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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
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
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
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
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
The wisdom of crowds
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
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”
An ExamplePrediction
• A random variable, e.g.
Will US go into recession in 2013?(Y/N)
An ExamplePrediction Market
• A random variable, e.g.
• Turned into a financial instrument payoff = realized value of variable
$1 if $0 if
I am entitled to:
Will US go into recession in 2013?(Y/N)
Recessionin 2013
No Recessionin 2013
2012
No
vem
ber
28
5:49
a.m
. E
T
2012
No
vem
ber
28
5:49
a.m
. E
T
Between 17.3% and 20.7% chance
http://www.predictwise.com/maps/2012president
11-05-2012 10:09AM
Design for Prediction
• Goals for trade– Efficiency (gains)– Inidiv. rationality– Budget balance– Revenue– Comp. complexity
• Equilibrium– General, Nash, ...
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
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
Why Liquidity?
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
Why Expressiveness?
Why Expressiveness?
Why Expressiveness?
Why Expressiveness?
Why Expressiveness?
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
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...
Industry Standard
• Ignore relationships:Treat them as independent markets
• Las Vegas sports bettingKentucky horseracingWall Street stock optionsHigh Street spread betting
NYSE 1926
http://online.wsj.com/article/SB10001424052748704858404576134372454343538.html
NYSE 1987
http://online.wsj.com/article/SB10001424052748704858404576134372454343538.html
NYSE 2006-2011•
2011 Deutsche Börse AG
•2007 Euronext
•2006 Archipelago, ipo
NYSE 7pm Sep 10, 2012
New Markets – Same CDA
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
Getting Greedy
• Design a marketfor information on exponentially many things
• “Combinatorial prediction market”
Combinatorial securities:More information, more fun
• Payoff is function of common variables,e.g. 50 states elect Dem or Rep
Combinatorial securities:More information, more fun
• Dem will win California
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
Combinatorial securities:More information, more fun
• There will be a path of blue from Canada to Mexico
OR
WA
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
Overview:Complexity results
Permutations Boolean Taxonomy
General Pair Subset General 2-clause Restrict
Tourney
General Tree
Auction-eer
NP-hard
EC’07
NP-hard
EC’07
Poly
EC’07
NP-hard
DSS’05
co-NP-complete
DSS’05
? ? ?
Market Maker
(LMSR)
#P-hard
EC’08
#P-hard
EC’08
#P-hard
EC’08
#P-hard
EC’08
Approx
STOC’08EC’12
#P-hard
EC’08
Poly
STOC’08
#P-hard
AAMAS‘09
Poly
AAMAS‘09
A research methodology
Design Build Analyze
HSXNFTSWSEXFXPS
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
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
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
Consistent pricing
0
1
0 1
A&B’&C
A&B&C
Independent markets
Consistent pricing
0
1
0 1
A&B’&C
A&B&C
Independent markets
Prices p
Consistent pricing
0
1
0 1
A&B’&C
A&B&C
Independent markets
Consistent pricing
0
1
0 1
A&B’&C
A&B&C B = 0.6
A = 0.8
C = 0.9
Independent markets
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
Consistent pricing
0
1
0 1
0.6 B = 0.6
0.4
0.8 A = 0.8
0.8
A&B’&C
A&B&C
0.9 C = 0.9
0.9
Consistent pricing
0
1
0 1
0.6 B = 0.6
0.4
0.8 A = 0.8
0.8
A&B’&C
A&B&C
0.9 C = 0.9
0.9
Approximate pricing
0
1
0 1
0.6 B = 0.6
0.4
0.8 A = 0.8
0.8
A&B’&C
A&B&C
0.9 C = 0.9
0.9
Approximate pricing
0
1
0 1
0.6 B = 0.6
0.4
Prices p0.8 A = 0.8
0.8
A&B’&C
A&B&C
0.9 C = 0.9
0.9
Approximate pricing
0
1
0 1
0.5 B = 0.5
0.5
Buy NotB
Prices p0.8 A = 0.8
0.8
A&B’&C
A&B&C
0.9 C = 0.9
0.9
Approximate pricing
0
1
0 1
0.5 B = 0.5
0.5
Prices p0.8 A = 0.8
0.8
A&B’&C
A&B&C
0.9 C = 0.9
0.9
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
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
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
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
Does it work?
Tested on over 300K complex predictions from Princeton study
Budget
10 States
Does it work?
Tested on over 300K complex predictions from Princeton study
Budget
Log Score
50 States
Does it work?
Tested on over 300K complex predictions from Princeton study
Revenue
Predictalot alpha
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
