Kelly Criterion
The Kelly criterion determines optimal bet size to maximize long-term wealth growth. Developed by John Kelly at Bell Labs in 1956, the formula emerged from information theory — Kelly was studying how to exploit noisy information channels. But the application to gambling and investing proved more influential.
The formula for even-money bets: stake the fraction of your bankroll equal to 2p - 1, where p is your probability of winning. If you have a 60% chance of winning, bet 20% of your bankroll. If you have a 51% edge, bet 2%. The math optimizes geometric growth rate, maximizing the expected logarithm of wealth.
The derivation follows from multiplicative dynamics. If you bet fraction f and win with probability p at odds b:1, your expected log wealth grows by:
E[log(wealth)] = p × log(1 + bf) + (1-p) × log(1 - f)
Maximizing this yields f* = p(b+1) - 1 / b, the Kelly fraction. For even money (b=1), this simplifies to f* = 2p - 1.
The criterion has a remarkable property: in the long run, Kelly betting almost certainly beats any other strategy. Any fixed fraction below Kelly grows slower. Any fraction above Kelly risks ruin. At the Kelly fraction, growth rate is maximized while ruin probability approaches zero.
Practical application requires modification. The formula assumes you know your true edge, which you never do. Overestimating edge leads to overbetting — and the cost of overbetting exceeds the cost of underbetting. Most practitioners use “fractional Kelly,” betting a quarter to half of the calculated amount.
Transaction costs matter. Frequent rebalancing to maintain Kelly fractions generates costs that can exceed gains. Liquidity matters — large Kelly bets may be impossible to execute at expected prices. Correlation matters — Kelly for multiple simultaneous bets requires portfolio optimization, not independent calculations.
The psychological challenge is severe. Kelly betting feels aggressive. A 60% edge means betting 20% of your bankroll each round. Losing streaks happen — a 60% edge means you still lose 40% of the time. Losing five in a row has probability 0.4^5 ≈ 1%. Happen once in a hundred trials, and your bankroll drops to 0.8^5 ≈ 33% of its peak.
Most people can’t stomach this variance. They quit, or override the system, or reduce bet size below Kelly. The mathematically optimal path requires tolerance for drawdowns that exceed most psychological limits. The formula is easy. Following it is hard.
Go Deeper
Books
- Fortune’s Formula by William Poundstone — Narrative history of Kelly, Shannon, Thorp, and the criterion’s journey from Bell Labs to casinos to Wall Street. Highly readable.
- Beat the Dealer by Edward Thorp — Thorp’s blackjack book that applied Kelly sizing to card counting.
- A Man for All Markets by Edward Thorp — Thorp’s memoir, including his work with Kelly and Shannon.
Essays
- John Kelly’s original 1956 paper “A New Interpretation of Information Rate” — The mathematical source.
- Ed Thorp’s “Understanding the Kelly Criterion” — Clear modern exposition of the mathematics and practical adjustments.
Practice
- Most practitioners use “fractional Kelly” — betting 25-50% of the calculated fraction to account for uncertainty in edge estimation.
Related: [[risk]], [[ergodicity]], [[fat-tails]], [[satisficing]]