the auction until the bidding reaches

the level of her valuation and then she drops out However in the cases

of partial information release and full information release ordinary bidders

might drop out b efore their valuations are reached With full information

release the ordinary bidder with the largest valuation will drop out of the

auction when the bidding reachesv With partial information release the

ordinary bidder with the largest valuation will drop out of the auction only

if al l bidders are active when the bidding reachesv Clearly if there is a

p ossibility that bidders drop out of the auction b efore their valuations are

reached this b enets the sp ecial bidder since in the event of winning the

auction she will not havetopay an amount equal to the valuation of the

highest ordinary bidder Therefore the sp ecial bidders exp ected payo is

greatest when all information is released and it is smallest when no in

formation is released The case of partial information release lies b etween

these two extremes This ranking of exp ected payos dictates the ranking

of maximum jump bids that the sp ecial bidder is willing to make For ex

ample since the sp ecial bidder exp ects the highest exp ected payo when all

information is released she is unwilling to make a large jump bid in this

case

In order to demonstrate that a jump bid equilibrium exists wemust

show that an ordinary bidder and a lowvaluation sp ecial bidder will not

mimic the jump bid of a high valuation sp ecial bidder To show this we

require the following technical conditions on Gw and F v to hold

Condition

Z

n

v

F G F

d

F v Gv F v

v

Z

n

v

F F v

v d

F v

v

and

Condition

Z

n

v

G F F

d

F v Gv F v

v

Note that Condition is a necessary condition for Condition Ba

sically b oth conditions state that for valuations b elowv the sp ecial

bidders distribution Gw is rstorder sto chastically dominated by the

ordinary bidders distribution F v This accords well with the comple

mentsubstitute justication given earlier for the split distribution of the

sp ecial bidder For example if the sp ecial bidder owns an ob ject whichis

a substitute for the one up for bid she mayvalue owning another one for

strategic purp oses or as a backup for the ob ject she already owns It is

likely that if a bidder values the ob ject for these reasons her valuation is

less than a bidder that requires the ob ject to undertake her core business



If the ob jects are sp ectrum licenses a rm maywant to acquire an additional license

in order to exclude comp etitors

The imp ortance of Conditions and for the subsequent analysis will b e

made clear b elow

The following prop osition characterizes equilibria where a high valuation

sp ecial bidder jump bids in the op ening round in order to signal information

to the other bidders

v a Prop osition Given a special bidder with a valuation w v

jump bid equilibrium does not exist when al l information is released In the

p

case of partial information release any can beusedtoconstruct

a jump bid equilibrium where

R

v

n

G F d

v

v

n

Gv F v

n

and Condition holds In the case of no information release any

can beusedtoconstruct a jump bid equilibrium if Condition holds

If all information is released the sp ecial bidder do es not b enet from

jump bidding In order to prevent a sp ecial bidder with a lowvaluation

from mimicking the jump bid the jump bid must b e set so large that a high

valuation sp ecial bidder earns the same exp ected prot as she would by not

jump bidding This result is similar in spirit to that of section

If there is partial information release then there is a range of p ossible

jump bids that constitute equilibria Since all jump bids within the interval

p

signal exactly the same information it seems appropriate for the

sp ecial bidder to select the jump bid that is least costly ie Since a

p

jump bid of gives the sp ecial bidder the same exp ected payo as that of

not jump bidding its clear that she b enets from jump bidding for jump

p

bids strictly less than

If no information is released the range of p ossible jump bids that consti

n

This range is p otentially larger than that asso ciated tute equilibria is

n p

with partial information release since Again there is one jump

bid that is least costly for the sp ecial bidder and it is identical to that of

the partial information case ie

The set of jump bid equilibria in the cases of no and partial information

release exist only if conditions and hold To understand the need for

these conditions consider the case of no information release and the sp ecial

bidder and the ordinary bidder with the largest valuation b oth havevalua

tions in the interval v v If the sp ecial bidder wins the auction without

jump bidding she will pay an amount equal to the largest ordinary bidders

valuation If the largest ordinary bidder wins she will pay the larger of the

second largest ordinary bidders valuation or the sp ecial bidders valuation

Condition is a sucient condition for the largest ordinary bidder to exp ect

a higher payo than the sp ecial bidder since it is likely that the ordinary

bidders valuation is smaller than that of the second highest ordinary bidder

ie F v stochastically dominates Gw forvaluations b elowv This

deters the highest valuation ordinary bidder from making large jump bids

n

ie

In the case of partial information release the condition needs to b e

strengthened to Condition This is b ecause the exp ected payo to the

sp ecial bidder from not jump bidding with partial information release in

increased relative to the scenario of no information release The sp ecial

bidders exp ected payo from not jump bidding increases b ecause there is

a small probability that she will pay less than the largest ordinary bidder

valuation if she wins the auction Recall this will only happ en if all ordinary

bidder valuations lie in the interval v v Because the exp ected payo

to the sp ecial bidder from not jump bidding is increased relative to the no

information release scenario the amount she is willing to jump bid decreases

To ensure that an ordinary bidder do es not have an incentive to mimic the

sp ecial bidders jump bid F v must sto chastically dominate Gw bya

sucient amount ie Condition must hold

The following corollory addresses the issue of the sellers exp ected rev

enue

Corol lory The seller exp ects less revenue if jump bid equilibria exist than

if jump bid equilibria do not exist

Regardless of the bidding strategy adopted by the sp ecial bidder the

ob ject is always acquired by the bidder with the largest valuation Therefore

jump bidding do es not eect auction eciency but only the amount that the

sp ecial bidder pays in the event she wins the auction Given the state where

the sp ecial bidders valuation is within the interval v v and the largest

ordinary bidders valuation is within the interval v v the sp ecial bidder

will receive a strictly greater payo from jump bidding than by not jump

bidding In all other states the sp ecial bidder receives the same exp ected

payo from jump bidding as from not jump bidding Since all states o ccur

with p ositive probabilityitfollows that the exp ected revenue to the seller

must clearly decrease when jump bid equilibria exist

A Numerical Example

To make our ideas more concrete we oer the following numerical example

Supp ose that the ordinary bidders valuations are uniformly distributed

over the unit interval while the sp ecial bidders distribution is given by

p

w for w v

Gw

for w

If the sp ecial bidder has a high valuation ie w she maywishto

signal this by means of a jump bid if either no information or only partial

information concerning the bidders is released If in a jump bid equilibriu m

the ordinary bidders observe a jump bid then they immediately drop out of

the auction since they eachhave a zero probability of winning the ob ject

Given this setup see the App endix holds if and only if Hv

where

p

R R

p

Z Z

v

n n

vd

n n

v

p

Hv d v d v

v

v

n

n n

v v

n n n n n

n n

v n v

nn nn n

It can b e shown that Hv is strictly concave with H andH

This implies that there exists a value v where for allvv H

and holds

The MultipleOb ject Case

The Mo del

In this section jump bidding is examined in the context of an auction for

multiple ob jects Tokeep the analysis as simple as p ossible we fo cus on the

case where there are two identical ob jects for sale As b efore we set the

sellers valuation for each ob ject to zero which implies the seller is willing

to accept any strictly p ositive bid

There is one bidder who we call the special bidder who wishes to pur

chase only one of the ob jects This bidder draws her valuation from the

distribution function Gw with a supp ort over the unit interval Weas

sume that Gw is strictly concave which implies that the density g w is

strictly decreasing In lo ose terms this means that the sp ecial bidder is

more likely to havealowvaluation for an ob ject rather than a high one

There is also one bidder who wecalltheordinary bidder who wishes

to purchase b oth ob jects up for bid This bidder draws her valuation from

the distribution F v with a supp ort over the unit interval

In a simultaneous auction bidders are confronted with a dicult strategic

decision If a bidder bids for all the ob jects she would like to win she runs

the risk of driving up prices on the ob jects she do es in fact end up winning

Auseb el and Cramton do cument this ineciency in multiunit auctions in

several dierent settings The problem confronting bidders who wantmore

than one ob ject is knowing when to stop bidding for several ob jects and

settle for fewer ob jects instead Web er provides anecdotal evidence of

rms strategically reducing their demands in the FCCs MTA broadband

auction

Consider the ordinary bidder who wants b oth ob jects Dene the func

tion sv as the p oint where an ordinary bidder with valuation v will stop

bidding for b oth ob jects and settle for one instead Ataninterior solution

this function is implicitly dened by

R

v

dG

Gv Gs

s

v s v

Gs Gv Gs

where the LHS is her payo from stopping at the bid level s to claim one

item and the RHS is her exp ected payo from continuing to bid for b oth

ob jects

Lemma The function sv is increasing over the interval v wherev

R

v

G d Over the interval v the function sv is is dened asv

equal to zero and s



Note that b ecause Gw is strictly concavev decreases as G w de

creases for all w

Equilibrium

We are lo oking for an equilibrium in the form of a couple b w where a

sp ecial bidder with a valuation equal to or greater than w jump bids to an

amount b If the sp ecial bidders valuation is less than w then instead of

jump bidding she bids in small increments

Consider the case where the sp ecial bidder has a valuation less than w

and therefore do es not jump bid and the ordinary bidder has a valuation



The fact that b oth bidders valuations are drawn from distribution s with the same sup

p ort is a useful simplicatio n Dierent supp orts could b e incorp orated into the analysis

without changing the fundamental results

greater than or equal to w In this case if the bidding reachesalevel s then

the ordinary bidder is indierentbetween continuing to bid for two ob jects

or settling for one instead if the following condition holds

R

w

dG

s

v s v

Gw Gs

The LHS is her payo if she quits bidding immediately and settles for one

ob ject while the RHS is her exp ected payo from continuing to bid for b oth

ob jects Up on rearrangement this condition b ecomes

R

w

dG

v s

s

Gw Gs

Given that Gw is strictly concave and vw itisclearby insp ection that

the LHS is always greater than the RHS for all s w This shows

that an ordinary bidder with a valuation equal to or greater than w will

always continue to bid for b oth ob jects if she infers that the sp ecial bidders

valuation is less than w

Now consider the case where b oth the ordinary and sp ecial bidders

valuations are less than w Dene the functions v w as the p oint where

the ordinary bidder will stop bidding for b oth ob jects and settle for one

instead Ataninterior solution the functions v w is implicitl y dened

by

R

v

dG

Gv Gs

s

v v s

Gw Gs Gv Gs

Lemma The functions v w is increasing over the interval v w w

R

vw

wherev w is dened byv w Gw G d The function

sv w is equal to zero over the interval vw ands v w w

Note that by insp ection of and it is evident thats v w sv

for all v w

Now consider the case where b oth bidders havevaluations greater than

w Up on observing the sp ecial bidders jump bid the ordinary bidder can

immediately drop out of the auction and receive one ob ject for a payo of

v orshecancontinue bidding for b oth ob jects from the p oint w Clearly

if sv w then the ordinary bidder will settle for one ob ject However

if sv w she mayormay not settle for one ob ject An ordinary bidder

with valuation v and sv w is indierentbetween continuing to bid

and quitting if the following holds

R

sv

v dG

Gsv

w

v v sv

Gw Gw

This condition can b e rewritten using the implicit denition of sv from

as

R

v

v dG

w

v

Gw

For a given w letv w denote the ordinary bidder with the largest valua

tion that would settle for one ob ject rather than continue to bid for b oth

Lemma The functionv w is increasing over the interval w where

R

w is dened by dG Gw Moreoverv v and

w

vw

Note that b ecause Gw is strictly concave and continuousw must b e



strictly greater than zero but less than Also as G w decreases for all

w w increases

Given the equilibria wehave describ ed a sp ecial bidder with valuation

w must b e indierentbetween jump bidding and not jump bidding This

condition is given by

Z

w

w sv dF w b F v w

wherev w is determined by This condition can b e rewritten as

R

w

sv dF

F w F w

b w

F v w F v w F w

Note that the jump bid b isaweighted average of w and the exp ected

value ofs v w given that the ordinary bidders valuation is less than w

This serves to demonstrate that the more information that is signaled by

way of the jump bid ie the larger the dierence b etween w andv w

the closer that b is to w Allw b pairs satisfying the ab ove relationship

constitute valid jump bid equilibria

Exp ected Revenue

In this section we examine whether the exp ected revenue of the seller de

creases when jump bid equilibria exist

Consider a sp ecial bidder with a valuation b elow w Her exp ected payo

in a nonjump bid equilibrium is given by

Z

w

w s dF

while in a jump bid equilibriu m it is given by

Z

w

w sv dF

Taking the dierence b etween and gives

Z

w

sv s dF

If in a jump bid equilibrium the sp ecial bidder has a lowvaluation

then she do es not jump bid which signals this unfavourable information to

the ordinary bidder whichmakes the ordinary bidder bid more aggressively

More aggressive bidding by the ordinary bidder aids economic eciency since

it makes it more likely that the ordinary bidder will win b oth ob jects when

in fact she values each one more than do es the sp ecial bidder Againin

eciency b enets the seller since she exp ects to sell the ob jects at a greater

price

Next consider a sp ecial bidder with a valuation greater than or equal to

w In a nonjump bid equilibrium her exp ected payo is given by Ina

jump bid equilibrium it is given by

Z

w

w b F v w w s dF

vw

Taking the dierence b etween and the sp ecial bidder receives a

greater exp ected payo from jump bidding if

R

vw

s dF

b

F v w

This states that a sp ecial bidder b enets from jump bidding for jump bids

less than the exp ected value of sv conditional on the ordinary bidders

valuation less thanv w Note that for the range of v ab ovev w there is

no dierence in the exp ected payos from jump bidding or not

Substituting for b from and rearranging terms gives

R

vw

s dF

F w

w

F v w F v w

R

w

sv dF

F w

F v w F w

If then a sp ecial bidder with a valuation equal to or greater than w

exp ects to prot from jump bidding

To determine if the sp ecial bidder prefers a jump bid equilibrium over a

nonjump bid equilibrium we need to consider her ex ante exp ected payo

her exp ected payo b efore she knows her valuation A jump bid equilibriu m

is preferred to a nonjump bid equilibrium if

Prfw w gE fjw w g Prfww gE fjww g

or up on substitution

Z Z

w w

Gw sv s dF dGw

Given the generality of the mo del mayorma y not hold in practice It

will dep end on the particular functional forms of Gw and F v

Given the zerosum nature of the auction if the sp ecial bidder prefers

a jump bid equilibriu m then the seller exp ects less revenue The exp ected

gain to the seller from the aggressive bidding of the ordinary bidder in the

case the sp ecial bidders valuation b eing less than w is exceeded by the

exp ected loss if the sp ecial bidders valuation is greater than w Note that if

the sp ecial bidder exp ects to b enet in a jump bid equilibrium the eciency

of the auction is also reduced This is b ecause it is more likely that each

bidder will win one ob ject despite the ordinary bidder valuing eachobject

more than the sp ecial bidder

A Numerical Example

To aid understanding we oer the following numerical example The sp ecial

bidder draws her valuation from the distribution Gw w w whichis

clearly strictly concave The ordinary bidder draws her valuation from the

distribution F v v where

Given the prop erties of Gw wehave

for v v

p p

sv

forvv v

and

for v vw

p

sv w

v w v w v forv w v w

p

p p p

wherev andv w w w

For ease of analysis wecho ose to fo cus on the jump bid equilibriu m

where w w so thatv w For this examplew

Consider rst the case where so that F v is uniform For this

parameter value the jump bid asso ciated with w is b

However the sp ecial bidder is worse o ex ante in this jump bid equilibriu m

compared to the nonjump bid equilibrium The calculation of shows that

it is slightly negative

Now consider the case where so that F v isconvex The jump

bid asso ciated with w in this case is b Moreover the

sp ecial bidder is b etter o ex ante in this jump bid equilibrium than in the

nonjump bid equilibriu m From

Conclusion

In this pap er we examine a bidders rationale for jump bidding Weshow

that in a oneob ject auction a bidder b enets from jump bidding only if

her distribution is discontinuous and if bidder identities are concealed The

more information that a seller conceals the higher the jump bid a bidder

is willing to make in order to signal her valuation and the larger is the

set of jump bid equilibria In the case of an auction for two ob jects we

show that a bidder who wants only one ob ject can exp ect a strictly greater

exp ected payo from jump bidding if the jump bid causes the other bidder

to reduce her demand Finallywe show that the exp ected revenue of the

seller decreases when a jump bid equilibrium exists

Wehave explored the role of jump bidding when bidders consider mak

ing a jump bid only in the rst round It is p ossible that other jump bid

equilibria exist that involve bidders making jump bids in later rounds but we

have not tried to characterize them The existence of equilibria constructed

from strategies of this typ e is left for future work Moreover our mo del

considers the case where bidder valuations are statistically indep endentand

private information An interesting avenue for future research is to explore

the interaction b etween jump bidding and information release in the context

of the paradigm

Throughout our analysis wehave fo cused on the role of jump bidding in

a single isolated auction If a sequence of auctions were scheduled a bidder

may jump bid in an early auction to signal that she is a very aggressive

bidder to her rivals This may allow her to gain a reputation for b eing a

tough bidder whichmay cause her rivals not to enter subsequent auctions

This p ossible role for jump bidding is very plausible and promises to b e an

interesting area for future research

References

Laurence M Auseb el and Peter C Cramton Demand reduction and in

eciency in multiunit auctions Technical rep ort University of Mary

land

Christopher Avery Strategic jump bidding in english auctions Review

of Economic Studies

Bhaskar Chakravorti William W SharkeyYossef Spiegel and Simon

Wilkie Auctioning the airwaves The contest for broadband p cs sp ec

trum Journal of Economics and Management Strategy

Peter C Cramton Money out of thin air The nationwide narrowband

p cs auction Journal of Economics and Management Strategy

Peter C Cramton The p cs sp ectrum auctions An early assessment

Journal of Economics and Management Strategy

Kent Daniel and David Hirshleifer A theory of costly sequential bid

ding Technical rep ort University of Chicago

David Kreps and Rob ert Wilson Reputation and imp erfect informa

tion Journal of Economic Theory

R Preston McAfee and John McMillan Analyzing the airwaves auction

Journal of Economic Perspectives



For a discussion of strategic b ehaviour to establish a reputation see Kreps and Wil

son

Paul R Milgrom and Rob ert J Web er A theory of auctions and com

p etitive bidding Econometrica

Rob ert J Web er Making more from less Strategic demand reduction

in the fcc sp ectrum auctions Journal of Economics and Management

Strategy

A Technical App endix

Proof of Proposition A bidder b enets from jump bidding if she exp ects

to receive a greater payo from using that strategy than from not jump

bidding If a bidder do es not jump bid and wins she exp ects to payan

amount equal to the second highest valuation conditional on her own b eing

the largest For a bidder with valuation v this is given by

R

v

n

dF

v

Av

n

F v

Weprove the prop osition in three stages First we show that no sepa

rating jump bid equilibria exist Second weshow that no p o oling equilibria

exist Finallywe show that no partially p o oling equilibria exists

i Supp ose that in equilibrium a bidder with valuation v makes a jump

bid of B v where B v is strictly increasing Therefore given a bidders

choice of jump bid all other bidders infer the bidders true valuation If

B v is the equilibriu m jump bid function then the exp ected prot of a

bidder with valuation v jump bidding to B v isgiven by

n

F v v B v

This must b e maximized at v v if B v is the equilibrium jump bid

function Taking the derivative with resp ect to v and setting v v in the

rstorder conditions gives

 n  n

v B v F v B v F v

n

Adding F v on b oth sides gives

n  n

F v v B v F v

Integrating b oth sides from v to v gives

Z

v

n n n

F v v B v F v v B v F v dv

v

n

It is straightforward to argue that F v so that the second term on

the LHS vanishes Solving for B v andintegrating by parts gives B v

Av Therefore this jump bid equilibrium is not b enecial to the buyer or

the seller Note that Av is also equal to the bidding function in a rstprice

sealedbid auction

ii Consider the p ossibility of a p o oling equilibrium In this equilibriu m

there exists a b where all bidders with valuations greater than or equal to b

jump bid to a Those bidders with valuations b elow b do not jump bid It is

clear from part i ab ove that a B b otherwise a bidder with a valuation

slightly smaller than b will cho ose to mimic the jump bid Toshow that

no jump bid equilibriu m exists we need to demonstrate that a bidder with

valuation v where vb do es not exp ect to prot from jump bidding This

implies that her exp ected prot from jump bidding must equal that from

not jump bidding This condition is given by

R

b

n

dF

v

n

F b v

n

F b

R

v

n

dF

n n

b

F v F b v

n n

F v F b

R

v

n

dF

v

n

F v v

n

F v

where the LHS is the exp ected prot from jump bidding Simplifying b oth

sides of this expression gives

Z Z Z

v b v

n n n

dF dF dF

v v b

which clearly holds with equality

iii Finally consider the p ossibility of a general partial p o oling equilib

rium Let v be the lowest valuation asso ciated with any jump bid of any

bidder Let bidder b e the bidder with the lowest jump bid valuation and

let the corresp onding jump bid b e b Bidder with valuations higher than

v may also jump bid to b Seeing a bid of b only bidders with valuations

lower than v will drop out since bidders with valuations higher than v still

haveachance to win

If bidder with valuation v wants to jump bid to b then the payo

from jump bidding must not b e lower than from not jump bidding This

means the following condition must hold

R

v

n

dF

v

n n

F v v b F v v

n

F v

which implies that

R

v

n

dF

v

b

n

F v

A bidder with a valuation lower than v sayv cho oses not to jump

bid This must mean that this bidders exp ected payo from jump bidding

do es not exceed that from not jump bidding This implies the following

condition must hold

R

v

n

dF

v

n n

F v v b F v v

n

F v

Solving for b gives

R

v

n

n n

dF

v F v F v

v

b

n n

F v F v

The RHS of this expression is increasing inv Asv approaches v the two

b ounds of b coincide which means that bidder with valuation v cannot

prot from jump bidding

p n

Proof of Proposition By direct insp ection it is clear that To

f p

complete the pro of we need to show that Using the expressions in

p

the text is greater than jumpF if the following condition holds

Z Z

v v

n n

dF v dF F v

v v

Z

v

n n

v dF vF v

v

This can b e rewritten as

Z Z

v v

n n

dF v dF F v

v v

Z

v

n

dF

v

Z

v

n n n

n F F F v vnF F v dF

v

Z

v

n

dF

v

Z

v

n n n

dF vnF F v vn F F F v

v

Z

v

n n

vF v v dF

v

This condition always holds provided

Proof of Proposition Toprove that a jump bid equilibri um exists weneed

to show that a lowvaluation sp ecial bidder and an ordinary bidder do not

have an incentive to mimic the jump bid



First consider a sp ecial bidder with a valuation w v v If she mimics

the jump bid her exp ected payo is given by

n 

F v w

while if she do es not jump bid her exp ected payo is given by

Z



w

 n

w dF

v

Note that the sp ecial bidders exp ected prot from using either strategy

do es not dep end on the amount of information released The sp ecial bidder



with valuation w will not jump bid provided the jump bid is suciently

high so as to make her exp ected payo from jump bidding less than from not

where a lowvaluation jump bidding This implies that there exists a p oint

sp ecial bidder would not mimic jump bids greater than this amount Setting

the sp ecial bidders exp ected payo from jump bidding equal to her exp ected

payo from not jump bidding and solving for the jump bid gives

R



w

 n

w dF

v



w

n

F v



Note that this expression is increasing in w and therefore is largest when

f  

This implies that a sp ecial w v Also note that if w v then

v cannot prot from jump bidder with a high valuation ie w v

bidding when all information is disclosed b ecause any jump bid strictly less

f

than will b e mimicked by a sp ecial bidder with valuationv When some

information is concealed a lowvaluation sp ecial bidder is unwilling to jump

f p n

bid as high as a high valuation sp ecial bidder ie

When some information is concealed wemust also consider the p ossi

bility of an ordinary bidder mimicking the jump bid First consider an

ordinary bidder with valuation v v v If she jump bids her exp ected

prot is

Z

v

n n

Gv F v v v dG F

v

while if she do es not jump bid its given by

Z

v

n

v dG F

v

She is indierentbetween jump bidding and not jump bidding for a jump

bid equal to

R

v

n

G F d

v

v

n

Gv F v

Note that this expression is indep endent of the ordinary bidders valuation



Next consider an ordinary bidder with valuation v v v If this

bidder mimics the jump bid her exp ected prot is

n 

Gv F v v

while if she do es not jump bid its given by

Z



v

 n

v dG F

v

She is indierentbetween jump bidding and not jump bidding for a jump

bid equal to

R R



v v

n  n

dG F v dG F



v v

n

Gv F v

By insp ection which means that if an ordinary bidder with a

valuation greater thanv cho oses not to jump bid one with a smaller

valuation will not as well

For a jump bidding equilibrium to exist with partial information release

p

is less than This is true if the following expression wemust show that

is satised

R R R

v v v

n n n

G F d F d v dF F v

v v v

n n

Gv F v F v

With some rearrangementitiseasytoshow that this expression holds if

n

Condition is satised Similarly it can b e shown that is less than if

Condition is satised

Proof of Lemma Since satises the conditions of the implicit function

theorem sv exists Obviously s Setting v in and re

arranging gives

R

dG

s

s

Gs

ClearlyifGw is strictly concave then this expression can never hold with

equalityforany s in the interval The RHS is always greater than the

LHS for all s This implies that s or that an ordinary bidder with the

highest p ossible valuation will always wish to keep bidding for two ob jects

Next consider the slop e of sv Equation can b e rewritten as follows

Z

v

G d v s Gs

s

Totally dierentiating this gives

Gs Gv ds

dv v sg s Gs

We show that dsdv is p ositiveby showing that b oth the numerator and

denominator of this expression are b oth negative

The denominator is negative since with Gw concavewehave

G Gs gs s

for all sIntegrating b oth sides from s to v gives

Z

v

v s

G Gs d g s

s

whichgives

Z

v

G d Gsv s g sv s

s

From wehave

v s Gs Gsv s Gsv s

Cancelling the term v s from b oth sides we obtain the denominator ab ove

and hence it must b e negative

To see that the numerator is negative substitute the expression for

Gsfrominto the numerator and rearrange The numerator is then

negativeif

Z

v

G d G v v s

s

which holds given that Gw is an increasing function and vs

Finallyifwe set s in and rearrange wehave

R

v

G d

v

The RHS is increasing in v and

R

v

Gd

lim

v 

v

and

R

v

G d

lim

v 

v

Therefore there exists a unique v callitv such that the expression holds

Clearly then sv is increasing for vv and equal to zero for v

v

Proof of Lemma Since satises the conditions of the implicit function

theorems v w exists

Obviouslys v Setting v w in and rearranging gives

R

w

dG

w s

s

w

Gw Gs

Clearlygiven that Gw is strictly concave the expression ab ove can never

hold with equality for s in the interval w The RHS is greater than the

LHS for all s This implies thats v w w

Next consider the slop e ofs v w Equation can b e rewritten as

follows

Z

v

G d v sGw Gs

s

Totally dierentiating gives

ds Gw Gs Gv

dv Gw v sg s Gs

We show that dsdv is p ositiveby showing that b oth the numerator and

denominator are negative The denominator is negative since with Gw

concavewehave

G Gs gs s

for all sIntegrating b oth sides from s to v gives

Z

v

v s

G Gs d g s

s

whichgives

Z

v

G d Gsv s g sv s

s

From wehave

v sGw Gs Gsv s Gsv s

Cancelling the term v s from b oth sides we obtain the denominator ab ove

and hence it is negative

To see that the numerator is negative substitute for Gw Gs

from into the numerator and rearrange The numerator is negative

if

Z

v

G d Gv v s

s

which holds given that Gw is an increasing function and that vs

Finallyifwe set s in and rearrange wehave

R

v

G

Gw

v

Since the RHS is increasing in v and

R

v

G d

lim

v 

v

and

R

v

G d

lim

v 

v

there exists a unique v callitv w such that the expression ab ove holds

Clearly thens v w is increasing for vvw and equal to zero for

v vw

Proof of Lemma By denitionv v Ifweset v in and solve

for w then the condition holds for a w callitw dened by

R

dG

w

Gw

Totally dierentiating gives

dv v w g w

 

dw v sv g sv s v Gsv Gw s v Gsv

Since sv w Gsv Gw The denominator will b e p ositiveif

Gsv

g sv

v sv

which holds b ecause Gw is strictly concaveand vsv The numerator is

p ositive since forv w itmust b e the case that vw otherwise the

ordinary bidder would not have a p ositive exp ected payo from continuing

to bid for b oth ob jects Thereforev w is an increasing function with

v v andv w