1. 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)

3. 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

4. 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--)

6. Objective: maximize revenue!!
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 i after normalizing the CPC by click through rate etc.. )
Allocate the query to one of the advertisers ( revenue = bij )
Objective: maximize revenue!!

7. 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)

8. Greedy Algorithm
Greedy: Give the query to the advertiser with the highest bid.

9. 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 $0
Book CD
Greedy: $100
B1 = B2 = $100
Bidder Bidder 2

10. 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 $0
Book CD
B1 = B2 = $100

11. Greedy is ½-competitive!
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 $0
Book CD
Greedy: $100
OPT: $199
B1 = B2 = $100
Bidder Bidder 2
Greedy is ½-competitive!

12. 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

13. Competitive factor: 1- 1/e
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 $0
Book CD
KP: $1.5
OPT: $2
B1 = B2 = $1
Bidder Bidder 2
Competitive factor: 1- 1/e

14. Our Algorithm Give query to bidder with max
bid (fraction of budget spent)

15. Use dual to get tradeoff function
Where does y come from?
New Proof for KP
Factor Revealing LP
Modify the LP
for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP

16. Use dual to get tradeoff function
Where does y come from?
New Proof for KP
Factor Revealing LP
Modify the LP
for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP

17. Step 1: Analyzing KP For a large k, define x1, x2, …, xk :
xi is the number of bidders who spent i/k of their money at the end of the algorithm
W.l.o.g. assume that OPT can exhaust everybody’s budget.
We will bound xi’s
Revenue:

18. Analyzing KP
OPT = N Revenue = Painted Area

19. Analyzing KP
Optimum Allocation

20. Analyzing KP
Optimum Allocation
Where did KP place these queries?

21. Analyzing KP
Optimum Allocation
Where did KP place these queries?

22. First Constraint:

23. First Constraint:

24. First Constraint:

25. First Constraint:
Second Constraint:

26. First Constraint:
Second Constraint:
In general:

27. Competitive factor of KP
Minimize
s.t.
Factor revealing LP JMS ’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

28. Use dual to get tradeoff function
Where does y come from?
New Proof for KP
Factor Revealing LP
Modify the LP
for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP

29. Recall: Our Algorithm The bids are arbitrary Algorithm:
Award the next query to the advertiser with max

30. Step 2: General Case
Can we mimic the proof of KP?
Bid =
Bid =

31. Step 2: General Case On a closer inspection
Considering all the queries:
Bid = i 1
Bid =

32. Use dual to get tradeoff function
Where does y come from?
New Proof for KP
Factor Revealing LP
Modify the LP
for arbitrary bids
Use dual to get tradeoff function
Tradeoff Revealing LP

33. Step 3: Sensitivity Analysis

34. 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.

35. 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

36. 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?

37. 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

38. 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 Google’s payment mechanism
Aggarwal, Goel, Motwani ’05
Edelman, Ostrovski, Schwarz ’05

39. Open Problem Search engine
The user’s 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

40. 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--)

41. 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
if p > bi

42. even if budgets are public knowledge!
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
LIE: (5,5,10)
Utility: 9
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!

43. 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?

44. Required properties
Observe supply limits – Auction never over-allocates.
Incentive compatibility – Bidder’s total utility is maximized by announcing her true utility and budget regardless of the strategies of other agents.
Individual rationality – Bidder’s 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.

45. 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

46. Open Problem Search engine
The user’s 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

47. THE END

48. Applications in other areas?
Circuit switching
Tradeoff revealing LP for other on-line and approximation algorithms

49. 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

50. Analyzing KP
1-1/e 1 2 3
N-1 N
1/(N-2)
1/(N-1)
1/N

51. Analyzing KP
1-1/e 1 2 3
N-1 N
REVENUE
= (1-1/e) N

52. Special case: On-line Matching
girls
boys
All budgets = 1
Bids are either 0 or 1
KVV: competitive factor of 1-1/e

53. Different bids and budgets?
Not so good ideas…
Highest bid then the highest budget
Bucket the close bids together break the ties based on the budgets in every bucket
We need to find a delicate trade-off between bid and budget