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 ID: 224719
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Slide1
Combinatorial Betting
Rick Goldstein and John LaiSlide2
Outline
Prediction Markets
vs
Combinatorial Markets
How does a combinatorial market maker work?
Bayesian Networks + Price Updating
Applications
Discussion
Complexity (if time permits)Slide3
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 consistentSlide4
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! outcomesSlide5
Two types of markets
Order matching
Risklessly
match buy and sell orders
Market maker
Price and accept any trade
Thin markets problem with order matchingSlide6
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?Slide7
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 worldSlide8
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
Slide9
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 matchingSlide10
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) eventsSlide11
Market Maker
P
rice securities efficiently
Logarithmic scoring ruleSlide12
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 hardSlide13
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 distributionSlide14
Bayesian Networks
ALCS
NLCS
World Series
Any distribution:
Bayes Net distribution:Slide15
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 graphSlide16
Bayesian Networks
Slide17
Market Maker
Idea: find trades whose implied probability distributions are simple Bayesian networks
Exploit properties of Bayesian networks to price and update efficientlySlide18
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 efficientlySlide19
Basic Lemmas
Slide20
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
Slide21
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 distributionSlide22
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
Slide23
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?Slide24
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(
)
Slide25
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)Slide26
=Slide27
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 makerSlide28
Discussion
Do you think these combinatorial markets are practical?Slide29
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 motionSlide30
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 2Slide31
More Criticism
Slide32
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
PredictalotSlide33
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): Inferred probability distribution not intuitiveSlide34
Tractable Pricing and Updates (optional)
Slide35
Complexity Result (optional)
Slide36
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