notes-econ-kellyCriterion

The Kelly criterion tells you how much, out of all the money you are willing to spend, you should bet on a particular bet.

One form of the Kelly criterion in general is "edge over odds", where edge is expected profit fraction, and odds is odds ratio. The output of this formula is the fraction of your total money that you should place on the bet.

Another is (bp - q)/b, where p is the probability of winning (your actual probability estimate, not the estimate consistent with the setting of the odds ratio), q is probability of losing (= 1-p), and b is the odds ratio (if you win the bet, you get back the odds ratio times the amount you put in; if you lose, you lose what you put in; b should be > 1). Another is (p - q/b).

Betting more than Kelly is worse than betting less.

In most cases your information is inaccurate. Since you want to error on the side of betting less, a popular strategy here is to bet "fractional Kelly", meaning multipy the result of the Kelly formula by a fixed fraction (less than 1) before using it. Common fractions are 1/2 to 1/6.

Note that:

• The expected value of the bet is (bp - q), the "edge". If this is negative, Kelly will tell you not to bet.
• Kelly never tells you to bet more than p (e.g. if you have an 80% chance of winning, you should certainly never bet more than 80% of your money).
• When the chance of losing is very small, then fractional Kelly is dominated by the fraction.