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Pricing Combinatorial Markets for Tournaments

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Title: Pricing Combinatorial Markets for Tournaments


1
Pricing Combinatorial Markets for Tournaments
  • Presented by Rory Kulz

2
Main Idea
  • Have seen
  • Combinatorial markets can offer a wider array of
    information aggregation possibilities than
    traditional prediction markets.
  • One way to implement these is with Hansons
    logarithmic market scoring rule (LMSR) market
    maker, but keeping the requisite distributions
    around may (must?) be computationally too hard.
  • Now
  • Look at a restricted case prove feasibility.

3
Outline
  • Preliminaries
  • P, etc. complexity classes
  • Bayesian networks
  • Context
  • Tournament problem
  • Betting languages and results
  • Open problems
  • Discussion

4
Preliminaries Complexity
  • P (Sharp P) is the class of counting problems
    associated to decision problems in NP.
  • 2-SAT
  • Given a boolean CNF formula ?, 2 variables per
    clause.
  • Is there a truth assignment that satisfies the
    formula?
  • 2-SAT
  • How many assignments exist satisfying the
    formula?
  • NP contained in P.
  • Count solutions. Is the count greater than zero?
  • Note that 2-SAT is in P, while 2-SAT is
    P-complete.

5
Preliminaries Bayesian Networks
  • Useful data structure to represent particular
    joint probability distributions P(X1, , Xn),
    especially where dependencies in the distribution
    are sparse.
  • DAG.
  • Each vertex corresponds to one random variable.
  • Edges encode conditional (in)dependence.
  • Key property P(X1,,Xn) ? P(Xiparents(Xi))

6
Preliminaries Bayesian Networks
  • At node representing Xi, store its probability
    conditioned on parents.
  • In general, computing a marginal probability P(Xi
    xi) from this is NP-complete.
  • For certain topologies, however, such as the one
    we will encounter, we can do better.
  • In fact, computing marginal and other conditional
    distributions for our topology can be done in
    O(n).

7
Context
  • Results for LMSR pricing complexity
  • Chen et al., 2008. The following are P-hard
  • Conjunctions or disjunctions of exactly two
    arbitrary events over an event space.
  • Ranking games with subset/pair betting languages.
  • As we saw in Bretts presentation, subset betting
    was P in a matching auction, so were not yet
    seeing tractability for the LMSR except in toy
    cases.

8
Tournaments
  • Elimination tournament, n teams.
  • So total of n 1 games played, which can be
    arranged into a tree structure (think NCAA
    brackets). First round teams picked at start,
    each subsequent round fed by previous rounds
    results.
  • 2n-1 possible outcomes (left or right child at
    each game node).

9
Still a Hard Problem
  • Can show that the pricing problem for tournaments
    for just monotone boolean 2-CNF formulas (a
    conjunction of clauses, each a disjunction of 2
    non-negated literals) is still P-hard.
  • Need to further restrict the betting language,
    i.e. the types of bets we can make.

10
Betting Language 1
  • Allow bets of the following three forms
  • (A) Team i will win game k.
  • (B) Team i will win game k, given that they make
  • it to that game.
  • (C) Team i beats team j, given that they face
    off.
  • How can we get at the pricing complexity?
  • General program to show P time find a Bayesian
    network that fits the problem.

11
Betting Language 1, Cont.
  • Consider Bayesian network arranged like the
    tournament tree, with nodes representing the
    outcome of each game.
  • Let edges go in direction opposite causality.
  • Show network supports uniform distribution and
    updates for bets of type (A).
  • Show network supports bets that are the
    conjunction of two type (A) bets.
  • Demonstrate how to construct assets (B), (C).

12
Betting Language 1 Complexity
  • Can show need to update O(n2) parameters on the
    Bayesian network.
  • Has been shown each update can be accomplished in
    linear time for this type of topology.
  • Hence have complexity O(n3).
  • In particular, polynomial time!

13
Introduced Dependencies
  • Unfortunately, can show for betting language 1,
    certain sequences of bets by players can
    introduce dependencies in the distribution where
    we would expect none.
  • Even between the marginal distributions for the
    outcomes of distinct first-round games!

14
Betting Language 2
  • Restrict to solely bets of type (C).
  • Can model using a Bayesian network where edges
    flow in a normal causal fashion.
  • Also can be priced in polynomial time without the
    dependency problems of the first betting
    language.
  • For this language though, an update phase is only
    O(n).

15
Open Questions
  • It seems like bets of type (A) introduce
    non-local effects into the distribution that
    might be responsible for the weird dependence
    behavior what about restricting to, say, bets
    (B) and (C)?
  • Relation to distribution aggregation problem?
  • Both betting languages offer access to
    polynomially many bets on n. Do there exist
    languages or other problems that are
    superpolynomial but can still be efficiently
    priced?
  • Fully characterize pricing problems for LMSR?

16
Questions/Discussion
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