Monday, February 11, 2008

Prediction Markets This Morning

I was feeling optimistic for the first time in a while about Senator Obama's chances for the Democratic party nomination, so I headed over to Intrade to see what the prediction markets were saying. I was a little surprised when I saw the following prices:
  1. Hillary Clinton as Democratic nominee: 29.9
  2. Hillary Clinton as General Election Winner: 21.5
This means that the Intrade market thinks that Clinton has a 29.9% chance to secure the Democratic Party nomination and a 21.5% chance to win the election. To see why these numbers are surprising, consider the following equation:
P(2) = P(2 | 1) P(1) + P(2|~1) P(~1)
That is, the probability of Clinton being elected is equal to the conditional probability of her winning in the event that she gets the nomination plus the conditional probability of her winning in the event that she doesn't. Since we know that she almost certainly can't win the election without getting the nomination, we know everything in the equation except for P(2|1). If we plug in the numbers we have from Intrade and solve, we find that the market's estimation of Clinton's chances of winning the election if she gets the nomination is 0.215 ÷ 0.299, or about 72%.

This seems amazingly high to me, in light of the fact that Clinton has not had a commanding lead in head-to-head polls. Moreover, McCain is trading at about 32% to win the general election, which suggests that one or both of them are overvalued (since 32% + 72% = 104%).

In practical terms this suggests that there is an opportunity for arbitrage in these markets. If a market were showing 32% and 72% for two mutually exclusive outcomes, for instance, you could bet against both of them for $96 and be guaranteed of winning $100, for a risk-free gain of $4. In practice this never happens because traders quickly recognize such absurdities and arbitrage them away. Unfortunately, I can't find a similarly sure-fire strategy for hedging irrational conditional probabilities. There are some strategies with positive expectation values, to be sure, but all of them have some scenarios in which they generate a loss (unlike the case above). This could mean one of (at least) two things:
  1. I haven't looked hard enough (entirely possible, since I have a lot of other stuff to do).
  2. I've botched the analysis (wouldn't be the first time).
  3. Prediction markets can support irrational values for conditional probabilities implied by related (but independently traded) claims.

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