Kelly criterion betting: what it assumes and how it fails

The Kelly criterion tells you what fraction of your bankroll to stake on a bet so that your bankroll grows as fast as possible over the long run, assuming you know the true probability of winning. That last assumption is where most of the risk lives. Kelly is a precise answer to a question bettors usually can't answer precisely.

What Kelly criterion betting optimizes

In 1956, J. L. Kelly Jr. showed that a bettor who stakes his whole bankroll every time maximizes his expected bankroll but is almost certain to go broke over a long series of bets. Betting a fixed fraction instead, and choosing the fraction that maximizes the bankroll's exponential growth rate, beats any other fixed fraction in the long run: that bettor "will, with probability one, eventually get ahead and stay ahead."^[1]

So Kelly isn't about the biggest expected profit on the next bet. It's about compounding: the stake that, repeated over many bets, grows the bankroll fastest.

The formula

For a bet at decimal odds d, with net odds b = d − 1, a win probability p and q = 1 − p:

Kelly fraction = (b × p − q) ÷ b

That's the same as your edge divided by the net odds, where edge is the expected value per unit staked, p × d − 1.^[2] If the edge is zero or negative, the Kelly stake is zero: don't bet. Convert American odds to decimal with the odds converter.

Try your own numbers with the Kelly calculator.

The assumption that breaks it: knowing the probability

Kelly takes your probability as fact. In betting, it's an estimate, usually from a no-vig line or a model (see Implied probability and no-vig odds, explained). Small errors matter a lot.

Take the -110 bet above. If you believe it wins 55% of the time, full Kelly says stake 5.5%. If the true probability is 52.5%, the right Kelly stake is about 0.25%, so you'd be betting roughly twenty times too much. At that size, the expected long-run growth of your bankroll is negative, even though every bet still has a slightly positive expected value.

Researchers who studied Kelly in depth put it this way: betting more than the Kelly amount increases risk and lowers long-run growth, and growth keeps falling the further you go past it. In continuous-time approximations, growth drops to roughly zero at about twice the Kelly stake.^[2]

Fractional Kelly

Because probability estimates are uncertain, many bettors stake a fixed fraction of the Kelly amount, such as half or a quarter. That deliberately trades some growth for more safety.^[2]

Half Kelly keeps about three-quarters of the growth while halving the stake. It doesn't rescue a badly wrong estimate, though: in the example above, half Kelly (2.75%) is still more than ten times the true Kelly stake, and its expected growth is still negative.

Other assumptions to keep in mind

• Bets are independent. Correlated bets (the same game, the same outcome in different markets) aren't separate chances, and staking each one at full Kelly overbets the combined position.
• Bets are placed one at a time. Kelly for several simultaneous bets needs a different calculation.
• It's a long-run strategy. Kelly maximizes growth over many bets, but in the short term it can be very risky; no staking rule is a sure way of winning over a finite run of bets.^[2]
• Real-world limits. Sportsbook maximum stakes and account limits can stop you placing the stake Kelly suggests (see Why sportsbooks limit winning bettors).