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Pricing Combinatorial Markets for Tournaments

Pricing Combinatorial Markets for Tournaments. Presented by Rory Kulz. Main Idea. Have seen: Combinatorial markets can offer a wider array of information aggregation possibilities than traditional prediction markets.

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Pricing Combinatorial Markets for Tournaments

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  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 Hanson’s 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(Xi|parents(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 Brett’s presentation, subset betting was P in a matching auction, so we’re not yet seeing tractability for the LMSR except in toy cases.

  8. Tournaments • Elimination tournament, nteams. • So total of n – 1games 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-1possible 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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