EC319: Auction TheoryMichaelmas Term

The first part of the course consists of an introduction to the game theoretical analysis of auctions. It presentsstandard auction formats and strategic behavior in such environments. Auctions will be analyzed both inprivate and interdependent value environments. Fundamental topics such as the revenue equivalence theorem,the optimal auction design problem and the linkage principle will be discussed extensively. Throughout thecourse we will consider departures from the standard model allowing for heterogeneity amongst players,budget constraints and resale. Time permitting a model of bidding rings will be presented. The focus of thecourse is mainly theoretical, but when possible empirical and experimental evidence supporting the formalmodels will be discussed with references to relevant work in the field.

Lecturer: Francesco Nava, f.nava@lse.ac.uk

Offi ce Hours: Wed 1.30pm-3.00pm, Building 32L, Offi ce 3.20

Course Website: moodle.lse.ac.uk

Time and Location: Thu 9.30-11.00am, Building 32L, Room LG.18

Class Work: The work for each week consists of one or more exercises taken from the textbook and someadditional problems. Answers to exercises will be posted on the website throughout the term. You areexpected to make a reasonable attempt at the classwork in advance of the class. Two written assignmentsare required before class in weeks 6 and 11 of MT. These assignments will be of similar scope and diffi cultyas exam questions. You are expected to be diligent about meeting the deadlines for submission.

Exams: The final exam will determine your grade on the course entirely. The two parts of the course willbe weighted equally. For each part of the course, you will have to answer two short questions and one longproblem. The exam will focus entirely on topics covered in lectures and classes.

Weekly Course Program

1. Review: Order Statistics & Game Theory [KA, KC, KF]

2. Independent Private Value Auctions [K1, K2]

3. Revenue Comparisons and Reserve Prices [K2]

4. Revenue Equivalence Principle [K3]

5. Extensions: Risk Aversion [K4]

6. Extensions: Budget Constraints [K4]

7. Mechanism Design [K5, 3]

8. Optimal Mechanisms [K5, 5]

9. Effi cient and Balanced Budget Mechanisms [K5, 6]

10. Interdependent Values [K6-7, 4]

Auction Theory Nava

Readings

Most materials covered can be found in the main textbook. The slides used in class will be posted. Some ofthe papers below may be suggested to deepen some topics.

Main Textbook: [K] Auction Theory, Vijay Krishna, Academic Press 2010

Related Papers [Not Required]:[0] Athey & Levine, “Comparing Open and Sealed Bid Auctions: Theory and Evidence”, Mimeo 2004

[1]Ausubel, “A Generalized Vickery Auction”, Econometric Society Conference, 2000

[2] Dasgupta & Maskin, “Effi cient Auctions”, Quarterly Journal of Economics, 2000

[3] Krishna & Perry, “Effi cient Mechanism Design”, Mimeo Penn State University, 1998

[4] Milgrom & Weber, “A Theory of Auctions and Competitive Bidding”, Econometrica 50, 1982

[5] Myerson, “Optimal Auction Design”, Journal of Operation Research 6, 1982

[6] Myerson & Satterwaite, “Effi cient Mechanisms for Bilateral Trading”, JET 28, 1983

[7] Reny & Perry, “An Ex-Post Effi cient Auction”, Econometrica 70, 2002

[8] Vickery, “Counterspeculation, Auctions and Competitive Sealed Traders”, Journal of Finance 7, 1961

[9] Wilson, “A Bidding Model of Perfect Competition”, Review of Economic Studies 44, 1977

[10] Zheng, “High Bids and Broken Winners”, Journal of Economic Theory 100, 2001

2

EC319 - Auction TheoryMichaelmas Term

Weekly Course Assignments

Several weekly exercises are taken from Krishna’s textbook. Problems assigned are discussed during classes andlag the topics covered in the lectures by approximately a week. Answers to exercises will be posted on the websitethroughout the term. You are expected to make a reasonable attempt at the classwork in advance of the class. Twoproblem sets have to be handed in as written assignments at the end of weeks 5 and 10 of Michealmas Term. Theseassignments will be of similar scope and diffi culty as exam questions. You are expected to be meet the deadlinesfor submission.

[2MT] Order Statistics

1. Consider n independent draws from the uniform probability distribution function f(x) = 1/k on [0, k].

(a) Compute the cumulative distribution function and probability distribution function of the first and secondorder statistic.

(b) Compute the probability distribution function of the second order statistic conditional on the first orderstatistic being equal to y.

(c) Compute the expected value of the first and the second order statistic.

(d) Compute the expected value of the second order statistic conditional on the first order statistic beingequal to y.

2. Consider 3 independent draws from the exponential probability distribution function f(x) = e−x on [0,∞).

(a) Compute the cumulative distribution function and probability distribution function of the first and secondorder statistic.

(b) Compute the probability distribution function of the second order statistic conditional on the first orderstatistic being equal to y.

[3MT] IPV Auctions

1. K2.1

2. Consider a first price auction with 3 buyers with values distributed according to f(x) = e−x on [0,∞). Findthe unique symmetric equilibrium.

3. [Extra] Consider a Dutch auction in which the auctioneer: starts the auction at a price so high no buyer iswilling to buy; and then lowers the price gradually until a buyer signals his willingness to buy. When thatoccurs, the buyer is sold the object at the last price quoted by the auctioneer. Prove that in an IPV setup thisDutch auction has an equilibrium that is equivalent to the symmetric equilibrium of the first price auction.

[4MT] IPV Auctions

1. Consider a first price auction with 3 buyers with values distributed according to f(x) = e−x on [0,∞).Compute the seller’s expected revenues.

2. K2.2

3. K2.4

[5MT] Revenue and Reserve Prices

1. Consider 3 buyers with values distributed according to f(x) = 1/2 on [0, 2].

(a) Compute the distribution of winning prices in the first and the second price auction.

(b) Compute the mean of the two distributions in part (a). How do these means relate to each other and torevenue in each of the two auctions?

Auction Theory Nava

(c) Compute the variance of the two distributions in part (a). How do these variances relate to each other?Can you explain this in light of the discussion in lecture?

2. Consider a first price auction with 3 buyers with values distributed according to f(x) = e−x on [0,∞).

(a) Compute the seller’s optimal reserve price.

(b) What are the seller’s expected revenues with optimal reserve price in place?

[6MT] IPV Auctions and Revenue Equivalence

4. K3.1

5. K2.5

6. Hand In Problem Set I

[7MT] Revenue Equivalence and IPV Auctions Extensions

1. K3.2

2. K4.1

[8MT] IPV Auctions Extensions and Mechanism Design

1. K4.2

2. Consider three buyers participating in a second price auction. Buyers are financially constrained, and theirvalue-wealth pairs (Xi,Wi) are identically and independently distributed according to a uniform on [0, 1]2.

(a) Show explicitly that bidding according to B(x,w) = min {x,w} is weakly dominant.(b) Compute the probability of winning with value-wealth pair (x,w).

(c) Find an expression for the expected payment of a buyer with value-wealth pair (x,w).

(d) Compute expected revenues of the seller.

3. [Extra] K4.5

[9MT] Revenue Maximizing Mechanisms

1. K5.1

2. K5.2

[10MT] Revenue Maximizing and Effi cient Mechanisms

1. Prove explicitly that if the design problem is regular and symmetric then the optimal mechanism reduces toa second price auction with reserve price.

2. K5.3

3. Consider 3 potential buyers with independent private values for an object. The values of bidders a and b areidentically distributed according to a uniform distribution on the interval [−1, 1], so that Fa(x) = Fb(x) =(x+ 1)/2. The value of bidder c is instead uniformly distributed on the interval [0, 2], so that Fc(x) = x/2.

(a) Define the Vickrey-Clarke-Groves (VCG) mechanism and specialize it to the scenario described above.

(b) Define effi ciency, incentive compatibility, and individually rationality. Show that the VCG mechanismsatisfies these properties.

(c) Compute the revenue of the VCG mechanism.

[11MT] Effi cient and Balanced Budget Mechanisms

1. K5.4

2. [Extra] K5.5

3. Hand In Problem Set II

2

Hand In Problem Set 1 Francesco Nava

EC319: Auction Theory Michaelmas Term

Please give your answers to your class teacher before class in week 6MT. Thank you!

1. Consider the following variant of a 1st price auction in an IPV setup. Sealed bids are collected. The highestbidder pays his bid, but receives the object only if the outcome of a toss of a fair coin is heads. If the outcomeis tails, the seller keeps the object and the high bidders bid. Assume bidder symmetry. [30 Marks]

(a) Find the unique symmetric equilibrium bidding function. [10 Marks]

(b) Do buyers bid higher or lower than in a 1st price auction? [5 Marks]

(c) Find an expression for the seller’s expected revenue. [10 Marks]

(d) Show that the seller’s expected revenue is exactly half that of a standard 1st price auction. [5 Marks]

2. [Harder] In a 2nd price auction with entry fee, buyers have to pay a fee E in order to participate in the auction.Buyers first choose whether to participate and pay the fee and then choose how much to bid. Consider anIPV setup with four buyers with values drawn from a uniform distribution on the interval [0, 1]. Suppose thatonly the seller knows how many buyers pay the fee. [40 Marks]

(a) Find the unique symmetric equilibrium of this auction and write down the expected payment of a buyerwith value x. Hint: Pay the fee only if xG(x) ≥ E. Explain. [15 Marks]

(b) What entry fee maximizes the equilibrium expected revenues of the seller? [15 Marks]

(c) Compare the revenues to those of a standard 2nd price auction. [5 Marks]

(d) Compare the revenues to those of a 2nd price auction with revenue maximizing reserve prices. [5 Marks]

3. Consider three buyers with independent and private values for an object at auction. The value of the objectat sale is uniformly distributed on [0, 1] for each buyer. Individuals know their valuation for the object, butignore that of their competitors. Buyers participate in an auction in which the highest bid wins the objectand in which everyone, but for the lowest bidder, has to pay his bid. [30 Marks]

(a) Compute the probability that a buyer wins the auction if he bids b and everyone follows a strictlyincreasing strategy β : [0, 1]→ R. [5 Marks]

(b) Use the revenue equivalence theorem to derive the symmetric equilibrium bidding strategy. [15 Marks]

(c) Compute the revenues of the auctioneer. [10 Marks]

Hand In Problem Set 2 Francesco Nava

EC319: Auction Theory Michaelmas Term

Please give your answers to your class teacher before class in week 11MT. Thank you!

1. Show that the expected value of the virtual valuation is always zero. [10 Marks]

2. Suppose that the buyers have values independently and identically distributed according to a uniform on [1, 3].Let there be 3 buyers.

(a) Derive the revenue maximizing auction for the seller. [10 Marks]

(b) Compute the seller’s expected revenue. [10 Marks]

3. Consider an environment with three buyers. Buyer 0 has a value that is drawn from a uniform distributionon the interval [0, 2], and every other buyer {1, 2} has a value that is independently drawn from a uniformdistribution on the interval [2, 3].

(a) Find the revenue maximizing mechanism amongst those that are IC, IR. [15 Marks]

(b) Find the revenue maximizing mechanism amongst those that are effi cient, IC, IR. Check that it is IC &IR. [15 Marks]

(c) Is there any mechanism that balances the budget and that is effi cient, IC, IR? If it exists, characterizeit. [15 Marks]

4. Consider a bilateral trade problem. Assume that the value of the object to be traded to the buyer is uniformlydistributed on [1, 3] and that the costs of the producer are uniformly distributed on [0, 2].

(a) Derive the VCG mechanism & check effi ciency IC & IR. [15 Marks]

(b) Is there any mechanism that balances the budget and that is effi cient, IC, IR? Explain. [10 Marks]

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heor

yR

evie

wE

C31

9–

Lec

ture

Not

esII

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

1/16

Com

pleteInform

ationGam

es

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

2/16

Not

es

Not

es

Com

plet

eIn

form

atio

n

Aco

mp

lete

info

rmat

ion

gam

eG

={ N, {

Ai,ui}

i∈N

} con

sist

sof

:

Nth

ese

tof

pla

yers

inth

ega

me;

Ai

pla

yeri’

sac

tion

set;

An

acti

onpr

ofile

ais

anel

emen

tth

ese

tA=×

j∈NAj;

For

con

ven

ien

ced

efin

eA−i=×

j∈N\iAj;

ui

:A→

Rp

laye

ri’

su

tilit

yfu

nct

ion

;

Map

pin

gac

tion

profi

les

top

ayoff

s.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

3/16

Nas

hE

quili

bria

Defi

niti

on(N

ash

Equ

ilibr

ium

NE

)

A(p

ure

stra

tegy

)N

ash

equ

ilibr

ium

ofa

gam

eG

con

sist

sof

ast

rate

gypr

ofile

a=

(ai,a −

i)∈A

such

that

ui(a)≥

ui(a′ i,

a −i)

for

anya′ i∈Ai

&i∈N

.

Pro

per

ties

:

stra

tegy

profi

les

are

ind

epen

den

t;

stra

tegy

profi

les

com

mon

know

led

ge;

pu

rest

rate

gyN

ash

equ

ilibr

iam

ayn

otex

ist.

Exa

mp

leon

eb

uye

rw

ith

valu

e2$

and

one

selle

r:

B\S

sale

no

sale

2$0,

20,

01$

1,1

0,0

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

4/16

Not

es

Not

es

IncompleteInform

ationGam

es

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

5/16

Inco

mpl

ete

Info

rmat

ion

An

inco

mp

lete

info

rmat

ion

gam

eΓ={N

, {Ai,

Xi,ui}

N,f}

con

sist

sof

:

Nth

ese

tof

pla

yers

inth

ega

me;

Ai

pla

yeri’

sac

tion

set;

Xi

pla

yeri’

sse

tof

pos

sib

lesi

gnal

s;

Apr

ofile

ofsi

gnal

sis

anel

emen

tx∈

X=×

j∈N

Xj;

For

con

ven

ien

ced

efin

eX−i=×

j∈N\i

Xj;

f∈

∆(X

)a

dis

trib

uti

onov

erth

ep

ossi

ble

sign

als;

ui

:A×

X→

Rp

laye

ri’

su

tilit

yfu

nct

ion

,ui(a|x).

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

6/16

Not

es

Not

es

Bel

iefs

and

Str

ateg

ies

Info

rmat

ion

stru

ctu

re:

Xi

den

otes

the

sign

alas

ara

nd

omva

riab

le;

Pla

yeri

obse

rves

only

Xi;

Con

dit

ion

alob

serv

ingXi=

x ih

isb

elie

fsar

e

f(X−i|x

i)=

f(X−i,x i)

∑x −

i∈X−if(x−i,x i)∈

∆(X−i)

.

Str

ateg

yP

rofi

les:

Ast

rate

gyco

nsi

sts

ofa

map

from

info

rmat

ion

toac

tion

sαi

:Xi→

Ai;

Ast

rate

gypr

ofile

isa

map

α(X

)=

(α1(X

1),

...,

αN(X

N))

.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

7/16

Pay

offs

&D

omin

ance

Th

eex

-post

,ex

-ante

andinterim

expec

tedutility

ofst

rate

gypr

ofile

αar

ere

spec

tive

lyd

efin

edby

:

ui(

α(x)|x)

:X→

R,

E[u

i(α(X

)|X)]=

∑Xui(

α(x)|x)f(x)∈

R,

E[u

i(α(X

)|X)|Xi=

x i]=

∑X−iui(

α(x)|x)f(x−i|x

i):X

i→

R.

Wea

kDominance

:

Str

ateg

yαi

wea

kly

dom

inat

esα′ i

iffo

ran

ya −

ian

dx

ui(

αi(x i),a −

i|x)≥

ui(

α′ i(x i),a −

i|x)

[str

ict

som

ewh

ere]

.

Str

ateg

yαi

isd

omin

ant

ifit

dom

inat

esan

yot

her

stra

tegy

α′ i.

Str

ateg

yαi

isu

nd

omin

ated

ifn

ost

rate

gyd

omin

ates

it.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

8/16

Not

es

Not

es

Dom

inan

tS

trat

egy

and

Bay

esN

ash

Equ

ilibr

ia

Defi

niti

ons

(Dom

inan

tS

trat

egy

Equ

ilibr

ium

DS

E)

ADominantStrategyeq

uilibrium

ofan

inco

mp

lete

info

rmat

ion

gam

eΓ

isa

stra

tegy

profi

leα

that

for

anyi∈N

,x∈

Xan

da −

i∈A−i

sati

sfies

ui(

αi(x i),a −

i|x)≥

ui(

α′ i(x i),a −

i|x)

for

any

α′ i

:Xi→

Ai.

Defi

niti

ons

(Bay

esN

ash

Equ

ilibr

ium

BN

E)

ABayes

Nash

equilibrium

ofan

inco

mp

lete

info

rmat

ion

gam

eΓ

isa

stra

tegy

profi

leα

that

for

anyi∈N

andx i∈

Xi

sati

sfies

E[u

i(α(X

)|X)|Xi=

x i]≥

E[u

i(a i

,α−i(X−i)|X

)|Xi=

x i]

for

anya i∈Ai.

Ifα

isa

DS

Eth

enit

isa

BN

E

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

9/16

Equ

ilibr

ium

Pro

per

ties

BN

Ere

qu

ires

forinterim

optimality.

Ex-ante

equ

ilibr

ium

isa

stra

tegy

profi

les.

t.fo

ran

yα′ i

:Xi−→

∆i

E[u

i(α(X

)|X)]≥

E[u

i(α′ i(Xi)

,α−i(X−i)|X

)].

Ex-post

equ

ilibr

ium

isB

NE

s.t.

for

anya i∈Ai

andx∈

X

ui(

α(x)|x)≥

ui(a i

,α−i(x −

i)|x).

DS

E⇒

Ex-

pos

t⇒

BN

E=

Inte

rim⇒

Ex-

ante

[by

inte

grat

ing]

.

Ex-

ante

opti

mal

ity⇒

Inte

rim

opti

mal

ity

alm

ost

sure

ly.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

10/16

Not

es

Not

es

BN

EE

xam

ple

I:E

xcha

nge

Ab

uye

ran

da

selle

rw

ant

totr

ade

anob

ject

:

Bu

yer’

sva

lue

for

the

obje

ctis

3$;

Sel

ler’

sva

lue

isei

ther

0$or

2$b

ased

onth

esi

gnal

,X

S={L

,H};

Bu

yer

can

offer

eith

er1$

or3$

top

urc

has

eth

eob

ject

;

Sel

ler

cho

ose

wh

eth

eror

not

tose

ll.

B\S

.Lsa

len

osa

leB\S

.Hsa

len

osa

le3$

0,3

0,0

3$0,

30,

21$

2,1

0,0

1$2,

10,

2

Th

isga

me

for

any

prio

rf

has

aB

NE

inw

hic

h

αS(L)=

sale

,αS(H

)=

no

sale

,αB=

1$.

Sel

ling

isw

eakl

yd

omin

ant

forS

.L.

Off

erin

g1$

isw

eakl

yd

omin

ant

for

the

bu

yer.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

11/16

BN

EE

xam

ple

II:

Dis

cret

eS

econ

dP

rice

Auc

tion

Tw

ob

uye

rsw

ant

tob

uy

obje

ctat

seco

nd

pric

eau

ctio

n:

Bu

yeri’

sva

lue

isei

ther

1$or

2$b

ased

onh

issi

gnal

,X

i={L

,H};

Bu

yers

cho

ose

wh

eth

erto

bid

1$or

2$;

Th

eh

igh

est

bid

win

sth

eob

ject

(tie

sar

ebr

oken

wit

hfa

irlo

tter

y);

Th

ese

con

dh

igh

est

bid

isp

aid

byth

ew

inn

er.

1.L\2

.L2$

1$1.L\2

.H2$

1$2$

-0.5

,-0.

50,

02$

-0.5

,00,

01$

0,0

0,0

1$0,

10,

0.5

1.H\2

.L2$

1$1.H\2

.H2$

1$2$

0,-0

.51,

02$

0,0

1,0

1$0,

00.

5,0

1$0,

10.

5,0.

5

Th

ega

me

has

aD

SE

inw

hic

hαi(L)=

1$an

dαi(H)=

2$fo

r∀i∈N

.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

12/16

Not

es

Not

es

BN

EE

xam

ple

III:

Dis

cret

eF

irst

Pri

ceA

ucti

on

Tw

ob

uye

rsw

ant

tob

uy

obje

ctat

seco

nd

pric

eau

ctio

n:

Bu

yeri’

sva

lue

isei

ther

1$or

2$b

ased

onh

issi

gnal

,X

i={L

,H};

Bu

yers

cho

ose

wh

eth

erto

bid

1$or

2$;

Th

eh

igh

est

bid

win

sth

eob

ject

(tie

sar

ebr

oken

wit

hfa

irlo

tter

y);

Th

eh

igh

est

bid

isp

aid

byth

ew

inn

er.

1.L\2

.L2$

1$1.L\2

.H2$

1$2$

-0.5

,-0.

5-1

,02$

-0.5

,0-1

,01$

0,-1

0,0

1$0,

00,

0.5

1.H\2

.L2$

1$1.H\2

.H2$

1$2$

0,-0

.50,

02$

0,0

0,0

1$0,

-10.

5,0

1$0,

00.

5,0.

5

Th

ega

me

has

aD

SE

inw

hic

hαi(L)=

αi(H)=

1$fo

r∀i∈N

.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

13/16

Extra:ContinuousGam

esExistence

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

14/16

Not

es

Not

es

Ext

ra:

Con

tinu

ous

Gam

es&

BN

EE

xist

ence

Are

fere

nce

for

pro

ofs

isA

they

Eco

nom

etri

ca20

01.

Pla

yers

’ac

tion

sets

Ai⊆

Ran

dar

eco

nve

x.

Pla

yers

’ty

pes

sets

Xi⊆

Ran

dar

eco

nve

x.

Let

Ui(a i

,α−i(S)|x i)=∫ x −

i∈Sui(a i

,α−i(x −

i)|x)f−i(x −

i|xi)dx −

i.

Let

Ui(a i

,α−i|x

i)=

Ui(a i

,α−i(

X−i)|x

i).

Defi

niti

on(R

egul

arit

yC

ondi

tion

sR

C)

Typ

esh

ave

ajo

int

den

sity

fw

hic

his

bou

nd

edan

dat

omle

ss.

Ui(a i

,α−i(S)|x)

exis

tsan

dis

fin

ite

for

anyS

con

vex

and

any

non

-dec

reas

ing

fun

ctio

nαj

:Xj→

Aj

forj6=

i.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

15/16

Sin

gle

Cro

ssin

g&

Exi

sten

ceB

NE

Defi

niti

on(S

ingl

eC

ross

ing

ofIn

crem

enta

lR

etur

nsS

CP

IR)

Th

efu

nct

ionh

:R2→

Rsa

tisfi

essi

ngl

ecr

ossi

ng

prop

erty

ofin

crem

enta

lre

turn

sif

for

anya>

aan

dx>

x

h(a

,x)>

h(a

,x)⇒

h(a

,x)≥

h(a

,x).

Defi

niti

on(S

ingl

eC

ross

ing

Con

diti

onS

CC

)

Th

esi

ngl

ecr

ossi

ng

con

dit

ion

for

gam

esw

ith

inco

mp

lete

info

rmat

ion

ism

etif

for

anyi∈N

,w

hen

ever

αj

:Xj→

Aj

isn

on-d

ecre

asin

gfo

ran

yj6=

i,p

laye

ri’

sob

ject

ive

fun

ctio

nUi(a i

,α−i|x

i)sa

tisfi

esS

CP

IRin

(ai,x i).

The

orem

(Exi

sten

ceof

BN

E)

Assumethat

SCCandRChold.Supposethat

ui(a|x)iscontinuou

sin

aforanyi∈N.Then

thereexists

aBNEin

non

-decreasingstrategies.

Nava

(LSE)

GameTheory

Review

MichaelmasTerm

16/16

Not

es

Not

es

Inde

pen

dent

Pri

vate

Val

ueA

ucti

onE

C31

9–

Lec

ture

Not

esII

I

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

1/29

The

Environment

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

2/29

Not

es

Not

es

Eco

nom

y

Con

sid

erth

efo

llow

ing

envi

ron

men

t:

1au

ctio

nee

r;

Np

oten

tial

bu

yers

;

1in

div

isib

leob

ject

own

edby

the

auct

ion

eer;

Bu

yer

i’s

valu

atio

nXi

for

the

obje

ct;

Xi

ind

epen

den

tan

did

enti

cally

dis

trib

ute

d;

Xicd

f ∼F

con

tin

uou

san

din

crea

sin

gon

[0,ω

];

Fad

mit

sa

con

tin

uou

sd

ensi

tyf

and

has

full

sup

por

t.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

3/29

Info

rmat

ion

&B

udge

ts

Pri

vate

Va

lues

How

mu

chp

laye

rva

lues

the

goo

dis

ind

epen

den

tof

oth

ers.

Info

rma

tio

n

Bu

yer

ikn

ows:

x ith

ere

aliz

atio

nof

Xi;

Xjcd

f ∼F

ind

epen

den

tly

acro

ssj6=

i.

Bu

dg

ets

No

bu

yer

face

san

yb

ud

get/

liqu

idit

yco

nst

rain

t.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

4/29

Not

es

Not

es

First

Price

Auction

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

5/29

Fir

stP

rice

Sea

led-

Bid

Auc

tion

Mec

ha

nis

m

All

pla

yers

sub

mit

seal

edb

ids

bi.

Hig

hes

tb

idw

ins

the

obje

ctan

dp

ays

the

amou

nt

bid

.

For

any

profi

leof

bid

sb={b

j}j∈

Np

ayoff

sar

e

Πi=

{ x i−

bi

ifbi>

max

j6=ibj

0if

bi<

max

j6=ibj

Tie

-bre

ak

ing

Ru

le

Eq

ual

prob

abili

tyof

gett

ing

obje

ct&

pay

ing

amon

gst

hig

hes

tb

idd

ers.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

6/29

Not

es

Not

es

FP

AE

quili

bria

&S

hadi

ng

Ast

rate

gypr

ofile{β

j}j∈

N,

βj

:[0,

ω]→

R+

any

j,is

sym

met

ric

if

βj=

βi=

βfo

ral

li,

j.

Wh

atis

the

opti

mal

bid

bby

bu

yer

iif

:

j6=

iad

opt

asy

mm

etri

c,in

crea

sin

g,d

iffer

enti

able

stra

tegy

β;

his

priv

ate

valu

eis

Xi=

x.

Not

eth

at:

Ab

idb=

xse

cure

sa

pay

off0;

An

yb

idb>

β(ω

)is

sub

opti

mal

;

β(0)=

0.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

7/29

FP

AE

quili

bria

&S

hadi

ng

Defi

ne

Y1=

max

j6=iXj=⇒

Y1cd

f ∼G

=FN−1.

Sin

ceβ

isin

crea

sin

gm

axj6=

iβ(X

j)=

β(Y

1).

Pla

yer

iw

ins

the

obje

ctif

b>

β(Y

1)⇔

β−1(b)>

Y1.

Exp

ecte

dp

ayoff

ofi

wh

ench

oos

ing

bth

us

amou

nts

to

G(β−1(b))(x−

b).

Max

imiz

ing

wit

hre

spec

tto

byi

eld

s

g(β−1(b))

β′ (

β−1(b))(x−

b)−

G(β−1(b))

=0.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

8/29

Not

es

Not

es

FP

AE

quili

bria

&S

hadi

ng

As

we

saw

,F

OC

amou

nt

to

g(β−1(b))

β′ (

β−1(b))(x−

b)−

G(β−1(b))

=0.

Su

chco

nd

itio

nat

asy

mm

etri

ceq

uili

briu

m,

b=

β(x),

bec

omes

d dx(G

(x)β(x))

=xg

(x).

Sin

ceβ(0)=

0,in

tegr

atio

nyi

eld

s

β(x)=

1

G(x)

∫ x 0yg

(y)d

y=

E[Y

1|Y

1<

x].

Nec

essa

ryn

otsu

ffici

ent

for

sym

met

ric

equ

ilibr

ium

exis

ten

ce.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

9/29

FP

AS

ymm

etri

cE

quili

bria

The

orem

Sym

met

ric

equ

ilibr

ium

stra

tegi

esin

afi

rst

pric

eau

ctio

nar

e:

β1(x)=

E[Y

1|Y

1<

x]

for

Y1

the

hig

hes

tof

N−

1in

dep

end

entl

yd

raw

nva

lues

.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

10/29

Not

es

Not

es

Pro

of.

Su

pp

ose

that

all

bid

der

sb

ut

one

follo

wth

est

rate

gyβ=

β1.

Let

xb

eh

isva

lue

and

bh

isb

idan

dd

efin

ez=

β−1(b)

[in

vert

ible

].O

nly

bid

sb≤

β(ω

)ar

eop

tim

al.

Exp

ecte

dpr

ofits

for

such

bid

der

are

Π(x

,b)=

G(z)[

x−

β(z)]=

=G(z)[

x−

E[Y|Y

<z]]=

=G(z)x−∫ z 0

yg(y)d

y=

=G(z)[

x−

z]+

∫ z 0G(y)d

y.

Th

us

rega

rdle

ssof

x≤

zor

x≥

zit

mu

stb

e

Π(x

,β(x))−

Π(x

,β(z))

=G(z)[

z−

x]−

∫ z xG(y)d

y≥

0.

Th

eref

ore

this

issy

mm

etri

ceq

uili

briu

m,

sin

ceb>

β(ω

)is

sub

opti

mal

.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

11/29

FP

AS

ymm

etri

cE

quili

bria

Plo

t

Th

ep

lot

bel

owsh

ows

wh

yit

mu

stb

eth

at

Π(x

,β(x))−

Π(x

,β(z))

=G(z)[

z−

x]−

∫ z xG(y)d

y≥

0.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

12/29

Not

es

Not

es

Deg

ree

ofS

hadi

ngan

dth

eU

nifo

rmC

ase

Sh

ad

ing

β1(x)=

1

G(x)

∫ x 0yd

G(y)=

=1

G(x)

[ [yG(y) ]x 0−∫ x 0

G(y)d

y

] =

=x−∫ x 0

G(y)

G(x)

dy=

x−∫ x 0

[ F(y)

F(x)

] N−1d

y.

Exa

mp

le

IfF(x)=

xis

un

ifor

mon

[0,1]:

β1(x)=

x−

1

xN−1

[ yN N

] x 0

=N−

1

Nx

.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

13/29

Uni

form

Exa

mpl

e

For

inst

ance

wit

h2

bid

der

s,ev

eryo

ne

bid

sh

alf

ofth

epr

ivat

eva

lue.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

14/29

Not

es

Not

es

SecondPrice

Auction

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

15/29

Sec

ond

Pri

ceS

eale

d-B

idA

ucti

on

Mec

ha

nis

m

All

pla

yers

sub

mit

seal

edb

ids

bi.

Hig

hes

tb

idw

ins

the

obje

ctan

dp

ays

the

seco

nd

hig

hes

t.

For

any

profi

leof

bid

sb={b

j}j∈

Np

ayoff

sar

e

Πi=

{ x i−

max

j6=ibj

ifbi>

max

j6=ibj

0if

bi<

max

j6=ibj

Tie

-bre

ak

ing

Ru

le

Eq

ual

prob

abili

tyof

gett

ing

obje

ct&

pay

ing

amon

gst

hig

hes

tb

idd

ers.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

16/29

Not

es

Not

es

Equ

ilibr

ium

Beh

avio

r

Lem

ma

Bid

din

gac

cord

ing

toβ2(x)=

xis

aw

eakl

yd

omin

ant

stra

tegy

ina

seco

nd

pric

eau

ctio

n.

Pro

of.

Let

pi=

max

j6=ibj.

Pla

yer

iby

bid

din

gx i

gets

ap

ayoff

Πi=

{ x i−

pi

ifx i

>pi

0if

x i<

pi

Ifh

ew

ere

tob

idbi<

x ih

ew

ould

still

earn

Πi=

{ x i−

pi

ifbi>

pi

0if

bi<

pi

Bu

tP

r(x i

>pi)>

Pr(

bi>

pi)

.S

imila

rly

bi>

x i.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

17/29

Revenue

Com

parisons

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

18/29

Not

es

Not

es

Rev

enue

Com

pari

son

Ina

1st

&2n

dpr

ice

auct

ion

the

exp

ecte

dp

aym

ent

ofb

idd

erw

ith

valu

ex

:

m1(x)=

Pr(

win)∗

1st b

id=

G(x)E

(Y1|Y

1<

x),

m2(x)=

Pr(

win)∗

E(2

ndb

id|1

stb

id=

x)=

G(x)E

(Y1|Y

1<

x).

Th

eex

pec

ted

ex-a

nte

pay

men

tby

ap

arti

cula

rp

laye

ris

for

k∈{1

,2}

E(m

k)=∫ ω 0

[ ∫ x 0yg

(y)d

y

] f(x)d

x=∫ ω 0

[ ∫ ω yf(x)d

x

] yg(y)d

y=

=∫ ω 0

[1−

F(y) ]

yg(y)d

y=∫ ω 0

y[1−

F(y) ]

f(N−1)

1(y)d

y.

Th

eref

ore

the

exp

ecte

dex

-an

tere

ven

ues

der

ived

byth

eau

ctio

nee

rar

e

E(R

k)=

N∗

E(m

k)=∫ ω 0

yf(N

)2

(y)d

y=

E(Y

(N)

2).

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

19/29

Rev

enue

Com

pari

son

Rev

enu

eE

qu

ival

ence

firs

tta

ke:

Lem

ma

Inan

IPV

setu

pth

eex

pec

ted

reve

nu

esfr

omb

oth

a1s

tan

da

2nd

pric

eau

ctio

nar

eth

esa

me.

Rev

enu

esin

bot

hfo

rmat

sh

ave

the

sam

em

ean

,b

ut

diff

eren

tva

rian

ce.

Mea

n-p

rese

rvin

gS

prea

d(M

PS

):

Defi

niti

on

Let

Xcd

f ∼F

and

Z|X

=x

cdf ∼

H(x)

soth

atE(Z|X

=x)=

0.If

Y=

[X+

Z]cd

f ∼G

,th

enw

esa

yth

atG

isa

mea

n-p

rese

rvin

gsp

read

ofF

.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

20/29

Not

es

Not

es

Rev

enue

Com

pari

son

Lem

ma

Inan

IPV

setu

pth

ed

istr

ibu

tion

ofeq

uili

briu

mpr

ices

ofa

2nd

pric

eau

ctio

nis

aM

PS

ofth

ed

istr

ibu

tion

ofeq

uili

briu

mpr

ices

ofa

1st

pric

eau

ctio

n.

Pro

of.

Rev

enu

esar

eas

ran

dom

vari

able

sre

spec

tive

lyR1=

β(Y

(N)

1)

&

R2=

Y(N

)2

thu

sit

mu

stb

eth

at

E(R

2|R

1=

p)=

E(Y

(N)

2|Y

(N)

1=

β−1(p))

=

=E(Y

(N−1)

1|Y

(N−1)

1<

β−1(p))

=β(β−1(p))

=p

.

Th

us

ther

eex

ists

Zsu

chth

atth

ed

istr

ibu

tion

ofR2

isth

esa

me

asth

atof

R1+

Zan

dE(Z|R

1=

p)=

0.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

21/29

Rev

enue

Com

pari

son

Exa

mpl

e

Au

nif

orm

dis

trib

uti

onex

amp

lew

ith

N=

2an

dω

=1.

FP

Ais

less

vola

tile

than

SP

A:

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

22/29

Not

es

Not

es

Reserve

Prices

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

23/29

Res

erve

Pri

ces

SP

A

Res

erve

Pri

ce

Sel

ler

rese

rves

the

righ

tn

otto

sell

ifw

inn

ing

pric

eis

bel

owr.

Th

ere

serv

epr

ice

isco

mm

onkn

owle

dge

amon

gst

pla

yers

.

2nd

Pri

ceA

uct

ion

Win

ner

pay

sre

serv

epr

ice

ifse

con

db

idis

bel

ow.

Mak

esn

od

iffer

ence

inb

ehav

ior

still

wea

kly

dom

inan

tto

bid

own

valu

e.

Th

eex

pec

ted

pay

men

tfo

rb

idd

erw

ith

valu

er

isrG

(r).

Ifx>

rth

en

m2(x

,r)=

rG(r)+∫ x r

yg(y)d

y.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

24/29

Not

es

Not

es

Res

erve

Pri

ces

FP

A

1st

Pri

ceA

uct

ion

:

Not

eth

at:

(a)

x<

rn

ever

gain

and

(b)

β(r)=

r.

Ifx≥

ras

bef

ore

we

can

get

that

β(x)=

E(m

ax{Y

1,r}|

Y1<

x)=

rG(r)

G(x)+

1

G(x)

∫ x ryg

(y)d

y.

Th

eex

pec

ted

pay

men

tb

ecom

es

m1(x

,r)=

rG(r)+∫ x r

yg(y)d

y.

Aga

inex

pec

ted

reve

nu

ein

bot

hfo

rmat

sco

inci

de.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

25/29

Res

erve

Pri

ceE

xam

ple

FP

A

Au

nif

orm

dis

trib

uti

onex

amp

lew

ith

N=

2an

dω

=1.

Bid

din

gis

mor

eag

gres

sive

wit

hre

serv

epr

ices

than

wit

hou

t:

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

26/29

Not

es

Not

es

Rev

enue

Eff

ects

and

Opt

imal

Res

erve

Pri

ces

Th

eex

-an

teex

pec

ted

pay

men

tis

for

k∈{1

,2}

E(m

k(r))

=r(

1−

F(r))

G(r)+∫ ω r

y(1−

F(y))

g(y)d

y.

Ifse

ller

atta

ches

valu

ex 0∈[0

,ω)

toth

eob

ject

and

sets

rese

rve

pric

er

his

exp

ecte

dp

ayoff

is

Π0(r)=

N∗

E(m

k(r))

+F(r)N

x 0.

For

λ(x)=

f(x)/

(1−

F(x))

(haz

ard

rate

),op

tim

alit

yre

qu

ires

dΠ

0

dr

=N[1−

F(r)−

rf(r)]

G(r)+

NG(r)f(r)x

0

=N[1−(r−

x 0)λ

(r)](1−

F(r))

G(r)=

0.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

27/29

The

Exc

lusi

onP

rinc

iple

Ifx 0

>0

then

dΠ

0(x

0)/

dr>

0th

us

r>

x 0.

Ifx 0

=0

then

dΠ

0(0)/

dr=

0,b

ut

it’s

loca

lm

inim

um

thu

sr>

x 0.

Hig

her

pay

offby

sett

ing

r>

x 0[E

xclu

de

som

etim

esto

rais

eb

ids]

.

Fir

stor

der

Nec

essa

ryco

nd

itio

nre

du

ces

to

(r∗−

x 0)λ

(r∗)

=1.

Ifλ

isin

crea

sin

g,th

eco

nd

itio

nal

sosu

ffici

ent.

En

try

fees

can

be

chos

ento

mim

icth

eeff

ects

ofre

serv

epr

ices

.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

28/29

Not

es

Not

es

Effi

cien

cyvs

Rev

enue

Co

mm

ents

on

Effi

cien

cy

Th

ough

are

serv

epr

ice

max

imiz

esre

ven

ues

toth

ese

ller

itm

ayb

ein

effici

ent.

Ifx 0

=0

and

r>

0th

ere

isa

pos

itiv

epr

obab

ility

that

the

selle

rre

tain

sth

eob

ject

.

Th

isis

ineffi

cien

tu

nle

ssal

lp

laye

rd

on

otva

lue

the

obje

ct(z

ero

prob

abili

ty).

Co

mm

itm

ent

Th

ere

nee

ds

tob

eco

mm

itm

ent

tore

serv

epr

ice

and

no

ex-p

ost

resa

le.

Sec

ret

rese

rve

valu

esaff

ect

beh

avio

r.

Nava

(LSE)

Indep

enden

tPrivate

ValueAuction

MichaelmasTerm

29/29

Not

es

Not

es

Rev

enue

Equ

ival

ence

Pri

ncip

leE

C31

9–

Lec

ture

Not

esIV

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

1/12

Revenue

Equivalence

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

2/12

Not

es

Not

es

Mai

nR

esul

t

Defi

niti

on

An

auct

ion

isstandard

ifth

ep

erso

nw

ho

bid

sth

eh

igh

est

amou

nt

isaw

ard

edth

eob

ject

.

Commen

t:A

lott

ery

inw

hic

hth

epr

obab

ility

ofw

inn

ing

dep

end

son

the

amou

nt

bid

isn

otst

and

ard

.

The

orem

(Rev

enue

Equ

ival

ence

)

Ifvalues

areiid

andbiddersrisk

neutral,then

anysymmetricincreasing

equilibrium

ofanystandardauctionsuch

that

theexpectedpaymentof

abidder

with

0valueis

0,yieldsthesameexpectedrevenue.

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

3/12

Pro

of.

Fix

any

stan

dar

dau

ctio

nfo

rmat

and

asy

mm

etri

ceq

uili

briu

mβ

.

Let

m(x)

be

the

exp

ecte

dp

aym

ent

wh

enth

eva

lue

isx

.

Let

βb

eso

that

m(0)=

0.S

up

pos

eth

atal

l,b

ut

1p

laye

rfo

llow

β.

Ifh

eb

ids

β(z)

he

win

sifz≥

Y1.

His

exp

ecte

dp

ayoff

is

Π(z|x)=

G(z)x−

m(z).

Inte

rim

opti

mal

ity

req

uir

esth

at

dΠ(z|x)/

dz=

g(z)x−

m′ (z)=

0.

Ifβ

iseq

uili

briu

mth

eeq

ual

ity

hol

ds

atz=

x.

Th

usm′ (y)=

g(y)y

and

m(x)=

m(0)+

∫ x 0yg

(y)dy=

G(x)E

(Y1|Y

1<

x).

Th

us

exp

ecte

dp

aym

ents

do

not

dep

end

onth

eau

ctio

nfo

rmat

.

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

4/12

Not

es

Not

es

Exa

mpl

e:U

nifo

rmR

even

ues

Un

ifor

md

istr

ibu

tion

defi

ned

on[0

,1].

F(x)=

xan

dG(x)=

xN−1

andg(x)=

(N−

1)x

N−2.

Exp

ecte

dp

aym

ents

amou

nt

to

m(x)=

∫ x 0(N−

1)y

N−1dy=

N−

1

NxN

.

Exp

ecte

dre

ven

ues

are

E(R

)=

N∫ 1 0

m(x)dx=

∫ 1 0(N−

1)x

Ndx=

N−

1

N+

1.

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

5/12

Applications

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

6/12

Not

es

Not

es

Thi

rdP

rice

Auc

tion

IfXiiid ∼F

and

max

Xi=

xw

eh

ave

that

:

F(N−1)

2(y|Y

(N−1)

1<

x)=

F(y)N−1+(N−

1)(F(x)−

F(y))F(y)N−2

F(N−1)

1(x)

,

f(N−1)

2(y|Y

(N−1)

1<

x)=

(N−

1)(F(x)−

F(y))f(N−2)

1(y)

F(N−1)

1(x)

.

Ifβ3

isa

sym

met

ric

BN

Eby

reve

nu

eeq

uiv

alen

cem

3(x)=

m2(x)

soth

at

F(N−1)

1(x)E

[β3(Y

(N−1)

2)|Y

(N−1)

1<

x]=

∫ x 0yg

(y)dy

,∫ x 0

β3(y)(N−

1)(F(x)−

F(y))f(N−2)

1(y)dy=

∫ x 0yf

(N−1)

1(y)dy

,∫ x 0

β3(y)(F(x)−

F(y))f(N−2)

1(y)dy=

∫ x 0yf(y)F

N−2(y)dy

.

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

7/12

Thi

rdP

rice

Auc

tion

Th

us

ifβ3

isin

crea

sin

gth

efo

llow

ing

con

dit

ion

hol

ds

∫ x 0β3(y)(F(x)−

F(y))f(N−2)

1(y)dy=

∫ x 0yf(y)F

(N−2)

1(y)dy

.

Tak

ing

der

ivat

ives

wit

hre

spec

tto

xim

plie

sth

at

0+

f(x)∫ x 0

β3(y)f

(N−2)

1(y)dy=

xf(x)F

(N−2)

1(x).

Sim

plif

yin

gf(x)

and

diff

eren

tiat

ing

agai

nim

plie

sth

at

β3(x)f

(N−2)

1(x)=

F(N−2)

1(x)+

xf(N−2)

1(x).

So

that

the

equ

ilibr

ium

bid

din

gst

rate

gym

ust

sati

sfy

β3(x)=

F(N−2)

1(x)

f(N−2)

1(x)+

x.

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

8/12

Not

es

Not

es

Unc

erta

inN

umb

erof

Bid

ders

N={1

,...

,N}

be

the

set

ofp

oten

tial

bid

der

s.

Sym

met

ric

prio

rpn=

Pr(|A|=

n+

1)

for

all

bid

der

s.

Con

sid

era

stan

dar

dau

ctio

nan

da

sym

met

ric

&in

crea

sin

geq

β.

Con

sid

era

bid

der

wit

hva

luex

wh

ob

ids

β(z)

inst

ead

.

Ifh

efa

cesn

bid

der

sh

ew

ins

ifz≥

Y(n)

1w

ith

prob

Gn(z)=

F(z)n

.

Th

epr

obab

ility

ofw

inn

ing

bec

omes

G(z)=

∑N−1

n=0pnGn(z).

Th

eex

pec

ted

pay

offth

us

bec

omes

Π(z|x)=

G(z)x−

m(z).

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

9/12

Unc

erta

inN

umb

erof

Bid

ders

2ndPrice

Auction

Sti

llw

eakl

yd

omin

ant

tob

idon

e’s

own

valu

e.

Exp

ecte

dp

aym

ent

isgi

ven

by

m2(x)=

∑N−1

n=0pnGn(x)E

(Y(n)

1|Y

(n)

1<

x).

1stPrice

Auction

Exp

ecte

dp

aym

ent

isgi

ven

bym

1(x)=

G(x)β(x).

By

the

reve

nu

eeq

uiv

alen

cem

1(x)=

m2(x)

and

thu

s

β(x)=

∑N−1

n=0pnGn(x)

G(x)E(Y

(n)

1|Y

(n)

1<

x).

Wei

ghte

dav

erag

eof

the

bid

sin

the

know

nn

um

ber

case

.

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

10/12

Not

es

Not

es

All

Pay

Auc

tion

orL

obby

ing

Hig

hes

tb

idw

ins

obje

ct,

all

bid

der

sp

ayth

eir

bid

.

Su

pp

ose∃β

Asy

mm

etri

ceq

uili

briu

msu

chth

atm

A(0)=

0.

Bec

ause

bid

der

sal

way

sp

ayan

dby

reve

nu

eeq

uiv

alen

ce

βA(x)=

mA(x)=

∫ x 0yg

(y)dy

.

To

solv

eth

eas

sign

edpr

oble

msh

owth

at:

Su

chst

rate

gyis

aB

NE

ofth

eau

ctio

n

Ch

eck

that

nob

od

yca

nb

enefi

tfr

omu

nila

tera

ld

evia

tion

s

... N

ava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

11/12

Sol

ving

Rev

enue

Equ

ival

ence

Pro

blem

s

To

solv

esu

chpr

oble

ms

and

corr

ectl

yw

rite

dow

nex

pec

ted

pay

offs

itis

inst

ruct

ive

toth

ink

ofth

efo

llow

ing

plo

tfo

rN

=3:

Nava

(LSE)

Reven

ueEquivalence

Principle

MichaelmasTerm

12/12

Not

es

Not

es

IPV

Ext

ensi

ons

EC

319

–L

ectu

reN

otes

V

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

1/26

Intr

odu

ctio

n

Ass

um

pti

ons

un

der

lyin

gth

ere

ven

ue

equ

ival

ence

prin

cip

le:

Ind

epen

den

ce

Ris

kn

eutr

alit

y

No

Bu

dge

tC

onst

rain

ts

Sym

met

ry

No

Res

ale

Exp

lore

con

seq

uen

ces

ofre

laxi

ng

the

last

fou

ras

sum

pti

ons.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

2/26

Not

es

Not

es

RiskAversion

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

3/26

Ris

kA

vers

ion

Ris

kn

eutr

alit

y⇒

pay

offlin

ear

inp

aym

ents⇒

reve

nu

eeq

uiv

alen

ce.

Ris

kav

ersi

on:

u:R

+→

Rsa

tisf

yin

gu(0)=

0,u′>

0&

u′′<

0.

Lem

ma

Inan

IPV

setu

pif

bid

der

sar

eri

sk-a

vers

ean

dsy

mm

etri

c,th

enth

eex

pec

ted

reve

nu

ein

a1s

tpr

ice

auct

ion

isgr

ater

than

ina

2nd

pric

eau

ctio

n.

Pro

of.

In2n

dpr

ice

auct

ion

bid

din

gon

e’s

valu

ere

mai

ns

wea

kly

dom

inan

t.T

he

exp

ecte

dpr

ice

isth

us

un

chan

ged

.

In1s

tpr

ice

auct

ion

ifal

lfo

llow

incr

easi

ng

(diff

)st

rate

gyγ

wit

hγ(0)=

0.O

pti

mal

ity

for

ab

idd

erw

ith

valu

ex

agai

nd

icta

tes

x∈

arg

max z

G(z)u(x−

γ(z))

.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

4/26

Not

es

Not

es

Ris

kA

vers

ion

Pro

of.

[Pro

ofco

nti

nu

ed]

Th

efi

rst

ord

erco

nd

itio

nev

alu

ated

atx

req

uir

es

γ′ (

x)=

u(x−

γ(x))

u′ (

x−

γ(x))

g(x)

G(x)

.

Wer

ep

laye

rsri

skn

eutr

alu(x)=

axth

isw

ould

amou

nt

to

β′ (

x)=

(x−

β(x))[g(x)/

G(x)]

.

Sin

ceu(0)=

0&

u′′<

0w

eh

ave

that

[u(y)/

u′ (

y) ]>

y&

thu

s

γ′ (

x)>

(x−

γ(x))[g(x)/

G(x)]

,

wh

ich

imp

lies

that

ifβ(x)>

γ(x)

then

γ′ (

x)>

β′ (

x).

Bu

tsi

nce

γ(0)=

β(0)=

0w

eh

ave

that

γ(x)>

β(x).

Exp

ecte

dp

aym

ent

in1s

tpr

ice

incr

ease

sw

hile

un

chan

ged

in2n

dpr

ice!

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

5/26

Exa

mpl

e:C

RR

AU

tilit

yan

d2

Bid

ders

Su

pp

ose

that

N=

2an

dth

atu

tilit

ysa

tisfi

esu(x)=

xα.

Ifso

,re

lati

veri

skav

ersi

onam

oun

tsto

xu′′ (

x)/

u′ (

x)=

1−

α.

Ob

serv

eth

atF

α=

F1

/α

isal

soa

cdf

&d

enot

eit

sp

df

byf α

=fF

1/

α−1/

α.

Th

e1s

tpr

ice

equ

ilibr

ium

stra

tegy

sati

sfies

γ(0)=

0an

d

γ′ (

x)=

(x−

γ(x))(f(x)/

αF(x))

⇒γ′ (

x)F

(x)1

/α+

γ(x)f(x)F

(x)1

/α−1/

α=

xf(x)F

(x)1

/α−1/

α

⇒γ(x)F

α(x)=∫ x 0

yfα(y)d

y.

Eq

uiv

alen

tto

risk

neu

tral

case

bu

tfo

rF

αin

stea

dof

F.

Sin

ceF

α≤

Fth

eex

pec

ted

reve

nu

eis

grat

erw

ith

risk

aver

sion

.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

6/26

Not

es

Not

es

Exa

mpl

e:C

RR

AU

tilit

yan

d2

Bid

ders

Inth

epr

evio

us

exam

ple

for

α=

1/

2an

dF(x)=

xw

ege

tth

at

γ(x)=

2x/

3≥

x/

2=

β(x).

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

7/26

BudgetConstraints

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

8/26

Not

es

Not

es

Bud

get

Con

stra

ints

Con

sid

erth

efo

llow

ing

sett

ing:

Age

nt

typ

eco

nsi

sts

ofa

valu

e-b

ud

get

pai

r(X

i,W

i).

Bid

der

wit

hty

pe(x

,w)

can

nev

erb

idm

ore

than

w.

Ifh

ed

oes

and

def

ault

s,a

big

pen

alty

isap

plie

d.

Age

nt

typ

esar

eiid

and(X

,W)pdf ∼

fon

[0,1]2

.

Ab

idco

nsi

sts

ofa

map

B: [

0,1]2→

R.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

9/26

Sec

ond

Pri

ceA

ucti

on

Lem

ma

Ina

2nd

pric

eau

ctio

nit

isa

wea

kly

dom

inan

tst

rate

gyto

bid

BII(x

,w)=

min{x

,w}.

Pro

of.

(a)

BII(x

,w)>

wis

dom

inat

ed.

Wh

enon

ew

ins:

ifth

ese

con

dh

igh

est

bid

isab

ove

w,

def

ault

and

pen

alty

follo

w,

ifth

ese

con

db

idd

oes

not

exce

edw

,b

idd

ing

ww

ins

asw

ell.

(b)

Ifx≤

wb

idd

eris

un

con

stra

ined

.

Th

esa

me

pro

ofof

clas

sn

otes

IIsh

ows

that

BII(x

,w)=

x.

(c)

Ifx>

wsa

me

argu

men

tsh

ows

that

BII(x

,w)<

wis

dom

inat

ed.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

10/26

Not

es

Not

es

SP

AC

ompu

ting

Rev

enue

Nex

tle

tu

sco

mp

ute

exp

ecte

dre

ven

ue:

For

x′′=

min{x

,w}⇒

BII(x

,w)=

BII(x′′ ,

1).

Th

ese

tof

pla

yers

bid

din

gle

ssth

anx′′

isgi

ven

by

LII(x′′ )

={ (X

,W)|

BII(X

,W)<

BII(x′′ ,

1)} .

Th

epr

obab

ility

that

typ

e(x′′ ,

1)

outb

ids

one

bid

der

is

FII(x′′ )

=∫ L

II(x′′ )

f(X

,W)d

Xd

W.

Th

epr

obab

ility

ofw

inn

ing

thu

sis

:G

II(x′′ )

=(F

II(x′′ ))N−1.

Exp

ecte

du

tilit

yfo

rty

pe(x′′ ,

1)

wh

enb

idd

ing

BII(z

,1)

is

GII(z)x′′−

mII(z

,1).

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

11/26

SP

AC

ompu

ting

Rev

enue

Exp

ecte

dp

aym

ents

inth

ese

con

dpr

ice

auct

ion

thu

sam

oun

tto

mII(x

,w)=

mII(x′′ ,

1)=∫ x′′ 0

ydG

II(y).

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

12/26

Not

es

Not

es

Fir

stP

rice

Auc

tion

Su

pp

ose

that

for

anin

crea

sin

gfu

nct

ion

βth

eeq

uili

briu

mst

rate

gyis

BI(x

,w)=

min{β

(x),

w}.

Itm

ust

be

that

β(x)<

xor

bid

der

x<

ww

ould

dev

iate

.

To

com

pu

tere

ven

ue,

we

pro

ceed

just

asb

efor

e:

For

x′

such

that

β(x′ )=

min{β

(x),

w}⇒

BI(x

,w)=

BI(x′ ,

1).

Th

ese

tof

pla

yers

bid

din

gle

ssth

anx′

isgi

ven

by

LI(x′ )={ (X

,W)|

BI(X

,W)<

BI(x′ ,

1)} .

Th

epr

obab

ility

that

typ

e(x′ ,

1)

outb

ids

one

bid

der

is

FI(x′ )=∫ L

I(x′ )

f(X

,W)d

Xd

W.

Th

epr

obab

ility

ofw

inn

ing

thu

sis

:G

I(x′ )=

(FI(x′ ))N−1.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

13/26

FP

AC

ompu

ting

Rev

enue

Exp

ecte

dp

aym

ents

inth

efi

rst

pric

eau

ctio

nth

us

amou

nt

to

mI(x

,w)=

mI(x′ ,

1)=∫ x′ 0

ydG

I(y).

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

14/26

Not

es

Not

es

BC

Rev

enue

Com

pari

son

Lem

ma

Ifb

idd

ers

are

fin

anci

ally

con

stra

ined

and

the

1st

pric

eau

ctio

nh

asan

equ

ilibr

ium

ofth

efo

rmB

I(x

,w)=

min{β

(x),

w},

then

the

exp

ecte

dre

ven

ue

inth

e1s

tpr

ice

auct

ion

exce

eds

that

ofth

e2n

dpr

ice

auct

ion

.

Pro

of.

Sin

ceβ(x)<

xfo

ran

yx

,it

mu

stb

eth

atLI(x)⊂

LII(x)

FI(x)≤

FII(x)

for

any

x∈(0

,1)⇒

FI

sto

chas

tica

llyd

omin

ates

FII

Exp

ecte

dre

ven

ues

are:

E(R

I)=

E(Y

I(N)

2)>

E(Y

II(N

)2

)=

E(R

II)

Sin

ceE(Y

a(N

)2

)=∫ 1 0

ma(x

,1)f

a(x)d

xfo

ran

ya∈{I

,II }

Bu

dge

tco

nst

rain

tsar

eso

fter

inth

e1s

tpr

ice

case

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

15/26

BC

Rev

enue

Com

pari

son

Th

efo

llow

ing

plo

tcl

earl

yd

isp

lays

the

reve

nu

era

nki

ng:

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

16/26

Not

es

Not

es

AsymmetricBidders

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

17/26

Asy

mm

etri

cB

idde

rsT

wo

Pla

yers

Uni

form

In2n

dpr

ice

bid

din

gon

e’s

valu

ere

mai

ns

opti

mal⇒

effici

ency

.

In1s

tpr

ice

stra

tegy

chan

ges

and

outc

ome

may

be

ineffi

cien

t.

1st

Pri

ceA

uct

ion

Xicd

f ∼Fi=

U[0

,ωi]

for

ap

laye

ri∈{1

,2}.

Str

ateg

ies

are

incr

easi

ng

&d

iffer

enti

able

fun

ctio

ns

βi

[φi=

β−1

i].

Itm

ust

be

that

βi(

0)=

0an

dβi(

ωi)=

bfo

ran

yi∈{1

,2}.

Exp

ecte

dp

ayoff

ofb

idd

eri6=

jof

bid

din

gb

is

Πi(

b,x

i)=

(xi−

b)F

j(φj(

b))

=(x

i−

b)φ

j(b)/

ωj.

Op

tim

alit

yan

dx i

=φi(

b)

imp

ly

φ′ j(

b)(

φi(

b)−

b)=

φj(

b).

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

18/26

Not

es

Not

es

Asy

mm

etri

cF

PA

By

rew

riti

ng,

add

ing

and

inte

grat

ing

for

φi(

0)=

0,

(φ′ j(

b)−

1)(

φi(

b)−

b)=

φj(

b)−

φi(

b)+

b,

d db(φ

1(b)−

b)(

φ2(b)−

b)=

2b,

(φ1(b)−

b)(

φ2(b)−

b)=

b2.

Th

ela

steq

uat

ion

imp

lies

that

b=

ω1ω

2/(ω

1+

ω2)

byφi(

b)=

ωi

φ′ i(

b)=

φi(

b)(

φi(

b)−

b)/

b2

byop

tim

alit

y

For

φi(

b)=

ξ i(b)b

+b

this

bec

omes

ξ′ i(

b)b

+ξ i(b)+

1=

ξ i(b)(

ξ i(b)+

1)

⇒ξ i(b)=

1−

k ib2

1+

k ib2

sin

ceξ′ i(

b)

ξ i(b)2−

1=

1 b.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

19/26

Asy

mm

etri

cF

PA

Th

eref

ore

φi(

b)=

2b/(1

+k i

b2)

and

k i=

(1/

ω2 i)−(1

/ω

2 j).

Inve

rtin

gon

ege

tsth

at

βi(

x)=

(1−√ 1−

k ix2)/

k ix

.

Ifω

1>

ω2

not

eth

atβ1(x)<

β2(x).

Th

ew

eake

rp

laye

rb

ids

mor

eag

gres

sive

ly.

Lem

ma

(Effi

cien

cy)

Wit

has

ymm

etri

cally

dis

trib

ute

dva

lues

a2n

dpr

ice

auct

ion

alw

ays

allo

cate

sth

eob

ject

effici

entl

y.A

1st

pric

eau

ctio

nw

ith

pos

itiv

epr

obab

ility

do

esn

ot.

Sin

ceβ1(x

+ε)

<β2(x)

for

ε>

0an

dsm

all

enou

gh.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

20/26

Not

es

Not

es

Asy

mm

etri

cR

even

ueC

ompa

riso

n

Con

sid

eran

exam

ple

wit

hω

1=

1/(1−

α)

and

ω2=

1/(1

+α).

Th

ed

istr

ibu

tion

ofse

llin

gpr

ices

in2n

dpr

ice

auct

ion

is,

for

p∈[0

,ω2],

LII α(p)=

Pr(

min{X

1,X

2}≤

p)=

F1(p)+

F2(p)−

F1(p)F

2(p)

=2p−(1−

α2)p

2

Th

ed

istr

ibu

tion

ofse

llin

gpr

ices

in1s

tpr

ice

auct

ion

is,

for

p∈[0

,1/

2],

LI α(p)=

Pr(

max{β

1(X

1),

β2(X

2)}≤

p)=

F1(φ

1(p))

F2(φ

2(p))

=(1−

α2)(

2p)2

/[ 1−

α2(2

p)4] .

LII α(p)

incr

ease

sw

ith

α,

LI α(p)

dec

reas

esw

ith

α&

EI 0(p)=

EII 0(p).

Th

us

EI α(p)>

EII α(p)

for∀α

>0.

[Un

ran

ked

inge

ner

al].

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

21/26

Resale

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

22/26

Not

es

Not

es

Res

ale

Tw

oA

sym

met

ric

Bid

ders

Will

bid

der

sal

way

sal

loca

teth

eob

ject

effici

entl

yif

resa

leis

per

mit

ted

?

Info

rma

tio

n:

the

win

ner

&al

lb

ids

are

reve

aled

bef

ore

resa

le.

Res

ale

:w

inn

erm

akes

ata

keit

orle

ave

itoff

er.

Ass

um

eth

atF16=

F2

bot

hon

[0,ω

].

Ifβ1,

β2

are

inve

rtib

le,

x 1<

x 2an

dβ1(x

1)>

β2(x

2).

Th

en1

rese

llsth

eob

ject

to2

atpr

ice

p=

x 2⇒

Effi

cien

cy.

Th

en

ext

lem

ma

show

sth

at:

1β1,

β2

are

not

inve

rtib

le;

2th

e1s

tpr

ice

auct

ion

wit

hre

sale

isin

effici

ent.

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

23/26

Lem

ma

Wit

has

ymm

etri

cIP

V,

the

1st

pric

eau

ctio

nfo

llow

edby

resa

led

oes

not

resu

ltin

effici

ency

.

Pro

of.

[Pro

of[1

/3]]

Rec

all

that

βi(

0)=

0an

dβi(

ω)=

bfo

ran

yi∈{1

,2}.

Su

pp

ose

that

βi

isin

vert

ible

and

that

2u

ses

β2⇒

Effi

cien

cy.

Pla

yer

1ex

pec

ted

pay

men

tif

he

bid

sas

ifh

isva

lue

was

z 1is

m1(z

1)=

β1(z

1)F

2(φ

2β1(z

1))−∫ φ 2

β1(z

1)

z 1m

ax{z

1,x

2}d

F2(x

2).

Sec

ond

term

isp

osit

ive

ifz 1

<φ2β1(z

1)

and

neg

ativ

eif

z 1>

φ2β1(z

1).

Typ

ex 1

exp

ecte

dp

ayoff

ifh

eb

ids

asif

his

valu

ew

asz 1

is

x 1F2(z

1)−

m1(z

1).

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

24/26

Not

es

Not

es

Pro

of.

[Pro

of[2

/3]]

Ifβ1

isin

vert

ible

opti

mal

ity

atz 1

=x 1

req

uir

esth

at

m′ 1(x

1)=

x 1f 2(x

1).

Op

tim

alit

yan

dm

1(0)=

0im

ply

that

m1(x

1)=∫ x 1 0

x 2d

F2(x

2).

Effi

cien

cy⇒

[1st

pric

e+

resa

le]∼ =

[2nd

pric

e]!

Ifβ1(ω

)=

b⇒

φ2β1(ω

)=

ω⇒

b=

E[X

2]

sin

ce

m1(ω

)=

bF2(ω

)=∫ ω 0

x 2d

F2(x

2).

Sw

itch

ing

pla

yers

one

gets

b=

E[X

1]6=

E[X

2].

Aco

ntr

adic

tion

!

Th

us

β1

&β2

can

not

be

bot

hst

rict

lyin

crea

sin

g!

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

25/26

Pro

of.

[Pro

of[3

/3]]

Ass

um

eb

oth

con

tin

uou

s,n

on-d

ecre

asin

g&

β2(x

2)=

b2

for

∀x2∈[x′ 2,

x′′ 2].

Cla

im:∃x

1∈(x′ 2,

x′′ 2)

such

that

β1(x

1)≥

b2.

Oth

erw

ise

x′′ 1=

[min

xx

s.t.

β1(x)=

b2]⇒

x′′ 1≥

x′′ 2

.

Als

ox′′ 1>

b2

byop

tim

alit

yof

typ

ex′′ 2

wh

ich

win

sat

mos

tx′′ 1

.

For

2ε<

x′′ 1−

b2,

typ

ex′′ 1−

εw

ould

profi

tby

bid

din

gb2+

ε.

At

infi

nit

esim

alco

stw

inag

ain

stan

yx 2∈[x′ 2,

x′′ 2]⇒

Aco

ntr

adic

tion

!

Th

us

let

x 1∈(x′ 2,

x′′ 2)

be

such

that

β1(x

1)≥

b2.

Ifx 2∈(x

1,x′′ 2]

bid

der

1w

ins

auct

ion

ineffi

cien

tly!

Op

tim

alit

yof

1in

resa

leim

plie

sth

atpr

ice

p>

x 1.

Th

us

wit

hp

osit

ive

prob

abili

typ>

x 2>

x 1⇒

Ineffi

cien

t!

Nava

(LSE)

IPV

Exten

sions

MichaelmasTerm

26/26

Not

es

Not

es

Mec

hani

smD

esig

nIP

VE

C31

9–

Lec

ture

Lec

ture

VI

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

1/29

Intr

odu

ctio

n

How

tod

esig

nse

llin

gm

ech

anis

ms

wit

hd

esir

able

prop

erti

es:

Rev

enu

eM

axim

izat

ion

Effi

cien

cy

Bal

ance

dB

ud

get

For

the

mom

ent

reta

inth

eIP

Vse

tup

:

Np

oten

tial

bu

yers

,1

auct

ion

eer

and

1in

div

isib

leob

ject

.

Bu

yeri’

sva

luat

ionXicd

f ∼Fi

on[0

,ωi]=

Xi

ind

epen

den

tly.

Th

eau

ctio

nee

rva

lues

the

obje

ctat

zero

.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

2/29

Not

es

Not

es

Mec

hani

sms

Rev

elat

ion

Pri

ncip

lean

dR

even

ueE

quiv

alen

ce

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

3/29

Mec

hani

sms

Am

ech

an

ism

con

sist

sof

trip

le(B

,π,µ

):

1M

essa

gesp

aces

Bi;

2A

nal

loca

tion

rule

π:B→

∆N={ p∈

RN +| ∑

N i=1pi≤

1} ;

3A

pay

men

tru

leµ

:B→

RN

.

Ina

1st

or2n

dpr

ice

auct

ion

,ex

cep

tfo

rti

es,

sati

sfy:

Bi=

Xi;

πi(b)=

I(b

i>

max

j6=ibj)

;

µI i(b)=

I(b

i>

max

j6=ibj)bi

for

FP

A;

µII i(b)=

I(b

i>

max

j6=ibj)

max

j6=ibj

for

SP

A.

Ast

rate

gyis

am

apβi

:Xi→

Bi.

Eq

uili

briu

mif

βi(x i)

max

imiz

esp

ayoff

sgi

ven

β−i

for

anyx i

andi.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

4/29

Not

es

Not

es

Rev

elat

ion

Pri

ncip

le

Ad

irec

tm

ech

an

ism

con

sist

sof

pai

r(Q

,M):

1A

nal

loca

tion

rule

Q:X→

∆N

;

2A

pay

men

tru

leM

:X→

RN

.

The

orem

(Rev

elat

ion

Pri

ncip

le–

Mye

rson

)

For

any

mec

han

ism

and

any

equ

ilibr

ium

for

that

mec

han

ism

,th

ere

exis

tsa

dir

ect

mec

han

ism

inw

hic

h:

1it

isan

equ

ilibr

ium

for

each

bu

yer

tore

por

th

isva

lue

tru

thfu

lly;

2ou

tcom

esar

eth

esa

me

asin

equ

ilibr

ium

ofth

eor

igin

alm

ech

anis

m.

Pro

of.

Defi

neQ(x)=

π(β(x))

andM(x)=

µ(β(x))

for

anyx∈

X.

Th

enco

nd

itio

ns

1an

d2

are

met

byd

efin

itio

nof

equ

ilibr

ium

.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

5/29

Ince

ntiv

eC

ompa

tibi

lity

Defi

niti

on

Con

sid

era

dir

ect

mec

han

ism

(Q,M

).

Th

ew

inn

ing

prob

abili

tyan

dth

eex

pec

ted

pay

men

tw

hen

rep

orti

ngz i

,if

oth

ers

are

sin

cere

,am

oun

tto

qi(z i)=∫ X−i

Qi(z i

,x−i)dF−i(x −

i),

mi(z i)=∫ X−i

Mi(z i

,x−i)dF−i(x −

i).

Th

eex

pec

ted

pay

offof

typ

ex i

isgi

ven

by:

x iqi(z i)−

mi(z i).

Th

em

ech

anis

mis

said

tob

ein

cen

tive

com

pa

tib

le(I

C)

if

x iqi(x i)−

mi(x i)=

max

z i∈X

i[xiq

i(z i)−

mi(z i)]≡

Ui(x i).

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

6/29

Not

es

Not

es

Ince

ntiv

eC

ompa

tibi

lity

Equ

ival

ence

Lem

ma

Ad

irec

tm

ech

anis

mis

ince

nti

veco

mp

atib

leif

and

only

ifU′ i(x i)=

qi(x i)

andU′′ i(x

i)≥

0fo

ral

lx i

and

alli.

Pro

of.

If:

IfIC

hol

ds

we

mu

sth

ave

that

Ui(z i)≥

z iqi(x i)−

mi(x i)=

Ui(x i)+(z

i−

x i)q

i(x i).

Th

esa

me

ineq

ual

ity

mu

sth

old

swit

chin

gx i

wit

hz i

and

thu

s

(zi−

x i)q

i(z i)≥

Ui(z i)−

Ui(x i)≥

(zi−

x i)q

i(x i).

(1)

Div

idin

gby

(zi−

x i)

and

taki

ng

limit

sas

z i→

x iyi

eld

s

limz i→

x i

Ui(z i)−

Ui(x i)

(zi−

x i)

=U′ i(x i)=

qi(x i).

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

7/29

Ince

ntiv

eC

ompa

tibi

lity

Equ

ival

ence

Pro

of.

Con

dit

ion

(1)

also

imp

lies

that

(zi−

x i)(qi(z i)−

qi(x i))≥

0,

wh

ich

iseq

uiv

alen

tto

q′ i(x i)≥

0,an

dth

usU′′ i(x

i)≥

0.

On

lyIf

:IfU′ i(x i)=

qi(x i),

the

exp

ecte

dp

ayoff

dep

end

sju

ston

Qsi

nce

:

Ui(x i)=

Ui(

0)+∫ x i 0

qi(t)dt.

Th

us,

for

anyx i

and

anyz i

,w

ege

tth

at

Ui(z i)−

Ui(x i)=∫ z i x i

qi(t)dt≥

(zi−

x i)q

i(x i),

wh

ere

the

ineq

ual

ity

hol

ds

asU′′ i(x

i)=

q′ i(x i)≥

0.B

ut

this

isIC

.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

8/29

Not

es

Not

es

Ince

ntiv

eC

ompa

tibi

lity

Com

men

ts

Th

eif

par

tof

the

lem

ma

sim

ply

amou

nte

dto

the

En

velo

pe

Th

eore

m.

Th

eex

pec

ted

chan

geis

pay

offUi(x i)−

Ui(z i)

isin

dep

end

ent

ofM

.

Pay

offUi(x i)

isco

nve

xas

itis

the

up

per

-en

velo

pe

oflin

ear

fun

ctio

ns:

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

9/29

Rev

enue

Equ

ival

ence

The

orem

(Rev

enue

Equ

ival

ence

–M

yers

on)

Ifth

ed

irec

tm

ech

anis

m(Q

,M)

isIC

,th

enfo

ran

yi

andx i

the

exp

ecte

dp

aym

ent

is

mi(x i)=

mi(

0)+

x iqi(x i)−∫ x i 0

qi(t)dt.

Pro

of.

Sin

ceUi(x i)=

x iqi(x i)−

mi(x i)

andUi(

0)=−m

i(0),

ICim

plie

sth

at

x iqi(x i)−

mi(x i)=−m

i(0)+∫ x i 0

qi(t)dt.

Th

us

exp

ecte

dp

aym

ents

inan

ytw

oIC

mec

han

ism

sw

ith

the

sam

eal

loca

tion

rule

are

equ

ival

ent

up

toco

nst

ant.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

10/29

Not

es

Not

es

Indi

vidu

alR

atio

nalit

y

Am

ech

anis

mis

said

tob

ein

div

idu

ally

rati

on

al

(IR

)if

Ui(x i)≥

0fo

ral

lx i

.

Wh

enIC

hol

ds,

IRsi

mp

lifies

tore

qu

irin

g

Ui(

0)≥

0.

Intu

rnth

isis

equ

ival

ent

to

mi(

0)≤

0.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

11/29

Opt

imal

Mec

hani

sms

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

12/29

Not

es

Not

es

Exp

ecte

dR

even

ue

Am

ech

anis

mis

op

tim

al

ifit

max

imiz

esre

ven

ues

sub

ject

toIC

&IR

.

Ifth

ese

ller

use

sa

dir

ect

mec

han

ism

exp

ecte

dp

aym

ents

are

E(m

i(Xi))=

mi(

0)+∫ ω i 0

x iqi(x i)dFi(x i)−∫ ω i 0

∫ x i 0qi(t)dtdFi(x i).

Not

ice

that ∫ ω i 0

∫ x i 0qi(t)dtdFi(x i)=∫ ω i 0

∫ ω i tdFi(x i)q

i(t)dt=

=∫ ω i 0

(1−

Fi(x i))qi(x i)

f i(x

i)dFi(x i).

Th

eref

ore,

the

selle

r’s

exp

ecte

dre

ven

ues

are

E(R

)=

∑i∈

N

[ mi(

0)+∫ X

[ x i−

1−

Fi(x i)

f i(x

i)

] Qi(x)dF(x)] .

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

13/29

Vir

tual

Val

uati

ons

Defi

ne

the

virt

ua

lva

lua

tio

nas

ψi(x i)=

x i−

1−

Fi(x i)

f i(x

i)=

x i−

1

λi(x i)

.

Vir

tual

valu

atio

nca

nb

ein

terp

rete

das

mar

gin

alre

ven

ue.

Als

oob

serv

eth

at E(ψ

i(Xi))=∫ ω i 0

[ y−

1−

Fi(y)

f i(y)

] dFi(y)

=E(X

i)−∫ ω i 0

∫ ω i yf i(x)dxdy

=E(X

i)−∫ ω i 0

∫ x 0dyf

i(x)dx

=E(X

i)−∫ ω i 0

xfi(x)dx=

0.

Ad

esig

npr

oble

mis

reg

ula

rif

ψ′ i>

0.[λ′ i>

0⇒

ψ′ i>

0]Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

14/29

Not

es

Not

es

Opt

imal

Mec

hani

sms

Th

eo

pti

ma

lm

ech

anis

mso

lves

max

(Q,M

)∑

i∈Nm

i(0)+∫ X

[ ∑i∈

Nψi(x i)Q

i(x) ]dF(x)

s.t.

(IC

)q′ i(x)≥

0an

d(I

R)

mi(

0)≤

0.

Am

ech

anis

mis

opti

mal

ifth

efo

llow

ing

two

con

dit

ion

sh

old

{ Qi(x)>

0⇔

ψi(x i)=

max

jψj(x j)≥

0(1

)M

i(x)=

Qi(x)x

i−∫ x i 0

Qi(t,x −

i)dt

(2)

Ob

serv

eth

at:

(1)

+re

gula

r⇒

IC,

(2)⇒

mi(

0)=

0⇒

IR,

(1-2

)m

axim

ize

reve

nu

e.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

15/29

Opt

imal

Mec

hani

sms

(for

Reg

ular

Pro

blem

s)

Defi

ne

the

smal

lest

valu

eth

atw

ins

agai

nst

x −i

by:

y i(x−i)=

inf{ z i|

ψi(z i)≥

max{m

axj6=

iψj(x j),

0}} =

=m

ax{m

axj6=

iψ−1

i(ψ

j(x j))

,ψ−1

i(0)}

.

Lem

ma

(Opt

imal

Mec

hani

sm–

Mye

rson

)

Ifth

ed

esig

npr

oble

mis

regu

lar

ther

eis

opti

mal

mec

han

ism

sati

sfyi

ng

Qi(x)=

{ 1if

x i>

y i(x−i)

0if

x i<

y i(x−i)

Mi(x)=

{ y i(x−i)

ifQ

i(x)=

10

ifQ

i(x)=

0

Pro

of.

By

sim

plf

yin

gth

epr

evio

us

expr

essi

ons

and

sin

ce∫ x i 0

Qi(t,x −

i)dt=

{ x i−

y i(x−i)

ifx i

>y i(x−i)

0if

x i<

y i(x−i)

.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

16/29

Not

es

Not

es

Opt

imal

Mec

hani

sms

Sym

met

yim

plie

s:ψi=

ψan

dy i(x−i)=

max{m

axj6=

ix j

,ψ−1(0)}

.

Lem

ma

(Sym

met

ric

OM

)

Ifth

ed

esig

npr

oble

mis

regu

lar

and

sym

met

ric

then

ase

con

dpr

ice

auct

ion

wit

hre

serv

epr

icer=

ψ−1(0)

isan

opti

mal

mec

han

ism

.

Th

eo

pti

ma

lm

ech

an

ism

isin

effici

ent

bec

ause

:

1ob

ject

isu

nso

ldw

hen

max

jψj(x j)<

0;

2x i

>y i(x−i)

do

esn

’tim

ply

x i≥

max

j6=ix −

j.

OM

favo

rsd

isad

van

tage

dp

laye

rsto

mot

ivat

eot

her

sto

bid

hig

her

.

IfF1

haz

ard

rate

dom

inat

esF2

[F1

sto

chas

tica

llyd

omin

ates

F2]

λ1(x)≤

λ2(x)⇒

ψ1(x)≤

ψ2(x).

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

17/29

Effi

cien

tM

echa

nism

s

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

18/29

Not

es

Not

es

Pro

per

ties

:B

alan

ced

Bud

get

&E

ffici

ency

Con

sid

erva

lues

inX

i=

[αi,

ωi]⊂

R.

Ap

aym

ent

rule

Mis

said

tob

eb

ala

ced

bu

dg

et(B

B)

if

∑i∈

NM

i(x)=

0.

An

allo

cati

onru

leQ

issa

idto

be

effici

ent

if

Q(x)∈

arg

max

π∈

∆N

∑j∈

Nπjx

j.

For

Qeffi

cien

td

efin

eth

em

axim

ized

soci

alva

lues

by

W(x)=

∑j∈

NQ

j(x)x

j,

W−i(x)=

∑j∈

N\iQ

j(x)x

j.

A2n

dpr

ice

auct

ion

wit

hou

tre

serv

epr

ice

iseffi

cien

tw

hen

αi=

0.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

19/29

VC

GE

ffici

ent

Mec

hani

sm

Defi

niti

on(V

icke

ryC

lark

eG

rove

s)

AV

CG

mec

han

ism

(Qv

,Mv)

con

sist

sof

aneffi

cien

tal

loca

tion

rule

and

ofa

pay

men

tru

leth

atsa

tisfi

es

Mv i(x)=

W(α

i,x −

i)−

W−i(x).

VC

Gis

ICsi

nce

tru

th-t

ellin

gis

opti

mal

byeffi

cien

cyof

Qv

,

Qv i(z

i,x −

i)x i−

Mv i(z

i,x −

i)

=Q

v i(z

i,x −

i)x i+

W−i(z i

,x−i)−

W(α

i,x −

i)

=∑

j∈NQ

v j(z

i,x −

i)x j−

W(α

i,x −

i)≤

W(x)−

W(α

i,x −

i).

VC

Gex

pec

ted

pay

offin

equ

ilibr

ium

isin

crea

sin

gan

dis

Uv i(x

i)=

E[W

(xi,X−i)−

W(α

i,X−i)].

VC

Gis

IRsi

nce

Uv i

isin

crea

sin

gan

dU

v i(α

i)=

0.Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

20/29

Not

es

Not

es

VC

GM

axim

izes

Rev

enue

s

Lem

ma

(Kri

shna

Per

ry)

Am

ong

all

mec

han

ism

for

allo

cati

ng

anob

ject

that

are

effici

ent,

ICan

dIR

,th

eV

CG

mec

han

ism

max

imiz

esth

eex

pec

ted

pay

men

tof

each

agen

t.

Pro

of.

Su

pp

ose

also

(Q∗ ,M∗ )

sati

sfies

the

thre

epr

oper

ties

.

Effi

cien

cyre

qu

ires

that

Q∗=

Qv

.

Sin

ceb

oth

mec

han

ism

sar

eIC

reve

nu

eeq

uiv

alen

ceth

eore

mh

old

s.

Th

us

pay

offs

diff

erby

no

mor

eth

ana

con

stan

tc i=

U∗ i(x

i)−

Uv i(x

i).

By

IRc i≥

0si

nce

Uv i(α

i)=

0.

Bu

tU∗ i(x

i)≥

Uv i(x

i)an

dQ∗=

Qv

imp

lyth

at

E[M∗ i(x

i,X−i)]≤

E[M

v i(x

i,X−i)].

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

21/29

AG

VB

alan

ced

Bud

get

Mec

hani

sm

Defi

niti

on(A

rrow

d’A

spre

mon

tG

erar

d-V

aret

AG

V)

An

AG

Vm

ech

anis

m(Q

a,M

a)

con

sist

sof

aneffi

cien

tal

loca

tion

rule

and

ofa

pay

men

tru

leth

atsa

tisfi

es

Ma i(x)=

1N−1

∑j6=

iE[W−j(x j

,X−j)]−

E[W−i(x i

,X−i)].

AG

Vex

pec

ted

pay

offin

equ

ilibr

ium

isin

crea

sin

gan

dis

Ua i(x

i)=

E[W

(xi,X−i)]−

1N−1

∑j6=

iE[W−j(X)]

.

AG

Vb

alan

ces

the

bu

dge

tsi

nce

∑i∈

N1

N−1

∑j6=

iE[W−j(x j

,X−j)]=

∑i∈

NE[W−i(x i

,X−i)].

AG

Vis

ICfo

rth

esa

me

reas

onV

CG

was

∑j∈

NQ

a j(z

i,x −

i)x j≤

∑j∈

NQ

a j(x)x

j=

W(x).

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

22/29

Not

es

Not

es

Effi

cien

t,B

B,

IC,

IRM

echa

nism

Lem

ma

(Kri

shna

Per

ry)

Th

ere

exis

tsan

effici

ent,

ICan

dIR

mec

han

ism

that

isB

Biff

the

VC

Gm

ech

anis

mre

sult

sin

ap

osit

ive

exp

ecte

dsu

rplu

s.

Pro

of.

Pre

viou

sle

mm

ash

ows

that

VC

G’s

pos

itiv

esu

rplu

sis

nec

essa

ry.

Su

ffici

ency

:B

yIC

&effi

cien

cy⇒

reve

nu

eeq

uiv

alen

ce⇒

form∈{v

,a}:

cm i=

E[W

(xi,X−i)]−

Um i(x

i).

Su

pp

ose

that

VC

Gru

ns

surp

lus

E[ ∑

i∈NM

v i(X

) ]≥

E[ ∑

i∈NM

a i(X

) ]=

0.

Th

isim

plie

s∑

i∈Ncv i≥

∑i∈

Nca i.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

23/29

Pro

ofC

onti

nued

Pro

of.

Defi

nedi=

ca i−

cv i

fori>

1an

dd1=−

∑i>

1di.

Not

ice

that

d1=

∑i>

1(c

v i−

ca i)≥

(ca 1−

cv 1).

Con

sid

erth

eB

Bp

aym

ent

rule

M∗

defi

ned

by

M∗ i(x)=

Ma i(x)−

di.

IfQ∗=

Qa,

the

mec

han

ism

(Q∗ ,M∗ )

iseffi

cien

t.

Itis

also

ICsi

nce

pay

offs

diff

erfr

om(Q

a,M

a)

byad

dit

ive

con

stan

t.

Ad

dit

ion

ally

(Q∗ ,M∗ )

isIR

bec

ause

U∗ i(x

i)=

Ua i(x

i)+

di≥

Ua i(x

i)+

ca i−

cv i=

Uv i(x

i)≥

0.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

24/29

Not

es

Not

es

Bila

tera

lT

rade

Pro

blem

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

25/29

Bila

tera

lT

rade

Pro

blem

1b

uye

rw

ith

valu

eV∼

FV

on[v

,v].

1se

ller

wit

hco

stsC∼

FC

on[c

,c].

Su

pp

osev<

can

dc≤

v[t

rad

eis

som

etim

esb

enefi

cial

].

Ifpr

od

uct

ion

&tr

ade

occ

ur

bu

yer

pay

sM

Van

dse

ller

rece

ives

MC

.

Inth

issc

enar

iop

ayoff

sb

ecom

eV−

MV

andM

C−

C.

BB

req

uir

esM

V=

MC

.

Effi

cien

cyre

qu

ires

trad

eto

take

pla

cew

hen

ever

V>

C.

Lem

ma

(Mye

rson

Sat

tert

hwai

te)

Inth

eb

ilate

ral

trad

epr

oble

m,

ther

eex

ists

no

mec

han

ism

that

iseffi

cien

t,IC

,IR

&B

B.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

26/29

Not

es

Not

es

Bila

tera

lT

rade

Pro

blem

Pro

of.

Con

sid

erth

eeffi

cien

tp

ayoff

sd

eriv

edfo

ran

ypr

ofile

x=

(v,−

c):

QV(x)=

1Q

C(x)=

1

uV(x)

v0

uC(x)

−c

0

To

char

acte

rize

the

VC

Gm

ech

anis

m,

not

eth

ateffi

cien

cyre

qu

ires

:

Qv V(x)=

{ 1if

v−

c>

00

ifv−

c<

0=

I(v

>c)

Qv C(x)=

{ 0if

v−

c>

01

ifv−

c<

0=

I(v

<c)

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

27/29

Pro

of.

For

such

defi

nit

ion

s,so

cial

wel

fare

and

resi

du

also

cial

wel

fare

amou

nt

to:

W(x)=

(v−

c)Q

v V(x)+

0Qv C(x)=

(v−

c)I

(v>

c)=

max{v−

c,0}

W−V(x)=−cQ

v V(x)

&W−C(x)=

Qv V(x)v

Hen

ce,

the

VC

Gal

loca

tion

and

pay

men

tssa

tisf

y:

QV(v

,c)=

I(v

>c)=

1−

QC(v

,c)

MV(v

,c)=

W(v

,−c)−

W−V(x)=

(v−

c)Q

v V(v

,−c)+

cQv V(x)

=(v−

c)I

(v>

c)+

cI(v

>c)=

I(v

>c)

max{c

,v}

MC(v

,c)=

W(v

,−c)−

W−C(x)=

(v−

c)Q

v V(1

,−c)−

vQv V(x)

=(v−

c)I

(v>

c)−

vI(v

>c)=−

I(v

>c)

min{v

,c}

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

28/29

Not

es

Not

es

Pro

of.

Th

eV

CG

exp

ecte

dp

ayoff

ofth

eb

uye

rof

clai

min

gto

bez

amou

nts

to:

E[(v−

max{C

,1})

I(z

>C)]=∫ z 0

vdFC(c)−∫ z 0

max{c

,v}d

FC(c)

=∫ z 0

vdFC(c)−∫ v 0

vdFC(c)−∫ z v

cdFC(c)

=vF

C(z)−

vFC(v)−∫ z v

cdFC(c).

Itis

effici

ent.

Itis

ICsi

nce

tru

thte

llin

gis

opti

mal

,

E[(v−

max{C

,v})

I(z

>C)]=

vFC(z)−

vFC(v)−∫ z v

cdFC(c)

[FO

C](v−

z)fC(z)≤

0⇒

v=

z.

Itis

IRsi

nce

Uv V(v)≥

0an

dU

v C(c)≥

0fo

ran

yv

&c

.

Itis

not

BB

sin

cem

ax{c

,v}−

min{v

,c}<

0ifv>

c.

Sin

ceV

CG

isre

ven

ue

max

imiz

ing

amon

gst

effici

ent,

ICan

dIR

mec

han

ism

s,th

ecl

aim

hol

ds

(pro

veas

inK

rish

na

Per

ryfi

rst

lem

ma)

.

Nava

(LSE)

Mechanism

DesignIPV

MichaelmasTerm

29/29

Not

es

Not

es

Inte

rdep

ende

ntV

alue

Auc

tion

sE

C31

9–

Lec

ture

Not

esV

II

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

1/24

Interdependent

Value

Models

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

2/24

Not

es

Not

es

Inte

rdep

ende

ntV

alue

s

Eac

hb

idd

erh

asso

me

info

rmat

ion

abou

tth

eva

lue

ofth

eob

ject

.

Th

ein

form

atio

nof

any

pla

yer

affec

tsva

luat

ion

sfo

ral

lot

her

s.

Let

Xi∈[0

,ωi]

den

ote

pla

yer

i’s

sign

alan

dX

=(X

1,.

..,X

N).

Pla

yer

i’s

valu

eVi

dep

end

son

all

sign

als

Vi=

v i(X

).

Ass

um

ev i

isn

on-d

ecre

asin

gin

Xj,

incr

easi

ng

inXi

&2-

diff

eren

tiab

le.

Inge

ner

aln

otal

lin

form

atio

nre

veal

edby

sign

als

&va

lues

are

E[V

i|X=

x]=

v i(x).

Ass

um

ev i(0

,...

,0)=

0,E(V

i)<

∞an

dri

skn

eutr

alit

y.

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

3/24

Com

mon

Val

ues

Inon

eex

trem

eca

seof

the

mo

del

valu

esar

epr

ivat

eif

v i(X

)=

Xi.

Inth

eop

pos

ite

extr

eme

valu

atio

ns

are

com

mon

(CV

)if

v i(X

)=

v(X

)=

V.

Th

eex

-pos

tva

lue

isco

mm

onto

all

and

un

know

nto

anyo

ne.

Asp

ecia

lca

seco

mm

only

use

dh

as:

V∼

F,

Xi|V

iid ∼

H(V

),

E[X

i|V=

v]=

v.

Th

ism

od

elis

use

dto

des

crib

eoi

ld

rilli

ng

leas

eau

ctio

ns.

[Min

eral

Rig

hts

]

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

4/24

Not

es

Not

es

The

Win

ner’

sC

urse

Con

sid

erth

esp

ecia

lca

seof

the

CV

mo

del

.

Ass

um

eb

idd

ers

pla

ya

1st

pric

eau

ctio

n.

i’s

esti

mat

eof

valu

eb

efor

eth

eau

ctio

nis

E[V|X

i=

x i].

Bu

tif

he

isan

nou

nce

dw

inn

erh

ises

tim

ate

bec

omes

E[V|X

i=

x i,Y

1<

x i]<

E[V|X

i=

x i].

Th

ough

each

sign

alis

un

bia

sed

E[X

i|V=

v]=

v,

the

max

isn

ot

E[m

axXi|V

=v]>

max

E[X

i|V=

v]=

v.

Th

ew

inn

er’s

curs

eco

nsi

stof

the

failu

reto

acco

un

tfo

rth

iseff

ect.

Sh

adin

gth

eb

ids

prev

ents

this

from

hap

pen

ing

ineq

uili

briu

m.

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

5/24

Affi

liate

dS

igna

ls

Let

x∨

x′={m

ax{x

i,x′ i}}

i∈N

and

x∧

x′={m

in{x

i,x′ i}}

i∈N

.

Sig

nal

s(X

1,.

..,X

N)

are

dis

trib

ute

dac

cord

ing

toa

join

td

ensi

tyf

.

(X1,.

..,X

N)

are

affi

liate

dif

fsa

tisfi

esfo

ran

yx

,x′∈

X

f(x∨

x′ )

f(x∧

x′ )≥

f(x)f(x′ )

.

Affi

liati

onim

plie

sth

at∂2

lnf

/∂x

i∂x j≥

0if

i6=

jan

dth

at:

1(X

1,Y

1,.

..,Y

N−1)

are

affilia

ted

for

Yk=

(k-h

igh

est

vari

able

inX−1);

2fo

rY1|X

1∼

G(X

1)

ifx 1

>x′ 1

g(y|x1)/

G(y|x1)≥

g(y|x′ 1)

/G(y|x′ 1)

;

3if

γis

anin

crea

sin

gfu

nct

ion

and

x 1>

x′ 1

E[γ(Y

1)|

X1=

x 1]≥

E[γ(Y

1)|

X1=

x′ 1]

.

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

6/24

Not

es

Not

es

Sym

met

ric

Mo

del

Sta

rtth

ean

alys

isof

inte

rdep

end

ent

valu

esby

assu

min

gsy

mm

etry

:

Val

uat

ion

map

isco

mm

onto

all

pla

yers

v i(X

)=

u(X

i,X−i)

.

and

sym

met

ric

inX−i.

Th

us

for

any

per

mu

tati

onπ

X−i

ofX−i

u(X

i,X−i)=

u(X

i,π

X−i)

.

Join

td

ensi

tyf

issy

mm

etri

cin

Xon

[0,ω

]Nan

dsi

gnal

sar

eaffi

liate

d.

Exp

ecte

dva

lue

tob

idd

eri

wit

hsi

gnal

xfa

cin

ga

hig

hes

tsi

gnal

ofy

is

v(x

,y)=

E[u(X

i,X−i)|X

i=

x,

Y1=

y].

Itis

non

-dec

reas

ing

iny

,in

crea

sin

gin

xan

dv(0

,0)=

0=

u(0).

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

7/24

SecondPrice

Auction

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

8/24

Not

es

Not

es

Sec

ond

Pri

ceA

ucti

on

Lem

ma

(Milg

rom

Web

er)

Sym

met

ric

equ

ilibr

ium

stra

tegi

esin

2nd

pric

eau

ctio

nar

eβ

II(x)=

v(x

,x).

Pro

of.

Let

any

j6=

1u

seβ=

βII

,p

ayoff

sof

pla

yer

1ty

pe

xof

bid

din

gb

are

Π(b|x)=∫ β−

1(b)

0[v(x

,y)−

β(y)]

dG(y|x).

For

Y1=

max

j6=1

Xj,

Y1|X

1∼

G(X

1)

and

β(x)=

v(x

,x).

For

ψ(b)=

β−1(b),

FO

Cfo

rb

imp

lyth

at

[v(x

,ψ(b))−

v(ψ

(b),

ψ(b))]g(ψ

(b)|

x)ψ′ (

b)=

0.

Wh

ich

intu

rnim

plie

sx=

ψ(b)

oreq

uiv

alen

tly

b=

β(x).

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

9/24

Rem

arks

onS

econ

dP

rice

Sym

met

ric

Equ

ilibr

ium

βII

issu

chth

atb

idd

er1

wit

hva

lue

xbr

eaks

even

byw

inn

ing

agai

nst

ah

igh

est

bid

βII(x)

sin

ce

E(V

1|X

1=

Y1=

x)=

v(x

,x)=

βII(x).

βII

isn

ota

wea

kly

dom

inan

tst

rate

gy.

βII

isth

eu

niq

ue

sym

met

ric

equ

ilibr

ium

.

Bu

tth

ere

may

be

diff

eren

tas

ymm

etri

ceq

uili

bria

[eve

nin

the

sym

met

ric

case

].

IPV

:2n

dpr

ice

and

En

glis

hau

ctio

ns

are

stra

tegi

cally

equ

ival

ent.

Not

anym

ore

sin

ced

rop

pin

gou

tre

veal

sin

form

atio

nab

out

valu

atio

ns.

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

10/24

Not

es

Not

es

EnglishAuction

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

11/24

Eng

lish

Auc

tion

[Op

enA

scen

ding

Pri

ce]

Th

efo

rmat

ofE

ngl

ish

Au

ctio

nco

nsi

der

edh

ere

has

:

the

auct

ion

eer

rais

ing

the

pric

egr

adu

ally

star

tin

gat

0;

bid

der

sle

avin

gth

ero

omw

hen

they

stop

par

tici

pat

ing

inth

eau

ctio

n;

bid

der

sn

otal

low

edto

reen

ter

the

auct

ion

once

they

qu

it;

all

bid

der

sin

the

room

obse

rvin

gat

wh

ich

pric

eex

its

occ

ur;

auct

ion

end

ing

wh

enon

eb

idd

erre

mai

ns;

the

pric

ep

aid

bei

ng

that

ofth

ela

stex

it.

Th

eeq

uili

briu

md

iscu

ssed

nex

tis

rob

ust

toa

chan

gein

form

atth

atal

low

sp

laye

rsto

call

out

pric

es[A

very

1998

].

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

12/24

Not

es

Not

es

Str

ateg

ies

inan

Eng

lish

Auc

tion

Str

ateg

yte

llw

hen

tod

rop

out

dep

end

ing

onse

vera

lva

riab

les:

1In

div

idu

alsi

gnal

;

2N

um

ber

ofp

laye

rsin

the

room

;

3P

rice

sat

wh

ich

the

rem

ain

ing

pla

yers

left

the

room

.

Con

sid

erth

efo

llow

ing

stra

tegy

for

k<

N:

βN(x)=

u(x

,...

,x),

βk(x

,pk+1,.

..,p

N)=

u(x

,...

,x,x

k+1,.

..,x

N),

(1)

for

x k+1

defi

ned

byβk+1(x

k+1,p

k+2,.

..,p

N)=

pk+1.

E.g

.βN−1(x

,pN)=

u(x

,...

,x,x

N)

for

βN(x

N)=

pN

.

Wh

ich

can

be

don

esi

nce

∂βk/

∂x>

0.

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

13/24

Sym

met

ric

Equ

ilibr

ium

ofE

nglis

hA

ucti

on

Lem

ma

(Milg

rom

Web

er)

Sym

met

ric

ex-p

ost

equ

ilibr

ium

stra

tegi

esin

the

En

glis

hau

ctio

nsa

tisf

y(1

).

Pro

of.

Con

sid

erb

idd

er1

wit

hsi

gnal

X1=

xfa

cin

gY1=

y 1>

...>

YN−1=

y N−1.

Ifx>

y 1on

ew

ins

the

obje

ctan

dp

ays

the

pric

eof

the

last

dro

pou

t:

p2=

β2(y

1,p

3,.

..,p

N)=

u(y

1,y

1,.

..,y

N−1).

On

eca

nn

otaff

ect

pric

e⇒

he

pref

ers

win

nin

gsi

nce

x>

y 1im

plie

s

u(x

,y1,.

..,y

N−1)−

u(y

1,y

1,.

..,y

N−1)>

0.

Ifx<

y 1h

elo

ses

bu

tis

bet

ter

offsi

nce

the

reve

rse

ineq

ual

ity

hol

ds.

So,

βis

ind

epen

den

tof

f,

and

isan

ex-p

ost

equ

ilibr

ium

.Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

14/24

Not

es

Not

es

First

Price

Auction

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

15/24

Fir

stP

rice

Auc

tion

Lem

ma

(Milg

rom

Web

er)

Sym

met

ric

equ

ilibr

ium

stra

tegi

esin

1st

pric

eau

ctio

nar

e

βI (

x)=∫ x 0

v(y

,y)d

L(y|x)

for

L(y|x)=

exp

( −∫ x y

g(t|t)

G(t|t)

dt) .

Pro

of.

[Pro

of1/

3](1

)L(x)

isa

dis

trib

uti

onon

[0,x]:

By

affilia

tion

g(t|t)/

G(t|t)≥

g(t|0)/

G(t|0)

and

−∫ x 0

g(t|t)

G(t|t)

dt≤−∫ x 0

g(t|0)

G(t|0)

dt=−∫ x 0

dln

G(t|0)

dt

dt=−

∞.

Tak

ing

exp

onen

tial

s⇒

L(0|x)=

0an

dn

otic

eL(x|x)=

exp(0)=

1.

Fin

ally

∂L(y|x)/

∂y=

L(y|x)g(y|y)/

G(y|y)≥

0

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

16/24

Not

es

Not

es

Pro

of.

[Pro

of2/

3](2

)β

Iis

incr

easi

ng

inx

:

Not

ice

that

∂L(y|x)/

∂x=−

L(y|x)g(x|x)/

G(x|x)≤

0.

Ad

dit

ion

ally

∂v(y

,y)/

∂y>

0is

incr

easi

ng

byas

sum

pti

on.

Inte

grat

ing

byp

arts

βI

get

βI (

x)=∫ x 0

v(y

,y)d

L(y|x)=

v(x

,x)−∫ x 0

L(y|x)d

v(y

,y).

Th

eref

ore:

∂β

I (x)

∂x=−∫ x 0

∂L(y|x)

∂x

∂v(y

,y)

∂yd

y≥

0.

(3)

β=

βI

isop

tim

alif

follo

wed

byot

her

s:

Sin

ceβ

isin

crea

sin

gth

ep

ayoff

ofa

bid

β(z)

afte

rob

serv

ing

xis

Π(z|x)=∫ z 0

(v(x

,y)−

β(z))

dG(y|x).

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

17/24

Pro

of.

[Pro

of3/

3]F

irst

ord

erco

nd

itio

ns

req

uir

e

∂Π(z|x)

∂z=

(v(x

,z)−

β(z))

g(z|x)−

β′ (

z)G

(z|x)=

0.

Not

ice

that

β′ (

x)=−∫ x 0

∂L(y|x)

∂xd

v(y

,y)=

g(x|x)

G(x|x)

∫ x 0L(y|x)d

v(y

,y)

=g(x|x)

G(x|x)[v(x

,x)−

β(x)]

.

Th

us

x>

z⇒

v(x

,z)>

v(z

,z)

&g(z|x)/

G(z|x)>

g(z|z)/

G(z|z)

∂Π(z|x)

∂z>

[ (v(z

,z)−

β(z))

g(z|z)

G(z|z)−

β′ (

z)] G

(z|x)=

0.

Ifx<

zsi

mila

rar

gum

ent

show

sth

at∂

Π(z|x)/

∂z<

0

Th

eref

ore

opti

mal

ity

req

uir

esth

atx=

zNava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

18/24

Not

es

Not

es

Revenue

Com

parisons

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

19/24

Rev

enue

s:E

nglis

hvs

Sec

ond

Pri

ceA

ucti

on

Lem

ma

Rev

enu

esin

anE

ngl

ish

auct

ion

are

not

smal

ler

than

thos

eof

a2n

dpr

ice.

Pro

of.

Fir

stn

otic

eth

atby

affilia

tion

ifx>

y

v(y

,y)=

E[u(X

)|X1=

Y1=

y]=

E[u(Y

1,Y

1,.

..,Y

N−1)|

X1=

Y1=

y]

≤E[u(Y

1,Y

1,.

..,Y

N−1)|

X1=

x,Y

1=

y].

Th

us

reve

nu

esin

2nd

pric

eau

ctio

nar

ew

eakl

ysm

alle

r

E[R

II]=

E[β

II(Y

1)|

X1>

Y1]=

E[v(Y

1,Y

1)|

X1>

Y1]

≤E[E[u(Y

1,Y

1,.

..,Y

N−1)|

X1=

x,Y

1=

y]|X

1>

Y1]

=E[u(Y

1,Y

1,.

..,Y

N−1)|

X1>

Y1]

=E[β

2(Y

1,.

..,Y

N−1)]=

E[R

En].

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

20/24

Not

es

Not

es

Rev

enue

s:F

irst

vsS

econ

dP

rice

Auc

tion

Lem

ma

Rev

enu

esin

a2n

dpr

ice

auct

ion

are

not

smal

ler

than

thos

eof

a1s

tpr

ice.

Pro

of.

Con

sid

erth

ed

istr

ibu

tion

K(y|x)=

G(y|x)/

G(x|x).

Not

eth

aty<

xim

plie

sL(y|x)≥

K(y|x)

byaffi

liati

onsi

nce

−∫ x y

g(t|t)

G(t|t)

dt≥−∫ x y

g(t|x)

G(t|x)

dt=−∫ x y

dln

G(t|x)

dt

dt=

ln

( G(y|x)

G(x|x)

) .

Pro

bab

ility

ofw

inn

ing

con

dit

ion

alon

xis

the

sam

ein

1st

&2n

dpr

ice.

Bu

tex

pec

ted

pay

men

tsco

nd

itio

nal

onx

diff

er[i

nte

grat

ion

byp

arts

]:

E[β

II(Y

1)|

Y1<

X1=

x]−

βI (

x)=∫ x 0

v(y

,y)d[K

(y|x)−

L(y|x)]=

=∫ x 0

[L(y|x)−

K(y|x)]

dv(y

,y)≥

0.

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

21/24

Equ

ilibr

ium

Bid

ding

and

the

Win

ner’

sC

urse

Cor

olla

ry

Wit

hsy

mm

etri

c,in

terd

epen

den

tva

lues

and

affilia

ted

sign

als

exp

ecte

dre

ven

ues

can

be

ran

ked

asfo

llow

s:E[R

En]≥

E[R

II]≥

E[R

I ].

In1s

tpr

ice

auct

ion

pla

yers

shad

eth

eir

bid

sto

acco

un

tfo

rw

inn

er’s

curs

e,

βI (

x)=∫ x 0

v(y

,y)d

L(y|x)≤∫ x 0

v(y

,y)d

K(y|x)

<∫ x 0

v(x

,y)d

K(y|x)=

E[V

1|Y

1<

X1=

x].

Ina

2nd

pric

eau

ctio

np

laye

rssh

ade

toac

cou

nt

for

win

ner

’scu

rse,

E[β

II(Y

1)|

Y1<

X1=

x]=∫ x 0

v(y

,y)d

K(y|x)<

E[V

1|Y

1<

X1=

x].

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

22/24

Not

es

Not

es

Effi

cien

cy

Au

ctio

ns

stu

die

dga

veob

ject

toh

igh

est

sign

aln

oto

hig

hes

tva

lue.

Bu

tth

isca

nb

ein

effici

ent.

Eg:

v i(x)=

x i+

2xj

for

i6=

j∈{1

,2}.

Val

uat

ion

ssa

tisf

yth

esi

ng

lecr

oss

ing

con

dit

ion

(SC

C)

iffo

ran

yi6=

jan

dan

yx

∂vi(

x)

∂xi≥

∂vj(

x)

∂xi

.

Inth

esy

mm

etri

cca

sev i(x)=

u(x

i,x −

i).

Let

u′ j

den

ote

the

par

tial

wrt

jth

argu

men

t.

Th

us

SC

Cre

du

ces

tou′ 1≥

u′ 2

bysy

mm

etry

assu

mp

tion

.

SC

Cen

sure

sth

ator

der

ofth

esi

gnal

sre

flec

tsth

atof

the

valu

es.

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

23/24

Lem

ma

(Milg

rom

Web

er)

Wit

hsy

mm

etri

cin

terd

epen

den

tva

lues

and

affilia

ted

sign

als,

ifS

CC

hol

ds

then

all

3au

ctio

nfo

rmat

sh

ave

aneffi

cien

tsy

mm

etri

ceq

uili

briu

m.

Pro

of.

Con

sid

erx i

>x j

and

let

α(t)=

t(x i

,xj,

x −ij)+(1−

t)(x

j,x i

,x−ij).

By

the

fun

dam

enta

lth

eore

mof

calc

ulu

san

dS

CC

,

u(x

i,x j

,x−ij)=

u(x

j,x i

,x−ij)+∫ 1 0∇

u(α(t))

α′ (

t)d

t.

By

SC

Can

dth

ed

efin

itio

nof

grad

ien

t,

∇u(α(t))

α′ (

t)=

u′ 1(α(t))(x

i−

x j)+

u′ 2(α(t))(x

j−

x i)

=(u′ 1(α(t))−

u′ 2(α(t)))(

x i−

x j)≥

0.

Th

eref

ore,

x i>

x jim

plie

su(x

i,x j

,x−ij)≥

u(x

j,x i

,x−ij).

Nava

(LSE)

Interdep

enden

tValueAuctions

MichaelmasTerm

24/24

Not

es

Not

es

The

Linkage

Principle

EC

319

–L

ectu

reN

otes

VII

I

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

1/10

Linkage

Principle

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

2/10

Not

es

Not

es

The

Linkage

Principle

Let

valu

esb

ein

terd

epen

den

tan

dsy

mm

etri

c.

Let

Ab

ea

stan

dar

dau

ctio

nw

ith

asy

mm

etri

ceq

uili

briu

mβA

.

WA(z|x)=

exp

ecte

dp

aym

ent

ofw

inn

erw

ith

sign

alx

&b

idβA(z).

Let

WA j(z|x)

the

par

tial

der

ivat

ive

wrt

j∈{z

,x}.

Not

ice

that

∂WA(x|x)/

∂x=

WA x(x|x)+

WA z(x|x).

Theorem

(Linkage

Principle–Milgrom

Weber)

Let

AandB

betwoauctionsin

whichthehighestbidder

winsandon

lyhe

paysapositiveam

ount.

Supposethat

each

has

asymmetricandincreasingequilibrium

such

that:

(i)W

A x(x|x)≥

WB x(x|x)foranyxand(ii)W

A(0|0)=

WB(0|0)=

0.

Then

theexpectedrevenuein

Aisnot

smallerthan

that

ofB.

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

3/10

Proof.

Inau

ctio

nc∈{A

,B}

bid

der

1w

ith

sign

alx

max

imiz

es

∫ z 0v(x

,y)dG(y|x)−

Wc(z|x)G

(z|x).

Op

tim

alit

yat

z=

xre

qu

ires

that

v(x

,x)g(x|x)−

Wc(x|x)g(x|x)−

Wc z(x|x)G

(x|x)=

0

⇒W

c z(x|x)=

[v(x

,x)−

Wc(x|x)]g(x|x)

G(x|x)

.

Let

∆(x)=

WA(x|x)−

WB(x|x)

and

not

ice

that

∆′ (x)=

[WA x(x|x)−

WB x(x|x)]+[W

A z(x|x)−

WB z(x|x)]

=[W

A x(x|x)−

WB x(x|x)]−

∆(x)[g(x|x)/

G(x|x)]

.

So,

∆(0)=

0&

[∆(x)≤

0⇒

∆′ (x)≥

0]

imp

ly∆(x)≥

0an

yx

.

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

4/10

Not

es

Not

es

Rem

arks

ontheLinkage

Principle

Exa

mp

le:

LP

imp

liesE[R

II]≥

E[R

I ]si

nce

byaffi

liati

on

WI (z|x)=

βI (z)

WII(z|x)=

E[β

II(Y

1)|X1=

x,Y

1<

z]

⇒W

II x(z|x)≥

WI x(z|x)=

0.

Corollary

IfLPassumption

sholdandsign

alsareindependentlydistributedthen:

E[R

A]=

E[R

B].

Proof.

Ifsi

gnal

sar

ein

dep

end

ent

only

the

bid

det

erm

ines

the

pay

men

t

WA x(x|x)=

WB x(x|x)=

0.

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

5/10

Applications

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

6/10

Not

es

Not

es

Pub

licInform

ation

Su

pp

ose

selle

rh

asso

me

info

rmat

ionS

affec

tin

gva

luat

ion

s

Vi=

v i(S

,X).

Ass

um

ev i(0)=

0an

dv i(S

,X)=

u(S

,Xi,X−i)

sym

met

ric

inX−i.

Als

oas

sum

e(S

,X1,.

..,X

N)

tob

eaffi

liate

d.

Ifin

form

atio

nis

not

reve

aled

agai

nle

t

v(x

,y)=

E[V

1|X

1=

x,Y

1=

y].

Ifin

form

atio

nis

reve

aled

inst

ead

let

v(s

,x,y)=

E[V

1|S

=s,X1=

x,Y

1=

y].

Cle

arly

vis

incr

easi

ng

inal

lva

riab

les

andv(0)=

0.

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

7/10

Pub

licInform

ationin

aFirst

Price

Auction

Ifin

form

atio

nis

not

reve

aled

agai

nW

I (z|x)=

βI (z).

Ifin

form

atio

nis

reve

aled

,th

ere

exis

tssy

mm

etri

ceq

uili

briu

m

βI (s,x)=∫ x 0

v(s

,y,y)d

L(y|s

,x)

&L(y|s

,x)=

exp( −∫

x yg(t|s,t)

G(t|s,t)dt) .

Th

eex

pec

ted

pay

men

tin

this

scen

ario

bec

omes

WI (z|x)=

E[β

I (S

,z)|X1=

x].

Th

us

byaffi

liati

onW

I x(z|x)≥

WI x(z|x)=

0.

LP

imp

lies

that

exp

ecte

dre

ven

ues

are

hig

her

ifin

fois

reve

aled

.

Sim

ilar

argu

men

tsfo

rE

ngl

ish

and

2nd

pric

eau

ctio

ns

wor

k.

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

8/10

Not

es

Not

es

Alternative

Linkage

Principle(LPA)

MA(z|x)=

exp

ecte

dp

aym

ent

ofp

laye

rw

ith

sign

alx

&b

idβA(z)

Theorem

(AllPay

Linkage

Principle–Krishna

Morgan)

Let

AandB

betwoauctionsin

whichthehighestbidder

wins.

Supposethat

each

has

asymmetricandincreasingequilibrium

such

that:

(i)M

A x(x|x)≥

MB x(x|x)foranyxand(ii)M

A(0|0)=

MB(0|0)=

0.

Then

theexpectedrevenuein

Aisnot

smallerthan

that

ofB.

Proof.

Inau

ctio

nc∈{A

,B}

bid

der

1w

ith

sign

alx

max

imiz

es∫ z 0

v(x

,y)dG(y|x)−

Mc(z|x).

Op

tim

alit

yat

z=

xre

qu

ires

Mc z(x|x)=

v(x

,x)g(x|x).

∆(x)=

MA(x|x)−

MB(x|x)⇒

∆′ (x)=

[MA x(x|x)−

MB x(x|x)]≥

0.

Th

us

∆(0)=

0im

plie

s∆(x)≥

0an

yx

.

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

9/10

AllPay

Auctionsor

Lobbying

LP

Ad

oes

not

req

uir

eth

atlo

sers

do

not

pay

.

Inan

all

pay

auct

ion

ever

yon

ep

ays

his

bid

MA

P(z|x)=

βA

P(z).

Ina

firs

tpr

ice

auct

ion

inst

eadM

I (z|x)=

βI (z)G

(z|x).

Th

us,

sin

ceby

affilia

tion

Gx(z|x)≤

0,

MI x(x|x)≤

MA

Px

(x|x)=

0.

By

theo

rem

LP

Ath

us:

E[R

AP]≥

E[R

I ].

Th

esy

mm

etri

ceq

uili

briu

mof

the

all-

pay

auct

ion

is

βA

P(x)=∫ x 0

v(y

,y)g(y|y)dy

.

Nava

(LSE)

TheLinka

gePrinciple

MichaelmasTerm

10/10

Not

es

Not

es

Bid

ding

Rin

gsE

C31

9–

Lec

ture

Not

esIX

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

BiddingRings

MichaelmasTerm

1/11

Intr

odu

ctio

n

Bid

din

gri

ngs

are

cart

els

tom

anip

ula

teth

eou

tcom

esof

the

auct

ion

.

Inve

stig

atin

gco

llusi

onin

auct

ion

sis

sub

stan

tial

anti

tru

stac

tivi

ty.

Bas

icS

etup

Con

sid

erth

eas

ymm

etri

cIP

Vm

od

elfo

rXi∼

Fi

on[0

,ω].

Let

I⊆

Nb

eth

ese

tof

bu

yers

inth

eb

idd

ing

rin

g.

and

N\I

be

the

set

ofb

uye

rsn

otin

the

bid

din

gri

ng.

Let

YS 1=

max

i∈S

Xi∼

GS=

∏i∈

SFi

for

any

sub

set

ofp

laye

rsS⊆

N.

Th

eri

ng

wan

tsth

eob

ject

togo

toth

em

emb

erva

luin

git

the

mos

t.

Nava

(LSE)

BiddingRings

MichaelmasTerm

2/11

Not

es

Not

es

Collusion

inaSecondPrice

Auction

Nava

(LSE)

BiddingRings

MichaelmasTerm

3/11

Col

lusi

onin

aS

econ

dP

rice

Auc

tion

Ass

um

eth

eri

ng

wor

ksan

dm

emb

ers

reve

alin

form

atio

n.

Rin

gpr

od

uce

spr

ofits

tom

emb

ers

byre

du

cin

gpr

ice

com

pet

itio

n.

Rin

gsu

bm

its

one

bid

toth

eau

ctio

nee

r[Y

I 1=

Xi

som

ei∈

I].

Con

dit

ion

alon

rin

gw

inn

ing

the

pay

men

tb

ecom

essm

alle

r

PI=

max{ Y

N\I

1,r} ≤

max{ Y

N\i

1,r} =

Pi.

Exp

ecte

dp

aym

ents

con

dit

ion

alx

are

smal

ler

wh

enth

eri

ng

isin

pla

ce.

Su

rplu

sm

ade

byri

ng

bu

yer

i∈

Ico

nd

itio

nal

onva

lue

xis

δ i(x)=

rGN\i(r)+∫ x r

ydG

N\i(y)−[r

GN\I(r)+∫ x r

ydG

N\I(y)]

GI\i (

x).

δ i(x)=

mi(

x)−

mi(

x)≥

0si

nce

pay

men

tsar

eal

way

slo

wer

.

Nava

(LSE)

BiddingRings

MichaelmasTerm

4/11

Not

es

Not

es

Gai

nsan

dL

osse

sfr

omC

ollu

sion

Th

eex

-an

teex

pec

ted

profi

tsfr

omth

eri

ng

bec

ome

δ I=

∑i∈

IE[δ

i(x i)]

.

Bid

der

sn

otin

the

rin

gd

on

otb

enefi

tor

suff

ersi

nce

for

i∈

N\I

:

Pi=

max{ Y

I 1,Y

N\I\i

1,r} =

max{ Y

N\i

1,r} =

Pi.

So

δ i(x)=

0an

yi∈

N\I

[rin

gex

erts

no

exte

rnal

ity

onou

tsid

ers]

.

Th

us

gain

sby

the

rin

gco

rres

pon

dto

loss

esby

the

selle

r.

Incr

easi

ng

the

rin

g’s

size

toJ⊃

Ib

enefi

tsm

emb

ers

atex

pen

seof

selle

rsi

nce

PJ=

max{ Y

N\J

1,r} ≤

max{ Y

N\I

1,r} =

PI.

Nava

(LSE)

BiddingRings

MichaelmasTerm

5/11

Effi

cien

tC

ollu

sion

An

effici

ent

collu

sio

nm

ech

anis

m:

I.al

loca

tes

the

righ

tto

repr

esen

tth

eri

ng

effici

entl

yan

dis

IC;

II.

det

erm

ines

pay

men

tsm

ade

byth

eri

ng

[pos

sib

lyn

egat

ive]

.

Lem

ma

(Mai

lath

Zem

sky)

Ina

2nd

pric

eau

ctio

nw

ith

rese

rve

pric

e,th

eex

pec

ted

gain

sfr

omeffi

cien

tco

llusi

onof

any

rin

gm

emb

erd

epen

don

the

iden

tity

,b

ut

not

onh

isva

lue.

Mor

eove

rth

eri

ng

do

esn

otaff

ect

outs

ider

s.

Pro

of.

Allo

cati

onin

2nd

pric

eau

ctio

nsa

tisfi

esQ

i(X)=

Pr(

Xi>

max{Y

N\i

1,r})

.

Wit

heffi

cien

tco

llusi

on( Q

,M) al

loca

tion

rem

ain

sth

esa

me

Q=

Q.

Sin

ceb

oth

mec

han

ism

sar

eIC

and

Q=

Q,

reve

nu

eeq

uiv

alen

ceim

plie

s:

δ i=

mi(

x i)−

mi(

x i)≥

0.

Nava

(LSE)

BiddingRings

MichaelmasTerm

6/11

Not

es

Not

es

Pre

-Auc

tion

Kno

ckou

ts(P

AK

T)

Rin

gco

nd

uct

san

auct

ion

tod

eter

min

ew

ho

repr

esen

tsth

eri

ng

[tic

ket]

.

Mem

ber

ssu

bm

itb

ids,

the

hig

hes

tis

awar

ded

the

tick

et.

Con

dit

ion

alon

win

nin

gth

eau

ctio

nh

ep

ays

toth

eri

ng

τ i=

Pi−

PI.

Rin

gm

akes

pay

men

tsof

δ i(d

efin

edab

ove)

toea

chb

idd

er.

Su

chm

ech

anis

mis

effici

ent,

ex-p

ost

IC,

IR,

BB

,b

ut

no

tex

-pos

tB

B.

Itis

n’t

ex-p

ost

BB

sin

ceit

mak

esp

aym

ents

δ iev

enw

hen

itlo

ses.

IfV

CG

isco

nd

uct

edto

allo

cate

tick

etit

run

sa

surp

lus.

Th

us

ther

eex

ists

aneffi

cien

t,IC

,IR

,ex

-pos

tB

Bm

ech

anis

m.

Su

chm

ech

anis

mis

no

tex

-pos

tIC

how

ever

[Kri

shn

aP

erry

Lem

ma]

.

Nava

(LSE)

BiddingRings

MichaelmasTerm

7/11

Collusion

andReserve

Prices

Nava

(LSE)

BiddingRings

MichaelmasTerm

8/11

Not

es

Not

es

Res

erve

Pri

ces

whe

nfa

cing

Col

lusi

on

Ifse

ller

know

sof

rin

gI

wh

atre

serv

epr

ice

shou

ldh

ese

t?

Th

ese

ller

face

sN−

I+

1b

uye

rsw

ith

valu

es{ Y

I 1,X

I+1,.

..,X

N

} .

ZIcd

f ∼H

Id

enot

esth

ese

con

dh

igh

est

ofsu

chva

lues

.

IfY

N 1>

rth

ese

ller

rece

ives

P=

max{ Z

I ,r} .

Th

eref

ore

the

exp

ecte

dse

llin

gpr

ice

is

r(H

I(r)−

GN(r))

+∫ ω r

ydH

I(y).

Op

tim

alit

yof

the

rese

rve

pric

erI

req

uir

es

HI(r

I)−

GN(r

I)−

rIgN(r

I)=

0.

Nava

(LSE)

BiddingRings

MichaelmasTerm

9/11

Lem

ma

(Mai

lath

Zem

sky)

Th

ead

dit

ion

ofa

bid

der

toa

bid

din

gri

ng

cau

ses

the

opti

mal

rese

rve

pric

efo

rth

ese

ller

tow

eakl

yin

crea

se.

Pro

of.

Con

sid

era

rin

gw

ith

J=

I+

1m

emb

ers.

Th

ese

ller

face

sN−

Ib

uye

rsw

ith

valu

es{ Y

J 1,X

I+2,.

..,X

N

} .

IfZ

Jcd

f ∼H

Jis

seco

nd

hig

hes

tof

such

valu

esth

enZ

J6=

ZI

iff

(1)

ZI=

XI+

1>

YN\J

1⇒

ZI>

YN\J

1=

ZJ,

(2)

ZI=

YI 1<

XI+

1⇒

ZI>

YN\J

1=

ZJ.

Bu

tZ

J≤

ZI

imp

lies

HJ(r)≥

HI(r)

for

any

ran

dth

us

HJ(r

I)−

GN(r

I)−

rIgN(r

I)≥

0,

wh

ich

imp

lies

that

rJ≥

rI.

Nava

(LSE)

BiddingRings

MichaelmasTerm

10/11

Not

es

Not

es

Uni

form

Exa

mpl

e:B

iddi

ngR

ing

Tw

ob

idd

ers

wit

hva

lues

un

ifor

mly

dis

trib

ute

din

[0,1].

Wit

hou

tri

ng

the

opti

mal

rese

rve

pric

eis

NFN−1(r)(

1−

F(r))−

rNf(r)F

N−1(r)=

0

⇒2r(1−

r)−

2r2=

0⇒

r=

1/

2.

Wit

han

all-

incl

usi

veri

ng

reve

nu

esof

the

selle

rb

ecom

e

r(1−

FN(r))

=r(

1−

r2).

Op

tim

alit

yth

enre

qu

ires

1−

3r2=

0⇒

r=

1/√

3>

1/

2.

Ina

1st

pric

eau

ctio

nco

llusi

onis

less

trac

tab

lean

dn

ot

self

-en

forc

ing

,b

ut

resu

lts

exis

tfo

rth

esy

mm

etri

cca

se.

Nava

(LSE)

BiddingRings

MichaelmasTerm

11/11

Not

es

Not

es

Mec

hani

smD

esig

nw

ith

Inte

rdep

ende

ntV

alue

sE

C31

9–

Lec

ture

Not

esX

Nav

a

LSE

Mic

hae

lmas

Ter

m

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

1/13

Intr

odu

ctio

n

Con

sid

erth

ese

tup

wit

hin

terd

epen

den

tva

lues

and

affilia

ted

sign

als.

Rev

elat

ion

prin

cip

leap

plie

sal

soin

this

con

text

.

Val

uat

ion

sv

sati

sfy

thesingle

crossingco

ndition,

if∀i6=

j

∂vi(

x)/

∂xi>

∂vj(

x)/

∂xi

atan

yx

such

that

v i(x)=

v j(x)=

max

k∈N

v k(x).

Ad

irec

tm

ech

anis

mco

nsi

sts

oftw

ofu

nct

ion

s

Q:X→

∆N

&M

:X→

RN

.

Th

ean

alys

ispr

oce

eds

asfo

llow

s:

1E

ffici

ent

dir

ect

mec

han

ism

s;

2R

even

ue

max

imiz

ing

dir

ect

mec

han

ism

s.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

2/13

Not

es

Not

es

Effi

cien

tM

echa

nism

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

3/13

Wit

hout

SC

Cno

Mec

hani

smm

ayb

eE

ffici

ent

Con

sid

era

scen

ario

inw

hic

hN

=2

and

v 1(x)=

x 1an

dv 2(x)=

x2 1.

Su

pp

ose

X1=

[0,2]

and

X2=

c[w

log]

.

Cle

arly

SC

Cis

viol

ated

sin

ce

∂v1(1

,c)/

∂x1=

1<

2=

∂v2(1

,c)/

∂x1.

Su

pp

ose

that

mec

han

ism

(Q,M

)is

effici

ent.

Effi

cien

cyan

dIC

for

pla

yer

1re

qu

ire

that

aty<

1<

z

y−

M1(y

,c)≥

0−

M1(z

,c),

0−

M1(z

,c)≥

z−

M1(y

,c).

Wh

ich

imp

lies

y≥

za

con

trad

icti

on!

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

4/13

Not

es

Not

es

SC

Cis

nece

ssar

yfo

rE

ffici

ency

Su

pp

ose

ther

eis

aneffi

cien

tm

ech

anis

mw

ith

anex

-pos

teq

uili

briu

m

[⇒th

ere

isa

dir

ect

effici

ent

mec

han

ism

wit

han

ex-p

ost

equ

ilibr

ium

].

Giv

enx −

iif

pla

yer

iw

ins

orlo

ses

rega

rdle

ssof

x ith

enS

CC

hol

ds.

Bu

yer

iispivotal

atx −

iif

ther

eex

ists

yan

dz

such

that

v i(y

,x−i)>

max

j6=iv j(y

,x−i)

,

v i(z

,x−i)<

max

j6=iv j(z

,x−i)

.

Bu

yer

i’s

ex-p

ost

ICat

yan

dz

req

uir

esth

at

v i(y

,x−i)−

Mi(

y,x−i)≥−

Mi(

z,x−i)

,

−M

i(z

,x−i)≥

v i(z

,x−i)−

Mi(

y,x−i)

.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

5/13

SC

Cis

nece

ssar

yfo

rE

ffici

ency

Rew

riti

ng

imp

lies

v i(y

,x−i)≥

Mi(

y,x−i)−

Mi(

z,x−i)≥

v i(z

,x−i)

.

Ex-

pos

tIC

imp

lies

mec

han

ism

ism

onot

onic

inva

lues

[if

iw

ins

wit

hsi

gnal

x ih

ew

ins

wit

han

ysi

gnal

big

ger

than

x i].

Effi

cien

cy⇒

i’s

valu

ere

mai

ns

the

hig

hes

tif

sign

alx i

incr

ease

s.

Th

us

ifv i(x)=

v j(x)

ach

ange

∂xi

ben

efits

mor

ei

than

j,

∂vi(

x)/

∂xi≥

∂vj(

x)/

∂xi.

Th

eref

ore

SC

Cis

nec

essa

ry.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

6/13

Not

es

Not

es

Gen

eral

ized

VC

GM

echa

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Con

sid

era

dir

ect

mec

han

ism

(Qv

,Mv)

sati

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ng

Qv i(x)=

{ 1if

v i(x)>

max

j6=iv j(x)

0if

v i(x)<

max

j6=iv j(x)

Mv i(x)=

{ v i(y

i(x −

i),x−i)

ifQ

v i(x)=

10

ifQ

v i(x)=

0

for

y i(x−i)=

inf{ z|v

i(z

,x−i)≥

max

j6=iv j(z

,x−i)} .

Su

chm

ech

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maw

ard

sob

ject

effici

entl

y&

gen

eral

izes

VC

G.

Itin

du

ces

tru

thte

llin

gby

chan

gin

gp

aym

ents

v i(y

i(x −

i),x−i)

.

Lem

ma

(Cre

mer

McL

ean)

Ifva

luat

ion

sv

sati

sfy

SC

C,

tru

thte

llin

gis

anex

-pos

teq

uili

briu

mof

the

VC

Gm

ech

anis

m(Q

v,M

v).

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

7/13

SC

Can

dth

eG

ener

aliz

edV

CG

Cra

mer

McL

ean

Pro

of.

Su

pp

ose

all

pla

yers

bu

ti

are

tru

thfu

l.

(I)

Ifv i(x)>

max

j6=iv j(x)⇒

x i>

y i(x−i)

.

Ifi

rep

orts

asi

gnal

z i>

y i(x−i)

he

win

san

dp

ays

v i(y

i(x −

i),x−i)<

v i(x).

sin

ceby

SC

Cv i(z

i,x −

i)>

max

j6=iv j(z

i,x −

i).

Ifi

rep

orts

az i≤

y i(x−i)

his

surp

lus

isze

roby

SC

C.

No

dev

iati

onz i6=

x iis

profi

tab

le.

(II)

Ifin

stea

dv i(x)<

max

j6=iv j(x)⇒

x i<

y i(x−i)

.

For

ito

win

byS

CC

he

wou

ldn

eed

tob

idz i>

y i(x−i)>

x i.

Bu

tp

aym

ents

wou

ldb

ecom

ev i(y

i(x −

i),x−i)>

v i(x).

Aga

inn

od

evia

tion

ispr

ofita

ble

.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

8/13

Not

es

Not

es

Com

men

tson

Gen

eral

ized

VC

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Wit

hpr

ivat

eva

lues

this

red

uce

sto

ord

inar

y2n

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ice

auct

ion

.

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the

effici

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equ

ilibr

ium

ofth

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ngl

ish

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ctio

n.

Th

eV

CG

mec

han

ism

isn

otd

etai

lfr

ee!

Itpr

esu

mes

that

the

auct

ion

eer

know

sth

eva

luat

ion

map

s.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

9/13

Opt

imal

Mec

hani

sm

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

10/13

Not

es

Not

es

Opt

imal

Mec

hani

sms

Ifb

uye

rs’

info

isco

rrel

ated

they

are

un

able

toga

inan

yre

nts

from

that

info

!

For

sim

plic

ity

con

sid

era

setu

pw

ith

dis

cret

esi

gnal

s

Xi={0

,∆,.

..,(

t i−

1)∆}.

Πd

enot

esth

ejo

int

prob

abili

tyd

istr

ibu

tion

onX

.

Πi=

[π(x−i|x

i)]

isi’

sb

elie

fsm

atri

xw

ith

t iro

ws

and×

i6=jtj

colu

mn

s.

Ifsi

gnal

sar

ein

dep

end

ent

all

row

sar

eeq

ual

ran

k(Π

i)=

1.

Aga

inas

sum

ev i(0)=

0an

dv i(x

j+

∆,x−j)≥

v i(x)

stri

ctif

i=

j.

SC

Cin

this

setu

pb

ecom

esfo

ran

yi6=

j

v i(x)≥

v j(x)⇒

v i(x

i+

∆,x−j)≥

v j(x

i+

∆,x−j)

.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

11/13

The

orem

(Cre

mer

McL

ean)

Ifsi

gnal

sar

ed

iscr

ete,

valu

atio

ns

vsa

tisf

yS

CC

and

Πi

isof

full

ran

kfo

ran

yi,

then

ther

eex

ist

am

ech

anis

min

wh

ich

tru

thte

llin

gis

the

effici

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ex-p

ost

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ium

inw

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hal

lb

uye

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tan

exp

ecte

dp

ayoff

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of.

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etr

uth

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exp

ecte

dp

ayoff

ofth

eV

CG

mec

han

ism

Uv i(x

i)=

∑x −

i∈X−i[v

i(x)Q

v i(x)−

Mv i(x)]

π(x−i|x

i).

Uv i

den

otes

the

t i-v

ecto

r,si

nce

Πi

isof

full

ran

k∃c

i=

[ci(

x −i)] x−i∈

X−i:

Πic

i=

Uv i

.

Con

sid

erm

ech

anis

m(Q

v,M

c)

wit

hp

aym

ents

Mc i(x)=

Mv i(x)+

c i(x−i)

.

Tru

thte

llin

gis

ex-p

ost

ICas

inV

CG

sin

cep

ayoff

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iffer

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ant.

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inte

rim

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ecte

dp

ayoff

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ral

lb

uye

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con

stru

ctio

n.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

12/13

Not

es

Not

es

Com

men

tsO

nC

rem

erM

cLea

nM

echa

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IRal

way

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old

sw

ith

equ

alit

ysi

nce

Uc i(x

i)=

∑x −

i∈X−i[v

i(x)Q

v i(x)−

Mv i(x)−

c i(x−i)]π(x−i|x

i)=

0.

Ex-

pos

tIR

isn

otm

etsi

nce

Qv i(x)=

0&

Mc i(x)>

0ca

nh

old

aton

ce.

CM

inte

rim

exp

ecte

dp

aym

ent

isV

CG

’sp

lus

that

ofa

lott

ery

one

mu

stp

arti

cip

ate

inc i(X−i)

.

Ifva

lues

are

priv

ate

CM

pay

men

tis

2nd

pric

ep

aym

ent

plu

sc i(x−i)

.

(I)

Op

tim

alit

yre

sult

hin

ged

onth

eco

rrel

atio

nam

ongs

tsi

gnal

s,b

ut

not

onin

terd

epen

den

cein

valu

es.

(II)

Effi

cien

cyre

sult

hin

ged

onth

ein

terd

epen

den

cein

valu

es,

bu

tn

oton

corr

elat

ion

amon

gst

sign

als.

Nava

(LSE)

Mechanism

DesignwithInterdep

enden

tValues

MichaelmasTerm

13/13

Not

es

Not

es

Note 1: Entry Fees Francesco Nava

EC319: Auction Theory Michaelmas Term

This note explains the comparison between entry fees and reserve prices. On page 24 of the textbook, it isclaimed that a reserve price r is equivalent to an entry fee:

e = rG(r)

For sake of simplicity consider a second price auction. In such auction it remains dominant to bid one’svaluation even if some players are not participating in the auction, because conditional on winning playerscannot affect the trading price. Thus, if only players with value x ≥ r participate the expected payment atauction of a type x ≥ r is:

m(x) =

∫ x

rydG(y)

Therefore a player with value x = r is just indifferent about whether to participate in the auction if theentry fee is e = rG(r), since:

u(r) = rG(r)−m(r) = rG(r) = e

Finally, buyers with value x′ < r strictly prefer not to participate in the auction:

u(x′) = x′G(r)−m(x′) < rG(r) = e

and buyers with value x′′ > r strictly prefer to participate:

u′(x′′) = G(x′′) > 0

Therefore an entree fee of e = rG(r) is equivalent to a reserve price of r since in both cases the interimexpected payoffs prior to the auction satisfy:

U(x) =

{xG(x)− rG(r)−

∫ xr ydG(y) if x ≥ r

0 if x < r

Note 2: Budget Constraints Francesco Nava

EC319:Auction Theory Michaelmas Term

This is an explicit proof of the argument made in the Extensions slides on page 10.

(1) Recall that since β(x) < x ≤ 1:

LI(x) = {(X,W )|min{β(X),W} < min{β(x), 1}} == {(X,W )|min{β(X),W} < β(x)}

LII(x) = {(X,W )|min{X,W} < min{x, 1}} == {(X,W )|min{X,W} < x}

I want to show that LI(x) ⊂ LII(x). To show this it suffi ces to argue that:

min{β(X),W} < β(x) ⇒ min{X,W} < x

But notice that since β′ > 0:

min{X,W} < x ⇔ β(min{X,W}) < β(x)

min{X,W} < x ⇔ min{β(X), β(W )} < β(x)

But since β(W ) < W :

min{β(X),W} < β(x) ⇒ min{β(X), β(W )} < β(x)

Which concludes the argument.

(2) The additional consideration on the revenues in the inequality. Notice that for any a ∈ {I, II}:

Ya(N)2 ∼ F a(N)2 (y) = F a(y)N +N(1− F a(y))F a(y)N−1

= NF a(y)N−1 − (N − 1)F a(y)N

We want to show that F I(x) < F II(x) implies F I(N)2 (x) < FII(N)2 (x) for any for any x ∈ (0, 1).

To do so notice that the expression NkN−1 − (N − 1)kN is strictly increasing in k for k ∈ (0, 1) since:

∂

∂k

[NkN−1 − (N − 1)kN

]= N(N − 1)[kN−2 − kN−1] > 0

Therefore since 0 < F I(x) < F II(x) < 1:

FI(N)2 (x)− F II(N)2 (x) = [NF I(x)N−1 − (N − 1)F I(x)N ]− [NF II(x)N−1 − (N − 1)F II(x)N ] < 0

Which in turn implies from integration by parts that:

E(RI) = E(YI(N)2 ) =

∫ 1

0ydF

I(N)2 (y) >

∫ 1

0ydF

II(N)2 (y) = E(Y

II(N)2 ) = E(RII)

Note 3: Regularity and IC Francesco Nava

EC319: Auction Theory Michaelmas Term

In the mechanism design slides on page 9, it is claimed that condition (1) and regularity imply incentivecompatibility. To prove this it suffi ces to show that condition (1) and regularity imply that q′i ≥ 0, sincethat was proven to imply IC. Notice that without ties condition (1) is equivalent to:

Qi(x) = 1⇔ ψi(xi) ≥ max{maxj 6=i ψj(xj), 0} (i)

1. For I(·) the indicator function, condition (i) implies that:

qi(xi) =

∫X−i

Qi(xi, x−i)dF−i(x−i)

=

∫X−i

I(ψi(xi) ≥ max{maxj 6=i ψj(xj), 0})dF−i(x−i)

= I(ψi(xi) ≥ 0)∫X−i

∏j 6=i I(ψi(xi) ≥ ψj(xj))dF−i(x−i)

= I(ψi(xi) ≥ 0)∏j 6=i

∫XjI(ψi(xi) ≥ ψj(xj))dFj(xj)

= I(ψi(xi) ≥ 0)∏j 6=i

∫XjI(ψ−1j ψi(xi) ≥ xj)dFj(xj)

= I(ψi(xi) ≥ 0)∏j 6=i Fj(ψ

−1j ψi(xi))

where in the second to last equality we used that by regularity ψ′j > 0 and thus ψj is invertible.

2. Notice that if ψi(xi) < 0 then q′i(xi) = 0. If instead ψi(xi) ≥ 0 we have that:

q′i(xi) = ψ′i(xi)∑k 6=i

fk(ψ−1k ψi(xi))

ψ′k(ψ−1k ψi(xi))

[∏j 6=i,k Fj(ψ

−1j ψi(xi))

]≥ 0

since any term in the summation is positive given regularity, ψ′k > 0, and since fk ≥ 0.

Which proves q′i(xi) ≥ 0 and therefore IC.

Note 4: Optimal Mechanism Francesco Nava

EC319: Auction Theory Michaelmas Term

Some more details on the example developed in class. So we had 2 players {a, b}. Player a with valuesuniformly distributed on [0, 2] and player b with uniform values on [1, 3]. In class we had shown:

ψa(xa) = 2xa − 2 and ψb(xb) = 2xb − 3

Therefore, the highest competing virtual valuations are:

ya(xb) = inf {za|ψa(za) > max {ψb(xb), 0}} == inf {za|za > xb − 0.5 ∩ za > 1} == max {xb − 0.5, 1}

yb(xa) = inf {zb|ψb(zb) > max {ψa(xa), 0}} == inf {zb|zb > xa + 0.5 ∩ zb > 1.5} == max {xa + 0.5, 1.5}

Thus, for these expressions as desired we get that OM satisfies for i ∈ {a, b}:

Qi(x) =

{1 if xi > yi(x−i)0 if xi < yi(x−i)

Mi(x) =

{yi(x−i) if xi > yi(x−i)0 if xi < yi(x−i)

Now with the correct expressions it can be easily noted that this mechanism can be ineffi cient for tworeasons:

1. The object is not sold. For instance, if (xa, xb) = (0.5, 1.25) since:

xa = 0.50 < ya(xb) = max {xb − 0.5, 1} = 1xb = 1.25 < yb(xa) = max {xa + 0.5, 1.5} = 1.5

2. The object is sold to the low value player. For instance, if (xa, xb) = (1.5, 1.75) since:

xa = 1.50 > ya(xb) = max {xb − 0.5, 1} = 1.25xb = 1.75 < yb(xa) = max {xa + 0.5, 1.5} = 2

Note 5: AGV Shifts & BB Francesco Nava

EC319: Auction Theory Michaelmas Term

Due to the algebra mishaps today during lectures let me establish again how to construct the shifts on theAGV mechanism that insure that the mechanism is IR.

1. First notice that by revenue equivalence the expected utility of both the VCG & the AGV mechanismscan differ at most by a constant. For m ∈ {v, a}, let:

cvi = E[W (xi, X−i)]− Uvi (xi)cai = E[W (xi, X−i)]− Uai (xi)

Suppose that VCG runs surplus. If so:

E[∑

i∈NMvi (X)

]≥ E

[∑i∈NM

ai (X)

]= 0 ⇔

⇔∑i∈N E [M

vi (X)] ≥

∑i∈N E [M

ai (X)] = 0 ⇔

⇔∑i∈N [c

vi − E[W−i(xi, X−i)]] ≥

∑i∈N [c

ai − E[W−i(xi, X−i)]] = 0 ⇔

⇔∑i∈N c

vi ≥

∑i∈N c

ai ⇔

⇔∑i>1(c

vi − cai ) ≥ (ca1 − cv1)

2. Define di = cai − cvi for i > 1 and d1 = −∑i>1 di. Notice that d1 ≥ (ca1 − cv1), by the last observation.

3. Consider the BB payment rule M∗ defined by:

M∗i (x) =Mai (x) + di

Such rule is trivially BB, since:∑i∈NM

∗i (X) =

∑i∈NM

ai (X) +

∑i∈N di = 0

4. If Q∗ = Qa the mechanism (Q∗,M∗) is effi cient. It is also IC since payoffs differ from (Qa,Ma) byadditive constant. Additionally (Q∗,M∗) is IR because:

U∗i (xi) = Uai (xi) + di ≥ Uai (xi) + cai − cvi = E[W (xi, X−i)]− cvi = Uvi (xi) ≥ 0

Note 6: Expectation and Maximum Francesco Nava

EC319: Auction Theory Michaelmas Term

In the interdependent value slides on page 4, it is claimed that for the pure common value model:

E[maxXi|V = v] > maxE[Xi|V = v] = v

Recall that the special case has:

• V ∼ F

• Xi|Viid∼ H(V )

• E[Xi|V = v] = v

Therefore we have that:

E[maxXi|V = v] =

∫ ωi

0ydH(y|v)N

maxE[Xi|V = v] = E[Xi|V = v] =∫ ωi

0ydH(y|v) = v

But notice G(y|v) = H(y|v)N is also a distribution and that it stochastically dominates H(y|v), sinceG(y|v) < H(y|v) for any y ∈ (0, 1). Thus we have that:

E[maxXi|V = v] =∫ ωi

0ydG(y|v) ≥

∫ ωi

0ydH(y|v) = maxE[Xi|V = v]

In general since the max is a convex operator we have that:

E[maxXi|V = v] ≥ maxE[Xi|V = v]

by Jensen inequality. That the max is convex can be seen from this plot:

0 1 20

1

2

x1

x2

The plot show that a lottery amongst two bundles having the same maximum is preferred to the maximumof the lottery. Indeed consider:

(x1, x2) =

{(2, 1) with probability 1/2(1, 2) with probability 1/2

Then immediately we have that:

E[maxXi] = 2 ≥ maxE[Xi] = 1.5