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 ifI am entitled to:

Will US go into recession in 2013?(Y/N)

Recessionin 2013

No Recessionin 2013

2012

Nov

embe

r 28

5:49

a.m

. ET

2012

Nov

embe

r 28

5:49

a.m

. ET

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 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

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

http://PredictWiseQ.com

http://PredictWiseQ.com

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 = 0.8C = 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

0.8A&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

0.8A&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

0.8A&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

0.8A&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 NotBPrices p0.8 A = 0.8

0.8A&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

0.8A&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

No really, does it work?

• http://PredictWiseQ.com

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