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