Prediction markets and the Kelly Criterion, part 1

Last week, on PredictIt, the “Yes” contract for Amy Barrett becoming Trump’s Supreme Court nominee was trading at an implied probability of 40%. Based on my own reasoning, I estimated her chances at closer to 20%. Put another way, the “No” contract was offered for $0.60, while I thought it was worth $0.80. So I decided to place a bet.

Question: How much should I bet?

I have learned that this is a surprisingly interesting question, one that once inspired Nobel laureates and billionaire investors to publish multiple academic papers calling each other morons.

Let me start with the answer. Well, the answer according to some. I found most expressions of this formula hard to remember, so I will (a) put it here up front where I can find it and (b) cast it in a simple form.

Define:

\[
\begin{align*}
p &= \textrm{your (estimated) probability of winning} \\
q &= \textrm{the opposite} = 1 – p \\
p’ &= \textrm{the market price (imputed probability)} \\
q’ &= \textrm{the opposite} = 1 – p’
\end{align*}
\]

Write down \(p-q\) and \(\frac{p’}{q’}\) next to each other without any parentheses:

\[p-q\frac{p’}{q’}\]

This is the fraction of your bankroll you should bet. Note that \(\frac{p’}{q’}\) is just the payoff on a winning bet, as in 1:1, 2:1, 10:1, or whatever. (Well, the reciprocal of the payoff.) This version of the formula directly applies to markets where winning contracts pay $1, like PredictIt.

So, for my example, \(p = 0.8\), \(q = 0.2\), \(p’ = 0.6\), \(q’ = 0.4\), and I should have bet \(0.8 – 0.2(\frac{0.6}{0.4}) = 0.8 – 0.3 = 0.5\), or half my bankroll.

This formula is called the Kelly Formula or Kelly Criterion. Describing where it comes from, some of its properties, and maybe a bit of its amusing history is the subject of this series. Which I might actually finish for a change.

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