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Kelly Criterion

Optimal bet sizing for long-term wealth growth

Seed · early Updated Jul 19, 2026 3 min read
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The Kelly criterion chooses a bet fraction to maximize expected logarithmic growth. John Kelly published it at Bell Labs in 1956 while connecting information rate to a repeated betting problem.1

For an even-money bet, the fraction is 2p - 1, where p is the chance of winning. A 60% chance gives 20%. A 51% chance gives 2%.

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:

For a one-sided bet that cannot take a negative position:

f* = max(0, (bp - q) / b)

Here q = 1 - p. When the bet has a positive edge, this is also (p(b + 1) - 1) / b. For even money, b = 1, so the positive-edge formula is 2p - 1.

The result needs strong assumptions: repeated independent bets, fixed known probabilities and payoffs, divisible stakes, and no outside cash needs. Under those assumptions, the fraction maximizes the long-run rate of log wealth. It does not maximize wealth over every finite run or promise that one bettor will beat every other bettor.

Betting above Kelly lowers expected log growth. Betting the whole bankroll can cause ruin after one loss. A smaller amount above Kelly does not make ruin automatic, but it can make drawdowns much worse.

Practical use requires care. The formula assumes that its inputs are right. In practice, probability and payoff estimates can be wrong or can change. Research on parameter uncertainty treats that gap as a central limit of direct Kelly sizing.2

Fractional Kelly means betting a set share of the calculated amount. It gives up some modeled growth to reduce exposure to estimation error and drawdown. One-quarter and one-half Kelly are common examples, not universal optima.

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.

One specified run of five losses has probability 0.4^5, about 1%. Five bets at 20% of the current bankroll leave 0.8^5, about 33% of the bankroll held before that run. This is not the chance of seeing any five-loss streak in a long sequence, and the starting point is not always the prior peak.

The log-growth optimum says nothing about whether a person can bear the path. A smaller fraction may fit a real drawdown limit better.

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.

Sources

  1. John L. Kelly Jr., “A New Interpretation of Information Rate”, Bell System Technical Journal 35, no. 4 (1956), 917–926. ↩

  2. Rose D. Baker and Ian G. McHale, “Optimal Betting Under Parameter Uncertainty: Improving the Kelly Criterion”, Decision Analysis 10, no. 3 (2013), 189–199. ↩