online algorithms for market clearing
DESCRIPTION
Online Algorithms for Market Clearing. Martin Zinkevich Joint Work with Avrim Blum and Tuomas Sandholm (SODA 2002). Introduction. E-Bay: Single Auction. Buy Bids: people wanting to buy Sell Bids: people wanting to sell. E-Bay: Single Auction. Buy Bids: people wanting to buy - PowerPoint PPT PresentationTRANSCRIPT
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Online Algorithms for Market Clearing
Martin ZinkevichJoint Work with Avrim Blum and Tuomas Sandholm(SODA 2002)
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Introduction
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E-Bay: Single Auction
Buy Bids: people wanting to buySell Bids: people wanting to sell
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E-Bay: Single Auction
Buy Bids: people wanting to buySell Bids: people wanting to sell
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E-Bay: Single Auction
Buy Bids: people wanting to buySell Bids: people wanting to sell
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Improvement #1: Double Auction
Buy Bids: people wanting to buySell Bids: people wanting to sell
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Graphical Version
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price Profit
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Improvement #2:Bidding For Different Intervals
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Temporal Bidding Problem
Buy Bids: people wanting to buySell Bids: people wanting to sell
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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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The Offline Problem
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Convert to a Graph
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Convert to a Graph
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Convert to a Weighted Graph
Buy Bids: people wanting to buySell Bids: people wanting to sell
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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Find a Matching of Maximum Weight
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A matching for a graph is a set of edges which do not share anyvertices
5
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Find a Matching of Maximum Weight
5
15
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Algorithms existto solve this in polynomial time
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Find a Matching of Maximum Weight
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Find a Matching of Maximum Weight
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
Matching ofmaximal weightmaximizes profit!
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Online Sees…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Two Online Difficulties
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Getting the Most Matches
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Getting the Most Per Match
Buy Bids: people wanting to buySell Bids: people wanting to sell
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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Also Maximal Numerically
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Offline: Convert to a Graph
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
No weightsneeded
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Convert to a Graph
Buy Bids: people wanting to buySell Bids: people wanting to sell
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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
A maximal matching on the graph maximizes liquidity
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Convert to a Graph Online…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
AliveExpired
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Convert to a Graph Online…
Price
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
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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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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The Greedy Algorithm…
AliveExpiredMatched
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A General Solution
Thinking aboutarbitrary graphs
Optimal:6 edges,12 vertices
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What If…
Thinking aboutarbitrary graphs
Optimal:6 edges,12 vertices
You:At least 6 vertices
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What If…
Thinking aboutarbitrary graphs
Optimal:6 edges,12 vertices
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
Thinking aboutarbitrary graphs
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When v2 Arrives…
Either v1 is unmatched…
v1
v2
Thinking aboutarbitrary graphs
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When v2 Arrives…
Or v1 is matched…
v1
v2
Thinking aboutarbitrary graphs
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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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
If we know whenthings expire,it helps to waitsometimes
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A Problem…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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A Problem…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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A Problem…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
Don’t know untilits too late!
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A Solution…Subsidy
Buy Bids: people wanting to buySell Bids: people wanting to sell
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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An Observation About Profit
$5
$1 $4
$8
$6
$2
+4 +4 -4 +4+4-4=4
Let’s forget time
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An Observation About Profit
$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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
No edge here
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An Incomplete Interval Graph
Time
Maintainexpirationtimes
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Defining A New Algorithm
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
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Defining A New Algorithm
Time
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
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Defining A New Algorithm
Time
Currentbid
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Defining A New Algorithm
Time
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Defining A New Algorithm
Time
Expiresfirst
Expiressecond
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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A New Algorithm
TimeAliveExpiredMatched
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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
Buy Bids: people wanting to buySell Bids: people wanting to sell
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
$$$/Ratio
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Threshold Algorithms
We can consider a family of online algorithms that wait for some threshold bid.There are two bad cases.
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Threshold Algorithms
One bad case is when the highest offer is $9.
Threshold
1 2 3 4 5 6 7 8 9
Offline 9 9 9 9 9 9 9 9 9 9
Value at 9
1 2 3 4 5 6 7 8 9
Ratio at 9
9 9/2
3 9/4
9/5
3/2
9/7
9/8
1
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Threshold Algorithms
The other is when the highest offer is just below the threshold.
Threshold
1 2 3 4 5 6 7 8 9
Offline atT-1
1 1 2 3 4 5 6 7 8
Value atT-1
1 1 1 1 1 1 1 1 1
Ratio at T-1
1 1 2 3 4 5 6 7 8
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Threshold AlgorithmsThreshold
1 2 3 4 5 6 7 8 9
Ratio at 9
9 9/2
3 9/4
9/5
3/2
9/7
9/8
1
Ratio at T-1
1 1 2 3 4 5 6 7 8
WC Ratio
9 9/2
3 3 4 5 6 7 8Occur when highest offer 9
Occur when highest offerthreshold-1
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What if he has $N?
Threshold
1 … …N
Ratio at N
N
Ratio at T-1
Ratio N … …N-1Occur when
highest offer NOccur when highest offerthreshold-1
N
N
1N
NN
N N
1NN
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What if he has $64?
Sell if more than $8.
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
Offline
Online
C.R.
Highest Offer
$$$/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
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Game Show Online Algorithms
There is a prize behind one door.
$1
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Game Show Online Algorithms
There is a prize behind one door.
$1
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Game Show Online Algorithms
There is a prize behind one door.
$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
$$$/Ratio
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In General
If your little brother has $N, the C.R. is log2 N.
Can do better: we can decrease the C.R. to ln N with a different exponential distribution.
N
xxT
ln
ln]Pr[
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Don’t like ice cream?
What happens if you don’t like ice cream?
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Relation to the Problem
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
pmax
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Don’t like ice cream?
Suppose you unconditionally refuse the first bid: competitive ratio of INFINITY!Deterministic strategy: take first bid.
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Don’t like ice cream?
Randomized strategy Choose k uniformly at random from
{0,1,2,3,4,5,6}. Choose a threshold of 2k.
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Don’t Like Ice Cream?
Choosing threshold at random.
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
Offline
Online
C.R.Max C.R. 7
Highest Offer
$$$/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
IntroductionSmall Problem #1: Maximizing LiquiditySmall Problem #2: The Ice Cream ProblemPutting it TogetherConclusionNew Results
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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
IntroductionSmall Problem #1: Maximizing LiquiditySmall Problem #2: The Ice Cream ProblemPutting it TogetherConclusionNew Results
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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
IntroductionSmall Problem #1: Maximizing LiquiditySmall Problem #2: The Ice Cream ProblemPutting it TogetherConclusionNew Results
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Break
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Bidding Truthfully
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
Why not lie?
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Bidding Truthfully
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
TimeWhy not lie?
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Bidding Truthfully
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
TimeWhy not lie?
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Profit
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
$0
Profit=$5
$5
$10
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Social Welfare
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
$0
Social Welfare=$10
$5
$10
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Profit
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
$0
Profit = $0$5
$10C.R.:
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Social Welfare
Buy Bids: people wanting to buySell Bids: people wanting to sell
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!
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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The Price is Right!
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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The Price is Right!
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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The Price is Right!
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Convert To A Graph…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Match Greedily…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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Match Greedily…
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Did We Win?
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Could We Win?
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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What Could We Win?
Buy Bids: people wanting to buySell Bids: people wanting to sell
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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
I am ALMOSTCERTAIN thishappens
HIGHLY unlikely
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The Gift Problem:Buyer View
Buy Bids: people wanting to buySell Bids: people wanting to sell
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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
Guess I was wrong!
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The Gift Problem:Worst Case
Buy Bids: people wanting to buySell Bids: people wanting to sell
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
Buy Bids: people wanting to buySell Bids: people wanting to sell
Price
Time
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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
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Questions?