The Kelly Criterion: How Much to Bet When You Have an Edge

In 2016, Victor Haghani and Richard Dewey gave 61 finance students and young professionals $25 each and thirty minutes to bet on a coin they were told lands heads 60% of the time. Bets were even money, and the payout was capped at $250. Every flip had a positive expected value.

Only 21% of the players reached the cap. 28% walked away with nothing, and the average payout was $91. A simple strategy that bets a constant 10–20% of the bankroll reaches the cap about 95% of the time.

That gap is the Kelly criterion. It is a formula not for what to bet, but for how much — and it is the difference between having an edge and keeping it.

Expected Value Is the Wrong Compass

Bet $1 on heads. With probability $0.6$ you win $1, and with probability $0.4$ you lose $1. On average the bet earns

$0.6 \cdot (+1) + 0.4 \cdot (-1) = 0.20.$

Every dollar wagered has a positive expected value of 20 cents. If expected value were the objective, the optimal strategy would be obvious: bet everything, every time. The expected value is maximized by the largest possible stake.

The catch is that wealth is not added, it is multiplied. Going all-in means one tails — which happens 40% of the time, every time — takes you to zero. The chance of surviving ten all-in flips is $0.6^{10} \approx 0.006$. A strategy with a positive expected value on every flip is almost certain to bankrupt you.

What we actually care about is the rate at which wealth compounds.

Compounding, Not Adding

Suppose you bet a fixed fraction $f$ of your current bankroll on each flip. A win multiplies your wealth by $1+f$; a loss multiplies it by $1-f$. After $W$ wins and $L$ losses — the order does not matter, because multiplication commutes — your bankroll is

$V = V_0\,(1+f)^W (1-f)^L.$

Taking logs turns the product into a sum:

$\log V = \log V_0 + W \log(1+f) + L \log(1-f).$

Over many flips, $W/N \to p$ and $L/N \to 1-p$, so the expected log-growth per flip is

$g(f) = p \log(1+f) + (1-p)\log(1-f).$

Maximize this. Differentiate and set the derivative to zero:

$$\frac{p}{1+f} - \frac{1-p}{1-f} = 0 \quad\Longrightarrow\quad p(1-f) = (1-p)(1+f) \quad\Longrightarrow\quad f^* = p - (1-p) = 2p-1.$$

For a 60/40 coin, $f^* = 0.20$. Bet 20% of your bankroll, whatever it currently is. With $25 the first bet is $5. Win and you have $30, so the next bet is $6. Lose and you have $20, so the next bet is $4. You never hit zero, because the stake shrinks with the bankroll.

Why 20% and Not 40%?

Why not bet more, since every flip has a positive expected value? Because losses compound more painfully than wins. After losing 40% of your money, you need a 67% gain just to get back to even.

Take a concrete run of 100 flips with exactly 60 heads and 40 tails — the typical outcome. At 20%, your money is multiplied by

$1.2^{60} \cdot 0.8^{40} \approx 7.49,$

so $25 becomes about $187. At 40%, the same 60 wins and 40 losses give

$1.4^{60} \cdot 0.6^{40} \approx 0.78,$

leaving you with about $19 — less than you started with, despite winning 60% of the flips.

Here is the whole picture around $p = 0.6$:

Growth rises, peaks at exactly 20%, and then falls back through zero just below 40%. Betting more than the Kelly fraction means accepting more risk in exchange for less growth. Past the zero crossing, you are paying for the privilege of losing money on average.

The Playground

The playground below plots the growth curve and runs 10,000 simulated players through the actual game. It uses the same $25 stake,$250 cap and even-money odds as the experiment.

A few things to try:

• The white line is your current fraction, the green dashed line is Kelly, and the red dashed line marks the point where expected growth hits zero. Drag the fraction past 40% and watch the growth curve go negative.
• Set flips to 100 and fraction to 20%. The typical multiple should be about 7.5×, matching the calculation above.
• Set the fraction to 40%. The growth rate turns negative, and the typical uncapped outcome drops to about $19 — worse than doing nothing, even though the coin still has a 20% edge. (The capped median stays high because the lucky players who reach $250 stop there.)
• Set the win probability to 55%. Kelly drops to 10%; the 20% line is now twice Kelly, and its growth is essentially zero.
• Switch the fraction between Kelly and half Kelly. Half Kelly gives up about a quarter of the growth for roughly half the volatility.

The paths are coloured by outcome: green paths reached the $250 cap, red paths were wiped out below $2, and grey paths ended somewhere in between. The two dashed vertical lines on the growth chart are the two most important numbers in the whole exercise: the peak and the zero.

Why the Curve Is a Parabola

Near $f=0$, the logarithms can be expanded:

$\log(1+f) \approx f - \frac{f^2}{2}, \qquad \log(1-f) \approx -f - \frac{f^2}{2}.$

Substituting,

$g(f) \approx (p-q)f - \frac{f^2}{2} = e f - \frac{f^2}{2},$

where $e = p-q = 2p-1$ is the edge. This little quadratic explains almost everything:

• The peak is at $f^* \approx e$, and the growth there is about $e^2/2$.
• At twice Kelly, $f = 2e$, growth is $e(2e) - (2e)^2/2 = 0$.
• The $-f^2/2$ term is the volatility drag. It is why a positive expected return does not automatically compound, and why sizing is a first-order decision rather than a detail.

For larger fractions the approximation breaks down, but the shape survives: the growth curve is a hill with a sharp peak and a long, punishing downside.

The Weak Spot: You Do Not Know $p$

Kelly assumes the edge is known exactly. Real edges are estimates, and estimates are wrong.

Suppose the coin is actually 55/45, but you believe it is 60/40 and bet 20%. The true Kelly fraction is 10%, so you are at twice Kelly — and, as the approximation says, your growth is roughly zero. The experiment's participants did not know the edge; they were told it. In markets, nobody tells you.

The standard defence is fractional Kelly. Bet half of Kelly, or a quarter. Half Kelly on a 60/40 coin grows at $1.50\%$ per flip instead of $2.01\%$ — about 75% of the growth — while cutting the volatility roughly in half:

$g(f^*/2) = 0.6\log(1.1) + 0.4\log(0.9) \approx 0.0150.$

Giving up a quarter of the growth for a much smoother ride is a trade most professionals happily make. It also buys robustness: if you have overestimated the edge, you are far less likely to be past the zero crossing.

What the Experiment Really Showed

Haghani and Dewey's subjects were not unsophisticated. They were economics students and young finance professionals. Yet:

• only 21% reached the payout cap;
• 28% went bust, finishing with less than $2;
• 18 of the 61 bet their entire bankroll on a single flip;
• 67% bet on tails at least once, even though heads was the side with the edge;
• only 5 had ever heard of the Kelly criterion.

Their average payout was $91. A constant 10–20% strategy would have produced close to $240 on average and reached the cap about 95% of the time. The authors' conclusion was not that the game was hard, but that the education was missing.

Kelly Beyond the Casino

The even-money formula generalises. If a winning bet pays $b$ per unit staked and a loss costs one unit,

$f^* = \frac{bp - q}{b} = p - \frac{q}{b},$

where $q = 1-p$. A 2:1 payout with $p=0.6$ gives $f^* = 0.6 - 0.4/2 = 0.4$. Better odds justify a larger fraction.

For continuous returns with mean $\mu$ and variance $\sigma^2$, the small-return approximation gives

$f^* \approx \frac{\mu}{\sigma^2}.$

This is the version that appears in portfolio management: size a position in proportion to its expected excess return and inversely to its variance. The growth rate at the optimum is approximately $\mu^2/(2\sigma^2)$ — the square of the Sharpe ratio, halved.

A few caveats matter:

• The edge must be known. In markets it is not, and it changes. Use fractional Kelly.
• The horizon and the cap matter. With one flip left and a cap far above your bankroll, betting everything maximizes expected value; with hundreds of flips and a cap within reach, a fraction a little below Kelly is better. Kelly optimizes the infinite-horizon growth rate, not every finite game.
• Log utility is an assumption. It is a sensible default for repeated bets, but it is not the only utility, and it does not describe everyone's tolerance for drawdowns.
• Never go all-in. Even Kelly can produce long, painful losing streaks. The formula is a ceiling, not a target.

Conclusion

A positive edge tells you which side to take. The Kelly criterion tells you how much to take — and it is the second question that decides whether you are still there to take the next bet. Maximize expected log wealth, bet a fixed fraction of what you have, and let the compounding do the work.

The uncomfortable part is that the formula's answer is usually smaller than intuition suggests: 20% on a coin with a 20% edge, and often half of that in practice. The experiment's 28% bust rate is what happens when intuition wins instead.

Further Reading

• J. L. Kelly, "A New Interpretation of Information Rate" (1956) — the original paper
• Edward O. Thorp, "The Kelly Criterion in Blackjack Sports Betting, and the Stock Market" — the classic practical treatment
• William Poundstone, Fortune's Formula — the history of Kelly, Shannon, and Thorp
• Victor Haghani & Richard Dewey, Rational Decision-Making Under Uncertainty (2016) — the coin-flip experiment