Ad Auctions: An Algorithmic Perspective

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Ad Auctions An Algorithmic Perspective

• Amin Saberi
• Stanford University
• Joint work with A. Mehta, U.Vazirani, and V.
  Vazirani

2
Outline

• Ad Auctions a quick introduction
• Search engines allocation problem
• Which advertisers to choose for each keyword?
• Our algorithm achieving optimal competitive
  ratio of 1 1/e(Mehta, S. Vazirani, Vazirani
  05)
• Incentive compatibility
• Designing auctions for budget constraint
  bidders(Borgs, Chayes, Immorlica, Mahdian, S.
  05)
• Auctions with unknown supply(Mahdian, S. 06)

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Keyword-based Ad

• Advertiser specifies
• bid (Cost Per Click) for each keyword(search
  engine computes the Click-Through Rate,expected
  value CPC CTR)
• total budget
• Search query arrives
• Search engine picks some of the Ads and shows
  them.
• charges the advertiser if user clicked on their
  Ad

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Online Ads

• Revolution in advertising
• Major players are Google, MSN, and Yahoo
• Enormous size, growing
• Helping many businesses/user experience
• An auction with very interesting characteristics
• The total supply of goods is unknown
• The goods arrive at unpredictable rate and should
  be allocated immediately
• Bidders are interested in a variety of goods
• Bidders are budget constrained

5
Outline

• Ad Auctions a quick introduction
• Search engines allocation problem
• Which advertisers to choose for each keyword?
• Our algorithm achieving optimal competitive
  ratio of 1 1/e(Mehta, S. Vazirani, Vazirani
  05)
• Incentive compatibility
• Designing auctions for budget constraint
  bidders(Borgs, Chayes, Immorlica, Mahdian, S.
  05)
• Auctions with unknown supply(Mahdian, S. --work
  in progress--)

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Our Problem

• N advertisers with budget B1,B2, Bn
• Queries arrive on-line
• bij bid of advertiser i for good j
• (More precisely bij is the expected revenue
  of giving the ad space for query j to advertiser
  iafter normalizing the CPC by click through rate
  etc.. )
• Allocate the query to one of the advertisers (
  revenue bij )
• Objective maximize revenue!!

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Competitive Factor

• a-competitive algorithm
• the ratio of the revenue of algorithm over
• the revenue of the best off-line algorithm
• over all sequences of input is at least a .
• Greedy ½-competitive
• Our algorithm 1 1/e competitive (optimal)

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Greedy Algorithm

• Greedy Give the query to the advertiser with the
  highest bid.

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Greedy Algorithm

• Greedy Give the query to the advertiser with the
  highest bid.
• It is not the best algorithm

Bidder 1
Bidder 2
Queries 100 books then 100 CDS
1 0.99
1 0
Book
CD
Greedy 100
Bidder 1 Bidder 2
B1 B2 100
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Greedy Algorithm

• Greedy Give the query to the advertiser with the
  highest bid.
• It is not the best algorithm

Bidder 1
Bidder 2
1 0.99
1 0
Book
CD
B1 B2 100
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Greedy Algorithm

• Greedy Give the query to the advertiser with the
  highest bid.
• It is not the best algorithm

Bidder 1
Bidder 2
Queries 100 books then 100 CDS
1 0.99
1 0
Book
CD
Greedy 100 OPT 199
Bidder 1 Bidder 2
B1 B2 100
Greedy is ½-competitive!
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History

• Known results(1 1/e) competitive algorithms
  for special cases
• Bids 0 or 1, budgets 1 (online bipartite
  matching)Karp, Vazirani, Vazirani 90
• bids 0 or e, budgets 1 (online
  b-matching)Kalyansundaram, Pruhs 96, 00
• Our result
• Arbitrary bids
• Mild assumption bid/budget is small.
• New technique Trade-off revealing LP

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KP Algorithm

• Special Case All budgets are 1 bids are either
  0 or e d
• Kalyansundaram, Pruhs 96 Give the algorithm to
  the interested bidder with the highest remaining
  money

Bidder 1
Bidder 2
Queries 100 books then 100 CDS
e e
e 0
Book
CD
KP 1.5 OPT 2
B1 B2 1
Bidder 1 Bidder 2
Competitive factor 1- 1/e
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Our Algorithm
Give query to bidder with max bid
(fraction of budget spent)
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Where does y come from?
Factor Revealing LP
New Proof for KP
Modify the LP for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP
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Where does y come from?
Factor Revealing LP
New Proof for KP
Modify the LP for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP
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Step 1 Analyzing KP

• For a large k, define x1, x2, , xk
• xi is the number of bidders who spent i/k of
  theirmoney at the end of the algorithm
• W.l.o.g. assume that OPT can exhaust everybodys
  budget.
• We will bound xis

Revenue
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Analyzing KP
OPT NRevenue Painted Area

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Analyzing KP

Optimum Allocation
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Analyzing KP

Optimum Allocation Where did KP place these
queries?
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Analyzing KP

Optimum Allocation Where did KP place these
queries?
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First Constraint

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First Constraint

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First Constraint

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First Constraint

Second Constraint
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First Constraint
Second Constraint
In general
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Competitive factor of KP
Minimize s.t.
Factor revealing LPJMS 02, MYZ 03,
We can solve it by finding the optimum primal and
dual. Optimal solution is and achieves a factor
of 1 1/e
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Where does y come from?
Factor Revealing LP
New Proof for KP
Modify the LP for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP
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Recall Our Algorithm

• The bids are arbitrary
• Algorithm
• Award the next query to the advertiser with
  max

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Step 2 General Case

• Can we mimic the proof of KP?

Bid
Bid
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Step 2 General Case

• On a closer inspection
• Considering all the queries

Bid
i
1
Bid
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Where does y come from?
Factor Revealing LP
New Proof for KP
Modify the LP for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP
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Step 3 Sensitivity Analysis
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Step 3 Modified Sensitivity Analysis
1 No matter what we choose, optimal dual
remains .
? Change in optimum
2 Choose so that the
change in the optimum is always
non-negative.
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End of Analysis

• Theorem There is a way to choose so that the
  objective function does not decrease.
  
• Corollary competitive factor remains 1 1/e.
• Remark We can show that our competitive factor
  is optimum

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More Realistic Assumptions

• Normalizing by click-through rate
• Charging the advertiser the next highest bid
  instead of the current bid
• Assigning a query to more than one advertiser
• When you have some statistical information about
  the queries?
• When the budget/bid ratio is small?

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Incentive Compatibility

• The bidders will find creative ways to improve
  their revenue
• Bid jamming
• Fraudulent clicks
• Aiming lower positions for an ad
• Incentive compatible mechanisms Provide
  incentives for advertisers to be truthful about
  their bids (and possibly budgets?)
• Some of the difficulties in designing truthful
  auctions
• Online nature of auction search queries arrive
  at unpredictable rates and they should be
  allocated immediately.
• Bidders are budget constrained

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A Few Abstractions

• Designing Auctions for budget constrained bidders
  (Borgs, Chayes, Immorlica, Mahdian, S. 05)
• Even in the off-line case, standard auctions
  (e.g. VCG) are not truthful.
• Designing truthful auctions is impossible if you
  want to allocate all the goods
• Optimum auction otherwise
• Auctions for goods with unknown supply(Mahdian,
  S. 06)
• Nash equilibria of Googles payment mechanism
• Aggarwal, Goel, Motwani 05
• Edelman, Ostrovski, Schwarz 05

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Open Problem

• The users perspective what are the right
  keywords/bids?
• The important factor for the customers is CPA
• What is the best bidding language?

User 1
Search engine
User 2
User n
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Outline

• Ad Auctions a quick introduction
• Search engines allocation problem
• Which advertisers to choose for each keyword?
• Our algorithm achieving optimal competitive
  ratio of 1 1/e(Mehta, S. Vazirani, Vazirani
  05)
• Incentive compatibility
• Designing auctions for budget constraint
  bidders(Borgs, Chayes, Immorlica, Mahdian, S.
  05)
• Auctions with unknown supply(Mahdian, S. --work
  in progress--)

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Auctions for budget constrained bidders

• Each bidder i has a value function and a budget
  constraint
• Bidder i has value vij for good j
• Bidder i wants to spend at most bi dollars
• The budget constraints are hard
• ui(S,p)
• All values and budget constraints are private
  information, known only to the bidder herself

?j 2 S vij p if p bi
-1 if p gt bi
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VCG mechansim

• Vickrey-Clarke-Grove mechanism
• (replace bids with minimum bid and budget)

Payment 2
Bidder 1 (v11, v12, b1) (10, 10, 10)
Welfare 10
Utility 18
Payment 1
Utility 9
LIE (5,5,10)
Bidder 2 (v21, v22, b2) (1, 1, 10)
Welfare 1
Payment 0
Total Welfare 11
VCG is not truthful, even if budgets are public
knowledge!
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Is there any truthful mechanism?

• Yes. Bundle all the items together and sell it as
  one
• item using VCG.
• Is there any non-trivial truthful mechanism?

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Required properties

• Observe supply limits Auction never
  over-allocates.
• Incentive compatibility Bidders total utility
  is maximized by announcing her true utility and
  budget regardless of the strategies of other
  agents.
• Individual rationality Bidders utility from
  participating is non-negative if she announces
  the truth.
• Consumer sovereignty A bidder can bid high
  enough to guarantee that she receives all the
  copies.
• Independence of irrelevant alternatives (IIA)
  If a bidder does not receive any copies, then
  when she drops her bid, the allocation does not
  change.
• Strong non-bundling For any set of bids from
  other bidders, bidder i can submit a bid such
  that it receives a bundle different than empty or
  all the items.

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A negative result

• Theorem There is no deterministic truthful
  auction
• even for allocating 2 items to 2 bidders that
  satisfies
• consumer sovereignty, IIA, and strong
  non-bundling.
• Proof idea Truthful auctions can be written as a
  set of threshold functions pi,j such that
  bidder i receives item j at price pi,j(v-i,b-i)
  if her bid is higher than thatrvalue
• Our assumptions impose functional relations on
  these thresholds. Then we can show that this set
  of relations has no solution

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Open Problem

• The users perspective what are the right
  keywords/bids?
• The important factor for the customers is CPA
• What is the best bidding language?

User 1
Search engine
User 2
User n
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THE END
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Applications in other areas?

• Circuit switching
• Tradeoff revealing LP for other on-line and
  approximation algorithms

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Keyword-based Ad

• Interesting characteristics of these auctions
• Online nature size and speed
• Search queries arrive at an unpredictable rate
• Ads should be allocated immediately (goods are
  perishable)
• Bidders are budget constrained

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Analyzing KP
1-1/e
1/(N-2)
1/(N-1)
1/N
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Analyzing KP
1-1/e
REVENUE (1-1/e) N
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Special case On-line Matching
girls
boys

• All budgets 1
• Bids are either 0 or 1
• KVV competitive factor of 1-1/e

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Different bids and budgets?

• Not so good ideas
• Highest bid then the highest budget
• Bucket the close bids togetherbreak the ties
  based on the budgetsin every bucket
• We need to find a delicate trade-off between bid
  and budget

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