Online Algorithms for Market Clearing

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

Introduction

E-Bay: Single Auction

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

Improvement #1: Double Auction

Graphical Version

Price Profit

Improvement #2:Bidding For Different Intervals

Temporal Bidding Problem

Introduction Time Expiration Time

Mark Toy’s R Us

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.

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.

Formalizing the Goal

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

The Maximal Matching

The Offline Problem

Convert to a Graph

Convert to a Weighted Graph

Find a Matching of Maximum Weight

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

Algorithms existto solve this in polynomial time

Matching ofmaximal weightmaximizes profit!

What Online Sees…

Two Online Difficulties

Getting the Most Matches

Getting the Most Per Match

Two Small Problems

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

Outline

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

Maximizing Liquidity

Simplified Matching

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

A Maximal Numeric Matching

Also Maximal Numerically

Offline: Convert to a Graph

No weightsneeded

Convert to a Graph

No edgehere

Convert to a Graph

Maximal Matching

Back To The Problem

A maximal matching on the graph maximizes liquidity

Convert to a Graph Online…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Online Graph…

AliveExpired

The Greedy Algorithm

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

The Greedy Algorithm…

AliveExpiredMatched

A General Solution

Thinking aboutarbitrary graphs

Optimal:6 edges,12 vertices

What If…

You:At least 6 vertices

What If…

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

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.

When v2 Arrives…

Either v1 is unmatched…

When v2 Arrives…

Either v1 is unmatched…

When v2 Arrives…

Or v1 is matched…

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.

Back To Bids

If we know whenthings expire,it helps to waitsometimes

A Problem…

Don’t know untilits too late!

A Solution…Subsidy

But at all times,have positivefunds

Negative!

An Observation About Profit

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

Let’s forget time

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

Let’s forget time

$2 5+8+2=15

1+4+6=11

Let’s forget time

Insert FavoriteMatching Here

Another Matching

Let’s forget time

+1+1+2=4

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.

A Matchable Set of Vertices

The red verticesare matchable

Not A Matchable Set

Must use the wholeset!

The red verticesare NOT matchable

Not A Matchable SetCan’t useverticesnot in the set

The red verticesare NOT matchable

A Solution…Subsidy

No edge here

An Incomplete Interval Graph

Maintainexpirationtimes

Defining A New Algorithm

Currentbid

Expiresfirst

Expiressecond

A New Algorithm

TimeAliveExpiredMatched

A New Algorithm

Outline

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

The Ice Cream Problem

Heh Heh Heh…I’ll

wash your car!

AND I’ll do your

homework!

AND I’ll clean your

AND I’ll give you my life

savings!

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.

Oh Well

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.

Relation to the Problem

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).

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!

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

Comparing Strategies

0 1 2 3 4 5 6 7 8 9 10

Offline

Online

Highest Offer

$$$/Ratio

Threshold Algorithms

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

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

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

Threshold AlgorithmsThreshold

1 2 3 4 5 6 7 8 9

Ratio at 9

Ratio at T-1

1 1 2 3 4 5 6 7 8

WC Ratio

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

Occur when highest offerthreshold-1

What if he has $N?

Threshold

1 … …N

Ratio at N

Ratio at T-1

Ratio N … …N-1Occur when

highest offer NOccur when highest offerthreshold-1

What if he has $64?

Sell if more than $8.

0 10 20 30 40 50 60 70

Offline

Online

Highest Offer

$$$/Ratio

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?

Game Show Online Algorithms

There is a prize behind one door.

Randomization

Choose a door at random.

1/3 1/3 1/3

Randomization

Case 1: the prize is behind the left door

Randomization

Case 2: the prize is behind the center door

Randomization

Case 3: the prize is behind the right door

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).

The Ice Cream Problem

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.

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

What if he has $64?

Choosing threshold at random.

0 10 20 30 40 50 60 70

Offline

Online(Expected)

C.R.Max C.R. 6

Highest Offer

$$$/Ratio

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.

ln]Pr[

Don’t like ice cream?

What happens if you don’t like ice cream?

Relation to the Problem

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

Randomized strategy Choose k uniformly at random from

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

Don’t Like Ice Cream?

Choosing threshold at random.

0 10 20 30 40 50 60 70

Offline

Online

C.R.Max C.R. 7

Highest Offer

$$$/Ratio

Bounds Still Good

Achieve a competitive ratio of (log2N)+1.

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

ln]Pr[

1]1Pr[

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:

ln]Pr[

1]1Pr[

Application To One Sell Bid

The competitive ratio of this algorithm is:

ln(pmax)+1

Outline

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.

A Helpful Guarantee

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

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.

Outline

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

Outline

Bidding Truthfully

Why not lie?

Bidding Truthfully

TimeWhy not lie?

Bidding Truthfully

TimeWhy not lie?

Profit

Profit=$5

Social Welfare

Social Welfare=$10

Profit

Profit = $0$5

$10C.R.:

Social Welfare

Social Welfare=$5

$10C.R.:2

Two Problems

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

New Objectives

Maximize Global WelfareHave Agents Bid Truthfully

The Price is Right!

Convert To A Graph…

Match Greedily…

What Did We Win?

What Could We Win?

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))

The Gift Problem

The Gift Problem:Buyer View

I am ALMOSTCERTAIN thishappens

HIGHLY unlikely

The Gift Problem:Buyer View

I am ALMOSTCERTAIN thishappens

HIGHLY unlikely

I’ll bid what I expect it to be worth

The Gift Problem:Worst Case

Guess I was wrong!

The Gift Problem:Worst Case

Guess I was wrong!

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.

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.

Lying Does Not Help

Other Results

Lower boundsMaximizing buy/sell volumeLearning a good threshold

Future Work

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

Questions?

Online Algorithms for Market Clearing

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