This one’s from 2019, but it’s worth reposting given recent interest in prediction markets.

The story starts with a post from economist Greg Mankiw, who wrote:

Who has the best chance of beating Donald Trump? A clue can be found using Bayes Theorem.

Here is the logic. Let A be the event that a candidate wins the general election, and B be the event that a candidate wins his or her party’s nomination. Predictit gives us the betting market’s view of P(A) and P(B). It is a safe assumption that P(B|A) = 1, that is, a candidate can win only if nominated. We can then use Bayes theorem to compute P(A|B), the probability that the candidate will win the general election conditional on being nominated.

So here are the results for P(A|B) as of now:

Buttigieg 0.80
Biden 0.77
O’Rourke 0.67
Sanders 0.65
Booker 0.60
Yang 0.60
Harris 0.57
Warren 0.44

That is, the betting markets suggest that Mayor Pete would be the strongest candidate if nominated, with Joe Biden close behind. (Of course, these numbers will bounce around as the prices in betting markets change.)

By the way, when I [Mankiw] did a similar calculation in 2006, Bayes liked Barack Obama.

I copied Mankiw’s post in its entirety, with the only change being that he wrote P(A / B) etc., and I changed the slash to the vertical bar, P(A|B). (Are there people who write conditioning using a slash rather than a vertical bar? I had no idea. P(A|B) is more standard, I believe. In the above post, Mankiw links to the wikipedia page which uses the P(A|B) notation. No big deal, it just seemed odd to me.)

Anyway, I think the above set of calculations is a great example for teaching conditional probability.

The next step is to push a bit: Do we really believe these numbers? There’s nothing wrong with the probability calculations, but I’m not sure we should be taking Predictit’s betting odds as actual win probabilities.

To start with, I looked at Mankiw’s list and wondered what Yang was doing on it. Yang’s a fringe candidate, right? I wrote my post in June, 2019, and Yang was polling at 0.8% on Real Clear Politics then. I went over to Predictit and it said you can buy a Yang contract for the Democratic nomination for the price of 9%. OK, sure, at 0.8% in the polls there’s room for improvement. But 9%??? Seems like a lot.

The next think I’m worried about, beyond bias in the online markets, is volatility.

Sure, Mankiw writes, “these numbers will bounce around as the prices in betting markets change,” but I think he’s not fully appreciating how noisy these numbers are!

Mankiw’s post is dated 27 Apr 2019. Predictit conveniently gives prices going back a few months, so I could do some Biden-Warren price comparisons of then to when I was writing my post:

27 Apr 12 Jun
Biden primary election 22 28
Biden general election 17 19
Warren primary election 9 19
Warren general election 4 13

Something weird was going on in April, when Biden’s price was 22 for the primary and 17 for the general election. This just can’t be right, and all I can conclude is that the betting markets here were thin enough that nobody was taking these numbers very seriously.

If you want to take the numbers as is, you’ll get the following:

27 Apr: Biden 17/22 = 0.77, Warren 4/9 = 0.44
12 Jun: Biden 19/28 = 0.68, Warren 13/19 = 0.68.

These numbers aren’t quite right, even if you take these betting markets seriously, because of rounding and the vig. If you add up all the prices on the “Who will win the 2020 Democratic presidential nomination?” page, you get something well over 100%. So you can’t directly interpret these prices as probabilities, even beyond the issues of bias and noise.

I discussed this with David Rothschild, who thinks a lot about elections and prediction markets (for example, here), and David responded as follows:

People ask me to compute this automatically on my blog, but I refrain, because it is so noisy this early. Here I compute the conditional probability range separately for Betfair and PredictIt, by diving the seller’s price of win / buyer’s price of nom & buyer’s price of win / seller’s price of nom. Betfair has advantage of being tighter by definition (PredictIt trades on the penny, but Betfair on the odds, which have more depth).

Here is a figure from an old paper I wrote with David Pennock about the 2012 election. As you can see, while informative, it can get quite noisy!

Anyway, my point here is not to criticize Mankiw but rather to thank him for putting out this fun example, and then to demonstrate how we can take it further by interrogating each step in the analysis. Which is how we do applied statistics in general.

P.S. In case you’re curious, based on the numbers when I wrote my post, where Biden’s implied electability is 19/28 = 0.68 and Warren’s is 13/19 = 0.68, we can look up Buttigieg. He was at 9/16 = 0.56, the least electable of the three. So, no, Bayes did not like Mayor Pete that day.

It’s a fun example, but when we look at the data more carefully, the original conclusion goes away.