combinatorial betting
Post on 25-Feb-2016
63 Views
Preview:
DESCRIPTION
TRANSCRIPT
Combinatorial BettingRick Goldstein and John Lai
OutlinePrediction Markets vs
Combinatorial MarketsHow does a combinatorial market
maker work?Bayesian Networks + Price
UpdatingApplicationsDiscussionComplexity (if time permits)
Simple MarketsSmall outcome space
◦ Binary or a small finite number Sports game (binary); Horse race (constant
number) Easy to match orders and price trades
Larger outcome space◦ E.g.: State-by-state winners in an election◦ One way: separate market for each state◦ Weaknesses
cannot express certain information “Candidate either wins both Florida and Ohio
or neither” Need arbitrage to make markets consistent
Combinatorial BettingDifferent approach for large outcome spacesSingle market with large underlying outcome
spaceElections (n binary events)
◦ 50 states, two possible winners for each state, 250 outcomes
Horse race (permutation betting)◦ n horses, all possible orderings of finishing,
n! outcomes
Two types of marketsOrder matching
◦ Risklessly match buy and sell ordersMarket maker
◦ Price and accept any tradeThin markets problem with order matching
Computational DifficultiesOrder matching
◦ Which orders to accept?◦ Is there is a non-null subset of orders we
can accept?◦ Hard combinatorial optimization question◦ Why is this easy in simple markets?
Market maker◦ How to price trades?◦ How to keep track of current state?◦ Can be computationally intractable for
certain trades◦ Why is this easy in simple markets?
Order MatchingContracts costs $q, pays $1 if event occursSell orders: buy the negation of the eventHorse race, three horses A, B, C
◦ Alice: (A wins, 0.6, 1 share)◦ Bob: (B wins, 0.3 for each, 2 shares)◦ Charlie: (C wins, 0.2 for each, 3 shares)
Auctioneer does not want to assume any riskShould you accept the orders?
◦ Indivisible: no. Example: accept all orders, revenue = 1.8, but might have to pay out 2 or 3 if B or C wins respectively
◦ Divisible: yes. Example: accept 1 share of each order, revenue = 1.1, pay out 1 in any state of the world
Order Matching: Details : (bid, number of shares, event) Is there a non-trivial subset of orders we can
risklessly accept?Let if : fraction of order to accept
Order Matching: PermutationsBet on orderings of n variablesChen et. al. (2007)Pair betting
◦ Bet that A beats B◦ NP-hard for both divisible and indivisible
ordersSubset betting
◦ Bet that A,B,C finish in position k◦ Bet that A finishes in positions j, k, l◦ Tractable for divisible orders◦ Solve the separation problem efficiently by
reduction to maximum weight bipartite matching
Order Matching: Binary Eventsn events, 2n outcomesFortnow et. al. (2004)Divisible
◦ Polynomial time with O(log m) events◦ co-NP complete for O(m) events
Indivisible◦ NP-complete for O(log m) events
Market MakerPrice securities efficientlyLogarithmic scoring rule
Market Maker
Pricing trades under an unrestricted betting language is intractable
Idea: reduction If we could price these securities, then we
could also compute the number of satisfying assignments of some boolean formula, which we know is hard
Market MakerSearch for bets that admit tractable pricingAside: Bayesian Networks
◦ Graphical way to capture the conditional independences in a probability distribution
◦ If distributions satisfy the structure given by a Bayesian network, then need much fewer parameters to actually specify the distribution
Bayesian NetworksALCS NLCS
World
Series
Any distribution:
Bayes Net distribution:
Bayesian NetworksDirected Acyclic Graph over the variables in
a joint distributionDecomposition of the joint distribution:
Can read off independences and conditional independences from the graph
Bayesian Networks
Market Maker Idea: find trades whose implied probability
distributions are simple Bayesian networksExploit properties of Bayesian networks to
price and update efficiently
Paper Roadmap1. Basic lemmas for updating probabilities
when shares are purchased on any event A2. Uniform distribution is represented by a
Bayesian network (BN)3. For certain classes of trades, the implied
distribution after trades will still be reflected by the BN (i.e. conditional independences still hold)
4. Because of the BN structure that persists even after trades are made, we can characterize the distribution with a small number of parameters, compute prices, and update probabilities efficiently
Basic Lemmas
Network Structure 1
Theorem 3.1: Trades of the form team j wins game k preserves this Bayesian Network
Theorem 3.2: Trades of the form team wins game k and team wins game m, where game k is the next round game for the winner of game m, preserves this Bayesian Network
Network Structure I Implied joint distribution has some strange
propertiesWinners of first round games are not
independentExpect independence in true distribution;
restricted language is not capturing true distribution
Network Structure II
Theorem 3.4: Trades of the form team i beats team j given that they meet preserves this Bayesian Network structure.
Bets only change distribution at a given node
Equal to maintaining separate, independent markets
Tractable Pricing and Updates
Only need to update conditional probability tables of ancestor nodes
Number of parameters to specify the network is small (polynomial in n)
Counting Exercise: how many parameters needed to specify network given by the tree structure?
Sampling Based MethodsAppendix discusses importance samplingApproximately compute P(A) for implied
market distributionCannot sample directly from P, so use
importance samplingSampling from a different distribution, but
weight each sample according to P()
ApplicationsPredictalot (Yahoo!)
◦Combinatorial Market for NCAA basketball “March Madness”
◦64 teams, 63 single elimination games, 1 winner
Predictalot allowed combinatorial bets◦Probability Duke beats UNC given they play◦Probability Duke wins more games than UNC◦Duke wins the entire tournament◦Duke wins their first game against Belmont
Status points (no real money)
=
Predictalot!Predictalot allows for 263 betsAbout 9.2 quintillion possible
states of the world2263 200,000 possible bets
◦Too much space to store all data◦Rather Predictalot computes
probabilities on the fly given past bets Randomly sample outcome space
Emulate Hanson’s market maker
DiscussionDo you think these combinatorial
markets are practical?
StrengthsNatural betting languagePrediction markets fully elicit beliefs of
participantsCan bet on match-ups that might not be played
to figure out information about relative strength between teams
Conditionally bettingBelieve in “hot streaks”/non-independence then
can bet at better rates that with prediction markets
Correlations Good for insurance + risk calculations
No thin market problemTrade bundles in 1 motion
CriticismDo we really need such an expressive
betting language?◦263 markets◦2263 different bets
What’s wrong with using binary markets?Instead, why don’t we only bet on known
games that are taking place?◦UCLA beats Miss. Valley State in round 1◦Duke beats Belmont in round 1
After round 1 is over, we close old markets and open new markets◦Duke beats Arizona in round 2
More Criticism
Even More Criticism64 more markets for tourney winner
◦Duke wins entire tourney◦UNC wins entire tourney◦Arizona State wins entire tourney
Need 63+64 ~> 2n markets to allow for all bets that people actually make
Perhaps add 20 or so interesting pairwise bets for rivalries?◦Duke outlasts UNC 50%?◦USC outlasts UCLA 5%?
Don’t need 263 bets as in Predictalot
Expressiveness v. TractabilityTradeoff between expressiveness and tractabilityAllow any trade on the 250 outcomes
◦ (Good): Theoretically can express any information◦ (Bad): Traders may not exploit expressiveness◦ (Bad): Impossible to keep track of all 250 states
Restrict possible trades◦ (Good): May be computationally tractable◦ (Good): More natural betting languages◦ (Bad): Cannot express some information◦ (Bad): Inferred probability distribution not intuitive
Tractable Pricing and Updates (optional)
Complexity Result (optional)
How does Predictalot Make Prices? (optional)
Markov Chain Monte Carlo◦Try to construct Markov Chain with
probabilities implied by past bets◦Correlated Monte Carlo Method
Importance Sampling◦Estimating properties of a distribution
with only samples from a different distribution
◦Monte Carlo, but encourages important values Then corrects these biases
top related