Kelly Criterion

The Kelly Criterion or as it is sometimes referred to as the Kelly formula is a formula used to maximize the long-term growth rate of repeated plays of a given gamble that has positive expected value. The formula specifies the percentage of the current bankroll to be bet at each iteration of the game. In addition to maximizing the growth rate in the long run, the formula has the added benefit of having zero risk of ruin, as the formula will never allow a loss of 100% of the bankroll on any bet. An assumption of the formula is that currency and bets are infinitely divisible, though this is met for practical purposes if the bankroll is large enough.The most general statement of the Kelly criterion is that long-term growth rate is maximized by finding the fraction f* of the bankroll that maximizes the expectation of the logarithm of the results. For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the following formula can be derived from the general statement:

   f* = (bp - q) / b
   where 
   f* = percentage of current bankroll to wager; 
   b = odds received on the wager; 
   p = probability of winning;
   q = probability of losing = 1 - p.

As an example, if a gamble has a 40% chance of winning (p = 0.40), but the gambler receives 2:1 odds on a winning bet, the gambler should bet 10% of her bankroll at each opportunity, in order to maximize the long-run growth rate of the bankroll.

For even-money bets (i.e. when b = 1), the formula can be simplified to:

   f* = 2p - 1

The Kelly Criterion was originally developed by AT&T Bell Laboratories physicist John Larry Kelly, Jr, based on the work of his colleague Claude Shannon, which applied to noise issues arising over long distance telephone lines. Kelly showed how Shannon’s information theory could be applied to the problem of a gambler who has inside information about a horse race, trying to determine the optimum bet size. The gambler’s inside information need not be perfect (noise-free) in order for him to exploit his edge. Kelly’s formula was later applied by another colleague of Shannon’s, Edward O. Thorp, both in blackjack and in the stock market.