Well, I sure don’t know. But thanks to the wonders of conditional probability and the data coming from prediction markets like Intrade, we can calculate those markets’ implied answer. Intrade markets separately give us the probability that a candidate will win his or her party’s nomination and that he or she will win the Presidency in the general election. From that, figuring out the conditional probability that the candidate will win the November 2012 election having already won the nomination—a proxy for electability—just requires division.

Consider former Alaska Governor Sarah Palin. As of this morning, one Intrade market gives her a 0.072 probability (or 7.2% chance) of being the GOP nominee, and a separate market gives her a 0.025 probability of being elected President in 2012. For the moment, grant me the assumption that the bettors are not envisioning her general election victory coming from a third-party candidacy (although if anyone could revive the Bull Moose Party…). Given that, her implied probability of winning the November election after having won the nomination is .025/.072, or .347. As compared to the generic Republican nominee’s .378 probability of victory in November, the markets put her at a disadvantage. The same is true of Michele Bachmann, whose probability of winning in November conditional on being nominated is .339.

Still, prediction markets don’t necessarily handle long-shot candidates well. So consider instead former Minnesota Governor Tim Pawlenty. With a .188 probability of winning the GOP nomination and a .071 probability of being elected President, his market-implied probability of winning the White House with the GOP nomination already in hand is 0.377. His conditional probability of victory in November is almost indistinguishable from the generic GOP nominee’s probability. It seems that New York Times blogger Nate Silver was right to call him a “league-average politician.”

Now let’s turn to Mitt Romney, another former Governor. With a .299 chance of winning the GOP nomination and a .123 chance of winning both the nomination and the general election, his implied probability of winning the Presidency given the GOP nomination is 0.411. But it is former Utah Governor Jon Huntsman that the markets see as the most powerful GOP contender for the fall of 2012. With a .150 probability of winning the GOP nomination and a .072 probability of winning the Presidency, the Intrade markets imply that Huntsman would have a 0.48 probability of taking the Presidency from the Republican National Convention’s stage in Tampa, Florida. In other words, to bettors, the GOP candidate (and the events correlated with each candidate’s fortunes) matters markedly. Conditional on his winning the nomination, they would give Huntsman almost even odds of becoming President in 2012.

I don’t know what the formula was here, but I’m still putting my money on Romney. He raised $10 million in a day. That’s more then enough to win New Hampshire and Iowa.

You’re stupid.

by way of follow-on to the recent post showing the lack of predictive power of polls 300 days out, i wonder what these markets looked like this far in advance of the last election.

if polling people in a certain circumstance is no good for predicting, by what mechanism would the market be any different (putting aside, i suppose, its greater emphasis on the opinions of people with disposable cash, which does seem relevant)? does the market handle “errors” differently than polls, thereby arriving at a “better” estimate?

This is fun, but I don’t buy the conversion of these prices into probabilities.

The flaw in this argument is that it assumes that the odds of someone getting the republican nomination is independent from the odds of their winning the presidency. This is clearly not the case. Rather, the winner of the nomination also signifies, indirectly, other changes in the political landscape which will also influence the odds of a candidate winning. For example, if Huntsman were to win the nomination , it would suggest a DECREASE in how influential the tea party is, relative to today, and this would be a confounding factor, whereas a scenario where Palin would win the nomination would suggest an INCREASE in how influential the tea party is. Thus if, miraculously, Palin were to win the nomination, and then, as a thought-experiment, Palin were to choose Huntsman to take her place, Huntsman would more likely lose, than a scenario where he had been actually chosen as the nominee by electors.

In reply to Salamander’s post, I don’t see where the original post makes any such independence assumption. It is entirely true that the set of events that might lead a certain candidate to win the nomination would also influence his or her probability of winning the general election. But as of today, if you are willing to interpret the prices from this particular market as probabilities, these conditional probabilities tell us what the Intrade market thinks the probability of each candidate winning the general election given the (potentially quite different) events that would lead that candidate to be nominated.

Dan: I don’t recommend that you interpret the prices as probabilities. I agree there’s some connection between prices and probabilities but I think it’s a mistake to interpret the prices as probabilities, except as an amusing exercise.

Salamander is correct that conditional probabilities don’t necessarily tell you electability. When you condition on a candidate winning the nomination, you’re assuming that all of the things that would need to happen for the candidate to win the nomination take place. There is uncertainty about the political landscape, as Salamander said, and also about the candidate’s qualities, and by only looking at the possible worlds where the candidate wins the nomination you are resolving this uncertainty in a particular way. (These statements should be probabilistic.)

A bad economy hurts Obama’s chances, so whichever Republican candidate benefits most in the primaries from a bad economy is going to have a high probability of winning the general election conditional on winning the primary, but that doesn’t mean that candidate is more electable. The Republicans won’t help their chances by picking that candidate. Similarly, if a candidate might have a fatal flaw (lack of campaigning skills, an embarrassing secret), that possible fatal flaw won’t reduce his conditional probability, because if it turns out that he is fatally flawed then he’ll be weeded out in the primary. Republicans who care about electability shouldn’t ignore his potential flaw and give him the nomination just because he still has a strong conditional probability.

A: The event that he is electable.

B: The event that he will be nominated.

I think you are using the Bayesian formula P(A|B) = P(A,B)/P{B).

If A and B are independent, P(A|B)=P(A). So if electability is independent of nominatability (I just made up a word), what you are computing is actually – to be redundant – electability Of course in this case the probability that he electable if he is nominated is the same as the probability that he is electable (whether he is nominated or not).

Since all you are claiming is that the number is the probability that he electable if he is nominated , the independence is not required but is not ruled out either.

Just to clear up the basics in light of some posts above.

If I understand your issue correctly, these markets are generating payoffs (or expected values) not probabilities. Otherwise, assuming your math is correct, it seems to me that the sum across all GOP candidates for the probability of being elected president should not exceed 1.0 (which they do).

except … based on the descriptions of the propositions in the original post, these are separate markets. the general election market is not looking at the other market before setting its prices (or maybe it is, but this is internalized in setting the general elkection price, as someone above notes), and

(most importantly, and again assuming the description as written) when it comes time to pay-out in the general pool, they won’t be refunding the money of all investors who bet on someone who didn’t even get the nomination. those people will be holding shares that went to zero sometime before the convention. the probability that s/he is electable if not nominated … is zero. the pricing market will eventually reflect that (these work just like stock options, IIRC). right now, the market is simply wrong in all but one case (and maybe in that case as well, but for other reasons). only one candidate will eventually have value in the second pool,, and the total value of all propositions in that pool (including obama’s) should add to one, at any given time (or the market is not efficient).

so, no matter what interpretation of price to probability you choose, or what interaction terms you introduce, it is not meaningful to multiply the prices together. at least, not based on the descriptions of the propositions offered in the original post.

ah, missed ben’s post, which makes a similar point to mine, only more efficiently.

Andrew: yup, the goal was nothing more than a fun exercise.

Vince, samsa, Salamander: that’s the math, thanks. I did add a parenthetical comment to the end of the post in light of Vince and Salamander’s caveats.

Ben: conditional probabilities which condition on different events are not restricted to sum to 1.

in re: using prices as probabilities

It depends on what kind of probability you’re talking about. It’s common to infer from the price at which someone regards a bet as fair the degree of belief they have in the proposition in question; if I would pay $1 for a bet that gives $1 if P is true, for example, then we can infer that I think P is .5 likely to occur (assuming that P is independent of whether I accept the bet on P). All we need to do make this inference is the assumption that I am rational and value dollars linearly (I place the same value on $1 if I am broke as if I have $1 million).

Now of course this just tells you what my subjective degree of belief in P is, it doesn’t tell you what the chance of P is. But chances and subjective degrees of belief aren’t unrelated; if I believed that the objective chance of P was .5 but didn’t have a subjective degree of belief in .5, I’d be being irrational (this is sometimes called the Principal Principle). So subjective degrees of belief are constrained by beliefs about the objective chances. As I understand it, the reason that people are willing to use betting markets like Intrade as a proxy for objective chance is something like this:

The price of a bet on P at Intrade reflects the subjective probability of many well-informed bettors though a market-pricing mechanism—if well-informed bettors think the bet is a good deal, they’ll buy it, and if it’s a bad deal they’ll sell. Although no individual bettor knows with certainty what the objective chance of P is, since they’re all trying to conform their credences to the objective chances, and they’re well-informed enough that they succeed at this task to some degree, the collective market-pricing mechanism should get us tolerably close to the actual objective chance of P.

So there is actually an argument to be made for interpreting prices as probabilities in this way. It does rely on four idealizing assumptions:

1. The individual bettors are rational.

2. The individual bettors value dollars linearly.

3. The individual bettors are relatively well-informed.

4. The market-pricing mechanism will produce a reasonably close approximation to our collective best guess at the objective chance.

I’m an epistemologist, not an economist, so the market price mechanism in (4) is the part that’s most mysterious to me. But that’s the argument, as I understand it.

Greg:

Yes, that’s the argument. I just don’t buy any of the four assumptions.

I expect someone who was more inclined to defend it than I would engage in more hand-waving around assumption #4, in order to suggest that market forces magically cancel out failures of rationality, information, etc.

But really the point of my post is that as long as we’re clear about what the assumptions are, and that we’re using these prices as a “collective best guess” at the chance of P, I think this is a pretty reasonable thing to use these prices for. That’s not to say we shouldn’t try to construct better predictive models, of course—because of failures of rationality and information, our collective best guess might not be very good.

It looks to me like the all Kolmogorov axioms hold, so what’s wrong with calling them probabilities?

I’m not sure what a quantifiable subjective degree of belief is supposed to actually look like. I don’t know any psychologist who believes in them, and I doubt strongly that any civilian (i.e., not economists, statisticians, probability theorists or data miners) will tell you they think Sarah Palin has 2.5% chance of winning in 2012. And I’m pretty sure that there will never be a statistically valid sample of independent, identically distributed US Presidential Elections in 2012 so that we can see what percent Sarah Palin wins. Ergo, I figure any set of numbers for the 2012 election that fit the Kolmogorov axioms ought to be blessed with the label “probability.” It’s not like there’s any falsifying test or practical difference it will make.

Greg: 4 looks like the conclusion that you want to draw from 1-3, not an assumption. One problem in making that work is that you have to assume that $1 received in a “democrat wins” state of the world has the same utility as $1 received in a “republican wins” world. For that matter, you could further subdivide by the present value depending on

whichrepublican you’re talking about. For example, if the next president is named Palin, I’m trading all my US$ for Canadian $ and moving north, so the value of $1 US is slightly less in that state of the world. A libertarian loon winning, promising to shut down all gov’t spending on anything redistributive (e.g., public education) almost certainly devalues the claim of $1 on the US economy’s future productivity. So it’s not a minor issue.Would one be better served to use the formula:

P(B)=P(B\A)P(A)+P(B\A^)P(A^)?

So for Romney we have

.123 divided by.299 times .299 plus (.123 divided by .41 times .41)

P(B)= .246

For Huntsman we get .144

So a 25 percent chance for Romney and 14 percent chance for Huntsman. Concidering where we are in the cycle these numbers look realistic.

as after the post last march on whether or not march madness pools constitute a “game,” i find myself wishing for a monkey cage intrade market.

Of course, nobody who is not a political wonk has any idea who Huntsman is. How is that factored in at this point? I think he has as much chance of getting the Republican nomination as Barney Frank does.

is intrade betting legal?

It’s not the amount they raise that makes them Prez. fool it VOTES. Unless you cheer for the “bought” president then you better be worried that some corporation owns your favorite GOP. W don’t need another Negligent dope in the white house bailing out every wealthy company in the US. Ron Paul.