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AI and Robotics

Nov 7, 2013 (5 years and 5 months ago)

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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

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

Market maker

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 keep track of current state?

C
an
be computationally

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

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
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

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!)

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

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

Allow any trade on the 2
50

outcomes

(Good): Theoretically can express any information

(Bad): Impossible to keep track of all 2
50

states

R

(Good): May be computationally tractable

(Good): More natural betting languages

I
nferred 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