CH

AP

TE

R14:

ALLO

CAT

ING

SC

AR

CE

RE

SO

UR

CE

S

MultiagentS

ystems

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Overview

•A

llocationofscarce

resourcesam

ongstanum

berof

agentsis

centraltom

ultiagentsystems.

•R

esourcem

ightbe:

–a

physicalobject–

therightto

useland

–com

putationalresources(processor,m

emory,...)

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

1

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•Ifthe

resourceisn’tscarce,there

isno

troubleallocating

it.

•Ifthere

isno

competition

forthe

resource,thenthere

isno

troubleallocating

it.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

2

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•In

practice,thism

eansw

ew

illbetalking

aboutauctions.

•T

heseused

tobe

rare(and

notsolong

ago).

•H

owever,auctions

havegrow

nm

assivelyw

iththe

Web/Internet

–Frictionless

comm

erce

•N

owfeasible

toauction

thingsthatw

eren’tpreviouslyprofitable:

–eB

ay–

Adw

ordauctions

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

3

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

4

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Whatis

anauction?

•C

oncernedw

ithtraders

andtheir

allocationsof:

–U

nitsofan

indivisiblegood;and

–M

oney,which

isdivisible.

•A

ssume

some

initialallocation.

•E

xchangeis

thefree

alterationofallocations

ofgoodsand

money

between

traders

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

5

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

LimitP

rice

•E

achtrader

hasa

valueor

limitprice

thattheyplace

onthe

good.

•A

buyerw

hoexchanges

more

thantheir

limitprice

fora

goodm

akesa

loss.

•A

sellerw

hoexchanges

agood

forless

thantheir

limit

pricem

akesa

loss.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

6

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•Lim

itpricesclearly

havean

effectonthe

behaviorof

traders.

•T

hereare

severalmodels,em

bodyingdifferent

assumptions

aboutthenature

ofthegood.

•T

hreecom

monly

usedm

odels:

–P

rivatevalue

–C

omm

onvalue

–C

orrelatedvalue

•T

heseare

them

odelsyou’llfind

mostoften

adoptedin

theliterature.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

7

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Private

value

•G

oodhas

anvalue

tom

ethatis

independentofwhat

itisw

orthto

you.

•Textbook

givesthe

example

ofJohnLennon’s

lastdollar

bill.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

8

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Com

mon

value

•T

hegood

hasthe

same

valueto

allofus,butwe

havediffering

estimates

ofwhatitis.

•W

inner’scurse

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

9

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Correlated

value

•O

urvalues

arerelated.

•T

hem

oreyou

areprepared

topay,the

more

Ishouldbe

preparedto

pay.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

10

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•A

marketinstitution

defineshow

theexchange

takesplace.

–D

efinesw

hatmessages

canbe

exchanged.–

Defines

howthe

finalallocationdepends

onthe

messages.

•T

hechange

ofallocationis

marketclearing.

•D

ifferencebetw

eenallocations

isnettrade.

–C

omponentfor

eachtrader

inthe

market.

–E

achtrader

with

anon-zero

componenthas

atrade

ortransaction

price.http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

11

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

–A

bsolutevalue

ofthem

oneycom

ponentdividedby

thegood

component.

•Traders

with

positivegood

componentare

buyers

•Traders

with

negativegood

componentare

sellers

•O

new

aytraders

areeither

buyersor

sellersbutnot

both.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

12

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Yes,butwhatis

anauction?

An

auctionis

am

arketinstitutionin

which

messages

fromtraders

includesom

eprice

information—

thisinform

ationm

aybe

anoffer

tobuy

atagiven

price,inthe

caseofa

bid,oran

offerto

sellatagiven

price,inthe

caseofan

ask—and

which

givespriority

tohigher

bidsand

lower

asks.

This

definition,asw

ithallthis

terminology,com

esfrom

Dan

Friedman.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

13

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

14

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

The

zoologyofauctions

•W

ecan

splitauctionsinto

anum

berofdifferent

categories.

•B

einggood

computer

scientists,we

drawup

ataxonom

y.

–T

hisgives

usa

handleon

allthekinds

therem

ightbe.

–Itsuggests

parameterization.

–Itcan

helpus

tothink

aboutimplem

entation.

•T

hisparticular

classificationis

abitzoological,butitis

agood

placeto

start.

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15

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Single

versusm

ulti-dimensional

•S

ingledim

ensionalauctions

–T

heonly

contentofanoffer

arethe

priceand

quantityofsom

especific

typeofgood.

–“I’llbid

$200for

those2

chairs”

•M

ultidimensionalauctions

–O

fferscan

relateto

many

differentaspectsofm

anydifferentgoods.

–“I’m

preparedto

pay$200

forthose

two

redchairs,

but$300ifyou

candeliver

themtom

orrow.”

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

16

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Single

versusdouble-sided

•S

ingle-sidedm

arkets

–E

itherone

buyerand

many

sellers,orone

sellerand

many

buyers.–

The

latteris

thething

we

normally

thinkofas

anauction.

•Tw

o-sidedm

arkets

–M

anybuyers

andm

anysellers.

•S

inglesided

markets

with

oneseller

andm

anybuyers

are“sell-side”

markets.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

17

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•S

ingle-sidedm

arketsw

ithone

buyerand

many

sellersare

“buy-side”.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

18

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Open-cry

versussealed-bid

•O

pencry

–Traders

announcetheir

offersto

alltraders

•S

ealedbid

–O

nlythe

auctioneersees

theoffers.

•C

learlyas

abidder

inan

open-cryauction

youhave

more

information.

•In

some

auctionform

syou

payfor

preferentialaccessto

information.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

19

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Single-unitversus

multi-unit

•H

owm

anyunits

ofthesam

egood

arew

eallow

edto

bidfor?

•S

ingleunit

–O

neata

time.

–M

ightrepeatifmany

unitsto

besold.

•M

ulti-unit

–B

idboth

priceand

quantity.

•“U

nit”refers

tothe

indivisibleunitthatw

eare

selling.

–S

inglefish

versusbox

offish.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

20

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Firstprice

versuskth

price

•D

oesthe

winner

paythe

highestpricebid,the

secondhighestprice,the

kthhighestprice?

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

21

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Single

itemversus

multi-item

•N

otsom

uchquantity

asheterogeneity.

•S

ingleitem

–Justthe

oneindivisible

thingthatis

beingauctioned.

•M

ulti-item

–B

idfor

abundle

ofgoods.–

“Two

redchairs

andan

orangecouch,or

apurple

beanbag.”–

Valuations

forbundles

arenotlinear

combinations

ofthevalues

oftheconstituents.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

22

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Standard

auctiontypes

•W

ew

illlookatthe

four“standard”

auctions:

–E

nglishauction

–D

utchauction

–F

irst-pricesealed

bidauction

–V

ickreyauction

•A

lsothe

so-calledJapanese

auction.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

23

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

English

auction

•T

hisis

thekind

ofauctioneveryone

knows.

•Typicalexam

pleis

sell-side.

•B

uyerscalloutbids,bids

increasein

price.

•In

some

instancesthe

auctioneerm

aycalloutprices

with

buyersindicating

theyagree

tosuch

aprice.

•T

heseller

may

setareserve

price,thelow

estacceptable

price.

•A

uctionends:

–ata

fixedtim

e(internetauctions);or

–w

henthere

isno

more

biddingactivity.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

24

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•T

he“lastm

anstanding”

paystheir

bid.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

25

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•C

lassifiedin

theterm

sw

eused

above:

–S

ingle-dimensional

–S

ingle-sided–

Open-cry

–S

ingleunit

–F

irst-price–

Single

item

•A

round95%

ofinternetauctionsare

ofthiskind.

•C

lassicuse

issale

ofantiquesand

artwork.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

26

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

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27

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Unlikely

tales

The

former

presidentofParke-B

enetreportsthata

dealerattending

asale

ofeighteenth-centuryFrench

furniturehad

arrangedto

unbuttonhis

overcoatwhenever

hew

ishedto

bid;buttoningthe

overcoatagainw

ouldsignalthathe

hadceased

bidding.T

hedealer,coatunbuttoned,

was

inthe

midstofbidding

fora

LouisX

VIsofa

when

hesaw

someone

outsideto

whom

hew

ishedto

speakand

suddenlyleftthe

room.

The

auctioneercontinued

tobid

forthe

dealerw

ho,when

hereturned

tothe

room,found

hehad

become

theow

nerofthe

sofaatan

unexpectedlyhigh

price.A

nargum

entthenfollow

edas

tow

hetheran

unbuttonedcoatnotin

theauction

roomis

thesam

eas

anunbuttoned

coatinthe

auctionroom

.

(Cassady,1969)

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28

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Dutch

auction

•A

lsocalled

a“descending

clock”auction

–S

ome

auctionsuse

aclock

todisplay

theprices.

•S

tartsata

highprice,and

theauctioneer

callsout

descendingprices.

•O

nebidder

claims

thegood

byindicating

thecurrent

priceis

acceptable.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

29

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•T

iesare

brokenby

restartingthe

descentfroma

slightlyhigher

pricethan

thetie

occurredat.

•T

hew

innerpays

theprice

atwhich

they“stop

theclock”.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

30

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•C

lassifiedin

theterm

sw

eused

above:S

ingle-dimensional;S

ingle-sided;Open-cry;S

ingleunit;F

irst-price;Single

item

•H

ighvolum

e(since

auctionproceeds

swiftly).

•O

ftenused

tosellperishable

goods:

–F

lowers

inthe

Netherlands

(eg.A

alsmeer)

–F

ishin

Spain

andIsrael.

–Tobacco

inC

anada.

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31

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

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32

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

33

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

34

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•T

heG

uardianstates

thattheA

alsmeer

auctiontrades

19m

illionflow

ersand

2m

illionplants

...everyday.

April23rd

2008(page

18–19)

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

35

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

First-price

sealedbid

auction

•In

anE

nglishauction,you

getinformation

abouthowm

ucha

goodis

worth.

•O

therpeople’s

bidstellyou

thingsaboutthe

market.

•In

asealed

bidauction,none

ofthathappens

–atm

ostyouknow

thew

inningprice

afterthe

auction.

•In

theF

PS

Bauction

thehighestbid

wins

asalw

ays

•A

sits

name

suggests,thew

innerpays

thathighestprice

(which

isw

hattheybid).

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36

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•C

lassifiedin

theterm

sw

eused

above:

–S

ingle-dimensional

–S

ingle-sided–

Sealed-bid

–S

ingleunit

–F

irst-price

•G

overnments

oftenuse

thism

echanismto

selltreasury

bonds.

–U

Kstilldoes.

–U

Srecently

changedto

SP

SB

.

•P

ropertycan

alsobe

soldthis

way

(asin

Scotland).

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37

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

The

Am

sterdamauction

•S

incem

edievaltime,property

inthe

lowcountries

hastraditionally

beensold

usingthe

“Am

sterdam”

auction.

•S

tartwith

anE

nglishauction.

•W

hendow

nto

thefinaltw

obidders,starta

Dutch

auctionstage.

•D

utchauction

startsfrom

twice

thefinalprice

oftheE

nglishauction.

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38

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Vickrey

auctions

•T

heV

ickreyauction

isa

sealedbid

auction.

•T

hew

inningbid

isthe

highestbid,butthew

inningbidder

paysthe

amountofthe

secondhighestbid.

•T

hissounds

odd,butitisactually

avery

smartdesign.

•Itis

inthe

bidders’interesttobid

theirtrue

value.

–incentive

compatible

inthe

usualterminology.

•H

owever,itis

notapanacea,as

theN

ewZ

ealandgovernm

entfoundout.

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

39

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•A

gain,classifiedas

above,itis:

–S

ingle-dimensional

–S

ingle-sided–

Sealed-bid

–S

ingleunit

–S

econd-price

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40

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Why

doesthe

Vickrey

auctionw

ork?

•S

upposeyou

bidm

orethan

yourvaluation.

–You

may

win

thegood.

–Ifyou

do,youm

ayend

uppaying

more

thanyou

thinkthe

goodis

worth.

–N

otsosm

art.

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41

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•S

upposeyou

bidless

thanyour

valuation.

–You

standless

chanceofw

inningthe

good.–

How

ever,evenifyou

dow

init,you

willend

uppaying

thesam

e.–

Notso

smart.

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42

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•S

o:there

isno

pointinbidding

aboveor

belowyour

valuation .

•O

fcourse,thisreally

assumes

thereare

alarge

number

ofbidders(see

theN

ewZ

ealandcase).

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

43

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Japanesefish

auction

•T

heauction

formused

tosellfish

inTokyo

isdifferent:

[The]distinctive

aspect[ofthisauction

form]is

thatallbidsare

made

byprospective

buyersat

thesam

etim

e,orapproxim

atelythe

same

time,

usingindividualhand

signsfor

eachm

onetaryunit.

...The

biddingstarts

assoon

asthe

auctioneergives

thesignal,and

thehighest

bidder,asdeterm

inedby

theauctioneer,is

awarded

thelot.

•T

hisis

thussim

ultaneousbidding

andrather

likean

FP

SB

auction.http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

44

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•T

iesare

“notuncomm

on[ly]”broken

byplaying

JanK

enP

on(or

‘paper,rock,scissors’).

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

45

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

46

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

47

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

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48

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Com

binatorialAuctions

•A

uctionsfor

bundlesofgoods.

•A

goodexam

pleofbundles

ofgoodare

spectrumlicences.

•F

orthe

1.7to

1.72G

Hz

bandfor

Brooklyn

tobe

useful,youneed

alicense

forM

anhattan,Queens,

Staten

Island.

•M

ostvaluableare

thelicenses

forthe

same

bandwidth.

•B

utadifferentbandw

idthlicence

ism

orevaluable

thanno

licensehttp://www.csc.liv.ac.uk/˜mjw/pubs/imas/

49

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•(T

heF

CC

spectrumauctions,how

ever,didnotuse

acom

binatorialauctionm

echanism)

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50

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•Let

Z={z1 ,...

,zm}

bea

setofitems

tobe

auctioned.

•W

egave

theusualsetofagents

Ag

={1

,...,n},and

we

capturepreferences

ofagentiw

iththe

valuationfunction:

vi:2Z7→

R

meaning

thatforevery

possiblebundle

ofgoodsZ⊆Z

,vi (Z

)says

howm

uchZ

isw

orthto

i.

•If

vi (∅)=

0,thenw

esay

thatthevaluation

functionfor

iis

normalised.

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51

Chapter

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nIntroduction

toM

ultiagentS

ystems

2e

•A

notherusefulidea

isfree

disposal:

Z1⊆

Z2

implies

vi (Z1 )≤

vi (Z2 )

•In

otherw

ords,anagentis

neverw

orseoffhaving

more

stuff.

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52

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•W

ealready

mentioned

theidea

ofanallocation.

•F

ormally

anallocation

isa

listofsetsZ

1 ,...Zn ,one

foreach

agentA

gi w

iththe

stipulationthat:

Zi⊆Z

andfor

alli,j∈

Ag

suchthat

i6=

j,we

haveZ

i ∩Z

j=∅.

•T

husno

goodis

allocatedto

more

thanone

agent.

•T

hesetofallallocations

ofZ

toagents

Ag

is:

alloc(Z,A

g)

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53

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•Ifw

edesign

theauction,w

egetto

sayhow

theallocation

isdeterm

ined.

•H

owshould

thisbe?

•O

nenaturalw

ayis

tom

aximize

socialwelfare.

–S

umofthe

utilitiesofallthe

agents.

•D

efinea

socialwelfare

function:

sw(Z

1 ,...,Zn ,v

1 ,...,vn )

=

n∑i=

1

vi (Zi )

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54

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•G

iventhis,w

ecan

definea

combinatorialauction.

•G

ivena

setofgoodsZ

anda

collectionofvaluation

functionsv1 ,...

,vn ,onefor

eachagent

i∈

Ag,the

goalisto

findan

allocation

Z∗1 ,...

,Z∗n

thatmaxim

izessw

,inother

words

Z∗1 ,...,Z

∗n=

argm

ax(Z

1,...,Z

n )∈alloc(Z

,Ag) sw

(Z1 ,...,Z

n ,v1 ,...,v

n )

•F

iguringthis

outisw

innerdeterm

ination.

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55

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•H

owdo

we

dothis?

•W

ell,we

couldgetevery

agentito

declaretheir

valuationv̂i

–T

hehatdenotes

thatthisis

whatthe

agentsays,notw

hatitnecessarilyis.

–T

heagentm

aylie!

•T

henw

ejustlook

atallthepossible

allocationsand

figureoutw

hatthebestone

is.

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56

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•O

neproblem

hereis

representation,valuationsare

exponential:vi

:2Z7→

R

–A

naiverepresentation

isim

practical.–

Ina

bandwidth

auctionw

ith1122

licensesw

ew

ouldhave

tospecify

21122

valuesfor

eachbidder.

•S

earchingthrough

themis

computationally

intractable.

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57

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Bidding

languages

•R

atherthan

exhaustiveevaluations,allow

biddersto

constructvaluationsfrom

thebits

theyw

anttom

ention.

•A

tomic

bids(Z

,p)w

hereZ⊆Z

.

•A

bundleZ′satisfies

abid

(Z,p)

ifZ⊆

Z′.

•In

otherw

ordsa

bundlesatisifes

abid

ifitcontainsat

leastthethings

inthe

bid.

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58

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•A

tomic

bidsdefine

valuations

vβ(Z′)

=

{

pif

Z′satisfies

(Z,p)

0oth

erwise

•A

tomic

bidsalone

don’tallowus

toconstructvery

interestingvaluations.

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59

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•To

constructmore

complex

valuations,atomic

bidscan

becom

binedinto

more

complex

bids.

•O

neapproach

isX

OR

bids

Bi=

({a,b}

,3)X

OR

({c,d},5)

•X

OR

becausew

ew

illpayfor

atmostone.

•W

eread

thebid

tom

ean:

Iwould

pay3

fora

bundlethatcontains

aand

bbutnot

cand

d.Iw

illpay5

fora

bundlethat

containsc

andd

butnota

andb,and

Iwillpay

5for

abundle

thatcontainsa,

b,c

andd.

•From

thisw

ecan

constructavaluation.

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60

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•T

hus:

vβ

1 ({a})=

0

vβ

1 ({b})=

0

vβ

1 ({a,b})

=3

vβ

1 ({c,d})

=5

vβ

1 ({a,b

,c,d})

=5

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61

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•M

oreform

ally,abid

likethis:

β=

(Z1 ,p

1 )XO

R...X

OR

(Zk ,p

k )

definesa

valuationvβ

likeso:

vβ(Z′)

=

{

0if

Z′doesn’tsatisfy

any(Z

i ,pi )

max{p

i |Zi⊆

Z′}

otherw

ise

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62

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•X

OR

bidsare

fullyexpressive,thatis

theycan

expressany

valuationfunction

overa

setofgoods.

•To

dothat,w

em

ayneed

anexponentially

largenum

berofatom

icbids.

•H

owever,the

valuationofa

bundlecan

becom

putedin

polynomialtim

e.

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63

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Winner

Determ

ination

•T

hebasic

problemis

intractable.

•B

utthisis

aw

orstcaseresult,so

itmay

bepossible

todevelop

approachesthatare

optimaland

runw

ellinm

anycases.

•C

analso

forgetoptimality

andeither:

–use

heuristics;or–

lookfor

approximation

algorithms.

•C

omm

onapproach:

codethe

problemas

aninteger

linearprogram

anduse

astandard

solver–

oftenw

orksin

practice.

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64

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

The

VC

GM

echanism

•In

generalwe

don’tknoww

hetherthe

v̂i aretrue

valuations.

•Life

would

beeasier

iftheyw

ere!

–W

ell,canw

em

akethem

truevaluations?

•Yes,in

ageneralization

oftheV

ickreyauction.

–V

ickrey/Clarke/G

rovesM

echanism

•M

echanismis

incentivecom

patible:telling

thetruth

isa

dominantstrategy.

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65

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•N

eedsom

em

orenotation.

•Indifferentvaluation

function:

v0(Z

)=

0

forallZ

.

•sw

−i is

thesocialw

elfarefunction

without

i:

sw−

i (Z1 ,...,Z

n ,v1 ,...

,vn )=

∑

j∈A

g,j6=

i vj (Zj )

•A

ndw

ecan

thendefine

theV

CG

mechanism

.

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66

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

1.Every

agentsimultaneously

declaresa

valuationv̂i .

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67

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

2.The

mechanism

computes:

Z∗1 ,...,Z

∗n=

argm

ax(Z

1,...,Z

n )∈alloc(Z

,Ag) sw

(Z1 ,...,Z

n ,v̂1 ,...,v̂

i ,...,v̂n )

andthe

allocationZ∗1 ,...,Z

∗nis

chosen.

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68

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

3.The

mechanism

alsocom

putes,foreach

agenti:

Z′1 ,...,Z

′n=

argmax

(Z1 ,...,Z

n )∈alloc(Z

,Ag) sw

(Z1 ,...,Z

n ,v̂1 ,...,v

0,...,v̂n )

theallocation

thatmaxim

isessocialw

elfarew

erethat

agenttohave

declaredv0

tobe

itsvaluation.

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69

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

4.Every

agentipays

pi ,w

here:p

=sw

−i (Z

′1 ,...,Z′n ,v̂

1 ,...,v0,...,v

n )

−sw

−i (Z

∗1 ,...,Z∗n ,v̂

1 ,...,v̂i ,...,v

n )

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70

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•O

nother

words,each

agentpaysoutthe

cost,toother

agents,ofithavingparticipated

inthe

auction.

•Itis

incentivecom

patiblefor

exactlythe

same

reasonas

theV

ickreyauction

was

before.

•Ifyou

bidm

orethan

yourvaluation

andw

in,wellyou

endup

payingback

whatthe

goodis

worth

toeveryone

else,which

ism

orethan

itisw

orthto

you.

•Ifyou

shadeyour

bid,youreduce

yourchance

tow

in,buteven

ifyouw

inyou

arestillpaying

whateveryone

elsethinks

itisw

orthso

youdon’tsave

money

byreducing

yoruchance

tow

in.http://www.csc.liv.ac.uk/˜mjw/pubs/imas/

71

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•S

ow

egeta

dominantstrategy

foreach

agentthatguarantees

tom

aximise

socialwelfare.

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72

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

eBay

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73

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•eB

ayruns

avariation

oftheE

nglishauction.

•V

ulnerableto

sniping.

•To

counterthis,eB

ayoffers

aautom

atedbidding

agent.

–R

educesthe

auctionto

aF

PS

B.

•M

anycom

paniesoffer

snipingservices.

•B

TW

,thereis

aneasy

fixto

sniping,buteBay

chosenotto

useit.

–A

ctivityrule

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74

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Adw

ordauctions

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75

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•To

decidew

hichads

getshown

inw

hichposition

forw

hichsearches,an

adword

auctionis

run.

•T

hisis

runin

realtime.

•(T

houghclearly

bidsare

placedbeforehand.)

•A

uctionis

avariation

onthe

Vickrey

auction.

•85%

ofGoogle’s

revenue($4.1

billion)in

2005cam

efrom

theseauctions.

•V

eryactive

areaofresearch.

–N

otclearw

hatthebestauction

mechanism

isfor

thisapplication.

–N

otclearw

hatthebestw

ayto

bidis.

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76

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

Sum

mary

•A

llocatingscarce

resourcescom

esdow

nto

auctions.

•W

elooked

atarange

ofdifferentsimple

auctionm

echanisms.

–E

nglishauction

–D

utchauction

–F

irstpricesealed

bid–

Vickrey

auction

•T

hew

elooked

atthepopular

fieldofcom

binatorialauctions.

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77

Chapter

14A

nIntroduction

toM

ultiagentS

ystems

2e

•W

ediscussed

some

oftheproblem

sin

implem

entingcom

binatorialauctions.

•A

ndw

etalked

abouttheV

ickrey/Clarke/G

rovesm

echanism,a

rareray

ofsunshineon

theproblem

sof

multiagentinteraction.

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78