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## September 25, 2004

### The Predictions of Prediction Markets

Manski has an *Econometrica* article about markets like the Iowa presidential market where people trade on probabilities of events happening. If the market price of a Bush contract paying $1 if he wins is .70, does that mean the market puts a 70% probability on a Bush victory? Sort of, but not really. Michael Stastny writes at MR:

On TradeSports a contract of George Bush in the winner-take-all market is currently selling for around $7. But what does this actually mean? Most traders and researchers would argue that 0.7 is the current "market probability" that the event "George Bush wins the 2004 presidential election" occurs. But this answer drives Charles Manski, an economist who recently also published an article in the September issue of

Econometrica, crazy.He says that under not-so-far-fetched assumptions, the price of a contract reveals nothing about the dispersion of traders' beliefs and partially identifies the central tendency of beliefs. A President.GWBush2004 contract trading at 70 reveals that 70 percent of traders believe the probability of the event "George Bush wins the 2004 presidential election" to be larger than 0.7. The mean subjective probability of this event lies somewhere in the open interval (0.49, 0.91) (price/mean belief region).

Alex Tabarrok at MR and posts at Bainbridgeand Deadparrots and probably elsewhere discuss this too. I haven't read it all, or even glanced at Manski's article, but here are some thoughts on how Manski's idea might work.

Suppose all traders have identical wealth and are risk neutral. Each will then bet his entire wealth based on his belief that Bush will win the election. Start with the simple case of two kinds of traders. Suppose 30% think Bush has 0 chance of winning, and 70% think he has 100% chance of winning. The mean market belief will be .70. When the dust settles, each side having bet all its wealth, the price of a George Bush contract in this market will be 70.

But this same result would have occurred if 30% think Bush has 0 chance of winning, and 70% think he has 70.001% chance of winning. For the Bush Optimists, a contract at 70 is still a good deal. But now the mean market belief is .49.

Or, suppose 30% think Bush has .6999 chance of winning, and 70% think he has 100% chance of winning. For the Bush pessimists, a contract at 70 is still a good deal (to sell--not to buy). But now the mean market belief is .91.

Notice that market behavior is based on *inequalities, not equalities*. What matters is whether the market price is lower than my subjective belief, not how much lower.

Thus, a market price of 70 just indicates a mean market belief between .49 and .91. That includes my first case, where the mean market belief is exactly .70, but also includes a lot of other ground.

This works because I have chosen the most extreme possible market belief dispersion-- lots of 0's or 100's. I've also assumed risk neutrality and identical wealths, but those are the reasonable assumptions. It is obvious that if all the 100% Bush Optimists have no wealth or are extremely risk averse then they will not bet even though they think they'd win, and so they would have no effect on the market price.

Posted by erasmuse at September 25, 2004 09:08 AM

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## Comments

you wrote:"Notice that market behavior is based on *inequalities, not equalities*."

this is quite incorrect,for it to be a trade the degree of belief of the bush buyer must be exactly the same [save the sign] than that of the kerry buyer and it is :for bush 0.7 and kerry 0.3 we have log(0.7/0.3)=-log(0.3/0.7)...

Posted by: jckommer at September 27, 2004 02:09 PM

In reality almost no one would invest their entire wealth in Bush contracts at .70 just because they think the actual chance of Bush winning is .70001. So Manski's model is unrealistic.

If you make the more reasonable assumption that the amount invested is proportional to the difference between the market price and the investors estimate of the correct price then the market clearing price is the weighted (by investor wealth) average of the investor's estimates of the correct price.

Posted by: James B. Shearer at September 27, 2004 07:19 PM

Here's how to get to a .70 price. Suppose I am risk neutral and I and people like me are more numerous than anybody else, and I can buy what I think is a .701 chance of winning $1 for .70. I will spend my entire wealth buying such contracts, if I can. Since I have more money than other people, though, the .70 price is not stable. A seller could tell me that he knows I have no alternative sellers and that he will hold out for .701. The Bush sellers can hold out for .701, and that is the only stable price. The beliefs of the sellers don't matter to this-- only, in this example, their lesser wealth.

James Shearer notes that the assumption of risk neutrality is unrealistic. Manski's result does not depend on it, though, I think. If all traders are equally risk averse and equally wealthy, wouldn't the market price be the same regardless of the level of risk aversion? Risk aversion would matter only if there is some asymmetry in the traders.

Posted by: Eric Rasmusen at September 28, 2004 09:59 PM