Pricing Combinatorial Markets for Tournaments

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Pricing Combinatorial Markets for Tournaments

• Presented by Rory Kulz

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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.

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Outline

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

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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.

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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))

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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).

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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.

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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).

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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.

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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.

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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).

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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!

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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!

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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).

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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?

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Questions/Discussion