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
S
ports 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, 2
50
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
◦
W
hich
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?
◦
C
an
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
(
𝑏
𝑖
,
𝑞
𝑖
,
𝐴
𝑖
)
: (bid, number of shares, event)
Is there a non

trivial subset of orders we can
risklessly
accept?
Let
𝐼
𝑖
(
𝑠
)
=
1
if
𝑠
∈
𝐴
𝑖
𝑥
𝑖
: 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, 2
n
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
P
rice securities efficiently
Logarithmic scoring rule
Market Maker
Pricing trades under an unrestricted betting language is
intractable
Idea: reduction
I
f 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
ALCS
NLCS
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
Paper Roadmap
1.
Basic
lemmas
for updating probabilities
when shares
are purchased on
any
event
A
2.
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
𝑗
1
wins game k
and team
𝑗
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
𝑛
2
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 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(
𝑋
𝑖
)
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)
=
Predictalot
!
Predictalot
allows for 2
63
bets
About 9.2 quintillion possible states of
the world
2
2
63
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
Trade bundles in 1 motion
Criticism
Do we really need such an expressive
betting language?
◦
2
63
markets
◦
2
2
63
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 2
63
bets as in
Predictalot
Expressiveness v. Tractability
Tradeoff between expressiveness and tractability
Allow any trade on the 2
50
outcomes
◦
(Good): Theoretically can express any information
◦
(Bad): Traders may not exploit expressiveness
◦
(Bad): Impossible to keep track of all 2
50
states
R
estrict possible trades
◦
(Good): May be computationally tractable
◦
(Good): More natural betting languages
◦
(Bad): Cannot express some information
◦
(Bad):
I
nferred 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
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