Computational Aspects of Prediction Markets

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Computational Aspects of Prediction Markets

• David M. Pennock, Yahoo! Research
• Yiling Chen, Lance Fortnow, Joe Kilian,Evdokia
  Nikolova, Rahul Sami, Michael Wellman

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Mech Design for Prediction

• Q Will there be a bird flu outbreak in the US in
  2007?
• A Uncertain. Evidence distributed health
  experts, nurses, public
• Goal Obtain a forecast as good as omniscient
  center with access to all evidence from all
  sources

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Mech Design for Prediction
possible states of the world
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A 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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http//intrade.com
Screen capture 2007/04/19
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Mech Design for Prediction

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

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

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

• Examples, research overview
• Some computational aspects of PMs
• Combinatorics
• Betting on permutations
• Betting on Boolean expressions
• Automated market makers
• Hansons market scoring rule
• Dynamic parimutuel market

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http//tradesports.com
http//intrade.com
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Play moneyReal predictions
http//www.hsx.com/
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http//us.newsfutures.com/
http//www.ideosphere.com
Cancercuredby 2010
Machine Gochampionby 2020
13
Yahoo!/OReilly Tech Buzz Game
http//buzz.research.yahoo.com/
14
Example IEM 1992
Source Berg, DARPA Workshop, 2002
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Example IEM
Source Berg, DARPA Workshop, 2002
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Example IEM
Source Berg, DARPA Workshop, 2002
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Does money matter? Play vs real, head to head

• Experiment
• 2003 NFL Season
• ProbabilitySports.com Online football forecasting
  competition
• Contestants assess probabilities for each game
• Quadratic scoring rule
• 2,000 experts, plus
• NewsFutures (play )
• Tradesports (real )
• Used last trade prices

• Results
• Play money and real money performed similarly
• 6th and 8th respectively
• Markets beat most of the 2,000 contestants
• Average of experts came 39th (caveat)

Electronic Markets, Emile Servan-Schreiber,
Justin Wolfers, David Pennock and Brian Galebach
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Does money matter? Play vs real, head to head
StatisticallyTS NFNF gtgt Avg TS gt Avg
20
Does it work?Yes...

• Evidence from real markets, laboratory
  experiments, and theory indicate that markets are
  good at gathering information from many sources
  and combining it appropriately e.g.
• Markets like the Iowa Electronic Market predict
  election outcomes better than pollsForsythe
  1992, 1999Oliven 1995Rietz 1998Berg
  2001Pennock 2002
• Futures and options markets rapidly incorporate
  information, providing accurate forecasts of
  their underlying commodities/securitiesSherrick
  1996Jackwerth 1996Figlewski 1979Roll
  1984Hayek 1945
• Sports betting markets provide accurate forecasts
  of game outcomes Gandar 1998Thaler
  1988Debnath EC03Schmidt 2002

21
Does it work?Yes...

• E.g. (contd)
• Laboratory experiments confirm information
  aggregationPlott 198219881997Forsythe
  1990Chen, EC-2001
• And field tests Plott 2002
• Theoretical underpinnings rational
  expectationsGrossman 1981Lucas 1972
• Procedural explanation agents learn from
  pricesHanson 1998Mckelvey 1986Mckelvey
  1990Nielsen 1990
• Proposals to use information markets to help
  science Hanson 1995, policymakers, decision
  makers Hanson 1999, government Hanson 2002,
  military DARPA FutureMAP, PAM
• Even market games work! Servan-Schreiber
  2004Pennock 2001

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Predicting Permutations

• Predict the ordering of a set of statistics
• Horse race finishing times
• 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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Market CombinatoricsBoolean
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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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Hansons Market Maker ILogarithmic Market
Scoring Rule
Thanks Yiling Chen

• n mutually exclusive outcomes
• Shares pay 1 if and only if outcome occurs
• Cost Function
• Price Function

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Hansons Market Maker IIQuadratic Market Scoring
Rule
Thanks Yiling Chen

• We can also choose different cost and price
  functions
• Cost Function
• Price Function

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Log Market Scoring Rule

• Market makers loss is bounded by b ln(n)
• Higher b ?more risk, more liquidity
• Level of liquidity (b) never changes as wagers
  are made
• Could charge transaction fee, put back into b
  (Todd Proebsting)
• Much more to MSR sequential shared scoring rule,
  combinatorial MM for free,... see Hanson 2002,
  2003, 2006

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Computational Issues
Source Hanson, 2002

• Straightforward approach requires exponential
  space for prices, holdings, portfolios
• Could represent probabilities using a Bayes net
  or other compact representation changes must
  keep distribution in the same representational
  class
• Could use multiple overlapping patrons, each with
  bounded loss. Limited arbitrage could be obtained
  by smart traders exploiting inconsistencies
  between patrons

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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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Open QuestionsCombinatorial Betting

• Usual hunt Are there natural, useful, expressive
  bidding languages (for permutations, Boolean,
  other) that admit polynomial time matching?
• Are there good heuristic matching algorithms
  (think WalkSAT for matching) logical reduction?
• How can we divide the surplus?
• What is the complexity of incremental matching?

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Open QuestionsAutomated Market Makers

• For every bidding language with polytime
  matching, does there exist a polytime MSR market
  maker?
• The automated MM algorithms are online
  algorithms Are there other online MM algorithms
  that trade more for same loss bound?

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http//buzz.research.yahoo.com

• Yahoo!,OReilly launched Buzz Game 3/05 _at_ETech
• Research testbed for investigating prediction
  markets
• Buy stock in hundreds of technologies
• Earn dividends based on actual search buzz
• API interface
• Exchange mechanism is dynamic parimutuel
  marketCross btw stock market and horse race
  betting

47
Yahoo!/OReilly Tech Buzz Game
http//buzz.research.yahoo.com/
48
Technology forecasts

• iPod phone

• Whats next?Google Calendar?
• Another Apple unveiling 10/12 iPod Video?

price
searchbuzz
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Analysis
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Tech Buzz Game
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An info market modelComputational properties

• From a computational perspective, we are
  interested in
• What can a market compute?
• How fast? (time complexity)i.e., What mechanisms
  or protocols lead to faster convergence to the
  rational expectations equilibrium?
• Using how many securities? (expressivity and
  representational compactness)i.e., What market
  structures require a minimum of securities yet
  still aggregate information quickly and
  accurately?

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Market computationFeigenbaum EC-2003

• General formulation
• Set up the market to compute some function
  f(x1,x2,,xn) of the information xi available to
  each market participant (e.g., we want the market
  to compute future interest rates given other
  economic variables)
• Represent f(x) as a circuit
• Questions
• How do we set up a marketto compute f?
• How quickly can the marketcompute f?

x1
x2
x3
x4
AND
XOR
OR
f(x1,x2,x3,x4) (x1?x2) ?(x3?x4)
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Market model

• Each participant has some bit of information xi
• There is a security F that pays off 1 if and
  only if f(x)1 at some future date, and 0
  otherwise.
• Trading occurs in synchronous rounds
• In each round, participants bid their true
  expectation
• Clearing price is determined using a simplified
  Shapley-Shubik trading model, yielding mean bid
• Questions we ask/answer
• Does the clearing price converge to a stable
  value?
• How fast does it converge (in how many rounds)?
• Does the stable price of F reveal the true value
  of f?

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Theorems

• For any prior distribution on x, if f(x) takes
  the form of a weighted threshold function
  (i.e.,f(x) 1 iff ?i wixi gt 1 for some weights
  wi), then the market price will ultimately
  converge to the true value of f(x) in at most n
  rounds
• If f(x) cannot be expressed as a weighted
  threshold function (i.e., f(x) is not linearly
  separable), then there is some prior on x for
  which the price of F is stuck at 0.5
  indefinitely, and does not reveal the true value
  of f(x)

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Example and interpretation
x1
x2
x3
x4
Interpretation of theory 1 security
supports computation of threshold fn only More
complex functions must utilize more securities
of securities required threshold circuit size
of f

• In the example, with onlya single security on f,
  themarket may not converge

AND
XOR
OR
f(x1,x2,x3,x4)
1 if (x1?x2) ?(x3?x4)
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Extensions, future work

• Dynamic information revelation and changes
• Overcoming false information
• Obtaining incentive compatibility
• Modeling agent strategies
• Modeling overlapping information sources
• Characterizing in terms of work/round
• Bayesian network representation of prior
• Dealing with limited-precision prices

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Open questions

• What is the relationship between our model and
  perceptron (neural network) learning?
• Perceptrons exactly compute threshold functions
• Could envision a system to learn smallest set of
  threshold functions to approximate desired
  function f, thereby minimizing the number of
  securities required
• Can alternate market protocols lead to faster
  convergence? Can subsidies speed convergence?
• What can other types of securities (e.g.,
  nonbinary securities) compute?

58
Does it work?Yes...

• Evidence from real markets, laboratory
  experiments, and theory indicate that markets are
  good at gathering information from many sources
  and combining it appropriately e.g.
• Markets like the Iowa Electronic Market predict
  election outcomes better than pollsForsythe
  1992, 1999Oliven 1995Rietz 1998Berg
  2001Pennock 2002
• Futures and options markets rapidly incorporate
  information, providing accurate forecasts of
  their underlying commodities/securitiesSherrick
  1996Jackwerth 1996Figlewski 1979Roll
  1984Hayek 1945
• Sports betting markets provide accurate forecasts
  of game outcomes Gandar 1998Thaler
  1988Debnath EC03Schmidt 2002

59
Does it work?Yes...

• E.g. (contd)
• Laboratory experiments confirm information
  aggregationPlott 198219881997Forsythe
  1990Chen, EC-2001
• And field tests Plott 2002
• Theoretical underpinnings rational
  expectationsGrossman 1981Lucas 1972
• Procedural explanation agents learn from
  pricesHanson 1998Mckelvey 1986Mckelvey
  1990Nielsen 1990
• Proposals to use information markets to help
  science Hanson 1995, policymakers, decision
  makers Hanson 1999, government Hanson 2002,
  military DARPA FutureMAP, PAM
• Even market games work! Servan-Schreiber
  2004Pennock 2001

60
Catalysts

• Markets have long history of predictive accuracy
  why catching on now as tool?
• No press is bad press Policy Analysis Market
  (terror futures)
• Surowiecki's Wisdom of Crowds
• Companies
• Google, Microsoft, Yahoo! CrowdIQ, HSX,
  InklingMarkets, NewsFutures
• Press BusinessWeek, CBS News, Economist,
  NYTimes, Time, WSJ, ...http//us.newsfutures.com/
  home/articles.html