Combinatorial Betting

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Combinatorial Betting

• David Pennock
• Joint with
• Yiling Chen, Lance Fortnow, Sharad Goel, Joe
  Kilian, Nicolas Lambert, Eddie Nikolova, Mike
  Wellman, Jenn Wortman

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Bet Credible Opinion
Hillary Clinton will win the election
I bet 100 Hillary will win at 1 to 2 odds

• Which is more believable?More Informative?
• Betting intermediaries
• Las Vegas, Wall Street, Betfair, Intrade,...
• Prices stable consensus of a large number of
  quantitative, credible opinions
• Excellent empirical track record

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Combinatorics ExampleMarch Madness
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Combinatorics ExampleMarch Madness

• Typical todayNon-combinatorial
• Team wins Rnd 1
• Team wins Tourney
• A few other props
• Everything explicit(By def, small )
• Every bet indep Ignores logical probabilistic
  relationships

• Combinatorial
• Any property
• Team wins Rnd kDuke gt UNC,NCSTACC wins 5
  games
• 2264 possible props(implicitly defined)
• 1 Bet effects related bets correctlye.g., to
  enforce logical constraints

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ExpressivenessGetting Information

• Things you can say today
• (43 chance that) Hillary wins
• GOP wins Texas
• YHOO stock gt 30 Dec 2007
• Duke wins NCAA tourney
• Things you cant say (very well) today
• Oil down, DOW up, Hillary wins
• Hillary wins election, given that she wins OH
  FL
• YHOO btw 25.8 32.5 Dec 2007
• 1 seeds in NCAA tourney win more than 2 seeds

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ExpressivenessProcessing Information

• Independent markets today
• Horse race win, place, show pools
• Stock options at different strike prices
• Every game/proposition in NCAA tourney
• Almost everything Stocks, wagers, intrade, ...
• Information flow (inference) left up to traders
• Better Let traders focus on predicting whatever
  they want, however they want Mechanism takes
  care of logical/probabilistic inference
• Another advantage Smarter budgeting

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A (Non-Combinatorial) Prediction Market

• Take a random variable, e.g.
• Turn it into a financial instrument payoff
  realized value of variable

Bird Flu Outbreak US 2007?(Y/N)
I am entitled to
Bird FluUS 07
Bird FluUS 07
1 if
0 if
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Why?

• Get information
• price ? probability of uncertain event(in
  theory, in the lab, in the field, ...next slide)
• Is there some future event youd like to
  forecast?A prediction market can probably help

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Does it work?
Thanks Yiling Chen

• Yes, evidence from real markets, laboratory
  experiments, and theory
• Racetrack odds beat track experts Figlewski
  1979
• Orange Juice futures improve weather forecast
  Roll 1984
• I.E.M. beat political polls 451/596 Forsythe
  1992, 1999Oliven 1995Rietz 1998Berg
  2001Pennock 2002
• HP market beat sales forecast 6/8 Plott 2000
• Sports betting markets provide accurate forecasts
  of game outcomes Gandar 1998Thaler
  1988Debnath EC03Schmidt 2002
• Market games work Servan-Schreiber 2004Pennock
  2001
• Laboratory experiments confirm information
  aggregationPlott 198219881997Forsythe
  1990Chen, EC01
• Theory rational expectations Grossman
  1981Lucas 1972

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http//intrade.com
Screen capture 2007/05/18
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http//tradesports.com
http//intrade.com
Screen capture 2007/05/18
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Intrade Election Coverage
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Combinatorics 1 of 2Boolean Logic

• Outcomes All 2n possible combinations of n
  Boolean events
• Betting languageBuy q units of 1 if Boolean
  Formula at price p
• General Any Boolean formula (22n possible)
• A not(B) ? (ACF) (DE)
• Oil rises Hillary wins Guiliani GOP nom
  housing falls
• Eastern teams win more games than Western in
  Tourney
• Restricted languages we study
• Restricted tournament languageTeam A wins in
  round i Team A beats B, given they meet
• 3-clauses A not(C) F

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Combinatorics 2 of 2Permutations

• Outcomes All possible n! rank orderings of n
  objects (horse race)
• Betting languageBuy q units of 1 if Property
  at price p
• General Any property of ordering
• A wins ? A finishes in pos 3,4, or 10th
• A beats D ? 2 of B,D,F beat A
• Restricted languages we study
• Subset bettingA finishes in pos 3-5 or 9 A,D,or
  F finish 3rd
• Pair bettingA beats F

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Predicting Permutations

• Predict the ordering of a set of statistics
• Horse race finishing times
• Number of votes for several candidates
• Daily stock price changes
• NFL Football quarterback passing yards
• Any ordinal prediction
• Chen, Fortnow, Nikolova, Pennock, EC07

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Market CombinatoricsPermutations

• A gt B gt C .1
• A gt C gt B .2
• B gt A gt C .1

• B gt C gt A .3
• C gt A gt B .1
• C gt B gt A .2

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Market CombinatoricsPermutations

• D gt A gt B gt C .01
• D gt A gt C gt B .02
• D gt B gt A gt C .01
• A gt D gt B gt C .01
• A gt D gt C gt B .02
• B gt D gt A gt C .05
• A gt B gt D gt C .01
• A gt C gt D gt B .2
• B gt A gt D gt C .01
• A gt B gt C gt D .01
• A gt C gt B gt D .02
• B gt A gt C gt D .01

• D gt B gt C gt A .05
• D gt C gt A gt B .1
• D gt C gt B gt A .2
• B gt D gt C gt A .03
• C gt D gt A gt B .1
• C gt D gt B gt A .02
• B gt C gt D gt A .03
• C gt A gt D gt B .01
• C gt B gt D gt A .02
• B gt C gt D gt A .03
• C gt A gt D gt B .01
• C gt B gt D gt A .02

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Bidding Languages

• Traders want to bet on properties of orderings,
  not explicitly on orderings more natural, more
  feasible
• A will win A will show
• A will finish in 4-7 A,C,E will finish in
  top 10
• A will beat B A,D will both beat B,C
• Buy 6 units of 1 if AgtB at price 0.4
• Supported to a limited extent at racetrack today,
  but each in different betting pools
• Want centralized auctioneer to improve liquidity
  information aggregation

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Auctioneer Problem

• Auctioneers goalAccept orders with
  non-negative worst-case loss (auctioneer never
  loses money)
• The Matching Problem
• Formulated as LP
• Generalization Market Maker ProblemAccept
  orders with bounded worst-case loss (auctioneer
  never loses more than b dollars)

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Example

• A three-way match
• Buy 1 of 1 if AgtB for 0.7
• Buy 1 of 1 if BgtC for 0.7
• Buy 1 of 1 if CgtA for 0.7

B
A
C
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Pair Betting

• All bets are of the form A will beat B
• Cycle with sum of prices gt k-1 gt Match(Find
  best cycle Polytime)
• Match /gt Cycle with sum of prices gt k-1
• Theorem The Matching Problem for Pair Betting is
  NP-hard (reduce from min feedback arc set)

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Subset Betting

• All bets are of the form
• A will finish in positions 3-7, or
• A will finish in positions 1,3, or 10, or
• A, D, or F will finish in position 2
• Theorem The Matching Problem for Subset Betting
  is polytime (LP maximum matching separation
  oracle)

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Market CombinatoricsBoolean

• Betting on complete conjunctions is
  bothunnatural and infeasible

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Market CombinatoricsBoolean

• A bidding language write your own security
• For example
• Offer to buy/sell q units of it at price p
• Let everyone else do the same
• Auctioneer must decide who trades with whom at
  what price How? (next)
• More concise/expressive more natural

1 if A1 A2
1 if A1A7
I am entitled to
I am entitled to
1 if (A1A7)A13 (A2A5)A9
I am entitled to
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The Matching Problem

• There are many possible matching rules for the
  auctioneer
• A natural one maximize trade subject tono-risk
  constraint
• Example
• buy 1 of for 0.40
• sell 1 of for 0.10
• sell 1 of for 0.20
• No matter what happens,auctioneer cannot
  losemoney

trader gets in stateA1A2 A1A2 A1A2 A1A2
0.60 0.60 -0.40 -0.40 -0.90 0.10
0.10 0.10 0.20 -0.80 0.20 0.20 -0.10
-0.10 -0.10 -0.10
1 if A1
1 if A1A2
1 if A1A2
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Complexity Results
Fortnow Kilian Pennock Wellman

• Divisible orders will accept any q ? q
• Indivisible will accept all or nothing
• Natural algorithms
• divisible linear programming
• indivisible integer programming logical
  reduction?

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Automated Market Makers
Thanks Yiling Chen

• A market maker (a.k.a. bookmaker) is a firm or
  person who is almost always willing to accept
  both buy and sell orders at some prices
• Why an institutional market maker? Liquidity!
• Without market makers, the more expressive the
  betting mechanism is the less liquid the market
  is (few exact matches)
• Illiquidity discourages trading Chicken and egg
• Subsidizes information gathering and aggregation
  Circumvents no-trade theorems
• Market makers, unlike auctioneers, bear risk.
  Thus, we desire mechanisms that can bound the
  loss of market makers
• Market scoring rules Hanson 2002, 2003, 2006
• Dynamic pari-mutuel market Pennock 2004

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Automated Market Makers
Thanks Yiling Chen

• n disjoint and exhaustive outcomes
• Market maker maintain vector Q of outstanding
  shares
• Market maker maintains a cost function C(Q)
  recording total amount spent by traders
• To buy ?Q shares trader pays C(Q ?Q) C(Q) to
  the market maker Negative payment receive
  money
• Instantaneous price functions are
• At the beginning of the market, the market maker
  sets the initial Q0, hence subsidizes the market
  with C(Q0).
• At the end of the market, C(Qf) is the total
  money collected in the market. It is the maximum
  amount that the MM will pay out.

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New Results in PipelinePricing LMSR market maker

• Subset betting on permutations is P-hard (call
  market polytime!)
• Pair betting on permutations is P-hard?
• 3-clause Boolean betting P-hard?
• 2-clause Boolean betting P-hard?
• Restricted tourney betting is polytime (uses
  Bayesian network representation)
• Approximation techniques for general case

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Overview Complexity Results
Permutations Permutations Permutations Boolean Boolean Boolean
General Pair Subset General 3 (2?) clause Restrict Tourney
Call Market NP-hard NP-hard Poly co-NP-complete ? ?
Market Maker (LMSR) P-hard ? P-hard P-hard? P-hard? Poly
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• March Madness bet constructor
• Bet on any team to win any game
• Duke wins in Final 4
• Bet exotics
• Duke advances further than UNC
• ACC teams win at least 5
• A 1-seed will lose in 1st round

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Dynamic Parimutuel MarketAn Automated Market
Maker
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What is a pari-mutuel market?

• E.g. horse racetrack style wagering
• Two outcomes A B
• Wagers

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What is a pari-mutuel market?
A
B

• E.g. horse racetrack style wagering
• Two outcomes A B
• Wagers

?
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What is a pari-mutuel market?
A
B

• E.g. horse racetrack style wagering
• Two outcomes A B
• Wagers

?
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What is a pari-mutuel market?

• Before outcome is revealed, odds are reported,
  or the amount you would win per dollar if the
  betting ended now
• Horse A 1.2 for 1 Horse B 25 for 1 etc.
• Strong incentive to wait
• payoff determined by final odds every is same
• Should wait for best info on outcome, odds
• ? No continuous information aggregation
• ? No notion of buy low, sell high no cash-out

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Pari-Mutuel MarketBasic idea
1
1
1
1
1
1
1
1
1
1
1
1
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Dynamic Parimutuel Market
C(1,2)2.2
.82
C(2,3)3.6
.78
C(2,2)2.8
.59
.87
.3
C(2,4)4.5
.4
C(3,8)8.5
.91
.49
.94
C(4,8)8.9
C(2,5)5.4
.96
C(5,8)9.4
0.97
C(2,6)6.3
C(2,7)7.3
C(2,8)8.2
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Share-ratio price function

• One can view DPM as a market maker
• Cost Function
• Price Function
• Properties
• No arbitrage
• pricei/pricej qi/qj
• pricei lt 1
• payoff if right C(Qfinal)/qo gt 1

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Mech Design for Prediction
Financial Markets Prediction Markets
Primary Social welfare (trade)Hedging risk Information aggregation
Secondary Information aggregation Social welfare (trade)Hedging risk
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Mech Design for Prediction

• Standard Properties
• Efficiency
• Inidiv. rationality
• Budget balance
• Revenue
• Truthful (IC)
• Comp. complexity
• Equilibrium
• General, Nash, ...

• PM Properties
• 1 Info aggregation
• Expressiveness
• Liquidity
• Bounded budget
• Truthful (IC)
• Indiv. rationality
• Comp. complexity
• Equilibrium
• Rational expectations

Competes withexperts, scoringrules,
opinionpools, ML/stats,polls, Delphi