Variance And Volatility In Gambling

The Kelly Criterion: Optimal Bet Sizing Under Uncertainty



The Kelly Criterion is a formula for determining the optimal size of a series of bets, originally developed by John L. Kelly Jr. at Bell Labs in 1956. While Kelly's paper addressed signal noise in telecommunications, the formula was quickly adopted by gamblers and investors seeking to maximize the long-term growth rate of their capital.



The basic Kelly formula for a simple bet with two outcomes is: f* = (bp − q) / b, where f* is the fraction of the bankroll to wager, b is the net odds received (decimal odds minus one), p is the probability of winning, and q is the probability of losing (1 − p). If a bettor estimates a 55% chance of winning a bet offered at even money (decimal 2.00), the Kelly fraction would be (1 × 0.55 − 0.45) / 1 = 0.10, suggesting 10% of the bankroll should be wagered.



The formula has several important properties. It maximizes the expected logarithm of wealth, which corresponds to the geometric mean of returns. It never recommends betting one's entire bankroll, and it recommends zero or negative stakes (meaning no bet) when no edge exists. However, full Kelly staking produces high volatility, which is why many practitioners apply fractional Kelly — typically wagering one-half or one-quarter of the recommended amount — to reduce the severity of drawdowns.



The Kelly Criterion extends beyond gambling into portfolio theory, where it has been compared to mean-variance optimization. Investors including Warren Buffett and Bill Gross have referenced Kelly-type reasoning in capital allocation decisions.



A Kelly Criterion calculator allows users to input their estimated probability and the offered odds to determine both the full Kelly stake and common fractional adjustments, along with projected growth rates and Risk of Ruin estimates.