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7 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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

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