The Kelly Criterion
A bet can have positive expected profit and still make the typical gambler poorer. On an even-money game won 60% of the time, staking 40% of capital earns an expected ₹8 per ₹100 wager cycle, yet compounds wealth downward by about 0.24% per play. John Kelly’s 1956 Bell Labs paper found the missing variable: bet size.
How it works
Kelly maximizes expected logarithmic wealth, which is the same as maximizing the long-run geometric growth rate. For a binary bet:
f* = (bp - q) / b
f* is the fraction of capital to stake, b is the net profit per ₹1 won, p is the probability of winning, and q = 1 - p. If bp - q is zero or negative, the prescription is to pass.
At even money, b = 1, so a 60% win probability gives f* = 0.60 - 0.40 = 0.20. Kelly stakes 20%, not the 100% suggested by maximizing expected wealth alone. One all-in loss ends compounding permanently.
The logarithm matters because repeated returns multiply. A 50% loss followed by a 50% gain does not restore wealth: ₹100 becomes ₹50, then ₹75. concept compounding rewards growth rates, not arithmetic averages.
The price of betting too much
For the same 60/40 even-money game:
| Fraction staked | Expected log growth per play | Reading |
|---|---|---|
| 0% | 0.00% | No growth |
| 5% | 0.90% | Quarter Kelly |
| 10% | 1.50% | Half Kelly |
| 20% | 2.01% | Full Kelly |
| 40% | −0.24% | Positive edge, negative compounding |
Kelly is not a safety rule. Full Kelly accepts deep drawdowns to reach the 2.01% growth rate in this example. Half Kelly gives about three-quarters of that growth while committing half as much capital, which is why fractional Kelly often survives contact with uncertain estimates.
From telephone noise to blackjack
Kelly was not studying casinos. His “gambler with a private wire” received imperfect information about chance events through a noisy channel. At fair odds, he proved that the extra growth obtainable from the signal equals its information rate. A bit became something that could be priced in compounded wealth.
Claude Shannon had defined channel capacity in 1948. Kelly supplied an economic interpretation eight years later, binding bankroll growth to concept information theory. Edward Thorp then carried the rule into blackjack, first publicly presenting his work in 1961 and publishing Beat the Dealer in 1962.
What's contested
The mathematics is settled under known probabilities, repeatable bets, stable odds, and divisible capital. Real decisions rarely grant all four.
Probability error is a central failure. If a bettor estimates p = 0.60 when the true probability is 0.52, the prescribed 20% stake compounds at roughly −1.23% per play. Correlated positions, changing markets, taxes, liabilities, and concept fat tails widen the miss.
Paul Samuelson’s 1979 objection goes deeper: maximizing long-run log growth is an objective, not a universal account of human preference. A person with a finite horizon or a fixed obligation may rationally prefer less growth and more certainty. Kelly answers “how fast can capital compound?” It does not answer “what is this capital for?”
Why this has to do with other realms
Kelly separates a good decision from a good outcome. A correctly sized bet can lose, while a reckless bet can win once. That distinction links bankroll management to concept poker decisions vs results and, less obviously, to concept karma yoga: judge the action by the information available when it was taken, then let repeated evidence revise the next action.
The harder bridge is concept bayesian inference. Kelly converts belief into exposure, so uncertainty about p should change the stake, not merely decorate the forecast. A probability without a position size is an opinion; a position size reveals how much confidence the probability deserves.
An open question
If the edge itself must be estimated from limited, shifting evidence, should the optimal fraction be calculated from the mean estimate of p, its full posterior distribution, or the cost of being wrong?
Key Sources
- John L. Kelly Jr., “A New Interpretation of Information Rate” (1956), Bell System Technical Journal 35, pp. 917–926. The original private-wire argument.
- Claude E. Shannon, “A Mathematical Theory of Communication” (1948), Bell System Technical Journal 27. The channel-capacity foundation Kelly reinterpreted.
- Edward O. Thorp, “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market” (1997; revised 1998). The practitioner’s bridge from gambling to markets.
- Paul A. Samuelson, “Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long” (1979), Journal of Banking & Finance 3(4), pp. 305–307. The objection to treating log growth as universal preference.
- Leonard C. MacLean, Edward O. Thorp, and William T. Ziemba, eds., The Kelly Capital Growth Investment Criterion: Theory and Practice (2011). A collection spanning the criterion’s mathematics and applications.
See Also
- concept compounding
- concept information theory
- concept bayesian inference
- concept fat tails
- concept poker decisions vs results
- concept karma yoga
Abhishek's take
I use Kelly less as a betting formula than as a test of whether confidence has consequences. A forecast that never changes exposure is decoration; a forecast that dictates full Kelly may be more precise than the evidence permits. Half Kelly is the compromise I keep returning to because estimation error is usually the hidden wager.
Tags: #kelly-criterion #position-sizing #probability #compounding #risk-of-ruin