Transcript Market Games for Mining Customer Information
Combinatorial Betting
David Pennock Joint with: Yiling Chen, Lance Fortnow, Sharad Goel, Joe Kilian, Nicolas Lambert, Eddie Nikolova, Mike Wellman, Jenn Wortman
Research
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
March Madness
Research
•
Combinatorics Example March Madness
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Typical today Non-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 k Duke > {UNC,NCST} ACC wins 5 games
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2 264 possible props (implicitly defined)
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1 Bet effects related bets “correctly”; e.g., to enforce logical constraints
Expressiveness: Getting Information
• Things you can say today: – (43% chance that) Hillary wins – GOP wins Texas – YHOO stock > 30 Dec 2007 – Duke wins NCAA tourney • Things you can’t 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
Expressiveness: Processing 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
Research
A (Non-Combinatorial)
•
Prediction Market
Take a random variable, e.g.
Bird Flu Outbreak US 2007?
(Y/N) •
Turn it into a financial instrument payoff = realized value of variable
I am entitled to: $1 if Bird Flu US ’07 $0 if Bird Flu US ’07
Research
Why?
• • Get information
price
probability of uncertain event (in theory, in the lab, in the field, ...next slide) Is there some future event you’d like to forecast?
A prediction market can probably help
[Thanks: Yiling Chen]
Does it work?
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, 1999][Oliven 1995][Rietz 1998][Berg 2001][Pennock 2002] HP market beat sales forecast 6/8 [Plott 2000] Sports betting markets provide accurate forecasts of game outcomes [Gandar 1998][Thaler 1988][Debnath EC’03][Schmidt 2002] Market games work [Servan-Schreiber 2004][Pennock 2001] Laboratory experiments confirm information aggregation [Plott 1982;1988;1997][Forsythe 1990][Chen, EC’01] Theory: “rational expectations” [Grossman 1981][Lucas 1972]
Research
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http://intrade.com
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Screen capture 2007/05/18
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http://intrade.com
http://tradesports.com
Screen capture 2007/05/18 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.
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Research
Intrade Election Coverage
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Combinatorics 1 of 2: Boolean Logic
• Outcomes: All 2 n events possible combinations of n Boolean • Betting language
Buy q units of “$1 if Boolean Formula” at price p
– General:
Any
Boolean formula (2 2 n possible) • A & not(B) (A&C||F) | (D&E) • Oil rises & Hillary wins | Guiliani GOP nom & housing falls • Eastern teams win more games than Western in Tourney – Restricted languages we study • Restricted tournament language Team A wins in round i ; Team A beats B,
Combinatorics 2 of 2: Permutations
• Outcomes: All possible n! rank orderings of n objects (horse race) • Betting language
Buy q units of “$1 if Property” at price p
– General:
Any
property of ordering • A wins 10th A finishes in pos 3,4, or • A beats D 2 of {B,D,F} beat A – Restricted languages we study • Subset betting A finishes in pos 3-5 or 9; A,D,or F finish 3rd • Pair betting
Research
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
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Chen, Fortnow, Nikolova, Pennock, EC’07
Research
Market Combinatorics
Permutations
• • •
A > B > C A > C > B B > A > C .1
.2
.1
• • •
B > C > A C > A > B C > B > A .3
.1
.2
Research
• • • • • • • • • • • •
Market Combinatorics
Permutations D > A > B > C D > A > C > B D > B > A > C A > D > B > C A > D > C > B B > D > A > C A > B > D > C A > C > D > B B > A > D > C A > B > C > D A > C > B > D B > A > C > D .01
.02
.01
.01
.02
.05
.01
.2
.01
.01
.02
.01
• • • • • • • • • • • •
D > B > C > A D > C > A > B D > C > B > A B > D > C > A C > D > A > B C > D > B > A B > C > D > A C > A > D > B C > B > D > A B > C > D > A C > A > D > B C > B > D > A .05
.1
.2
.03
.1
.02
.03
.01
.02
.03
.01
.02
Research
• • • •
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 A>B” 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
Research
Auctioneer Problem
• • •
Auctioneer’s goal: Accept orders with non-negative worst-case loss (auctioneer never loses money)
The Matching Problem
Formulated as LP
•
Generalization: Market Maker Problem: Accept orders with bounded worst-case loss (auctioneer never loses more than b dollars)
Research
Example
• • • •
A three-way match Buy 1 of “$1 if A>B” for 0.7
Buy 1 of “$1 if B>C” for 0.7
Buy 1 of “$1 if C>A” for 0.7
B A C
Research
• • •
Pair Betting
All bets are of the form “A will beat B” Cycle with sum of prices > k-1 ==> Match (Find best cycle: Polytime) Match =/=> Cycle with sum of prices > k-1
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Theorem: The Matching Problem for Pair Betting is NP-hard (reduce from min feedback arc set)
Research
•
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)
Research
Market Combinatorics
Boolean
I am entitled to: $1 if A1&A2&…&An I am entitled to: $1 if A1 &A2&…&An I am entitled to: $1 if A1& A2 &…&An I am entitled to: $1 if A1&A2&…& An I am entitled to: $1 if A1 &A2&…& An I am entitled to: $1 if A1& A2 &…& An I am entitled to: $1 if A1 & A2 &…&An I am entitled to: $1 if A1 & A2 &…& An •
Betting on complete conjunctions is both unnatural and infeasible
Research
• •
Market Combinatorics
Boolean A bidding language: write your own security
I am entitled to: $1 if Boolean_fn | Boolean_fn
For example
I am entitled to: $1 if A1 | A2 I am entitled to: $1 if A1& A7 • • • • I am entitled to: $1 if (A1& A7) ||A13 | (A2|| A5 )&A9
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
Research
• • • •
The Matching Problem
There are many possible matching rules for the auctioneer A natural one: maximize trade subject to no-risk constraint Example:
• • • trader gets $$ in state: A1A2 A1 A2 A1 A2 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
No matter what happens, auctioneer cannot lose money
-0.10 -0.10 -0.10 -0.10
Research
Fortnow; Kilian; Pennock; Wellman • •
Complexity Results
Divisible orders: will accept any q*
Indivisible: will accept all or nothing q
LP reduction from X3C # events O(log n) O(n) divisible polynomial co-NP-complete indivisible NP-complete 2 p complete • reduction from SAT
Natural algorithms
• •
divisible: linear programming indivisible: integer programming; logical reduction?
reduction from T BF
Research
[Thanks: Yiling Chen]
Automated Market Makers
• • •
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]
Research
[Thanks: Yiling Chen] • • • • • • •
Automated Market Makers
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 ΔQ shares trader pays C(Q+ ΔQ) – C(Q) to the market maker; Negative “payment” = receive money Instantaneous price functions are
p i
(
Q
)
C
(
Q
)
q i
At the beginning of the market, the market maker sets the initial Q 0 , hence subsidizes the market with C(Q 0 ). At the end of the market, C(Q the MM will pay out.
f ) is the total money collected in the market. It is the maximum amount that
Research
• • • • • •
New Results in Pipeline: Pricing 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
Overview: Complexity Results
Permutations 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
•
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 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.
Dynamic Parimutuel Market: An Automated Market Maker
Research
What is a pari-mutuel market?
A B • • •
E.g. horse racetrack style wagering Two outcomes: Wagers: A B
Research
What is a pari-mutuel market?
A B • • •
E.g. horse racetrack style wagering Two outcomes:
A Wagers: B
Research
What is a pari-mutuel market?
A B • • •
E.g. horse racetrack style wagering Two outcomes:
A Wagers: B
Research
• •
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
Research
Pari-Mutuel Market
Basic idea
1 1
Research
Dynamic Parimutuel Market
C(
1
,
2
)=2.2
C(
2
,
3
)=3.6
C(
2
,
2
)=2.8
C(
2
,
4
)=4.5
C(
3
,
8
)=8.5
C(
4
,
8
)=8.9
C(
2
,
5
)=5.4
C(
5
,
8
)=9.4
C(
2
,
6
)=6.3
C(
2
,
8
)=8.2
C(
2
,
7
)=7.3
Research
• •
Share-ratio price function
One can view DPM as a market maker Cost Function:
C
(
Q
)
i n
1
q i
2 • •
Price Function:
p i
(
Q
)
Properties
• • • •
No arbitrage price i /price j = q i /q j price i < $1 payoff if right = C(Q final )/q o > $1
q i j n
1
q j
2
Research
Mech Design for Prediction
Primary Secondary Financial Markets Social welfare (trade) Hedging risk Information aggregation Prediction Markets Information aggregation Social welfare (trade) Hedging risk
Research
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 with: experts, scoring rules, opinion pools, ML/stats, polls, Delphi