Pl Plan fo for T Today Fair division basics Intro to auctions to - - PowerPoint PPT Presentation

SLIDE 1

Pl Plan fo for T Today

— Fair division basics — Intro to auctions

SLIDE 2

SLIDE 3

2people

who

want to split

a heterogeneous

divisible good

Player 1

is

told to divide

the

cake into

2 pieces

she values

equally

Player 2

picks

their favorite

Motel

good is

G D

V

s is the

value

that player i

has

fur

piece

• fcake

S

Vico D

I

value is

additive

• ndisjoint intervals

1

1

values

are divisible

CE Loi

and

X

there

is

a

YinX

set

via

evi

X

SLIDE 4

Arethe

players

incentivized

to follow

rules

• f

Iat

Uchoose

protocol

An

allocation

A

Ai Aa

An

Ai

piece hatplayeri gets is

Pareto optimal

YF

no

alternative allocam

B

Bi ibn

sit

everyaggert is

atleast

as

happy

some

agent

is strictly happier

Int

ya

choose

not Paeto

• ptimal

give the

whole

cake

1

is

PO

How to

define fairness bothplayers

are equally happy

Icut

y

choose doesn't satisfy

this

allocation

is proportional

y

v

Ai

th

ti

Aryan

envy free

if

fi j

v

Ai

rifAj

EF

proportional

viCAj

s

v CA

V j

h

l

vifAj

envifai

v

Ai

th

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I

proportional but

10

23

not

EF

EET

SLIDE 5

Proportioned

Allocah

Claim

Procedure is proportional

Proof

whenever there are

K players left

remaing

Cake is

worth

at least

In

to each ofthem

Base case

k

n

Inductive step

k left

R

IH

holds

SLIDE 6

Ej

msn.im

sL

Eval fxy3

returns

Vi

x y

Cut

x

c

returns

y set

Vila yD

e n

queries for

Moving knife 7 simple Doe

conquer

Cfnbgn

queries

Wgn queries

are

necessary

SLIDE 7

SLIDE 8

SLIDE 9

4

1.5

4

OH

SLIDE 10

SLIDE 11

1960

2h

3 1995

CB TJ

2016

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h

4

203 cuts all

n

n n

n

n

n

ch B

n

et na

SLIDE 12

Auctions

Singh item

ancan

n

bidders

Bidders

private

Fh

bidder has

a value

vi b

rules specify

who wins

IEQ

D

m

gangampahpa

First price

auction

highest

bidder

wins

pay their

bid

2nd price

anchor

highestbidder wins

pay

2nd

highest bid

All pay

anch

highest bidder

wins

every

pays whattheybid

bidder utility

Vi

pi

ythyoewqtp.fm

p

iting poggi

80

60

t

I

55

2

SLIDE 13

Thin

In

a 2nd price archer

its

a dominant

strategy

to

bed truthfully

PI

Fx

bids g

everyone

else

B maxghese

Max

Vi

B

bi

Vi

d Intfully

individually rational

bid

who values

it

most

welfare maximizing

no

• ther

allocatn

can

have higher total

utility tothe bidders

auctioneer

IIi

a

ts

Ts

another

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i

each

bidder only wants

• ne

Design

auction

that

is tmtnful

IRT

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1

wegaemaximr.mg

g

change everyone

second highest

bid X

not IR

ry

Change

the

itn highest bidder

the

Dst

bid

SLIDE 14

SO

8

5

3

T

Charge

the winners

the

tDsthghest

bid

1st

priori

for every player

pi

mapping from rakes

to bids for player i

assure

that

each

player's value

Vi

N

F

and those probability

dishes

are public

2 players

Vi

Va

Uniffe too

yonepheynlij

p Va

Va

How should

you

bid

when

you

value

is

V

PG

b

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b

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VI

v

b

PrfVzL2b

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food

SLIDE 15

max

v

b b

b

Bayes Nash Equilibrium

BME

pi Pa

pre

is

a

BNR

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v

pi

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revenue

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too

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auctioneer revenue

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mmµby

T

SLIDE 16

Thin

Revenueequivalency

Set

• f

buyers

values

all drawn i

idf

Than

any

auction

mechanism

• n which

the

• bject goes to

the bidder with

the highest

Value

and the

bidder

win

min

possible

value has

ably

has

the

sane

expected

revenue

exp

papjgtmofbfdndnaw.ly Effvanchon

2players

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arch

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