Cosmic Variance’s Presidential Prediction Contest

Sean Carroll:

I feel compelled to offer up another round of predictions, now that we’ve narrowed the field to two major candidates. By why not make it more fun and have a prediction contest? Anyone can join in, just by leaving your prediction the comments. Entries that appear before the end of June will officially count.

But to make things somewhat science-y, let’s use equations to judge who will win. Each prediction consists of two numbers: the fraction f of the total popular vote cast for the two major candidates that goes to Barack Obama, but also the standard deviation σ of your prediction for that percentage. We are thus ignoring the electoral college entirely, and dealing with the annoyance of third-party candidates by concentrating exclusively on McCain vs. Obama. And we are assuming for purposes of misleadingly-precise quantification that each prediction follows a normal (Gaussian) distribution:

displaystyle P(x) = frac{1}{sigma sqrt{2pi}} expleft(-frac{(x-f)^2}{2sigma^2}right) ,.

And here is the rub:  the winner is not the one whose fraction f is closest to the final answer, but the one whose value of P(x) is the highest, when x is equal to the fraction of votes Obama actually does win.  The smaller your standard deviation is, the higher your P(x) will be for x very close to your predicted value f , but the faster it will die off as you get further away. So if you are extremely confident, you can ensure victory by choosing an appropriately tiny standard deviation on your prediction. Contrariwise, if you choose a large standard deviation, you might get lucky if none of the confident folks comes close to the actual result. Cool, eh?