### 7-slides

```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
◦ 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 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

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
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 keep track of current state?
◦ Can be computationally intractable for certain trades
◦ Why is this easy in simple markets?

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
Order Matching: Details




( ,  ,  ): (bid, number of shares, event)
Is there a non-trivial subset of orders we can risklessly
accept?
Let  () = 1 if  ∈
: fraction of order to accept
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

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

Market Maker
Price securities efficiently
 Logarithmic 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 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

Bayesian Networks
NLCS
ALCS
World
Series

Any distribution:

Bayes Net distribution:
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
Bayesian Networks
Market Maker


Idea: find trades whose implied probability distributions
are simple Bayesian networks
Exploit properties of Bayesian networks to price and
update efficiently
1.
2.
3.
4.
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
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 1 wins game k
and team 2 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 properties
 Winners of first round games are not independent
 Expect 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


2
separate, independent markets



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 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( )

Applications

Predictalot (Yahoo!)
◦ Combinatorial Market for NCAA basketball
◦ 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 bets
 About 9.2 quintillion possible states of
the world
63
2
2
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
Discussion

Do you think these combinatorial
markets are practical?
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
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
More Criticism
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
Expressiveness v. Tractability
 Allow any trade on the 250 outcomes
◦ (Good): Theoretically can express any information
◦ (Bad): Impossible to keep track of all 250 states
◦ (Good): May be computationally tractable
◦ (Good): More natural betting languages
◦ (Bad): Cannot express some information
◦ (Bad): Inferred probability distribution not intuitive

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
```