an expressive auction design for online display advertising€¦ · online display advertising...

An Expressive Auction Design for Online Display Advertising

AUTHORS: Sébastien Lahaie, David C. Parkes, David M. Pennock

Li PU & Tong ZHANG

Motivation• Online advertisement – allow advertisers to specify what

kinds of users they are targeting: geographical, user profiles, behavioral

• Targeting slots → targeting users

2

Motivation

• Bidding language:

• “Give me a number”• “Please tell me your values of all outcomes”

• Expressive about the preferences of advertisers, and succinct as well

3

Motivation• Allocation and Pricing algorithms:

• Computing efficient allocation in combinatorial auctions can be NP‐hard

• Good design of bidding language leads to tractable allocation and pricing algorithms

4

• Incentive‐compatibility:

• Minimize the incentives of bidders to manipulate

Goals

• Efficient and scalable allocation algorithm

• Unique market‐clearing prices that minimize the bidders’ incentives to game

• Efficient, combinatorial algorithm for computing market‐clearing prices

• Market‐clearing: be cleared of all surplus and shortage (supply = demand)

5

Overview• Advertisers are charged by impressions• At the beginning of a period, all advertisers report their bids for different types of impressions

Advertisers

Supply EstimatorScheduler

AllocationAlgorithm

PricingAlgorithm

6

prices

Supply estimation

Definition of Impression

A = fA1; :::; Asg

A1 = fCA; MA; NY; :::; undefinedgAs = f0 : 00 » 1 : 00; :::; undefinedg

ha1; :::; asi 2 M

hCA; :::; 7 : 00 » 8 : 00i

m = jM j =sY

t=1

jAtj

² attributes

² values

² impressions

² number of all impressions

7

Impressions and Prices

xi = [xi(1) xi(2) ::: xi(m)] 2 ZM+

xi(j) 2 f0; 1; 2; :::g number of times an impression was displayed

p ¢ xi =Xj2M

p(j)xi(j)

² bundle of impressions

² total price of bidder i :

² price for each impression p = [p(1) p(2) ::: p(m)]

² utility of bidder i : ui(xi) = vi(xi)¡ p ¢ xi

8

Allocation of Impressions

² supply of impressions z = [z(1) z(2) ::: z(m)]

² feasible: allocated · supplynX

i=1

xi(j) · z(j) ;8j 2 M

² feasible allocation x = [x1 x2 ::: xn] 2 ¡

² e±cient : x 2 arg maxy2¡

nXi=1

vi(yi)

² why e±cient? long term goals9

Market‐clearing Prices

² market-clearing prices are also called

Di(p) = arg maxxi

vi(xi)¡ p ¢ xi

competitive equilibrium (CE) prices

² demand set:

CE = hx; pi a) xi 2 Di(p) ;8i 2 N

b) p(j) = 0 ifXi2N

xi(j) < z(j) ;8j 2 M

² if p are CE prices

x is e±cient , hx; pi is a CE

10

Bid Tree

$.0

$.0 $.2

$.5

$‐.1 $.1

CA FL

{auto, sports}

news blog

$.0

$.0 $.2

$.5

$‐.1 $.1

CA FL

{auto, sports}

news blog

value = $.0+$.0+$.5+$.1 = $.6

fashion

$.0

$.0 $.2

$.5

$‐.1 $.1

CA FL

{auto, sports}

news blog

$.0

50

0

capacity

capacity

each node of the tree is a subset Tk μM , Ti = fTkg11

Properties of Bid Tree² Each node of the tree is a subset Tk μM , Ti = fTkg

² Ti is laminar : 8 T; T 0 2 Ti, either T μ T 0, T 0 μ T ,or T \ T 0 = ;

fashion

$.0

$.0 $.2

$.5

$‐.1 $.1

CA FL

{auto, sports}

news blog

$.0

50

0 ciT

biT 0

viT (r) =

(biTr ; if 0 · r · ciT

¡1 ; otherwise

vi(xi) =XT2Ti

viT

ÃXj2T

xi(j)

!

hCA; sports; blog; 7 : 00 » 8 : 00i

hCA; autoihCA; sports; 7 : 00 » 8 : 00i

12

Example of Bidder Value

fashion

$.0

$.0 $.2

$.5

$‐.1 $.1

CA FL

{auto, sports}

news blog

$.0

50

0

hCA; sports; news; nighti : 1000

hCA; auto; nighti : 500

hFL; auto; newsi : 40

500£ $:5 = 250

1000£ $:5 + 1000£ $¡ :1 = 400

40£ $:2 = 8

Total value of bidder i : vi(xi) = 250 + 400 + 8 = 658

Bidding tree Ti of bidder i

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Queries: Value and Demand• Value: input – allocation and bidding tree

output – value of the bidder

• Demand: input – prices of each type of impression

and bidding tree

output – allocation to maximize utility

xi, Ti

vi(xi; Ti)

xi

p, Ti

Demand Query Algorithm:

1. Compute marginal surplus

2. Pick out the impressions that have positive surplus (profit) and sort them in descending order

3. Assign a number as large as possible to the current most profitable impression, update the bidding tree and repeat

¼i(j) =P

fT2Tijj2Tg biT ¡ p(j)

14

Example of Demand Query

M =

26666664hCA; sportsihCA; fashionihCA; autoihFL; sportsihFL; fashioni

37777775 p =

26666664$:1

$:05

$:05

$:1

$:05

37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

50

0100

200

¼i =

26666664$:1 + $:4¡ $:1

$:1¡ $:05

$:1¡ $:05

$:2¡ $:1

$:2¡ $:05

37777775 =

26666664$:4

$:05

$:05

$:1

$:15

37777775

¼i(j) =P

fT2Tijj2Tg biT ¡ p(j)

! (sort) ! M =

26666664hCA; sportsihFL; fashionihFL; sportsihCA; fashionihCA; autoi

3777777515


M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

50

0100

200 xi =

266666640

0

0

0

0

37777775

M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

50

00

100 xi =

26666664100

0

0

0

0

3777777516


M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

50

00

100 xi =

26666664100

0

0

0

0

37777775

M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

0

00

100 xi =

26666664100

50

0

0

0

3777777517


M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

0

00

100 xi =

26666664100

50

0

0

0

37777775

M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

0

00

100 xi =

26666664100

50

0

0

0

3777777518


M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

0

00

100 xi =

26666664100

50

0

0

0

37777775

M =


37777775fashion

$.0

$.1 $.2

$.4

CA FL

sports

$.0

0

00

0 xi =

26666664100

50

0

0

100

3777777519

Allocation² Take into account forecasted supply z(j). The allocationproblem is a linear program:

maxx

Xi2N

XT2Ti

Xj2T

biTxi(j)

s.t.

8>>>>><>>>>>:

Xj2T

xi(j) · ciT (T 2 Ti; i 2 N)Xi2N

xi(j) · z(j) (j 2 M)

xi(j) ¸ 0 (i 2 N; j 2 M)

² Proposition 1 : When agents submit their valuations asbid trees, the corresponding allocation LP has an integeroptimal solution.

20

Allocation

² Use subgradient method on the dual of the allocation LP:

pk+1 = pk + ¯kgk

gk = yk ¡P

i xki , where yk is a revenue-maximizing

allocation at price pk.

min¼;p

Xi2N

XT2Ti

¼iT ciT +Xj2M

p(j)z(j)

s.t.

8>>><>>>:X

fT2Tijj2Tg

¼iT ¸X

fT2Tijj2Tg

biT ¡ p(j) (i 2 N; j 2 M)

¼iT ¸ 0 (i 2 N;T 2 Ti)

p(j) ¸ 0 (j 2 M)

² Dual of the allocation LP:

21

Pricing and Incentives² Proposition 2 : If x is an optimal primal solution to theallocation LP and p is an optimal dual solution, then hx; piis a competitive equilibrium.

² Proposition 3 : The set of competitive equilibrium pricesis a lattice when valuations are described by bid trees.

² unique minimal element p ! most possible surplus² unique maximal element ¹p ! most possible revenue

² VCG payment is not feasible because of linear pricing

² Choose p to minimize the incentives of misreporting(minimize the improvement of payo® when misreporting)

22

Proposition 1 & 2

23

LP solution <x,p> is a CE

A fractional CE exists

M‐concave valuations: impressions are “substitutes”

A fractional CE → an integer CE

Write down the dual LP

By complementary slackness conditions, p(j)>0 implies all supply are allocated → revenue maximized

xi(j)>0 and πiT(j)>0 implies x maximize bidder i’s utility at price p

Proposition 1 Proposition 2

Bidder Feedback

² Proposition 4 : Suppose bidder i increases biT to biT + ¸in its bid tree, and leaves the bids in the other nodes un-changed. Then there exists an e±cient allocation, withrespect to the new pro¯le of bid trees, in which i receivesat least diT units of impressions from T μM .

² If advertiser i wants to get at least diT impressions fromsites in T μM , he has to ensure that ciT ¸ diT andincrease the bid biT to get the desired volume.

² Add a new constraint :P

j2T xi(j) ¸ diT

Let ¸ be the optimal value of the dual variablecorresponding to this constraint.

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Conclusion• A bidding language (bid tree) that allows advertisers to specify values for different types of impressions, and ignore the irrelevant attributes

• Scalable allocation and pricing algorithms

• Bidder feedback to help advertisers assess the value of some nodes to get desired number of impressions

• Since the number of impressions may be very large, what attributes the seller should choose?

• Since the mechanism is not incentive‐compatible, what is the dynamics of a series of bidding?

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