The most common prediction markets we see are binary markets; those with two possible outcomes. Will Candidate X be reelected, which team will win a sports contest, will so-and-so be convicted, etc. The next most common is determining the outcome from a list of possibilities known before the event: winner of a multi-candidate election, the World Cup, or the Super Bowl. This post talks about another kind of market: predicting the value of a continuous variable, as in how much snowfall will NYC get this winter, how many cases of flu will there be in Iowa next month, the level of a company's sales, the value of the Dow-Jones Industrial Average. Even "When will social security reform be passed?" and "When will the new product be shipped?" can be expressed in this format: the date of occurrence is the predicted value.
There are three common approaches to predicting continuous variables. I call them price bands, price ladders, and scaled claims. In the price ladder representation, a series of securities is offered, and each one is phrased as "the value will be lower than X". A series of securities with different values of X can cover all the possibilities (as long as the sequence is capped by a security that includes all other values). Price bands phrase each security's claim as "the value will be between X and Y". (These need to be capped on both ends by "W or less" and "Z or more".) Scaled claims represent the same kind of question with a single security that pays off a variable amount determined by the outcome.
Bands and Ladders are duals: any bet that can be made in one system can be made in the other with a structured bet. If you believe that the outcome will be between X and Y, and the market offers only ladders, then you buy X and sell Y. If the market offers bands and you want to bet that the price will be above (or below) some value, you buy all the securities at or above that value. Scaled claims don't offer these options.
The question to answer in designing these markets is which approach is more convenient to the trader, and more amenable to analysis. (If the output of these markets are predictions, then we want usable predictions.) In order to offer markets in continuous outcomes, the market operator has to decide what a plausible range of outcomes would be, and decide what possibilities are reasonable choices. That helps determine how many bands and how wide they should be. If the consensus view expects a value to be between 1200 and 1500, someone offering bands or ladders wants to set things up so there are choices ranging from 1100 to 1600, with some choices within the consensus forecast. With variable payouts, if the value is going to change very much, the outcomes of interest should take up as much of the probability space as possible. On FX (where design of the securities is a subject of public discussion) there is often a discussion about what outcomes are most likely in order to choose a wide active range that the price will move around in. If the market operator can't tell where the contentious issue will lie, they are forced to choose between many securities, most of which will be uninterested, or a few wide bands, and risk that there's little disagreement on the outcome (to the resolution the claim provides) for much of the claim's lifetime.
It isn't necessary that the bands be of equal sizes. When the magnitude of the outcome is highly variable, a log scale can be used. On FX, log scales are often considered for death tolls for epidemics or other disasters, though I couldn't find any claims that ended up using it. It can also make sense to have narrower bands in the region of the most likely outcomes, and wider bands further away. Robin Hanson suggested another approach: split up the bands as the consensus changes. In order to be fair to the traders, the securities should cover all the possibilities (i.e. no gaps or excluded ranges; most easily done by having the lowest range be "< X" and the highest be "> Y".) If one band has a high percentage of the interest, split it into sub-ranges, and give each investor who owns shares in the range being split a corresponding number of each of the new shares. Reverse splits can't be done cleanly, so it's important to not split too soon or too often if the UI doesn't handle lots of claims well.
Another weakness of scaled claims is that the market only produces one price, so you can't tell when investors' opinions are bimodal. For instance, if many people think that there will either be a huge epidemic or a low effect, with medium epidemics being unlikely.
The market will be more liquid if people can express their view without having to buy multiple securities to cover their expectations. So to some extent the best choice depends on whether people are more likely to think they know the most likely value, or more likely to think "the value will be at least X" (or "at most X"). I think people usually find it easier to decide on a maximum or minimum value than a most-likely range. On the other hand, price bands make it easier to understand the market's forecast. The calculations of implied prices based on the prices of puts and calls in the stock market are complicated, and deriving a prediction from prices of laddered securities should be just as involved.
If the securities are built as price bands, the approach to improving liquidity in N-way markets that I described in a previous article would be applicable. The same trick doesn't apply to laddered securities, since those assets can't be expressed as linear combinations of one another.
TradeSports had a market in snowfall in New York City last winter, and many of their financial bets are for continuous variables. They also offered their bets on capture of Saddam Hussein and Osama bin Laden as well as the date of passage of Social Security reform as a series of deadline dates, which have the same semantics as ladders.
FX has scaled claims, and this form is also used by IEM for judging the popular vote (as opposed to the winner-take-all market based on the elector college outcome.) IEM's local flu markets (currently inactive?) were done as price bands as well (different colors represented different observed rates of flu.)
HedgeStreet is currently using overlapping bands in some of their markets (50-60, 55-65, 60-70). This gives more choices to the customer, but that means it splits the liquidity. Doubling the number of outcomes investors have to pay attention to cuts liquidity approximately in half. I can't think of an advantage of this choice. HedgeStreet also has price bands that aren't capped, so it's possible that none of the bands will include the outcome.
NewsFutures isn't currently offering any contracts on continuous variables that I could find. I don't remember any in the past either. HSX's stocks are open-ended continuous payout securities. Yahoo! Tech Buzz has discreet outcomes with payouts proportinal to the search measure.
Previous articles in this series:
- PM intro: basic formats (2005-12-30)
- PMs with Open-ended Prices (2006-01-05)
- Looking at Both Sides (2006-04-17)
- Book and Market Maker (2006-04-28)
- Liquidity in N-Way claims (2006-07-19)