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

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)

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

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

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--)

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!!

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)

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

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 1 Bidder 2

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

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 1 Bidder 2 Greedy is ½-competitive!

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

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 1 Bidder 2 Competitive factor: 1- 1/e

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

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

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

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:

Analyzing KP OPT = N Revenue = Painted Area

Analyzing KP Optimum Allocation

Analyzing KP Optimum Allocation Where did KP place these queries?

Analyzing KP Optimum Allocation Where did KP place these queries?

First Constraint:

First Constraint:

First Constraint:

First Constraint: Second Constraint:

First Constraint: Second Constraint: In general:

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

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

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

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

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

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

Step 3: Sensitivity Analysis

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.

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

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?

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

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

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

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--)

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 > bi

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!

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?

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.

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

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

THE END

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

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

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

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

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

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