ad auctions: an algorithmic perspective

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)


(Borgs, Chayes, Immorlica, Mahdian, S. ‘05)

Auctions with unknown supply(Mahdian, S. --work in progress--)

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!!

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)

Greedy Algorithm

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

Greedy Algorithm


It is not the best algorithm:

$1 $0.99

$1 $0

Book

CD

Bidder 1 Bidder 2

B1 = B2 = $100

Queries: 100 books then 100 CDS

Bidder 1 Bidder 2

Greedy: $100

Greedy Algorithm



$1 $0.99

$1 $0

Book

CD

Bidder 1 Bidder 2

B1 = B2 = $100

Greedy Algorithm



$1 $0.99

$1 $0

Book

CD

Bidder 1 Bidder 2

B1 = B2 = $100


Bidder 1 Bidder 2

Greedy: $100OPT: $199

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 , 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

KP Algorithm

$ $

$ $0

Book

CD

Bidder 1 Bidder 2

B1 = B2 = $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


Bidder 1 Bidder 2

KP: $1.5OPT: $2

Competitive factor: 1- 1/e

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

Our Algorithm

Where does come from?

New Proof forKP

Factor Revealing LP

Modify the LPfor 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 theirmoney 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 = NRevenue = Painted Area

Analyzing KP

Optimum Allocation

Analyzing KP

Optimum AllocationWhere did KP place these queries?

First Constraint:

First Constraint:

Second Constraint:

First Constraint:

Second Constraint:

In general:

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 LPJMS ’02, MYZ ’03, …


New Proof forKP

Factor 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 =

1

i

Bid =


New Proof forKP

Factor Revealing LP




Step 3: Sensitivity Analysis

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 =

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

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

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)


(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

-1 if p > bi

j 2 S vij – p if p ≤ bi

VCG mechansim

Vickrey-Clarke-Grove mechanism

(replace bids with minimum bid and budget)

Utility: 9

Payment: 1 Utility: 18

Payment: 2Bidder 1:

(v11, v12, b1) = (10, 10, 10)

Bidder 2: (v21, v22, b2) = (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!

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.

Theorem: There is no deterministic truthful auctioneven for allocating 2 items to 2 bidders that satisfiesconsumer 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

A negative result:

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

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

123N-1N

1-1/e

1/N

1/(N-1)

1/(N-2)

Analyzing KP

REVENUE= (1-1/e) N

1-1/e

123N-1N

Special case: On-line Matching

All budgets = 1 Bids are either 0 or 1

KVV: competitive factor of 1-1/e

girls boys

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 budgetsin every bucket

We need to find a delicate trade-off between bid and budget

ad auctions: an algorithmic perspective

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