Will Lieberman Remain in Office?
Tuesday, August 8th, 2006The latest poll shows the race tightening:
Challenger Ned Lamont … held a lead of 51 percent to 45 percent over Lieberman among likely Democratic voters. The sampling error margin was plus or minus 3.5 percentage points.
The race has tightened in recent days, with Lamont’s lead cut from 13 points.
The prediction markets tell a very different story. Here are the most recent prices for the Tradesports contract that pays $10 if Lieberman wins the primary, $0 if he loses. First, the past month’s data:

And here are today’s prices:

Lieberman has been losing ground steadily for the past month, though he appears to be taking a slight uptick this morning. Right now the market gives him only around a 32% chance of victory in the primary.
Things are looking bleak, but this isn’t the whole story. Lieberman says that if defeated in the primary, he’ll run as an independent. Let’s have a look at another Tradesports contract, one that pays out if a someone other than a Republican or Democrat wins the November senate election. Essentially, this measures the probability that Lieberman will win as an independent since there are no other strong independent or third party candidates. Here’s the graph of lifetime closing prices:

Right now, this contract is trading for $3.01, putting the probability at about 30% that Lieberman will win as an independent.
Let’s use some basic probability to see whats going on here. First a little notation. Let:
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Joe = the event that Lieberman wins the Democratic Primary
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Ned = the event that Lamont wins the Democratic Primary
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Other = the event that someone else wins the Democratic Primary
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I = the event that neither a Democrat nor a Republican wins in November
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Stays = the event that Lieberman stays in office
By the law of total probability, we have:
P(I) = P(I|Joe)P(Joe) + P(I|Ned)P(Ned) + P(I|Other)P(Other)
Of course, there are no real Democratic challengers besides Lamont, so we make the simplifying assumption that P(Other) is approximately zero:
P(I) = P(I|Joe)P(Joe) + P(I|Ned)P(Ned)
Adding in the most recent price for that second Tradesports contract gives:
0.3 = P(I|Joe)P(Joe) + P(I|Ned)P(Ned)
Clearly P(I|Joe) is near zero. If Lieberman wins the Democratic nomination there is less than a snowball’s chance in hell that he will lose the November election. Since we assumed no other Democratic challengers, P(Ned)=1-P(Joe). Adding in the price of the first contract:
0.3 = P(I|Ned)P(Ned) = P(I|Ned)[1-P(Joe)] = P(I|Ned)(0.68)
Hence, P(I|Ned) is approximately equal to 0.44. One more application of the Law of Total Probability will give us Lieberman’s overall chance of staying in the Senate based on the recent Tradesports prices. Assuming that P(Stays|Joe) is about 1, we have:
P(Stays) = P(Stays|Joe)P(Joe) + P(Stays|Ned)P(Ned)
= P(Joe) + P(Stays|Ned)P(Ned)
As mentioned above, P(I) is essentially the probability that Lieberman wins as an independent, which is an equivalent event to (Stays|Ned). In conclusion:
P(Stays) = 0.32 + (0.44)(0.68) = 0.62
The market is betting that Lieberman remains in office.

