### Al Gore defies the laws of probability

As if the huge success of An Inconvenient Truth followed by winning the Nobel Peace Prize wasn't enough, now Al Gore is defying the laws of probability! At least in terms of probable outcomes of the U.S. Democratic nomination and the presidential election itself.

Intrade, the leading commercial prediction market lets you bet on political outcomes. For example, "Barack Obama to win 2008 US Presidential Election" is currently trading at 47.5, meaning the market believes there's a 47.5% chance Obama will be the next president.

What's the probability that Al Gore will win the Democratic nomination? As I write this, the market believes it's 1.7%. Seems fair enough ...

... until you look at the probability that Al Gore will be the next president. Curiously, the market believes that's 2.1%. How could that be? Don't you have to win the nomination before you have a

Let's do a few probability calculations (note that the vertical bar | means "given than"):

Of course there's no such thing as a probability above 100%, which suggests that something is amiss. Perhaps there is a belief that Gore might run as an independent, which would invalidate my assumption that he would have to win the Democratic nomination in order to become president. Or perhaps at any given time the prediction market isn't strictly coherent in the Bayesian sense that it should be consistent with the axioms of probability.

My interpretation is that Gore remains sufficiently attractive as a presidential candidate that if he were nominated, the chances of him winning the election are very high. But not quite 124%.

Intrade, the leading commercial prediction market lets you bet on political outcomes. For example, "Barack Obama to win 2008 US Presidential Election" is currently trading at 47.5, meaning the market believes there's a 47.5% chance Obama will be the next president.

What's the probability that Al Gore will win the Democratic nomination? As I write this, the market believes it's 1.7%. Seems fair enough ...

... until you look at the probability that Al Gore will be the next president. Curiously, the market believes that's 2.1%. How could that be? Don't you have to win the nomination before you have a

*chance*of becoming the president? (My 13-year-old daughter points out that perhaps Gore could run as an independent, but hey, we're Canadians, so this is a great mystery to us. So for now, let's leave this possibility aside.)Let's do a few probability calculations (note that the vertical bar | means "given than"):

Prob(Gore pres.)Therefore

= Prob(Gore pres. & Gore nom.)

= Prob(Gore pres. | Gore nom.) * Prob(Gore nom.)

Prob(Gore pres. | Gore nom.)So

= Prob(Gore pres.) / Prob(Gore nom.)

Prob(Gore pres. | Gore nom.)Therefore, if Gore is nominated, the probability he'll be elected president is 123.5%!

= 2.1%/1.7% =123.5%

Of course there's no such thing as a probability above 100%, which suggests that something is amiss. Perhaps there is a belief that Gore might run as an independent, which would invalidate my assumption that he would have to win the Democratic nomination in order to become president. Or perhaps at any given time the prediction market isn't strictly coherent in the Bayesian sense that it should be consistent with the axioms of probability.

My interpretation is that Gore remains sufficiently attractive as a presidential candidate that if he were nominated, the chances of him winning the election are very high. But not quite 124%.

Labels: Al Gore, Barack Obama, De Finetti, politics, prediction markets, U.S. Presidential election

## 17 Comments:

Nick, I fear this post might be a classic.

Not sure if that's a good or a bad thing! But I've long been fascinated by probabilities greater than one, and "110% effort".

Gore is NUTS!

"My interpretation is that Gore remains sufficiently attractive as a presidential candidate that if he were nominated, the chances of him winning the election are very high. But not quite 124%."

Some thing is missing big time on your interpretation of the election in the United States!

I feel Ralph Nadar has a higher chance of winning the Presidential Election than Sen. Al Gore. And remember working with Congress as an Independent won't be much fun

and like dancing with yourself.

Sorry Al, you time has past your by... Maybe in the Next World!

Hank Dunckel

google me today

I think their idea is that he has a slim chance of winning the presidency as an independant. so he's got a 1.7% chance of winning the democratic ticket, and let's say a 1.4% chance of winning the presidency as a democrat. But he also has a .7% chance of winning the presidency as an independant. Hence, 2.1% total.

Nick, the phenomenon you cite has a lot more to do with how markets work than statistics. Sports bookmakers move the spreads or payoffs depending on how the money flows to each side(the initial quotation does involve probabilities, to be sure).

You actually pointed out an example of market inefficiency. Assuming that Gore must be the Democratic nominee in order to win the election, you could sell the wins-it-all position and buy the nominee position and lock in a profit. As with all arbitrage opportunities, this situation was temporary; check this morning's quotes on Intrade.

I don't really know much about statistics.

I do know that this post is hilarious.

Well done.

I just read about this on another site. It would be interesting to have Gore as a candidate. I like your interpretation of this possible outcome for the candidates

Renaissance Man: the probabilities don't add up like that because the events are not mutually exclusive (to win the presidency as a Democrat you first have to win the Democratic ticket). But I think your intuition is more or less what I had in mind.

Stephen: Thanks for your insightful comments. My knowledge of economics is limited, but I did think of arbitrage when I wrote this. As you suggest, one would expect that the opportunity for arbitrage would be brief, and so I was skeptical about whether that's what I was seeing. I don't know much about market (in)efficiency, but I suspect that for a prediction market it's related to the "coherence" of probabilities that I referred to.

I may be wrong, but I don't think Intrade pays interest on the investment when you "sell" a contract. So if we assume that the probability of gore winning the nomination and presidency are zero, the contract would not go to zero. Nobody would sell at zero because they could invest in a savings account and earn actual money. Instead the contract would remain at a small value and trend toward zero as the contract approaches expiration.

The closer a contract gets to expiration the closer to zero it approaches.

This explains the difference in probabilities!!

The higher value for the presidency contract represents the higher opportunity cost of money associated with a contract that expires at a later date.

I hope I've explained this in a way that is not terribly confusing.

Hi Tony, interesting point. Since I don't have a background in economics or finance, I'm not sure that I completely follow your argument, but I think you might be on to something. One question: since the prevailing interest rate is fairly low, wouldn't this effect be fairly small?

Tony's idea may be contributing to the effect but I believe that there are two other possibilities. First, one would expect little trading in these two contracts. The "last" trade for each of these markets may have been at very different times. In your analysis, you are assuming that you could trade each of these contracts at the same time.

Second, illiquid markets are typically characterized by wide bid-offer spreads. The bid is the price at which you could sell, while the offer is the price at which you could buy. The first trade price that you are seeing may be caused by someone "lifting an offer" (i.e., buying at the higher price), while the second price is due to "hitting the bid" (i.e., someone selling at the lower price). These prices, once again, would be inappropriate to use in your model.

Intrade also has higher transaction costs at the margin, to avoid the long-shot bets I suppose.

I would imagine the probabilities would likely line up if transaction costs approached zero.

More evidence that tells me these prediction markets are bogus. With all uncertainty in the Democratic primary this election cycle, this somehow wouldn't seem all that unlikely...

Highgamma wrote:

one would expect little trading in these two contracts. The "last" trade for each of these markets may have been at very different times. In your analysis, you are assuming that you could trade each of these contracts at the same time.Good points. If I recall correctly, "Al Gore to be the Democratic Presidential Nominee in 2008" was in the "Most actively traded" category. But "Al Gore to be President" was probably less actively traded. It makes me curious about how to quantify some of these characteristics, like frequency of trading and bid-offer spread.

Finance Monk: Could you explain what you mean by "higher transaction costs at the margin"? My knowledge is rather, um, marginal.

Matth.Russell: It seems to me that the prediction markets are not at all bogus. Their past performance has been impressive. The Iowa Electronic Market and in Canada, the UBC Election Stock Market, have both given fairly accurate predictions.

By the way, check out the Iowa Health Prediction Market, which has markets for seasonal influenza activity.

Al gore might have gone over the edge now. Whatever hes taking.....ill take two!!!

Intrade would like you to believe that it's about probability. Really, it's about paramutual betting.

123.5% looks suspiciously close to 6/5. A racetrack would assign 6 to 5 odds to a horse of mediocre prospects, but with name recognition, in order to reduce the betting load on a favorite who is attracting too much attention.

This way, the track --- er, I mean Intrade, pardon me, --- can reduce its payouts and increase it's income.

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