Auctions with resale markets : an exploratory model of treasury bill markets

tA

e

">

WORKING

PAPER

ALFRED

P.

SLOAN

SCHOOL

OF

MANAGEMENT

AUCTIONS WITH RESALE MARKETS:

AN

EXPLORATORY MODEL

OF

TREASURY BILL MARKETS

Sushil

Bikhchandani

and

Chi-fu

Huang

June

1988

Revised June

1989

WP

#

3070-89-EFA

MASSACHUSETTS

INSTITUTE

OF

TECHNOLOGY

50

MEMORIAL

DRIVE

CAMBRIDGE,

MASSACHUSETTS

02139

AUCTIONS

WITH

RESALE MARKETS:

AN

EXPLORATORY

MODEL

OF

TREASURY BILL

MARKETS

Sushil

Bikhchandani

and

Chi-fu

Huang

June

1988

Revised June

1989

SEP

1

2

1989

RECEIVED

AUCTIONS

WITH

RESALE MARKETS:

AN EXPLORATORY

MODEL

OF

TREASURY

BILL

MARKETS*

Sushil

Bikhchandani^and

Chi-fu

Huang^

June

1988

Revised

June

1989

Abstract

Thispaper developsa

model

ofcompetitive bidding witha resalemarket.

The

primary

market

ismodelledasa

common-

value auction,inwhich biddersparticipateforthepurpose ofresale. After the auction the winning bidders sell the objects in a secondary

market

and

the buyers on the secondary

market

receive information

about

the bids submitted

in the auction.

The

effect of this information linkage

between

the

primary

auction

and

the secondary

market

on bidding behaviour in the primary auction is examined.

The

auctioneer's expected revenues

from

organizing the primary

market

as a discriminatory auction versus a uniform-price auction are

compared, and

plausible sufficient conditions

under

which

the uniform-price auctionyields higher expected revenues are obtained.

An

example

ofour model, withtheprimary

market

organized as adiscriminatory auction,is

the U.S.Treasurybill market.

*WethankJohnCoxforkindlingourinterest in Treasurybillauctions and MargaretMeyerforhelpful conversations.

We

aregratefultoChiang SungoftheChemicalBank foransweringmanyofour questions onthe organizationof Treasury billauctions.

We

wouldalso like toacknowledge helpfulcommentsfrom seminarparticipants atCity Collegeof

New

York, Princeton, Stanford and

UCLA.

Thesecond authoris

gratefulforsupportundertheBatterymarchFellowshipProgram.

^Anderson GraduateSchoolofManagement,UniversityofCalifornia,Los Angeles,

CA

90024 'Sloan SchoolofManagement,MassachusettsInstitute ofTechnology,Cambridge,

MA

02139

1 Introduction 1

1

Introduction

One

ofthe

major

achievementsofeconomic theory in the pastdecade has been a deeper understanding oftheconduct

and

design of auctions. Recent surveys ofthe literature

on

auctions are

McAfee and McMillan

(1987),

Milgrom

(1987),

and

Wilson (1988). Auctions accountfora large

volume

of

economic and

financialactivities.

The

U.S. Interior

Depart-ment

uses asealed-bidauctiontosellmineralrightsonfederally

owned

properties. Auction houses regularly conduct auctionsof antiques, jewelry,

and works

ofart. Every

week

the U.S. Treasury

Department

uses a sealed-bid auctiontosell Treasurybills

worth

billionsof dollars.

Many

other

economic

and

financialtransactions,although notexplicitlyconducted asauctions, canneverthelessbe thoughtof as implicitly carriedoutthrough auctions; see, forexample, an analysis ofsalesofseasoned

new

issuesin Parsons

and Raviv

(1985),

and

the

market

forcorporatecontrol in

Tiemann

(1986).

One

featureoften sharedby suchfinancial activitiesisthat there exist active resale or secondary markets for the objects for sale. This is true, for example, for Treeisury bills

and

for seasoned

new

issues.

One

may

argue that

when

there exists the possibility of resale,theauctionwill be

common-value

withthe resale pricebeingthe

common

valuation

among

allthe participating bidders.'

Thus

itmight be arguedthat the theoryof

common-valueauctions developed by Wilson (1969),

and

Milgrom and

Weber

(1982a) isapplicable.

The

observation that an auction with a resale

market

is

common-value

is certainly true.

However,

there are situations that

make

existing theory inapplicable.

A

case in point is

Treasurybill auctions.

InTreasury bill auctions, there areusually about forty bidders or

primary

dealers

who

participate in the weekly auction.

These

primary dealers are large financial institutions.

They submit

competitive sealed bids that are price-quantity pairs. Others, usually indi-vidual investors,can

submit

noncompetitive sealedbids that specifyquantity (less then a prespecified

maximum).

The

noncompetitivebids, usually small in quantity,always win.

The

primary dealers

compete

for theremaining billsina discriminatory auction.

That

is,

the

demands

ofthe bidders, startingwiththe highest pricebidder

down,

are

met

untilall

thebillsare allocated.

The

winning competitive bidders

pay

the unit pricetheysubmitted. Allthe noncompetitive bidderspay the quantity-weighted averagepriceofallthewinning competitive bids. After the auction, the Treasury

department announces some

summary

1 Introduction 2

statisticsabout the bidssubmitted. These include • total tender

amount

received;

• total tender

amount

accepted;

-• highest

winning

bid; • lowestwinning bid;

• quantity weighted averageofwinningbids;

and

• thesplit

between

competitive

and

noncompetitive bids.

The

Treasurybillsare thendelivered to thewinning bidders

and

can beresold atanactive secondary market.

Sinceprimary biddersare large institutions,theytendtohaveprivateinformationabout the

term

structure of interest rates thaiisbetterthantheinformation possessedbyinvestors inthe secondary markets.

The

primary dealerssubmitbids in the auctionbased

on

both informationthatispublicly availableatthe time,

and

their private information.

The

buyers

on

the resale

market

haveaccessonly to public information, includinginformationrevealed bytheTreasuryaboutthe bidssubmittedinthe auction.

To

theextentthat bidssubmitted reveal the private information ofthe primary dealers, the resale price in the secondary

market

will be responsive to the bids. Thiscreates anincentive for the

primary

dealers tosignal theirprivateinformationtothesecondary

market

participants. This information linkage

between

the actionstakenbythe biddersintheauction

and

the resale priceisabsent

in existing

models

of

common-value

auctions

and

is theprimary focusofthispaper.

^

In Section 2,

we

develop a

model

of competitive bidding with a resale market.

The

primary

dealers orbiddersare risk-neutral

and

haveprivateinformation

about

the true value oftheobjects.

We

assume

that the bidders' private signals

and

the true value areaffiliated

random

variables, i.e., roughly speaking, higher realizations of a bidder's private signal

imply thathigherrealizations ofthe true value,

and

ofthe other bidders' private signals, are

more

likely. Aftertheauctiontheauctioneerpubliclyannounces

some

information (the prices paid

by

thewinning bidders,for example) aboutthe auction.

The

winning

bidders ^Foran analysisofbiddingwith aresalemarketwhenthe valuations of the bidders are

common

knowledge,

1 Introduction 3

thensell the objects inthesecondary market,at a priceequal totheexpectedvalue of the object conditionalon allpublic information.^

Although

the primary motivation ofthis

model

is the Treasury bill market, there are

many

institutional details,

which

areabsent. For instance,

we

require that competitive bidders

demand

at

most

oneunit ofthe object, instead ofbeing allowedtochoosequantity.

And

we

do not

model

theeffectofany forwardcontracts,

and

closesubstitutes (such as\ast

week's Treasurybills)

owned

byprimary biddersontheirbiddingstrategies. In thispaper

we

focuson one aspectoftheTreasurybill

market

—

theinformationallinkage ofthe resale

market and

theprimary auction.

The

results ofthispaper

may

alsobehelpfulin analyzing other types ofauctionswith resalemarkets, such as artauctions,inwhichbiddershavecorrelated values. Ifa painting

by

Van

Gogh

is auctioned at a price

much

higher than expected, then one might expect this

and

other paintingsby

Van

Gogh

tobesold athigherpricesin future.

Insection 3

we

analyze discriminatory auctions. It is

assumed

that thewinningbids

and

the highest losing bids are revealed at theendof the auction.

We

provide sufficient condi-tionsforthe existence of aunique

symmetric Nash

equilibrium innondecreasingstrategies in the auction. Unlikethe

model

in

Milgrom and Weber

(1982a)

where

biddersparticipate forthepurposeof

consumption and

thereis noresalemarket,the affiliationproperty alone

is not sufficient for the existence of anequilibrium.

The

equilibrium bids

we

obtain axe higherthanthosederivedin

Milgrom and

Weber

(1982a),because

primary

biddershave

an

incentive to signal.

A

keyinsightgained

from

thetheoryof

common-valuation

auctionswithoutresale

mai-ketsisthat the greater the

amount

ofinformation (aboutthe true value of the objectsfor sale)revealed inan auction,the greater the expected revenue forthe auctioneer.

Any

re-ductionintheuncertaintyabout the true value

weakens

thewinners'cursewhichin turns causesthebidders tobid

more

aggressively.

Thus

iftheauctioneerhatsprivateinformation

affiliated with the bidders' valuations, he can increase his expected revenue

by

precom-mitting to

announcing

his private information before the auction.

When

there isa resale

market

and

there existsan incentive forthe bidders to signal, the auctioneer

may

reduce the bidders' incentive to signal

and

thusdecreasehisexpected revenue ifhe announceshis ^Riley (1988) investigates amodel in whichconditionalupon winning, each bidder'spaymentdepends uponallthe bidssubmitted. Inour model, the expected valueforeachprimarybidderdepends onthe bids submitted.

1 Introduction ₄

private information. Sufficient conditionsforthe public

announcement

ofthe auctioneer's privateinformationto increasehisexpected revenueareprovided.

In Sections 4

and

5

we

consider auniform-priceauction, thatis an auctionin

which

the rulefordeterminingthewinningbiddersisidenticaltotheoneinthediscriminatoryauction, butthewinning bidders

pay

auniformpriceequaltothe highest losing bid.

The

existence ofan equilibrium

depends

in part

on

the kind ofinformation about the auction publicly revealedbythe auctioneer. If,as

we assumed

fordiscriminatory auctions, thewinningbids are

announced

then

we show

by

example

that there

may

existanincentiveforthe bidders to

submit

arbitrarily largebidsinorder to deceive thesecondary

market

buyers. Biddersin adiscriminatoryauction

do

not havesuch anincentivesince,

upon

winning, they

must pay

what

they bid. However,a

symmetric

Nash

equilibrium alwaysexistsin theuniform-price auction,provided that onlythe price paidby thewinning biddersis

announced

(and thus biddershave

no

incentive tosignal).

Next,

we

turntothequestionofthe auctioneer's revenues.

The

keyinsightgained

from

thetheoryofauctionswithout resalemarketsmentioned aboveisalsouseful here.

Without

resalemarkets, uniform-price auctionsyield higher revenues than discriminatoryauctions, sinceintheformerthe priceislinked totheinformationofthehighest losing bidder.

When

there are resalemarketsinwhichthebuyers

draw

inferencesabout the truevalue

from

the bids

and

thewinningbids are announced, thereisanadditional factor

which works

in the

same

direction.

Namely

that in a uniform-price auctionit ischeaper tobidhigh in order to signalahigh realization of privateinformation, since conditional

upon

winning a bidder does not

pay what

he bids. Therefore, inour

model

as well,uniform-price auctionsresult in greaterexpected revenuesforthe auctioneer,

when

there existsanequilibrium.

We

also provide plausible sufficient conditions under which the auctioneer's revenues

when

theprimary auction isorganized as auniform-priceauction

and

only the pricepaid by

winning

bidders is

announced

isgreater than under adiscriminatory auction

when

the winning bids are announced. Section 6 contains concluding remarks. Allproofs are in an appendix.

The

revenue

maximizing

mechanism

for sellingTreasurybills

was

a subject ofdebatein the early 1960s.

Friedman

(1960)proposedthat theTreasury should switch

from

a discrim-inatoryauctiontoauniform-price auctionforthesaleofTreasurybills.

Apart from

the fact that uniform-price auctions

would

inducebidders to revealtheirtrue

demand

curves,

2

The

model

5

bidders

from

participating.

Both

Goldsein (1960)

and

Brimmer

(1962)disputed Friedman's contention.

Smith

(1966),

on

the basis of a

mathematical

model, concluded that uniform-price auctionsyield greater revenues. Unlike Smith'smodel, our

model

isgame-theoretic

in thateach bidder's beliefsabout the others' bids areconfirmed in an equilibrium,

and

we

model

the informationlinkage

between

theprimary auction

and

thesecondary market. Like Smith, our analysis provides support for Friedman's proposal that the Treasury bill

auction should be uniform-price.

2

The

model

Considera

common-value

auctionin

which

nrisk-neutralbidders (the dealers

who

submit

competitivebidsinTreasurybillauctions) bidforkidentical,indivisibleobjects,with

n

>

k.

The

truevalueofthe objectsisthe

same

forallthe bidders,

and

is

unknown

to

them

atthe time they

submit

bids.

Each

bidderprivatelyobserves a signalaboutthe true value,based

on which

hesubmitsabid.

We

cissumethat there areno noncompetitivebids. InSection6

we

indicate

how

noncompetitive bids can be incorporated in our model.

Throughout

we

assume

thateach bidder

demands

(oris allowed) at

most

oneunit ofthe object.

The

primary

dealers' interest in the objects being auctioned is solely for the purpose of resale in the secondary market.

We

assume

that the primary dealers' personal (con-sumption) valuations of the object are alwayssufficiently lowerthan the valuations of the resale

market

buyersthatthey

would

prefertoresellthe objects ratherthen

consume

them. Forinstance, theprimary bidders

may

havecapital constraints so that unlessthey sellthis

week'sTreasurybills,they

may

not beable to participateinnext week'sauction. Since, as

we

willestablish,primarybidders

make

positiveexpectedprofits inthe auction,they

might

prefer to sell theTreasury bills at their expected value conditionalon all publicly

known

information.

Ifinstead one

assumes

that theprimary bidders'personal value of the object is atthe szLmelevel as that of the resale buyers, then ifallthe private informationof thewinning primary biddersis not revealed after theprimary auction,the primary bidderswillnever

sell the object if the expected value of the object conditional

on

all public information

and

their privately

known

information isgreaterthanthe resale price,

and

willalwayssell

when

theirexpected valueisstrictly lessthanthe resaleprice. Thereforethe resale

market

buyerswill

make

strictlynegative expectedprofits

and

the resale

market

will break

down.

2

The

model

6

Hence we

preclude this possibility. Forsimilar reasons

we

assume

thatprimary bidders can participateinthe resale

market

only assellers,not asbuyers.

Aftertheauctiontheauctioneerpubliclyannounces

some

information

about

the auction. Forsimplicity

we assume

that thatthewinningbids

and

the highest losing bidsubmittedin theauctionarepublicly announced. Inthe case oftheuniform-priceauction,there

may

not exist an equilibriumifthewinningbids are revealed attheendofthe auction. Therefore,

we

alsoanalyze uniform-price auctions

when

thewinning bids arenot

announced and

only the pricepaidbythewinning biddersis announced.

We

allow the possibility that

some

additionalinformationaboutthe value of the objects forsale

may

become

publicly availableafterthe bids aresubmitted, butbeforetheopening ofthe secondary market.

The

k winners in the primary auction then sell the objects to risk-neutral buyers on the secondary markets.

The

buyers on the secondary markets

do

nothaveaccess to

any

privateinformation

about

the true value.

They

infer

what

they can

from

the informationreleased bytheauctioneer about theprimaryauction,

and any

other publicly available information. Thus, regardless ofthesecondary

market

mechanism

—

an auctionorapostedprice

market

—

the resale pricewillbetheexpectedvalue of the object conditionalonall publicly availableinformation.^

The

n

risk-neutralbidderswillbe indexed byi

=

1,2,...,n.

The

true value ofeachobject being auctioned is a

random

variable,

V

.

Each

bidder i has a

common

prior

on

V, and

observesa private signal, X,,aboutthe true value. Let

P

denote

any

otherinformation that

becomes

publicaftertheauctionisoverbutbefore the resale

market

meets.

We

willassume, except

when

otherwisestated, that given

P

the bidders' signals arenotuninformative about the true value, thatis,

BlVlP]

^

E[V\Xi,X2,

,^„,P]. Ifthiscondition isviolated,for

example

when

P =

V

, our

model

reduces to the usual

common-value

auction without a

resalemarket.

Let f{v,p,x) denote the joint density function of

V

, P,

and

the vector of signals

X

=

{Xi,X2,

.,Xn).

It is

assumed

that/ is

symmetric

in the Icist

n

arguments. Let [v,v\ X

[p,p]X[x,i]"bethesupport of/,

where

[z,i]"denotesthe n-foldproductof[x,x].

Note

that

we

do

notruleout the possibility that the support of the

random

variablesis

unbounded

either

from

above or

from

below. Further, itis

assumed

that all the

random

variablesin *Theparticipantsinthesecondary market do nothavetoberisk neutral. In the caseofTreasurybill

auctions, the "true" valueofabillforaprimarybidder that pays say$1 in182dayswillbeE[m|/], where

m

denotes therandommarginalrate ofsubstitutionbetween consumption 182daysfrom nowandthatof

today and whereIdenotes the information revealed byallthe bids submitted. This true valuewillbe the secondary marketpricewhetherornot thesecondarymarketparticipantsare riskneutral.

3 Discriminatory auction 7

this

model

are affiliated.

That

is,forailx,x'

G

[^,^1",v, v'

E

{v,v],

and

p,p'

e

{p,p],

f{iv,p,x)

V

{v',p',x'))f{{v,p,x)

A

{v',p'y))

>

fiv,p,x]f{v',p',x'),

where

V

denotes the

componentwise

maximum,

and

A

denotes the

componentwise

mini-mum.

Affiliationissaid tobestrictiftheabove inequalityisstrict. Affiliationimpliesthat

if

H

is anincreasing^ functionthen

E[H{V

,P,Xi,X2,

...,Xn)\ci

<

Xi

<

di, i

=

l,...,n]

is an increasingfunction ofc,, d,.

The

reader is referred to

Milgrom and Weber

(1982a)

for other implications ofaffiliation.

We

further

assume

forsimplicity that if

H

is contin-uously differentiable then

'E\H{V ,P,Xi,

X^,..,Xn)\ci

<

X,-

<

d,, i

<

n] is continuously differentiable in c,

and

d,, for all c,,(i,

£

[x,x], with the convention that the derivative at Xisthe right-hand derivative

and

at x isthe left-hand derivative. Moreover,

we

shall

assume

that {V

,P,Xi,

...,A''„) are strictly affiliated so that if

H

is strictly increasing in

a.ny

o{{V,P,Xi,...,X„),say

in

Xi,thenE[H{V,P,Xi,X2,...,Xn)\ci

<

Xi

<

di, t7^ 1] is

strictlyincreasinginc,-

and

d, for allCi,di

G

\x,x).

3

Discriminatory auction

In a discriminatory auction, thebidders submitsealedbids

and

thekhighest bidders

win

the auction.

A

winning bidder paysthe pricethatheor shebids. In thissection

we show

that

when

the bidders' private signals areinfoTmationcomplements^,ina sense tobedefined later,there exists a

symmetric

Nash

equilibrium withstrictly increasing strategies in the bidding

game

among

the primary dealers. Unlikethe auctions

examined

in

Milgrom and

Weber

(1982a), the affiliation property alone is not sufficient for the existence of a

Nash

equilibrium. Intuitively,

when

themotiveoftheprimary biddersistoresell inasecondary

market

in

which

thebuyers

know some

orallof theirbids (ora

summary

statisticbased

on

their bids), there existsanincentivefortheprimarybidders to bid

more

than they otherwise

would and

thus signal theirprivateinformation. This isbecause,by affiliation, the resale valueisresponsivetothe bidssubmittedtotheextentthe bids reveal the privateinformation received by the

primary

bidders. If each bidder's incentive to signal increeises with his information realization,thenthere existsan equilibriuminstrictlyincreasing strategies. It istheinformationcomplementarityofthe bidders' signalswithrespect to the true value that ^Throughout this paper, we willuse weak relations. For example, increasing means nondecreasing, positivemeansnonnegative,etc. Ifarelationisstrict,wewill say, forexample,strictlyincreaaing.

'Thereader willsee thatour notionofinformationcomplementarityisdifferentfromthatin Milgrom andWeber (1982b)

3 Discriminatory auction 8

ensures that the bidders' incentive to signal increases with theirinformation realizations

and

enables

them

to sort themselves in a separating equilibrium.

We

also

show

that if

secondary

markets

participant'sbeliefs are

monotone,

in a sense tobe defined,then there existsa unique

symmetric

equilibrium.

Ina

model where

bidders participateinan auctionfor final

consumption

ofthe objects,

Milgrom and

Weber

(1982a)

show

that the auctioneer'sexpected revenuecan be increased

ifhe

precommits

totruthfullyreporting his privateinformation about the objectsfor sale before the auction, providedthat hisprivateinformation isaffiliatedwiththe bidders' pri-vateinformation. Thisfollows sincebypublicly

announcing

hisinformation,theauctioneer introducesan additionalsourceofaffiliation

among

theprimary bidders' private informa-tion

and

thus

weakens

the winners'curse.

Hence

the bidders

compete

more

aggressively

and

theexpectedsellingpriceisincreased. However,inour

model

witha resale

market

this resultisnotnecessarilytrue.

A

portionof the bidsubmittedbyabidderisattributed tohia incentiveto signal tothe resale

market

participants. Ifthe auctioneer's privateinformation

isa "substitute" forthe bidders'information,

announcing

thatinformation willreduce the responsiveness ofthe resale price to the bidders' information. This in turn reduces the incentiveforthebiddersto signal

and

may

causetheexpected sellingprice tofall.

On

the other hand,

when

the auctioneer's private information is a

"complement"

to that of the bidders',itisalwaysbeneficial fortheauctioneerto

announce

hisprivateinformation.

3.1

Existence

of

a

symmetric

Nash

equilibrium

By

thehypothesis that /(v,p,x) is

symmetric

in itslast n arguments,this isa

symmetric

game.

Thus

it is natural to investigate the existence of a

symmetric

Ncish equilibrium.

We

examine

the

game

from

bidder I'spoint of view.

The

analysis

from

theotherbidders' viewpoint issymmetric.^

Note

that at the time

when

bidder 1 submits his bid, he only observes his private information Xi.

Thus

a strategy for bidder I is a function ofXi. Bidder i'sstrategy isdenoted 6, : [x,x\

—

9?.

We

beginour analysisby deriving the

first-order necessary conditions for an n-tuple (6,...,6) to be a

Nash

equilibrium in strictly increasing

and

differentiable strategies,

when

buyersin thesecondary

market

believethat

(6,..

.,6) arethe strategies followedinthe bidding.

^Tosimplifythe analysiswearbitrarilyassume throughoutthatincaseofatiethewinnerisnotchosen randomly. Rather, bidder 1 isdeclared the winner. This assumptionisinconsequential. Theequilibrium strategywillremainunchangedifweassumethatincaseof atie,the winner(s) is(are)chosenfromthetied

3 Discriminatory auction 9

Since buyersin thesecondary

market

do not have access to privateinformation about the true value, the resale priceistheexpectation

ofV

conditional

on

allpublicinformation.

As

mentioned

earlier,tosimplify the analysis

we assume

that theauctioneer

announces

the pricespaidby

winning

bidders(i.e.,thewinningbids)

and

thehighest losing bid.*

Suppose

thatbidders t

=

2,...,n adopt the strategy b,bidder 1 receives information

Xi

=

x

and

submits abidequal to6.

Then

ifbidder 1wins with a bidbthe resale pricewillbe

r''(6-i(6),y'i,...,n,P)

=e\v

X,

=

b-'{b),b-HKy^)),---,f>-\biY,),P\

Xi

=

b-\b),Y,,...,Y,,p],

where

b~^ denotes the inverse^ of b

and

Yj is the j-th order statistic of {X2,. .,Xn).

Note

that if

P

=

V,

r'^[b~^{b),Yi,...,Yk,P)

=

V,

and

our

model

reduces toan ordinary

common-

valueauction withouta resale market. Define

v^[x',x,y)^-E[r'^{x',Yu...,n,P)\Xi

= x,n =

y]- (2)

By

our hypothesis about strict affiliation,both r

and

v arestrictly increasingin eachof their

arguments

(provided that given P, the bidders' signals are not uninformative about V).

The

expected profit forbidder 1

when

Xi

=

x

and

hesubmitsa bid equal tobis

n'{b\x)

^

B[{r'{b-'{b)Si,---,y.,P)-b)l^,^i^y^^}\x^

=

x]

=

E

[e

[[r'{b-\b),Yu...,n,P) -

b) l{,>i^Y,)y\Xi,n] \x,

=

x]

=

E

[{v'{b-'{b),XuY,)

-

b) 1{,>S(K0}|^1

=

^] .

=

/

{v'{b-\b),x,y)-b)f,{y\x)dy, JX

where

thesecondequality followsfromthelawof iterative expectations,

and

fk{y\x)denotes the conditional density function ofy^given Xi. Takingthefirstderivative ofn'^(6|z) with respect tob gives

^n^

₌

(v^(6-'(6),x,6-'(6))

-

b)h{b-Hb)\x){b'{b-'{b)))-^

-

F,{b-Hb)\x)

+

{b'{b-'{b)))-'fp'Ki{b-\b),x,y)My\x)dy,

where

b'{x) is the derivative of 6(1), Ft(t/|i) is theconditional distribution function ofYk given Xi,

and

vf is the partial derivative of v'^ with respect to its first argument. For

'ifsomeoftheother losing bids,orsomefunctionofthem,arealso announced alltheresults remain unchanged.

*If6<b{z) thenfc~'(fc)=iandif6 >6(r) then6"'(ii)

=

1. Thusweonly need toconsider valuesofb

3 Discriminatory auction 10

[b,...,b) to be a

Nash

equilibrium,it is necessary that 6 be a best response for bidder 1

when

bidderst

=

2,...,nadoptstrategy b

and

the resale

market

participants believe that allbiddersadoptb.

That

is,relation(3)

must

be zero

when

6

=

b{x):

=

_^^[^j(^)

=

{A^,x,x) -

6(i))A(i|x)(6'(x))-i

-

F,{x\x)

Rearranging (4) givesan ordinary differentialequation:

6(x)

=

(v (i,i,x)

-fc(i))-—

—

-+

/

vi(i,i,y)-—

—

dy. (5)

rk{x\i) Jx ^k[^\^)

Note

that, bythe definition ofv'^

and

thelawof iterativeexpectations,

v''(x,x,y)

=

E[y|j^i

= x,n-y].

(6)

Besides(5),there are

two

other necessaryconditions thatb

must

satisfy: (i)v (x, x,x)

>

b{x),

Vi e

[x,x];

and

(ii) b[x)

=

u'^(x, x, x). Condition (i) follows sinceexpected profit for

bidder1 hastobe positiveinequilibrium. Condition (ii) follows

from

(i)

and

the fact that

if6(x)

<

t; (i, X,x),then by slightlyincreasingthe bid to 6(i)

+

e

when Xi

=

x, expected profitcan beraised

from

zero to

some

strictly positive

amount.

The

solution to (5) withthe

boundary

condition6(x)

=

v'^[x,x,x)is

6(x)

=

v^{x,x,x)

-

r

L{u\x)dt{u)

+

r

J^dL{u\x),

(7) Jx Jx

hW^)

where

L(u|x)

=exp{-i:^ds},

t{u) =v<^(u,u,u), (8)

H^)

=

Ix*^i("' "' y)

A(y|")'^y-Note

thatL{ii\x)

and

t(u) areincrecisingfunctions ofu

and

thusare meaisureson [x,i].

We

will

show

in

what

followsthat b{x) of(7) alsosatisfiescondition (i),

maximizes

expected profitunderthehypothesisthat{Xi,..

.,

Xn, P)

are information

complements

withrespect

to

V

,

and

is strictly increasing.

Definition

1

Random

variables, (Zi,.. .

,Zm),aresaidtobeinformation

complements

with

respect toanother

random

variable

T

if

^\\

'

""'

_>0,

_{^i^j,Vzu...,Zm,}

aziOZj where

Zm\-3 Discriminatory auction 11

Thus,the

random

variables(A'l,...,Xn,

P)

are information

complements

with respect

to

V

ifthemarginalcontribution to the conditionalexpectationofV" ofa higher realization of

Xi

islargerthehigherthe realization ofanyother

Xj

orP. This information

complemen-tarityconditionissatisfied bya largeclassof distributions. Forexample, if_<^(2i,.. .,Zm) is

linearinthe 2,'s, then (Zi,..

.,

Zm)

are informationcomplements.

Thus

if(T, Zi,.. .,

Zm)

aremultivariatenormallydistributed,then [Zi,.. .

,

Zm)

areinformation

complements

with

respect toT.

We

give threeexamplesofstrictly affiliated

random

variablesthat also satisfy theinformationcomplementarity condition.

Example

1 Let [T, Zi,...

,Zm)

be multivariate normally distributed withdensityfunction

g{t,zi,..

.,Zm)- Let

S

bethe variance-covariance matrix of these

random

variables

and

as-sume

that

E~^

exists

and

has strictly negative off-diagonal elements. It is easily verified that

d^ln

g/dtdzi

>

and

d^Ing/dzidzj

>

for i

_^

j.

Theorem

1 of

Milgrom and

We-ber (1982a) thenimplies that {f,Zi,...,

Zm)

are strictly affiliated. Since E[f\Zi, ...,Zm]

is linear in the Zi's, [Zi,..

.,

Zm)

areinformation

complements

with respect toT.

Besidesthe multivariatenormallydistributed

random

variables,thereisa large class of distributionswithlinearconditional expectations.

The

followingisan example.

Example

2 LetZi,i

=

1,.. .

,

m

beindependentconditionalon

T

and

distributedaccording

gamma

distributiongiven

T =

t:

( \A ) rf^'^r^e-^'/' ifZi

>

0,

gi[Zi\t)

=

(

n^

«

.

I otherwise,

where

a >

0, t

>

0,

and

T isthe

gamma

function. Let

\/T

also be distributedaccordingto

gamma

distribution with a density

Ml/0

=

I

^(')'"^~^^'

'^'>0'

I otherwise,

where 7

>

and

a

>

0. Using

Theorem

1 of

Milgrom and Weber

(1982), one verifiesthat {T,Z\,...,

Zm)

<"£ strictly affiliated. Directcomputation yields

-F"ri7

7

1

-

£i=i:?Li_^

t,^!lZi,...,Zm\

-

-ma

+

7

—

1

3 Discriminatory auction 12

Note

that the prior distribution of

T

in

Example

2 is an element of the family of "conjugate distributions" of

gamma

distribution; see

DeGroot

(1970,

Chapter

9).

Other

distributionswith linearconditional expectations can be constructed similarly. Interested readersshouldconsult Ericson(1969)

and

DeGroot

(1970).

The

following

example

gives

random

variables that arestrictinformationcomplements.

Example

3 LetZi,i

=

1,2,...,m; be independent conditionalon

T

with density

The

density

ofT

is

[ otherwise,

y otherwise.

It is easily verified that d^ \T\gi{z{\t)/dzidt

>

Vz,t

€

(0,1).

Theorem

1 of

Milgrom and

Weber

(1982) implies that p, satisfies the strict affiliation inequality.

The

same

theorem also

shows

that

,, ^ ,

,x_

( /»(onr=i5.(2.10 t72i,22,...,2,.,<e(o,i)

p(i,zi,Z2,...,Zm)-|

otherwise

is strictly affiliated. Direct com.putation yields

[,Z2...,Zm)

='E\T\Zi

=

Zi,Z2

=

Z2,...,Zm

=

Zm\

=

forzi,Z2,...,2m

£

(0)1)- Finally, one can also verify thatd^(p[zi, Z2, . ,Zm)/dzidzj

>

for alii :^j

ifae

(0,1).

The

following

lemma

is a direct consequence ofthe definition ofinformation comple-mentarity. Allproofs areinan appendix.

Lemma

1 Suppose that {Xi,...,X„,

P)

are information

complements

with respect to

V.

Then

Vi{x,x',y) isstrictlyincreasinginx'

and

y, wherevf denotesthepartialderivative of

v"* with respect to itsfirstargument.

The

following proposition

shows

that if(A'l,..

.

,A'„,

P)

are information

complements

3 Discriminatory auction 13

Proposition

1 Suppose that (A'l,..

.

,A'„,

P)

areinformation

complements

with respect to

V.

Then

v'^{x,x,x)

>

b{x),

Vi €

[z,r]

and

the inequality is strictfor x

>

x, where b is

definedin (7).

A

corollary ofProposition 1 is that b isstrictly increasing.

Thus

our assumption that resale

market

buyerscan invert the primarybids toobtain the bidders' signal realizations

isjustified.

Corollary

1

The

strategyb defined in (7) isstrictlyincreasing.

Before proceeding,

we

firstrecord two

lemmas

that are directconsequencesofthe defi-nitionofaffiliation.

Lemma

2

(Milgrom

and Weber

(1982a))

fjk(y|i)//t(y|i) isdecreasing in x.

Lemma

3 Let x'

>

x

>

y.

Then

Fk{y\x')/Fi,{x\x')

<

Fk{y\x)/Fk{x\x). That is, the distributionfunctionFic{-\x')/Fk{x'\x') dominatesthe distributionfunction Fk{-\x) /Fic{x\x) in the senseoffirst order stochasticdominance.

The

main

resultofthissectionis

Theorem

1

The

n-tuple (6,...,6), with b as defined in (7), is a

Nash

equilibrium of the discriminatory auction providedthat [Xi,...,Xn, P) areinformation

complements

with

re-spectto

V

.

When

bidders in the discriminatory auction participate only for the purpose of con-sumption,

Milgrom and

Weber

(1982a) haveidentified a

symmetric

Nash

equilibriumwith abiddingstrategy

6^(z)

=

v''(x,x,i)

-

r

L{u\x)dt{u), (9)

where

L{u\x)

and

t{u) are as defined in(8).

The

biddingstrategyforthepurposeof resale identified in

Theorem

1 isstrictlyhigher thanb foreveryx

G

{x,x] by an

amount

equal to

3 Discriminatory auction 14

the

magnitude

ofwhich

depends

onv^,theresponsivenessofthe resale value to the

submit-ted bid.^° It isthisinformational linkbetween the resale value

and

the bidssubmitted by biddersinthediscriminatory auctionthat gives thebiddersanincentive tosignal.

Of

course, sincethebisstrictlyincreasing,the resalebuyerscaninvertthe bids

announced

bythe auc-tioneer toobtain the private informationofthebiddersand,as in Ortega-Reichart(1968)

and

Milgrom and

Roberts (1982),inequilibrium

no

onegetsdeceived.

It is

worth emphasisng

that theprimary bidders benefitifthe resale

market

buyers are better informedabout the true value. If, for instance,

P =

V' or if

F

"centains" all the relevantinformation in

Xi,X2,

..

.

,X„,,thenthere

would

be

no

signalling incentiveforthe

bidders

and

theexpected price(s) in the auction

would

be lower.

One

might expect the "informativeness" of

P

to increasewith the length of the time period

between

the

end

of theprimary auction

and

the start ofthe resalemarket.

Next

we

establish that the

symmetric

equilibriumidentified

above

isthe unique

sym-metric equilibriumifsecondary

market

buyersbeliefssatisfy amonotonicitycondition.

The

beliefsofthesecondary

market

buyersabout the bidders' privateinformationare said tobe

monotone

ifthey are nondecreasing in the bidssubmitted. For instance, ifthe secondary

market

buyersbelieve thatbidder I's private information liesin the interval [xj(6),i"(6)]

when

bidder 1bidsb,thenthemonotonicity condition

on

beliefs

would

implythatx[{-)

and

i"(-)arenondecreasingfunctions.^^ _{Sincebidders withhigherrealizations of their private}

information

would

expect higher resale prices, it isnatural toexpect

them

to bid more. Thereforemonotonicity ofbeliefsseemsto be a natural restriction to impose.

Of

course,

when

bids are in the range ofthe equilibrium bidding strategies, beliefs are obtained

by

inverting thebiddingstrategies.

Sincea

symmetric

equilibriumisa naturalfocalpointina

game

with

symmetric

players,

and

as

shown

below (6,...,6) isthe only

symmetric

equilibrium innondecreasing strategies

when

the resalebuyers' beliefsare

monotone,

we

willusethisequilibrium

when

comparing

expected revenues

between

adiscriminatory auction

and

auniform-priceauction.

Theorem

2 Ifthe secondary marketbuyers have

monotone

beliefs then (6, 6,...,6), where bisas defined in (7), istheunique

symmetric

equilibrium innondecreasingstrategies.

'"If

P

= V

thereisnosignallingmotive andtheequilibrium strategyisasspecified in (9),sincer''()

=

V

andthustif(•)

=

andh{)

=

0. Also,when

P

is"veryinformative"about

V

the signallingmotiveisweak. For exampleifweletP„

= V +

€„ where£„is,say,uniformly distributedon[—^, ^]thenasnincreaaes the

resalepricebecomeslessresponsivetotheplayers'bidsandinthelimitthe incentivetosignaldisappears.

"InthesymmetricequilibriumofTheorem1,theresalebuyersbeliefsaremonotone(sincebisincreasing) withz'i(6)

=

x'{b)

=

r'(6). For6>6(1),x\{b)

=

xt{b)

=

x,andforb<6(1),i',(6)

=

itC-)

=

Z-3 Discriminatory auction 15

3.2

Public

announcement

of

the auctioneer's

information

Suppose

that the auctioneer has private information about

V,

represented by a

random

variable Xq.

We

will consider the impact of

announcing

Xq

before the auction

on

the expectedselling price. Let b[-;xo)be a

symmetric

equilibriumbiddingstrategyconditional

on

Xq

=

Xq. It is

assumed

that 6(-;xo) isincreasing

and

differentiablein x. Ifbidder 1 bids b

and

wins then the resale price inthiscasewill be

p''Cb~\b;xo),Y\,...

,n,P]Xo)

^e[v\x, =

r\b-xo),Yu.

..

,n,P,Xo

=

xo],

where

b(b {b;xo)]Xo)

=

b. Putting

u;''(i',i,y;xo)

=

E

[p^{x'

,Yu

,Yk,

P;xo)\Xi

= x,n =

y,^o

=

^o],

itisstraightforwardto

show

that b{x;xo)

must

satisfy

V{x;xo)

=

{rv'^{x,x,x;xo)-b{x;xo))j!, ,

' °.

+

/ u)f{x,x,y;Xo)fk{y\x-xo)dy. (10) ^t(2;|x;xoJ Jx

where

/t(y|x;xo) denotes the conditional density ofY\ given

Xi

=

x

and

Xq

=

xq. In addition, the

boundary

condition b[r,xo)

=

w'^[x, x, x;xq)

must

be satisfied.

The

solution to (10)with this

boundary

conditionis

6(i;io)

=

it;'^(x,i,x;xo)

-

/ L{u\x;xo)dt[u;xo)

+

/ . .' -dL{u\x;xo), ₍₁₁₎

Jx Jx fk(u\u;xo)

where

L{u\x;xo)

=

exp<^-/

—

—

-ds\,

t(u;xo)

=

uj (u,u,u;xo), /i(u;xo)

=

/ w'}{u,u,y;xo)fk{y\u;xo)dy. Jx

Next,

we

define conditionalinformationcomplements.

Definition

2

Random

variables, [Z\,..

.,Zm), "resaidtobeinformation

complements

con-ditionalon

random

variable

Y

with respectto

random

variable

T

if

d^4>{zi,...,zm,y)

_w-^

•

_W

_W

OZiOZj where

3 Discriminatory auction 16

If {Xi,A'2,..

.,

Xn, P)

are information

complements

conditionalon

Xq

with respect to

V

,thenaproofidentical tothat of

Theorem

1

shows

thatbdefined in (11)is a

symmetric

equilibriumstrategy. Thisisstatedwithout proofinthe following proposition.

Proposition

2

The

n-tuple(b{-;xo),...,b{-;xo)), withb{-;xo) as definedin (11),isa

Nash

equilibrium of thediscriminatory auction

when

the auctioneer

announces

Xq

=

xq,provided that [Xi,...

,Xn,P)

are information

complements

conditionalon

Xq

with respect to

V

and

the resale marketparticipants believe thatallthe biddersfollow strategy 6(-;io).

Note

that the existence ofan equilibrium does not

depend

on

whether

(XcXi,

...

,X„,P)

are information

complements

with respect to

V

.

There

exists a

Nash

equilibrium aslong

as

{X\,X2,-

,Xn,P)

areinformation

complements

conditionalon

Xq

with respect to

V.

Our

main

resultinthissubsectionwillbethat theexpectedsellingpriceunderthe policy of always reportingA'ocannot be lowerthan that under anyother reporting policy provided that[Xo,Xi,...

,Xn,P)

are information

complements

withrespect to

V

.^^

We

firstshow,

inthenext proposition,that6(1;zq) is a increasing function ofxq.

Proposition

3 Supposethat

{V ,Xo,Xi,.

..

,Xn,P)

areaffiliated

and

that{Xq,Xi,...

,Xn,P)

are information

complements

with respect to

V.

Then

b{x,Xo) is an increasingfunction of XQ.

The

main

resultofthissection is

Theorem

3 Suppose that

{V ,Xo,Xi,

...,Xn, P) areaffiliated

and

that _(A'o,Ai,.. .,

X„,P)

are information

complements

with respect to

V

.

A

policy of publicly revealing the seller's

information cannotlower,

and

may

raise, the expectedrevenuefor the seller in a discrimi-natory auction.

Given

Proposition3, theproofof

theorem

3

mimics

that of

Milgrom and Weber

(1982a,

Theorem

16),

and

is omitted.

The

interested reader is referred to

Bikhchandani and

Huang

(1988)fordetails.

Theorem

3

depends

criticallyonthe fact thatunderitshypothesis 6(1;xq)isincreeising

in iQ.

When

{Xo,Xi,...,Xn,

P)

are not information complements, 6(r;io)

may

not

be

'^Note thatthisassumption isstrongerthan the conditional information complementarityrequired for

4 Uniform-price auction 17

anincreasingfunction ofxq,

and

revealing

Xq may

reduce the bidders' incentive tosignal.

Thisinturn

may

lowertheexpected revenueforthesellereven

though

(>Yo,Xi,...

,Xn,P)

areaffiliated.

4

Uniform-price auction

In a uniform-price auction the bidders submitsealed bids

and

the k highest bidders

win

the auction.

The

pricethey

pay

isequal tothe {k-f-l)sthighest bid. Initially

we

obtaina

candidatefora

symmetric Nash

equilibriumundertheassumptionthat after theauctionthe auctioneerrevealsthewinningbids

and

the highest losing bid,thatis,the pricepaidbythe winningbidders. Ifanyofthelowerbids are also revealed,ourresults

remain

unchanged.

We

first obtainstrategies thatsatisfy first-order necessary conditions for a

symmetric

Ncish equilibrium.

Next

we

show

that

when

there exists a

symmetric

equilibrium in the uniform-price auction, the auctioneer's expected revenues at this equilibrium are greater than at the

symmetric

equilibrium ofthe discriminatory auction.

The

intuition behind this resultisas follows.

From

thetheoryofauctionswithoutresalemarkets

we know

that

compared

withadiscriminatoryauction, a greater

amount

ofinformation isrevealedduring auniform-priceauction. This

weakens

thewinners'curse intheuniform-price auction

and

resultsin greater revenues for the auctioneer. In addition

when

the primary auction

and

the resalemarketsare informationally linked, asinour model,it ischeaperto

submit

higher bids in the uniform-price auction in order to signal to the resale

market

buyers. This in

turn further increasestheexpected revenues

from

uniform-priceauctions.

However,thesecondofthese

two

factors

—

thefactthatin auniform-price auction the pricepaid

by

a bidderconditional

upon

winning does notincreasecishisbidisincreased

—

can result innonexistence ofequilibriumin a uniform-price auction. Ifthe resale priceis

very responsiveto the bidssubmittedthere

may

existanincentiveforthebiddersto

submit

arbitrarily largebids

and

upset any purportedequilibrium. Inthelast part ofthissection

we

show

by

example

thatthisisindeedpossible,

and more

generally

show

that

when

)t

=

1,

and

P

is a constant there does not exist any

Nash

equilibriumin strictly increasingpure strategies.

4 Uniform-price auction 18

4.1

Necessary

conditions

for

a

symmetric

equilibrium

As

inthediscriminatory auction,each primarybidder's strategy isa function

from

[x,x\ to the real line.

Suppose

that [bo,bo, ...,to) isa

Nash

equilibrium instrictly increasing

and

differentiable strategies,

when

buyersin thesecondary

market

believethat allbidders use

fco- Ifallother biddersuse strategy bo,bidder 1 receives information

Xi

=

x

and

submits a bidequal tob,then ifbidder 1winsthe resale priceis

V\X,

=

bo'{b),boHbo{Yi)),..

.,boHho{n),P]

v\xi

=

bo\b),Yi,...,n,P].

r"{6o-i(6),n,...,n,P)

=E

=

E

r" isstrictly increasingin allits arguments.

Note

that r"(-)

=

r'^(-). Ifbidder 1wins the auction, the expectedresalepriceconditionalon

Xi and

Yk is

v"(6o-i(6),i,y)

=

E

[r"(6o-i(6),y,,...,n,P)|Xi

= x,n =

v] (13)

By

strict affiliation,v" is strictlyincreasing in itsarguments.

From

the definition of v it

followsthat

v"{x',x,y)

—

v {x',x,y) Vi',i,y. ₍₁₄₎

We

show

below that6o

must

begiven bythe followingequation:

bo{x)

=

v^{x,x,x)

+

-^

(15)

where

h{x)isas defined in(8).

U

Xi

=

X

and

bidder 1bidsb,hisexpectedprofitis

n"(6|x)

=

E[(r"(6o^(6),F,,...,n,P)-6o(n))l^i>,„(y,)}|Xi

=

i]

=

E

[e [(r"(6o-^(6),yi,...,n,p)

-

6o(n))

i{6>6„(fo}|^i'^*J1^^

=

_^

=

E

[(v"(6o^(6),

A'i,n)

-

6o(n))

l{6>6o(n)}|^i

=

^1

/

Jx

(v"(6o'(6),x,y)-6o(y))A(y|x)(iy. (16)

Taking

thefirstderivative ofn"(6|i) withrespect tobgives

'-^^

=

{v"{boHb),^,l>o'{b))

-

b)hibo\b)\x]{b'o{bo\b)))-'

+ mbo\b)))-'

jf^'\:nboHb),x,y)Uy\x)dy,

4 Uniform-price auction 19

where

6o(i) isthe derivative of bo[x)

and

n"is the partial derivative of v" with respect to

its first argument. For (60,•••.^o) to be a

Nash

equilibrium, it is necessary that relation (17)be zero

when

b

=

bo[x).

That

is,

=

6[,(a;)^"'(/l

'n ^

_{={v"(x,x,x)-bo(x))h{x\x)}

+

/|t;!j'(i,i,y)A{y|i)(iy.

where

we

usethe eissumptionthat60 isstrictlyincreasing. Rearranging termsimpliesthat 60(1) isasdefinedin (15).

Inauniform-price auctionwithoutresalemarkets

Milgrom and

Weber

(1982a)

show

that the

symmetric

Nash

equilibriumbiddingstrategyisfc"(i)

=

v"[x,x,x).

As

indiscriminatory auctions,thebiddingstrategy60isstrictlyhigherthan6", forevery x

G

(1,1],by an

amount

which depends on

v",theresponsivenessofthe resale value to thesubmitted bid.

4.2

Revenue comparison

with

the

discriminatory

auction

Next

we

show

that

when

there exists a

symmetric

equilibrium in the uniform-price auc-tion,it generatesstrictlygreater expected revenuesthanthe

symmetric

equilibriumofthe discriminatory auction.^'

Theorem

4

When

(60.

,^0)

"

asymmetric

Nash

equilibriumintheuniform-priceauction, the expected revenues generated at this equilibrium are strictly greater than the expected revenue atthe

symmetric

equilibrium ofthe discriminatory auction.

4.3

PossibiUty

of

nonexistence

of

equiUbrium

In thissubsection

we

illustrate the possibility thatstrongsignalling incentives

on

the part ofthe bidders

may

lead to nonexistence ofa pure strategy

Nash

equilibrium

when

win-ning bids are

announced

inauniform-priceauction. First

we

present an

example

in

which

max{A'i,X2,

...,Xn,

P}

isasufficient statisticof(Ari,vf2,• ,Xn,

P)

forthe posterior

den-sity of

V

.

Although

in this

example

the

random

variables are only

weakly

affiliated, it

illustrates thedifficultiesthatarise

when

the resale priceisvery responsive tothewinning bids.

'^AnargTiment Bimilartothat inTheorem 2establishesthat 60isthe only candidate forasymmetric equilibriumwhentheresalemarketbuyershavemonotonebeliefs.Thisremarkalsoappliestothesymmetric equilibriumwewillobtaininSection 5.

4 Uniform-price auction 20

Example

4 Suppose that all the bids are

announced

after a uniform-price auction.

The

priormarginaldensity

ofV

isuniformwithsupport[0, l].

The random

variables{Xi, X2,.. .,

Xn, P)

are identically distributed, are independent conditional on

V

,

and

their conditional

den-sity is uniform on [0,V^]. Let

Z =

ma\{Xi,X2,

,Xn,P}-

It is readily confirmed that

Y;[V\Xi,X2,...,X„,P]

=

BlVlZ]. Clearly, the resaleprice will be equal<oEjV'lZ],

and

E[V\Z]

>

Z

E[V\Z=l]

=

1. - (19)

Let {bi,b2, ..,bn) be a candidate

Nash

equilibrium, with 6, strictly increasing.

We

limit our attention to weakly

undominated

strategies

and

thus

<

6,(X,)

<

1. Let n"(6|i) be bidderI's expectedpayoff

when

he bidsb

and

Xi

=

x.

Then

itiseasily verifiedthatifx

<x

then

n"(6i(x)|x)

<

n"(6i(i)|i), (20) since ifbidder1 bids bi{i), hewill always win wheneverhe would have with a bid0/61(1)

and

pay the

same

price. In addition he will also win whenever thek-th highest bid of the others' bid isin {bi{x),bi{x)). Moreover, since bi is strictly increasing, (19) implies that the resale price ifhe bids 61(1) is equalto one which isat least as large as the resaleprice ifhe winswith a bid 0/61(1).

The

inequality in (20) isstrict aslong as there is a nonzero probability that the k-th highest bidisin theinterval(61(1),61(1)).

Thus

the only candidate

Nash

equilibrium appears to be a

somewhat

degenerate one in

which atleast one ofthe bidders, say bidder 1, always bids one,

and

atleast n

—

k bidders bidsufficiently low so thatthey never win,

and

then

—

k lowest bidis low enough so that bidder1 always

maximizes

his profitsby bidding one.

But

ifthere is anystrictly positive cost of participating in the auction (such as anentry fee or a bidpreparationcost)then the n

—

k bidders

who

neverwin at this equilibrium willnot participate in the auction.

Next

we

show

thatiik

=

1

and

if

P

istotallyuninformative

about

V

,then theredoes

notexista

symmetric

Nash

equilibriumin strictly increasing strategies. Essentially

when

there isonly oneobject,

and no

otherinformation

becomes

public afterthe auction, there exists a large incentivefor the bidders to submit high bids, since the resale price is very responsiveto thewinningbid.

5 Uniform-price auction without signalling 21

Proposition

4 Suppose that k

=

1

and

that

P

isindependent of

V

.

Then

ifthe winning bid

and

the highest losing bid are

announced

there does not exist a

Nash

equilibrium in strictlyincreasingstrategies.

However, there existotherexamplesforwhich an equilibriumexists. For instanceifV'

is uniform

on

[0,1]

and

Xi

is uniformon [V',!], then t;i(-,-,-)

=

0,

and

the equilibrium strategy is to bid 6o(x)

=

v"(i,i,i).

Whether

there exist intuitivesufificient conditions under

which

an equilibriumexistsremains an

open

question.

5

Uniform-price auction

without

signalling

Since there exists a possibility ofnonexistence ofequilibrium in a uniform-price auction with signalling, in this section

we

analyze uniform-price auctions

when

the

winning

bids are not announced.

Thus

the bidders

do

not have an incentive to signal their private information, since if they

win

their bids are not revealed.

Even

without the signalling incentive,

we

are able to

show

thatinat leasttwoscenariostheexpected revenue generated

by

this uniform-price auction ishigher than thatgenerated by thediscriminatory auction discussedin Section 3.

We

believe that thefirstscenario isa plausibleoneforthe case of the Treasurybillmarket.

5.1

Existence

of

a

symmetric

Nash

equilibrium

when

the

winning

bids are

not

announced

We

show

below that there exists a

symmetric

Nash

equilibrium in strictly increeising

and

difFerentiable strategies, {b',b',...,b'),

when

buyersin thesecondary

market

believe that allbiddersuse6*.

Suppose

that biddersi

=

2,..

.,n adoptthestrategy6*,bidder1receives

information

Xi

=

x,

and

submitsa bidequaltob. Ifbidder 1wins the resale priceis

=

b[v\xi

>n,n,p],

where

Zj isthe j-thorderstatisticof {Xi,X2,...,Xn)-

The

equality follows

from

the fact

thatthe signals are identically distributed, r" is strictlyincreasing inboth its arguments.

Ifbidder 1winsthe auction, theexpected resaleprice conditionalon V^

and

Xi

is

5 Uniform-price auction without signalling 22

By

strict affiliation, 0"isstrictlyincreasinginitsarguments. Thus,\{

Xi

=

x

and

bidder1

bidsb,hisexpected profitis

n"{6|x)

H

E[(f"(n,p)-6-(y',))i{,>,.(^^,j|xi

=

z]

=

E

[e

[(f"(n,p)

-

6*(n))

i(.>t.(n)}|^i.n] |^i

=

x]

=

E[(e"(Xx,n)-6'(n))i{,>,.(^^)}|xi

=

i]. ₍₂₁₎ Define

6*(i)

=

t)"{i,i). (22)

Note

thatb' isstrictly increasing.

We

show

that [b',b',...,6*)isanequilibrium.

Theorem

5

The

n-tuple {b',b'...,6') is a

Nash

equilibriumin the uniform-price auction providedthat resale market buyersbelieve thatallthe bidders follow the strategy 6*.

Milgrom and

Weber

(1982a) have

shown

that the pricepaid by

winning

bidders in a uniform-priceauction

when

bidders participateforthepurposesof

consumption

isEiVjA'i

=

Yk,yk]-

We

show

inthenext

lemma

that the pricepaid by winning biddersinthe uniform-priceauctioninour

model

isgreaterthanthis. Thisistrueeven

though

theprimary bidders

do

not havea signallingmotive.

Lemma

4 With probability one, the pricepaidby winningbidders in auniform-price auc-tion with a resale market is greater than that in a uniform-price auction without resale markets (in whichthe bidders participate for consumption). That is

b'{Y,)>E[v\Xi

=

Y,,Y,]

with probability one.

The

"true value" ofthe object forthe primary biddersisthe resale price. Thus, if

no

additional information

becomes

available after the auction, that is if

P

is constant, the winners'curseon theprimary bidders isweakened. Since thereisnosignallingmotive,one

would

expectthe bidsintheprimary auctiontoincrease

when

P

isconstant(or

when

P

is

independento(

{V ,X\,X2,.

. ,Xn)). Thisisprovedin thenext

lemma.

Lemma

5 Withprobability one, the bidsintheuniform-priceauctionstrictlyincrease

when

P

is constant, that is

when

noadditionalinformation (other than about the bids submitted

5 Uniform-price auction withoutsignalling 23

5.2

Revenue comparison

with

the

discriminatory

auction

We

obtain

two

setsofsufficientconditionsunder whichtheexpected revenues generatedat the

symmetric

equilibriumoftheuniform-price auction obtainedintheprevioussubsection, aregreaterthan theexpected revenues atthe

symmetric

equilibriumofthediscriminatory auctionofSection 3.

The

firstsetseemsplausibleforthe case of Trecisury billauctions.

The

following

theorem

statesthat ifthe publicinformation,P, isnot very informative about the true valueofthe objects, the uniform-price auction generates higher expected revenuethan thediscriminatory auction.

Theorem

6

There exists a scalar

M

>

such that if

df"{y,p)/dp

<

M

for all _y,p

E

[x,x]

X

[p,p\, then theuniform-price auction without signallinggeneratesstrictlyhigher ex-pectedrevenue than thediscriminatoryauction withsignalling, attheir respective

symmetric

equilibria.

In words.

Theorem

6 says that if the ex post public information

P

has little impact

on

theresalepriceconditionalontheinformationreleased

from

the uniform-priceauction, then the auctioneer's expected revenue is higher in the uniform-price auction even

when

winningbids arenotannounced. Thisistrueeven

though

thereis

no

signallingeispect inthe uniform-price auction. Inthe case of the Treasury billauction, bids aresubmitted before

1:00pm

every

Monday.

The

resultsofthe auction are

announced around

4:30pm and

the resale

market

comes

intoplay.

One

would

expectthatanypublicinformationthatnormally arrives

between 1:00pm and

4:30pm would

notbeveryinformativeabout

V

conditional

on

the results of theearlierauction.

The

following

theorem

givesanalternative scenariounder

which

once againthe uniform-priceauction generates higher revenues. Essentiallyitsays that ifthe signalling motiveof thebiddersisnotstrong,thentheuniform-price auction generateshigher expectedrevenue.

Theorem

7 Supposethat

[V

,Xi,..

.,A'„) arestrictly affiliated. There exists ascalar

M

>

such thatifdr'^{x,yi,...,yk,p)/dx

<

M

forallx,yi,...,yk,p

€

[z,!]*"*"^x_[£,p], then the

uniform-price auction without signalling generatesstrictlyhigher expected revenue than the discriminatory auction withsignalling, at their respective

symmetric

equilibria.

A

scenario

where

Theorem

7 is applicable is

when

the public information

P

is very informative

about

the truevalue

V

.

Then

theimpactof

Xi

onthe resale pricewillbesmall

6 Concluding remarks 24

when

bidder 1 wins. Thisscenario, however, does not

seem

tobe plausible in the case of Treasury billauctions.

6

Concluding remarks

Thispaperisan exploratorystudyofcompetitive bidding

when

there exists a resale

market

which

isinformationallylinked to the bidding.

We

have

shown

that there exists a

symmet-ric

Nash

equilibriumin discriminatory auctions

when

winningbids

and

the highest losing bid are

announced

providedthat the relevant variables are affiliated

and

are information

complements.

Thisis theonly

symmetric

equilibrium

when

the resale

market

buyers have

monotone

beliefs. Ifhisinformationis

complementary

tothat of the bidders, theauctioneer

will increasehis expected revenue by precommitting to

announce

his privateinformation beforetheauction.

We

know

littleaboutthe general

impact

ofthe ex postinformation

P

on

the signallingmotiveof bidders. Forthe case ofTreasurybillmarkets thisisnot important since

we

believe that very little additionalinformation

becomes

publicly available in the short time period

between

the closing of the Treasury bill auction

and

the openingofthe resalemarket.

A

relatedquestion

which

isofgreater importanceforTreasurybillauctions

is

whether

there exist plausible scenarios in

which

the auctioneer can increase expected revenue by

announcing

hisprivate informationaftertheauction(andbefore thesecondary

markets

convene)rather thanbeforethe auction.

These

warrantfurther investigation.

We

establishedthe possibility ofnonexistenceofan equilibriuminauniform-price auc-tion

when

winningbids areannounced. However,

when

there existsa

symmetric

equilibrium,

itgenerates greaterexpected revenues than the

symmetric

equilibriumin adiscriminatory auction.

We

also

showed

that there existsa

symmetric

Nash

equilibriumin auniform-price auction

when

onlythe highest losing bidisannounced,sothat thereis

no

signallingincentive for the bidders.

Two

scenarios are provided

where

the uniform-priceauction without sig-nallinggeneratesstrictlyhigher expected revenuefortheauctioneerthanthediscriminatory auction.

Some

implications ofour

model

deserve attention. First,it iseaisytoincorporate non-competitive bids in our model, provided the total

amount

ofnoncompetitive bids, j, is

common

knowledge

beforethe auction.

We

can

model

theprimary auction asone with n bidders

and

k

+

jobjects,jofwhichare

awarded

tononcompetitive biddersattheaverage

noncom-6 Concludingremarks 25

petitive bidders is positive

and

equal to the ex ante expected profits of the competitive bidders.

There

is a prespecified

minimum

and

maximum

quantity for each

noncompeti-tive bid,

which

may

be the reason that resale

market

buyers

do

not

buy

Treasury bills

through noncompetitivebids;

and

itrnay alsoexplain

why

competitive bidders

do

not sub-mit only noncompetitivebidswhile avoiding(presumable costly)information collection.^* Second,the fact that ex anteexpected profitsofthebiddersis strictly positive(except in

uniform-price auctionswithoutsignalling,

when

P

is uninformative aboutV'') implies that in expectation, theaverageprice in theauction is strictly less than the resale price. This comparision has beenempirically

documented

by

Cammack

(1986). Third,as pointed out

insection 3.1, theprimarybidders benefit ifthe resale

market

buyers arebetterinformed aboutthetrue value, sinceitdecreases the bidders'signalUngincentive.

There

are several possible extensionsofourmodel. First,in the Treaisurybill auction, the

primary

bidders

submit

price-quantity pairs

and

demand

more

than

one

unit ofthe Treasurybill. Thisfeature ismissinginour model. Second,

two

weeks

before theconduct oftheweekly Treasury billauction,forwardcontracts of theTreasury billstobe auctioned are traded

among

the primary bidders.

The

relationship

among

theforward prices, bids submitted,

and

the resale priceneeds tobeinvestigated. Third, there existsawidevariety of close substitutes ofTreasury billscarried as inventories byprimarybidders.

These

close substitutes

may

havea significanteffectonthe interplay betweentheforward markets

and

theweeklyauction.

We

hope

to investigate

some

of these issuesin ourfuture research.

'*Byrenormalizing theunits,wecanaasumethateach competitive bidderdemandsexactly

m

>1or zero

units. Thisallows uj tokeep the prespecified

maximum

demand ofnoncompetitive biddersatalevel less

7

Appendix

26

7

Appendix

Proof

of

Lemma

l:

The

jointdensity of (V'.P.A'i,?^!,...,¥„_!) is

(n

-

_{l)!/(v,p,iiyi,...,yn-i)l{„i>v2>... >„„_,}•}

As

a consequence,the conditional density ofV' given {P,Xi,Yi,...,^,,-1) is

f{v,P,x,yi,...,yn-i)

f{p,x,yu...,y.-i)

'^"^'^^•^'"-'>-Thus {P,Xi,Yi,

....,Yk) are information complements. Let rf denote the derivative ofr'^ ,

which

isdefined in (l), with respect to its first argument. It is then easily verified that fj(i,j/i,...

,j/jt,p) is astrictly increasingfunction ofp, j/y,Vj".

Next

notethat vf{x,x',y)

=

E[Tf{x,Yi,...,n,P)\Xi

=

x',Y,

=

y]-Theorem

5of

Milgrom and Weber

(1982a) then implies,byaffiliation,that vf{x,x',y) isa

strictlyincreasingfunction ofi'

and

y. I

PROOF

OF Proposition

i:

We

first write

v'{x,X,x)

-

b{x)

=

_{rx L[u\x)dt{u)}

-

_rx

j^^dL{u\x)

(23)

=

_{/| L{u\x)}{vfiu,u,u)

+

viiu,u,u)

+

viiu,u,u)

-

^)

dn.

By

thehypothesisthat [Xi,.. .

,A'„,

P)

areinformation

complements

withrespect to

V

and

Lemma

1,Vi(u,u,u)

>

Vi(u,ti,y) fory

<

u. Itfollows that

Fk[n\u) Jx Fk{u\vL)

Substitutingthisrelation into (23) gives

v'^{x,x,x)

-

b{x)

>

r

L{u\x)(v^(u,u,ti)

+

vi{u,u,u))

du

>

0.

Note

thatthe

above

inequality isstrict for x

€

{x,x] since Vj

>

and

^3

>

by

strict affiliation. I

Proof

of

Corollary

l:

We

will

show

thatb'{x)

>

O

Vi

>

z.

From

Proposition 1

we

have v'^{x,x,x)

-

b{x)

>

Vx >

z.

The

proofis completed by insertingthis in _(5),

and

notingthat vf

>

0. I