online algorithms for market clearing

Online Algorithms for Market Clearing

Martin ZinkevichJoint Work with Avrim Blum and Tuomas Sandholm(SODA 2002)

2

Introduction

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E-Bay: Single Auction

Buy Bids: people wanting to buySell Bids: people wanting to sell

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5



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Improvement #1: Double Auction


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Graphical Version


Price Profit

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Improvement #2:Bidding For Different Intervals

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Temporal Bidding Problem


Price

Time

Introduction Time Expiration Time

Jane

Mark Toy’s R Us

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How to Make Money

Buy the PS2 from one person for some money, and sell it to someone else for more than you bought it.Keep no inventory, have sellers mail PS2 directly to buyers.

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Types of Algorithms

One can imagine two types of algorithms: Offline: An algorithm which knows the

whole sequence before making decisions.

Online: An algorithm which finds the type, price, and expiration time of a bid as it is introduced, and must act on a bid before it expires.

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Formalizing the Goal

Find a matching of concurrent buy and sell bids that maximizes profit

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The Maximal Matching


Price

Time

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The Offline Problem


Price

Time

15

Convert to a Graph


Price

Time

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Convert to a Graph


Price

Time

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Convert to a Weighted Graph


Price

Time

5

15

25

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Convert to a Weighted Graph

5

15

25

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Find a Matching of Maximum Weight

5

15

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A matching for a graph is a set of edges which do not share anyvertices

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A matching for a graph is a set of edges which do not share anyvertices

5

25

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5

15

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Algorithms existto solve this in polynomial time

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Price

Time

Matching ofmaximal weightmaximizes profit!

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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What Online Sees…


Price

Time

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Two Online Difficulties


Price

Time

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Getting the Most Matches


Price

Time

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Getting the Most Per Match


Price

Time

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Two Small Problems

Maximizing Liquidity: maximizing the number of matchesThe Ice Cream Problem: maximizing profit of one trade

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Outline

IntroductionSmall Problem #1:Maximizing LiquiditySmall Problem #2:The Ice Cream ProblemPutting it TogetherConclusionNew Results

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Maximizing Liquidity

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Simplified Matching

Let’s try to maximize the number of trades with positive profit on each trade.

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A Maximal Numeric Matching


Price

Time

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Also Maximal Numerically


Price

Time

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Offline: Convert to a Graph


Price

Time

No weightsneeded

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Convert to a Graph


Price

Time

No edgehere

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Convert to a Graph

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Maximal Matching

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Maximal Matching

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Back To The Problem


Price

Time

A maximal matching on the graph maximizes liquidity

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Convert to a Graph Online…


Price

Time

AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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Price

Time


AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Online Graph…

AliveExpired

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The Greedy Algorithm

If you see an edge you can use, use it.

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The Greedy Algorithm…

AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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AliveExpiredMatched

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A General Solution

Thinking aboutarbitrary graphs

Optimal:6 edges,12 vertices

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What If…



You:At least 6 vertices

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What If…



You:At least 6 verticesImplies at least 6/2=3 edges

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Proof(Everyone Remain Calm)

Of each edge in the optimal matching, we get at least one endpoint in our matching.Call the first vertex to be introduced for the edge v1 and the second vertex v2.

v2

v1

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v2

When v2 Arrives…

Either v1 is unmatched…

v1


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When v2 Arrives…

Either v1 is unmatched…

v1

v2


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When v2 Arrives…

Or v1 is matched…

v1

v2


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Proof Summary

We get at least one endpoint from each edge in the optimal matching.This guarantees us at least half of the edges in the optimal matching.

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Back To Bids


Price

Time

If we know whenthings expire,it helps to waitsometimes

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A Problem…


Price

Time

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A Problem…


Price

Time

114

A Problem…


Price

Time

Don’t know untilits too late!

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A Solution…Subsidy


Price

Time

But at all times,have positivefunds

Negative!

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An Observation About Profit

$5

$1 $4

$8

$6

$8

+4 +4 +2 +4+4+2=10

Let’s forget time

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$5

$1 $4

$8

$6

$2

+4 +4 -4 +4+4-4=4

Let’s forget time

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$5

$1 $4

$8

$6

$2 5+8+2=15

1+4+6=11

+4

Let’s forget time

Insert FavoriteMatching Here

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Another Matching

$5

$1 $4

$8

$6

$2

Let’s forget time

+1

+1

+2

+1+1+2=4

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How To Track

If there exists any matching(each pair positive profit) on the bids we have seen so far, then the total profit is positive.From now on, we will commit to bids instead of trades, and select vertices instead of edges.

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A Matchable Set of Vertices

The red verticesare matchable

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Not A Matchable Set

Must use the wholeset!

The red verticesare NOT matchable

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Not A Matchable SetCan’t useverticesnot in the set

The red verticesare NOT matchable

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A Solution…Subsidy


Price

Time

No edge here

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An Incomplete Interval Graph

Time

Maintainexpirationtimes

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Defining A New Algorithm

Time


Price

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Time


Price

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Time

Currentbid

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Time

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Time

Expiresfirst

Expiressecond

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A New Algorithm

TimeAliveExpiredMatched

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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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A New Algorithm


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Outline

IntroductionSmall Problem #1: Maximizing LiquiditySmall Problem #2: The Ice Cream ProblemPutting it TogetherConclusionNew Results

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The Ice Cream Problem

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Heh Heh Heh…I’ll

wash your car!

AND I’ll do your

homework!

AND I’ll clean your

room!

AND I’ll give you my life

savings!

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Whoops!

At some point in the future, Mom is going to walk in the room, and get more ice cream for little brother from the freezer.

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Oh Well

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OK, Let’s Formalize

Your little brother offers you increasing dollar amounts for your ice cream, stopping at some price not exceeding $9(his life savings).If you do not accept any bid, you eat the ice cream, which is worth $1.

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Relation to the Problem


Price

Time

pmax

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Offline/Online

Offline knows all the bids your brother will offer in the future.Online does not know what bids your brother will offer in the future, and must act on a bid as it is offered(before the next bid).

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Solution Concept

Try to do at least as well as some factor of the optimal offline algorithm, called the competitive ratio.Max(Offline/Online) over all scenariosBigger is BAD!

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Comparing Strategies

Offline Strategy: Take the highest offerOnline Strategy: Take $3 if offered

H. Offer

1 2 3 4 5 6 7 8 9

Offline 1 2 3 4 5 6 7 8 9

Online 1 1 3 3 3 3 3 3 3

Ratio 1 2 1 4/3

5/3

2 7/3

8/3

3

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Comparing Strategies

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Offline

Online

C.R.

Highest Offer

$$$/Ratio

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Threshold Algorithms

We can consider a family of online algorithms that wait for some threshold bid.There are two bad cases.

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One bad case is when the highest offer is $9.

Threshold

1 2 3 4 5 6 7 8 9

Offline 9 9 9 9 9 9 9 9 9 9

Value at 9

1 2 3 4 5 6 7 8 9

Ratio at 9

9 9/2

3 9/4

9/5

3/2

9/7

9/8

1

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The other is when the highest offer is just below the threshold.

Threshold

1 2 3 4 5 6 7 8 9

Offline atT-1

1 1 2 3 4 5 6 7 8

Value atT-1

1 1 1 1 1 1 1 1 1

Ratio at T-1

1 1 2 3 4 5 6 7 8

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Threshold AlgorithmsThreshold

1 2 3 4 5 6 7 8 9

Ratio at 9

9 9/2

3 9/4

9/5

3/2

9/7

9/8

1

Ratio at T-1

1 1 2 3 4 5 6 7 8

WC Ratio

9 9/2

3 3 4 5 6 7 8Occur when highest offer 9

Occur when highest offerthreshold-1

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What if he has $N?

Threshold

1 … …N

Ratio at N

N

Ratio at T-1

Ratio N … …N-1Occur when

highest offer NOccur when highest offerthreshold-1

N

N

1N

NN

N N

1NN

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What if he has $64?

Sell if more than $8.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Offline

Online

C.R.

Highest Offer

$$$/Ratio

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Mixing Thresholds

“Best” threshold gets Low thresholds do better if the highest offer is lower.Higher thresholds do better if the highest offer is higher.How can we mix these together?

N

166

Game Show Online Algorithms

There is a prize behind one door.

$1

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$1

168



$1

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Randomization

Choose a door at random.

1/3 1/3 1/3

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Randomization

Case 1: the prize is behind the left door

$1

1/3

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Randomization

Case 2: the prize is behind the center door

$1

1/3

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Randomization

Case 3: the prize is behind the right door

$1

1/3

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Game Show Problem

No deterministic algorithm can achieve a reward in the worst case.A randomized algorithm receives a mean reward of $1/3 dollars in each case.For every case, the competitive ratio is the same(3).

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The Ice Cream Problem

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Better Business ThroughRandomization

Choose k uniformly at random from {1,2,3,4,5,6}(roll a six-sided die). Choose a threshold of 2k.Now we compare the expected reward of the online algorithm to the offline algorithmStill worst case scenario for brother’s valuation.

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What if he has $64?

Choosing threshold at random.Off On C.R. 2 4 8 16 32 641 1 1 1 1 1 1 1 12 1.17 1.7 2 1 1 1 1 13 1.17 2.6 2 1 1 1 1 14 1.67 2.4 2 4 1 1 1 15 1.67 3 2 4 1 1 1 16 1.67 3.6 2 4 1 1 1 17 1.67 4.2 2 4 1 1 1 1

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What if he has $64?

Choosing threshold at random.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Offline

Online(Expected)

C.R.Max C.R. 6

Highest Offer

$$$/Ratio

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In General

If your little brother has $N, the C.R. is log2 N.

Can do better: we can decrease the C.R. to ln N with a different exponential distribution.

N

xxT

ln

ln]Pr[

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Don’t like ice cream?

What happens if you don’t like ice cream?

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Relation to the Problem


Price

Time

pmax

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Suppose you unconditionally refuse the first bid: competitive ratio of INFINITY!Deterministic strategy: take first bid.

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Randomized strategy Choose k uniformly at random from

{0,1,2,3,4,5,6}. Choose a threshold of 2k.

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Don’t Like Ice Cream?

Choosing threshold at random.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Offline

Online

C.R.Max C.R. 7

Highest Offer

$$$/Ratio

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Bounds Still Good

Achieve a competitive ratio of (log2N)+1.

Again, we can get a (ln N)+1 with a different exponential distribution.

1)(ln

ln]Pr[

N

xxT

1)(ln

1]1Pr[

NT

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Application to One Sell Bid

Buy/Sell bids in [0,pmax]

See a sell bid at almost zero.Choose a profit threshold T at random according to:

1ln

ln]Pr[

max

p

xxT

1ln

1]1Pr[

max

pT

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Application To One Sell Bid

The competitive ratio of this algorithm is:

ln(pmax)+1

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Outline


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Putting it Together

The ice cream problem represents well when we only have one sell bid.The matching problem allows us to maximize volume, but does not consider the profit that we will achieve with each edge.

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A Helpful Guarantee

What if we had a guarantee that every edge in the graph corresponded to a high profit?

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The Final Algorithm

Choose a threshold according to a specific exponential distribution.When you convert the auction to a graph, only include edges that have a profit above that threshold.Run the matching algorithm on the graph as it is observed.

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Outline


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Conclusion

On the temporal bidding problem, there exists an algorithm which can achieve a competitive ratio of:

ln(pmax)+1

This is the best possible competitive ratio for an online algorithm

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Outline


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Break

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Bidding Truthfully


Price

Time

Why not lie?

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Bidding Truthfully


Price

TimeWhy not lie?

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Bidding Truthfully


Price

TimeWhy not lie?

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Profit


Price

$0

Profit=$5

$5

$10

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Social Welfare


Price

$0

Social Welfare=$10

$5

$10

200

Profit


Price

$0

Profit = $0$5

$10C.R.:

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Social Welfare


Price

$0

Social Welfare=$5

$5

$10C.R.:2

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Two Problems

Agents not motivated to bid truthfullyProfit and social welfare competitive ratios different

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New Objectives

Maximize Global WelfareHave Agents Bid Truthfully

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The Price is Right!


Price

Time

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The Price is Right!


Price

Time

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The Price is Right!


Price

Time

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The Price is Right!


Price

Time

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Convert To A Graph…


Price

Time

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Match Greedily…


Price

Time

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Match Greedily…


Price

Time

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What Did We Win?


Price

Time

212

What Could We Win?


Price

Time

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What Could We Win?


Price

Time

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The Price Is Right!

Bids in range [pmin, pmax]Choose a price p at random between pmin and pmax according to an exponential distribution.Greedily match those buy bids above to sell bids below that price.Worst-case competitive ratio for true bids is below 2(ln(pmax)-ln(pmin))

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The Gift Problem

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The Gift Problem:Buyer View


Price

Time

I am ALMOSTCERTAIN thishappens

HIGHLY unlikely

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The Gift Problem:Buyer View


Price

Time

I am ALMOSTCERTAIN thishappens

HIGHLY unlikely

I’ll bid what I expect it to be worth

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The Gift Problem:Worst Case


Price

Time

Guess I was wrong!

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The Gift Problem:Worst Case


Price

Time

Guess I was wrong!

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Where it gets REAL ugly…

Buyer: well, I REALLY think my son would like a Playstation.Seller: well, I REALLY think my daughter wouldn’t like a Playstation.

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Restricting the Valuation

Agents have a point in time where they precisely discover their valuations: no probabilities.Under this model, the mechanism is incentive-compatible.

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Lying Does Not Help


Price

Time

223

Other Results

Lower boundsMaximizing buy/sell volumeLearning a good threshold

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Future Work

Multiple item bids(“I want 50 kegs, no less!”)Combinatorial auctionsSupport of more general valuations

225

Questions?

online algorithms for market clearing

Documents

buysell bids

single auctionbuy bids

weighted graphbuy bids

double auctionbuy bids

sellgraphical versionbuy

online algorithms

matching of maximum

matching of concurrent