a tractable combinatorial market maker using constraint generation
DESCRIPTION
A tractable combinatorial market maker using constraint generation. MIROSLAV DUDÍK, SEBASTIEN LAHAIE, DAVID M. PENNOCK Microsoft Research. Thanks: David Rothschild, Dan Osherson , Arvid Wang, Jake Abernethy, Rafael Frongillo , Rob Schapire. - PowerPoint PPT PresentationTRANSCRIPT
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A tractable combinatorial market maker using constraint generation
MIROSLAV DUDÍK, SEBASTIEN LAHAIE,DAVID M. PENNOCKMicrosoft Research
Thanks: David Rothschild, Dan Osherson, Arvid Wang, Jake Abernethy, Rafael Frongillo, Rob Schapire
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A combinatorial question:How pivotal was Ohio?
• Day before the election:• 83.1% chance that whoever wins Ohio will win
the election• If Obama wins Ohio, 93.9% chance he’ll win
the election• If Romney wins Ohio, 53.2% chance he’ll win
the election
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More fun election-eve estimates
• 22% chance Romney will win in Iowa but Obama will win the national election
• 75.7% chance the same party will win both Michigan and Ohio
• 48.3% chance Obama gets 300 or more Electoral College votes
• 12.3% chance Obama will win between 6 and 8 states that begin with the letter M
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More fun election-eve estimates
• 22% chance Romney will win in Iowa but Obama will win the national election
• 75.7% chance the same party will win both Michigan and Ohio
• 48.3% chance Obama gets 300 or more Electoral College votes
• 12.3% chance Obama will win between 6 and 8 states that begin with the letter M
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Where did you get these numbers?
• A: We crowdsourced them• http://PredictWiseQ.com• A fully working beta example of our technical
paper in ACM EC’12
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The wisdom of crowds
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The wisdom of crowds
More:http://blog.oddhead.com/2007/01/04/the-wisdom-of-the-probabilitysports-crowd/http://www.overcomingbias.com/2007/02/how_and_when_to.html
Ignore crowd:if you’re in the99.7th percentile
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Can we do better?
• model it - baseline• model it - baseline++• poll a crowd - mTurk• pay a crowd - probSports contest• pay a crowd - Vegas market• pay a crowd - TradeSports market
• guess
“Prediction market”
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An ExamplePrediction
• A random variable, e.g.
Will US go into recession in 2013?(Y/N)
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An ExamplePrediction Market
• A random variable, e.g.
• Turned into a financial instrument payoff = realized value of variable
$1 if $0 ifI am entitled to:
Will US go into recession in 2013?(Y/N)
Recessionin 2013
No Recessionin 2013
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2012
Nov
embe
r 28
5:49
a.m
. ET
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2012
Nov
embe
r 28
5:49
a.m
. ET
Between 17.3% and 20.7% chance
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http://www.predictwise.com/maps/2012president
11-05-2012 10:09AM
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Design for Prediction
• Goals for trade– Efficiency (gains)– Inidiv. rationality– Budget balance– Revenue– Comp. complexity
• Equilibrium– General, Nash, ...
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Design for Prediction
• Goals for trade– Efficiency (gains)– Inidiv. rationality– Budget balance– Revenue– Comp. complexity
• Equilibrium– General, Nash, ...
• Goals for prediction– Info aggregation– 1. Liquidity– 2. Expressiveness– Bounded budget– Indiv. rationality– Comp. complexity
• Equilibrium– Rational expectations
Competes with:experts, scoring rules, opinion pools, ML/stats, polls, Delphi
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Design for Prediction
• Goals for trade– Efficiency (gains)– Inidiv. rationality– Budget balance– Revenue– Comp. complexity
• Equilibrium– General, Nash, ...
• Goals for prediction– Info aggregation– 1. Liquidity– 2. Expressiveness– Bounded budget– Indiv. rationality– Comp. complexity
• Equilibrium– Rational expectations
Competes with:experts, scoring rules, opinion pools, ML/stats, polls, Delphi
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Why Liquidity?
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Why Liquidity?
Low liquidity takes the prediction out of marketshttp://blog.oddhead.com/2010/07/08/why-automated-market-makers/
Between 0.2% and 99.8% chance
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Why Expressiveness?
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Why Expressiveness?
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Why Expressiveness?
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Why Expressiveness?
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Why Expressiveness?
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Why Expressiveness?
• Call option and put options are redundant• Range bets require four trades
(“butterfly spread”)• Bid to buy call option @strike 15 can’t match
with ask to sell @strike 10• Can’t set own strike• Bottom line: Lacks expressiveness
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Why Expressiveness?
• Dem Pres, Dem Senate, Dem HouseDem Pres, Dem Senate, GOP HouseDem Pres, GOP Senate, Dem HouseDem Pres, GOP Senate, GOP House...
• Dem PresDem HouseDem wins >=270 electoral votesDem wins >=280 electoral votes...
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Industry Standard
• Ignore relationships:Treat them as independent markets
• Las Vegas sports bettingKentucky horseracingWall Street stock optionsHigh Street spread betting
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NYSE 1926
http://online.wsj.com/article/SB10001424052748704858404576134372454343538.html
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NYSE 1987
http://online.wsj.com/article/SB10001424052748704858404576134372454343538.html
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NYSE 2006-2011
•2011 Deutsche Börse AG
•2007 Euronext
•2006 Archipelago, ipo
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NYSE 7pm Sep 10, 2012
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New Markets – Same CDA
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A Better Way(Or,... Bringing trading into digital age)
• Expressiveness– Linear programming– Bossaerts, Fine, Ledyard: Combined Value Trading
Fortnow et al.: Betting Boolean Style– http://bit.ly/multipm
• Expressiveness + Liquidity– Automated market maker– Always quote a price on anything– Downside: requires subsidy/risk
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Getting Greedy
• Design a marketfor information on exponentially many things
• “Combinatorial prediction market”
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Combinatorial securities:More information, more fun
• Payoff is function of common variables,e.g. 50 states elect Dem or Rep
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Combinatorial securities:More information, more fun
• Dem will win California
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Combinatorial securities:More information, more fun
• Dem will lose FL but win election• Dem will win >8 of 10 Northeastern states• Same party will win OH & PA
OH
PA
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Combinatorial securities:More information, more fun
• There will be a path of blue from Canada to Mexico
OR
WA
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Some Counting
• 54 “states”: 48 + DC + Maine (2), Nebraska (3)• 254 = 18 quadrillion possible outcomes• 2254 1018008915383333485 distinct predictions
More than a googol, less than a googolplex• NOT independent
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Overview:Complexity results
Permutations Boolean Taxonomy
General Pair Subset General 2-clause RestrictTourney
General Tree
Auction-eer
NP-hardEC’07
NP-hardEC’07
PolyEC’07
NP-hardDSS’05
co-NP-completeDSS’05
? ? ?
Market Maker(LMSR)
#P-hardEC’08
#P-hardEC’08
#P-hardEC’08
#P-hardEC’08ApproxSTOC’08EC’12
#P-hardEC’08
PolySTOC’08
#P-hardAAMAS‘09
PolyAAMAS‘09
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A research methodology
Design Build Analyze
HSXNFTSWSEXFXPS
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Examples
Design
• Prediction markets– Dynamic parimutuel– Combinatorial bids– Combinatorial
outcomes– Shared scoring rules– Linear programming
backbone• Ad auctions• Spam incentives
Build Analyze
• Computational complexity
• Does money matter?
• Equilibrium analysis
• Wisdom of crowds: Combining experts
• Practical lessons
• Predictalot• Yoopick• Y!/O Buzz• Centmail• Pictcha• Yootles
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Automated Market MakerExchange Market Maker
Independent TractableNo riskNo info propagationIndustry standard
TractableExponential loss boundNo info propagation
Combinatorial NP-hardNo riskFull info propagationMajor liquidity problem
#P-hardLinear/Const loss boundFull info propagation
• Info propagation Reward traders for information, not computational power
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Automated Market MakerExchange Market Maker
Independent TractableNo riskNo info propagationIndustry standard
TractableExponential loss boundNo info propagation
Our approach TractableGood loss boundSome info propagation
Combinatorial NP-hardNo riskFull info propagationMajor liquidity problem
#P-hardLinear/Const loss boundFull info propagation
• Info propagation Reward traders for information, not computational power
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Consistent pricing
0
1
0 1
A&B’&C
A&B&C
Independent markets
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Consistent pricing
0
1
0 1
A&B’&C
A&B&C
Independent markets
Prices p
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Consistent pricing
0
1
0 1
A&B’&C
A&B&C
Independent markets
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Consistent pricing
0
1
0 1
A&B’&C
A&B&C B = 0.6
A = 0.8C = 0.9
Independent markets
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Consistent pricing
0
1
0 1
0.6 B = 0.6
0.8 A = 0.8
A&B’&C
A&B&C
0.9 C = 0.9
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Consistent pricing
0
1
0 1
0.6 B = 0.6
0.4
0.8 A = 0.8
0.8A&B’&C
A&B&C
0.9 C = 0.9
0.9
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Consistent pricing
0
1
0 1
0.6 B = 0.6
0.4
0.8 A = 0.8
0.8A&B’&C
A&B&C
0.9 C = 0.9
0.9
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Approximate pricing
0
1
0 1
0.6 B = 0.6
0.4
0.8 A = 0.8
0.8A&B’&C
A&B&C
0.9 C = 0.9
0.9
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Approximate pricing
0
1
0 1
0.6 B = 0.6
0.4
Prices p0.8 A = 0.8
0.8A&B’&C
A&B&C
0.9 C = 0.9
0.9
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Approximate pricing
0
1
0 1
0.5 B = 0.5
0.5
Buy NotBPrices p0.8 A = 0.8
0.8A&B’&C
A&B&C
0.9 C = 0.9
0.9
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Approximate pricing
0
1
0 1
0.5 B = 0.5
0.5
Prices p0.8 A = 0.8
0.8A&B’&C
A&B&C
0.9 C = 0.9
0.9
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Approximate pricing
0
1
0 1
0.8
0.5
A = 0.8
B = 0.55
0.5 0.8
Prices p
A&B’&C
A&B&C
0.9 C = 0.9
0.9
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For Election
• Create 50 states – initialize with prior• Create all groups of 2 – init as indep• For conjunctions of 3 or more, group with it
opposite disjunction:A&B&C, A’|B’|C’
• Each group is indep MM – fast• In parallel:
Generate, find, and fix constraints
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Microsoft Research, New York City
Arbitrage and Constraints
• Possibility of risk-free profit:
• Execute trades:– Buy x shares of A– Buy x shares of B– Sell x shares of A B
Prob[A] + Prob[B] ≥ Prob[A B]
Price[A] + Price[B] − Price[A B] ≤ 0
September 26, 2012
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Constraints
• Clique lower boundP(L1|...|Lm) ≥ΣC P(Li) –ΣC P(Li&Lj)
• Spanning tree upper boundP(L1|...|Lm) ≤Σ P(Li) –ΣT P(Li&Lj)
• Threshold constraints TBA• Choosing constraints is key!
– Depends on bets (unlike Monte Carlo)– An art
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Does it work?
Tested on over 300K complex predictions from Princeton study
Budget
10 States
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Does it work?
Tested on over 300K complex predictions from Princeton study
Budget
Log Score
50 States
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Does it work?
Tested on over 300K complex predictions from Princeton study
Revenue
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Predictalot alpha
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Further reading
Blog post on PredictWiseQhttp://blog.oddhead.com/2012/10/06/predictwiseq/
Gory details: What is (and what good is) a combinatorialprediction market?http://bit.ly/combopm
Guest post on Freakonomicshttp://bit.ly/combopmfreak
Our paper in ACM EC’12http://research.microsoft.com/apps/pubs/default.aspx?id=167977