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Ad Auctions: An Algorithmic Perspective Amin Saberi Stanford University Joint work with A. Mehta, U.Vazirani, and V. Vazirani

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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. ‘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

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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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Our Problem: N advertisers: with budget B 1,B 2, …B n Queries arrive on-line; b ij : bid of advertiser i for good j (More precisely: b ij 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 = b ij ) Objective: maximize revenue!!

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Competitive Factor 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 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: $1 $0.99 $1$0 Book CD Bidder 1Bidder 2 B 1 = B 2 = $100 Queries: 100 books then 100 CDS Bidder 1 Bidder 2 Greedy: $100

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Greedy Algorithm Greedy: Give the query to the advertiser with the highest bid. It is not the best algorithm: $1 $0.99 $1$0 Book CD Bidder 1Bidder 2 B 1 = B 2 = $100

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Greedy Algorithm Greedy: Give the query to the advertiser with the highest bid. It is not the best algorithm: $1 $0.99 $1$0 Book CD Bidder 1Bidder 2 B 1 = B 2 = $100 Queries: 100 books then 100 CDS Bidder 1 Bidder 2 Greedy: $100 OPT: $199 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 , 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 $$ $ $ $ $0 Book CD Bidder 1Bidder 2 B 1 = B 2 = $1 Special Case: All budgets are 1; bids are either $0 or $ alyansundaram, Pruhs ’96: Give the algorithm to the interested bidder with the highest remaining money Queries: 100 books then 100 CDS Bidder 1 Bidder 2 KP: $1.5 OPT: $2 Competitive factor: 1- 1/e

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

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

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Where does come from? New Proof for KP Factor Revealing LP 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 x 1, x 2, …, x k : x i 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 x i ’s Revenue:

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Analyzing KP OPT = N Revenue = 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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Second Constraint:

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First Constraint: Second Constraint: In general:

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We can solve it by finding the optimum primal and dual. Optimal solution is and achieves a factor of 1 – 1/e Competitive factor of KP Minimize s.t. Factor revealing LP JMS ’ 02, MYZ ’ 03, …

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Where does come from? New Proof for KP Factor Revealing LP 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 =

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

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Where does come from? New Proof for KP Factor Revealing LP 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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2 : Choose so that the change in the optimum is always non-negative. 1: No matter what we choose, optimal dual remains. Step 3: Modified Sensitivity Analysis Change in optimum =

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

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Open Problem 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 User 2 User n Search engine

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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 v ij for good j Bidder i wants to spend at most b i dollars The budget constraints are hard u i (S,p) = All values and budget constraints are private information, known only to the bidder herself -1 if p > b i j 2 S v ij – p if p ≤ b i

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VCG mechansim Vickrey-Clarke-Grove mechanism (replace bids with minimum bid and budget) Utility: 9 Payment: 1 Utility: 18 Payment: 2 Bidder 1: (v 11, v 12, b 1 ) = (10, 10, 10) Bidder 2: (v 21, v 22, b 2 ) = (1, 1, 10) “Welfare”: 10 “Welfare”: 1 Total “Welfare”: 11 Payment: 0 LIE: (5,5,10) 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 – 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.

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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 {p i,j } such that bidder i receives item j at price p i,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 A negative result:

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Open Problem 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 User 2 User n Search engine

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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 123N-1N 1-1/e 1/N 1/(N-1) 1/(N-2)

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Analyzing KP REVENUE = (1-1/e) N 1-1/e 123N-1N

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

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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

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