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(X 1, …, X n ), especially where dependencies in the distribution are sparse. DAG. – Each vertex corresponds to one random variable. – Edges encode conditional (in)dependence. Key property: P(X 1,…,X n ) = ∏ P(X i |parents(X i ))

6. Preliminaries: Bayesian Networks At node representing X i, store its probability conditioned on parents. In general, computing a marginal probability P(X i = x i ) 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, 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. 2 n-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(n 2 ) 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(n 3 ). 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