## The Kelly Betting Criterion

The Kelly betting criterion is an interesting idea from probability. It’s best explained by example. Suppose you can make bets (or investments in economic terms) that have a positive expectation (meaning you’ll win more than half the time). Each bet is independent. How much of your bankroll should you bet each play in order to maximize your winnings?

If you bet too much, you run the risk of a streak of bad luck and you’d lose all your money. If you bet too little, you won’t be getting the maximum value out of each winning bet.

Suppose your probability of winning is p = 0.60 so your probability of losing is q = 0.40. Expressed as odds, odds of winning is b = (0.60 / 0.40) to 1, which is 1.5 to 1. Now suppose your payoff, coincidentally, is also 1.5 to 1 meaning for every \$1 bet, if you win you get \$1.50 (pretty nice when you have a greater than 0.50 chance of winning!)

The Kelly criterion says that in order to maximize your profit you should bet (bp – q) / b percent of your bankroll each time.

Suppose you start with \$1000 and p = 0.60 and q = 0.40 and b = 1.5 as above. You should bet a fraction

f = (1.5)(0.60) – 0.40 / 1.5
= (0.90 – 0.40) / 1.5
= 0.50 / 1.5
= 0.33

So you’d bet \$333.33 on your first bet, and in general 1/3 of your bankroll every time. You’d lose everything if you lost three times in a row. The chances of that happening are (0.40)(0.40)(0.40) = 0.064 or 64 times in a thousand. In practice, you’d be wise to be a bit conservative and bet less than 1/3 of your bankroll each play.