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Combinatorial Betting

Combinatorial Betting. Rick Goldstein and John Lai. Outline. Prediction Markets vs Combinatorial Markets How does a combinatorial market maker work? Bayesian Networks + Price Updating Applications Discussion Complexity (if time permits). Simple Markets. Small outcome space

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Combinatorial Betting

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  1. Combinatorial Betting Rick Goldstein and John Lai

  2. Outline • Prediction Markets vs Combinatorial Markets • How does a combinatorial market maker work? • Bayesian Networks + Price Updating • Applications • Discussion • Complexity (if time permits)

  3. Simple Markets • Small 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

  4. Combinatorial Betting • Different approach for large outcome spaces • Single market with large underlying outcome space • Elections (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

  5. Two types of markets • Order matching • Risklessly match buy and sell orders • Market maker • Price and accept any trade • Thin markets problem with order matching

  6. Computational Difficulties • Order 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?

  7. Order Matching • Contracts costs $q, pays $1 if event occurs • Sell orders: buy the negation of the event • Horse 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 risk • Should 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

  8. 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

  9. Order Matching: Permutations • Bet on orderings of n variables • Chen et. al. (2007) • Pair betting • Bet that A beats B • NP-hard for both divisible and indivisible orders • Subset 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

  10. Order Matching: Binary Events • n events, 2n outcomes • Fortnow 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

  11. Market Maker • Price securities efficiently • Logarithmic scoring rule

  12. 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

  13. Market Maker • Search for bets that admit tractable pricing • Aside: 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

  14. Bayesian Networks ALCS NLCS • Any distribution: • Bayes Net distribution: World Series

  15. Bayesian Networks • Directed Acyclic Graph over the variables in a joint distribution • Decomposition of the joint distribution: • Can read off independences and conditional independences from the graph

  16. Bayesian Networks

  17. Market Maker • Idea: find trades whose implied probability distributions are simple Bayesian networks • Exploit properties of Bayesian networks to price and update efficiently

  18. Paper Roadmap • Basic lemmas for updating probabilities when shares are purchased on any event A • Uniform distribution is represented by a Bayesian network (BN) • For certain classes of trades, the implied distribution after trades will still be reflected by the BN (i.e. conditional independences still hold) • 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

  19. Basic Lemmas

  20. 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

  21. Network Structure I • Implied joint distribution has some strange properties • Winners of first round games are not independent • Expect independence in true distribution; restricted language is not capturing true distribution

  22. 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

  23. 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?

  24. Sampling Based Methods • Appendix discusses importance sampling • Approximately compute P(A) for implied market distribution • Cannot sample directly from P, so use importance sampling • Sampling from a different distribution, but weight each sample according to P()

  25. Applications • Predictalot (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)

  26. =

  27. Predictalot! • Predictalot allows for 263 bets • About 9.2 quintillion possible states of the world • 2263 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

  28. Discussion • Do you think these combinatorial markets are practical?

  29. Strengths • Natural betting language • Prediction markets fully elicit beliefs of participants • Can bet on match-ups that might not be played to figure out information about relative strength between teams • Conditionally betting • Believe in “hot streaks”/non-independence then can bet at better rates that with prediction markets • Correlations • Good for insurance + risk calculations • No thin market problem • Trade bundles in 1 motion

  30. Criticism • Do 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

  31. More Criticism

  32. Even More Criticism • 64 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

  33. Expressiveness v. Tractability • Tradeoff between expressiveness and tractability • Allow 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

  34. Tractable Pricing and Updates (optional)

  35. Complexity Result (optional)

  36. 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

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