# Combinatorial Betting

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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 - PowerPoint PPT Presentation

### Text of Combinatorial Betting

Combinatorial Prediction Markets

Combinatorial BettingRick Goldstein and John LaiOutlinePrediction Markets vs Combinatorial MarketsHow does a combinatorial market maker work?Bayesian Networks + Price UpdatingApplicationsDiscussionComplexity (if time permits)Simple MarketsSmall outcome spaceBinary or a small finite numberSports game (binary); Horse race (constant number)Easy to match orders and price tradesLarger outcome spaceE.g.: State-by-state winners in an electionOne way: separate market for each stateWeaknessescannot express certain informationCandidate either wins both Florida and Ohio or neitherNeed 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 outcomesHorse race (permutation betting)n horses, all possible orderings of finishing, n! outcomesTwo types of marketsOrder matchingRisklessly match buy and sell ordersMarket makerPrice and accept any tradeThin markets problem with order matchingComputational DifficultiesOrder matchingWhich orders to accept?Is there is a non-null subset of orders we can accept?Hard combinatorial optimization questionWhy is this easy in simple markets?Market makerHow to price trades?How to keep track of current state?Can be computationally intractable for certain tradesWhy 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, CAlice: (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 respectivelyDivisible: yes. Example: accept 1 share of each order, revenue = 1.1, pay out 1 in any state of the worldOrder Matching: Details

Order Matching: PermutationsBet on orderings of n variablesChen et. al. (2007)Pair bettingBet that A beats BNP-hard for both divisible and indivisible ordersSubset bettingBet that A,B,C finish in position kBet that A finishes in positions j, k, lTractable for divisible ordersSolve the separation problem efficiently by reduction to maximum weight bipartite matchingOrder Matching: Binary Eventsn events, 2n outcomesFortnow et. al. (2004)DivisiblePolynomial time with O(log m) eventsco-NP complete for O(m) eventsIndivisibleNP-complete for O(log m) eventsMarket MakerPrice securities efficientlyLogarithmic scoring rule

Market MakerPricing trades under an unrestricted betting language is intractableIdea: reductionIf 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 NetworksGraphical way to capture the conditional independences in a probability distributionIf distributions satisfy the structure given by a Bayesian network, then need much fewer parameters to actually specify the distribution

Bayesian NetworksALCSNLCSWorld SeriesAny distribution:

Bayes Net distribution:

14Bayesian 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 MakerIdea: find trades whose implied probability distributions are simple Bayesian networksExploit properties of Bayesian networks to price and update efficientlyPaper RoadmapBasic lemmas for updating probabilities when shares are purchased on any event AUniform 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 efficientlyBasic Lemmas

Network Structure 1

Network Structure IImplied joint distribution has some strange propertiesWinners of first round games are not independentExpect independence in true distribution; restricted language is not capturing true distributionNetwork Structure II

Tractable Pricing and UpdatesOnly need to update conditional probability tables of ancestor nodesNumber 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 MethodsApplicationsPredictalot (Yahoo!)Combinatorial Market for NCAA basketballMarch Madness64 teams, 63 single elimination games, 1 winnerPredictalot allowed combinatorial betsProbability Duke beats UNC given they playProbability Duke wins more games than UNCDuke wins the entire tournamentDuke wins their first game against BelmontStatus points (no real money)

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Predictalot!Predictalot allows for 263 betsAbout 9.2 quintillion possible states of the world2263 200,000 possible betsToo much space to store all dataRather Predictalot computes probabilities on the fly given past betsRandomly sample outcome spaceEmulate Hansons market maker

27DiscussionDo 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 teamsConditionally bettingBelieve in hot streaks/non-independence then can bet at better rates that with prediction marketsCorrelationsGood for insurance + risk calculationsNo thin market problemTrade bundles in 1 motion

CriticismDo we really need such an expressive betting language?263 markets2263 different betsWhats wrong with using binary markets?Instead, why dont we only bet on known games that are taking place?UCLA beats Miss. Valley State in round 1Duke beats Belmont in round 1After round 1 is over, we close old markets and open new marketsDuke beats Arizona in round 2More CriticismEven More Criticism64 more markets for tourney winnerDuke wins entire tourneyUNC wins entire tourneyArizona State wins entire tourneyNeed 63+64 ~> 2n markets to allow for all bets that people actually makePerhaps add 20 or so interesting pairwise bets for rivalries?Duke outlasts UNC 50%?USC outlasts UCLA 5%?Dont need 263 bets as in Predictalot32Expressiveness 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 statesRestrict 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 CarloTry to construct Markov Chain with probabilities implied by past betsCorrelated Monte Carlo MethodImportance SamplingEstimating properties of a distribution with only samples from a different distributionMonte Carlo, but encourages important valuesThen corrects these biases

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