Last year while working out a few thoughts on arbitrage opportunities in basketball tournament prediction markets at Inkling, it occurred to me that the Inkling pricing mechanism was just a little bit off for such applications. The question is whether something better can be done. An answer comes from the folks at Yahoo Research: yes.

Inkling’s markets come in a couple of flavors, so far as I know all using an automated market maker based on a logarithmic market scoring rule (LMSR). In the multi-outcome case – for example, a market to pick the winner of a 65-team single elimination tournament – the market ensures that all prices sum to exactly 100. If a purchase of team A shares causes its share price to increase by 5, then the prices of all 64 other team shares will decrease by a total of 5.

The logic of the LMSR doesn’t tell you exactly how to redistribute the counter-balancing price decreases. In Inkling’s case they appear to redistribute the counter-balancing price movements in proportion to each team’s previous share price (so, for example, a team with an initial price of 10 would decrease twice as much as a team with a previous price of 5). While for generic multi-outcome prediction markets this approach seems reasonable, it doesn’t seem right for a tournament structure. (I raised this point in a comment posted here at Midas Oracle last September, and responses in that comment thread by David Pennock and Chris Hibbert were helpful.)

The problem arises for pricing tournament markets because the tournament structure imposes certain relationships between teams that the generic pricing rule ignores. Incorporating the structure into the price rule in principle seems like the way to go. Robin Hanson, in his original articles on the LMSR, suggests a Bayes net could be used in such cases. Now three scientists at Yahoo Research have shown this approach works.

In “Pricing Combinatorial Markets For Tournaments,” Yiling Chen, Sharad Goel and David Pennock demonstrate that the pricing problem involved in running a LMSR-based combinatorial market for tournaments is computationally tractable so long as the shares are defined in a particular manner. In the abstract the authors report, “This is the first example of a tractable market-maker driven combinatorial market.”

An introduction to the broader research effort at Yahoo describes the “Bracketology” project in a less technical manner:

Fantasy stock market games are all the rage with Internet users…. Though many types of exchanges abound, they all operate in a similar fashion.

For the most part, each bet is managed independently, even when the bets are logically related. For example, picking Duke to win the final game of the NCAA college basketball tournament in your online office pool will not change the odds of Duke winning any of its earlier round games, even though that pick implies that Duke will have had to win all of those games to get to the finals.

This approach struck the Yahoo! Research team of Yiling Chen, Sharad Goel, George Levchenko, David Pennock and Daniel Reeves as fundamentally flawed. In a research project called “Bracketology,” they set about to create a “combinatorial market” that spreads information appropriately across logically related bets.…

In a standard market design, there are only about 400 possible betting options for the 63-game [sic] NCAA basketball tournament. But in a combinatorial market, where many more combinations are possible, the number of potential combinations is billions of billions. “That’s why you’ll never see anyone get every game right,” says Goel.…

At its core, the Bracketology project is about using a combinatorial approach to aggregate opinions in a more efficient manner. “I view it as collaborative problem solving,” Goel explains. “This kind of market collects lots of opinions from lots of people who have lots of information sources, in order to accurately determine the perceived likelihood of an event.”

Now that they know they can manage a 65-team single elimination tournament, I wonder about more complicated tournament structures. For example, how about a prediction market asking which Major League Baseball teams will reach the playoffs? Eight teams total advance, three division leaders and a wild-card team from the National League and the same from the American League. The wild-card team is the team with the best overall record in the league excepting the three division winners.

In principle the MLB case seems doable, though it would be a lot more complicated that a mere 65-team tournament that has only billions of billions of possible outcomes.

[NOTE: A longer version of this post appeared at Knowledge Problem as “At the intersection of prediction markets and basketball tournaments.”]