The Kelly Criterion: How Much to Stake (and Why Fractional)

By · author of the MIT-licensed teachersbettextbook library ·

You found a price that is wrong. Now the question that actually decides whether you keep your money: how much of it goes on?

Meet your classmates

Same five. Today the question is how much, not whether.

Betting big does not work the way you think

Lose half your money, then gain half back. You are not level. You are at 75 cents on the dollar.

That is the whole problem. A big loss digs a hole a same-sized win cannot fill, and the bigger you bet, the deeper each hole gets. Past a certain stake, no edge on earth is big enough to climb out of it.

So Bookie Brian does not need Average Joe to be wrong. He just needs Joe to keep betting big.

Kelly is the size that grows fastest

There is one stake that grows a bankroll faster over the long run than any other. Bet more than it and you grow slower and risk more, which is not a trade-off, it is just worse. Bet less and you leave growth on the table, which is a real cost but a survivable one.

f∗=b p−qb,q=1−pf^{*} = \dfrac{b\,p - q}{b}, \qquad q = 1 - p

Read it as: how big the edge is, divided by what the price pays. Bigger edge, bigger bet. Longer odds, smaller bet, because you are going to be wrong more often on the way there.

Then bet half of it anyway

Look at the chart again. Full Kelly finished ahead of Evan's half. That is not a mistake in the drawing, and it is not going to change: when you know the win probability exactly, full Kelly grows fastest by definition. Nothing beats it.

Here is the catch. You never know the probability exactly. You have an estimate, and Kelly is savage about an estimate that is too high: think you have 57% when you really have 55% and full Kelly quietly becomes an over-bet, which is Joe's panel, just slower.

Half Kelly gives up about a quarter of the growth and roughly halves the swings, and it stays sane even when your edge turns out to be smaller than you thought. That is the trade almost everybody should take, and it is what this site defaults to.

No edge, no bet. When the price is fair or worse, Kelly says $0 and it means it.

Watch it happen

Joe and Evan back the same hamster in the same races. Joe fires a quarter of his money at each one. Evan puts on about a twentieth. That is the only difference between them.

Average Joe Bookie Brian +EV Evan Joe, a quarter of it Evan, half Kelly, 5% $20 $10 $0 $0.33 $14.16 200 races. Same edge, same results, two stake sizes. $0.95 $17.38 200 races. Same edge, same results, two stake sizes. $0.98 $17.38 200 races. Same edge, same results, two stake sizes. $0.31 $14.16 200 races. Same edge, same results, two stake sizes. $0.53 $15.74 200 races. Same edge, same results, two stake sizes. Both of them back the same hamster, every race. Joe bets big. Evan bets small. Nothing else differs. Joe goes home hungry. He was right just as often.
A real simulation, not a drawing, and one season out of five picked at random. Both bettors start on the dashed $10 line and back the same hamster in the same 200 races, 55% to win at even money. The only difference is the size of the bet. Joe stakes a quarter of whatever he has left, which is more than twice the Kelly stake, and that alone takes him to nothing while he is right exactly as often as Evan is. Brian takes the bets and never places one. Full Kelly, drawn to the top of its own hill, would finish ahead of Evan's half here; the Nerd Corner below works out why Evan still bets half.

Same edge, same races, same luck. The only difference was how much went on each one, and that alone decided who still had money in March.

Want more math?the Nerd Corner

Nerd Corner

Advanced material. Nothing above depends on it.

A real stake, end to end

Worked example: a real Kelly stake

Edge: p=55%,even odds (b=1.00, q=0.45)\text{Edge: } p = 55\%, \quad \text{even odds } (b = 1.00,\ q = 0.45)
f∗=1.00×0.55−0.451.00=0.10  ⇒  10% of bankrollf^{*} = \frac{1.00 \times 0.55 - 0.45}{1.00} = 0.10 \;\Rightarrow\; 10\%\text{ of bankroll}
Half Kelly: 0.10×0.5=5%\text{Half Kelly: } 0.10 \times 0.5 = 5\%
Full Kelly says fire 10% at it. On a real, noisy edge, half-Kelly’s 5% is the bet that’s still standing next season.

Where the formula comes from

The shape of the answer

The growth curve and joint sizing

One function underneath it all. Stake fraction ff of bankroll on every bet with win probability pp and net odds bb; expected log-growth per bet:

g(f)=p ln⁡(1+b f)+(1−p) ln⁡(1−f)g(f) = p\,\ln(1 + b\,f) + (1 - p)\,\ln(1 - f)

A win multiplies bankroll by 1+b f1 + b\,f, a loss by 1−f1 - f; the probability-weighted average of the logs is the compound growth rate. Kelly’s f∗f^{*} is where the curve peaks.

The hill is lopsided: left of the peak growth fades gently; right of it the ln⁡(1−f)\ln(1 - f) term falls off a cliff.

The hill in numbers (p = 55%, even odds)

g(0.10)=0.55 ln⁡1.10+0.45 ln⁡0.90≈+0.50% / bet (full Kelly)g(0.10) = 0.55\,\ln 1.10 + 0.45\,\ln 0.90 \approx +0.50\%\text{ / bet (full Kelly)}
g(0.05)≈+0.375% / bet  ⇒  half the stake keeps ∼75% of the growthg(0.05) \approx +0.375\%\text{ / bet} \;\Rightarrow\; \text{half the stake keeps } \sim 75\%\text{ of the growth}
g(0.20)=0.55 ln⁡1.20+0.45 ln⁡0.80≈0% / bet (double Kelly)g(0.20) = 0.55\,\ln 1.20 + 0.45\,\ln 0.80 \approx 0\%\text{ / bet (double Kelly)}
At double Kelly, growth is zero. Past it, a real edge shrinks your bankroll from sizing alone. That’s the whole argument for fractional Kelly, in three lines.

Set the derivative to zero and the peak is the formula from the top, in edge-over-odds form:

f∗=b p−qb  ≡  p−1−pbf^{*} = \frac{b\,p - q}{b} \;\equiv\; p - \frac{1 - p}{b}

The numerator b p−qb\,p - q is expected profit per dollar, the EV. Kelly is edge over odds: the same edge at longer odds gets a smaller stake, because losses land more often.

The harder case: several bets at once. The single-bet formula assumes each bet is the only claim on the bankroll; stacked per-bet stakes double-count shared losing outcomes. Optimize the stakes jointly, over every way the slate can land:

max⁡fi≥0, ∑ifi≤1 ∑ωP(ω) ln⁡ ⁣(1+∑ifiXi),Xi={+biwin−1loss\max_{f_i \ge 0,\ \sum_i f_i \le 1}\ \sum_{\omega} P(\omega)\,\ln\!\Big(1 + \sum_i f_i X_i\Big), \qquad X_i = \begin{cases} +b_i & \text{win} \\ -1 & \text{loss} \end{cases}

where ω\omega runs over all 2n2^{n} win/loss combinations of the nn bets. The app’s joint sizer computes exactly this.

Why summing per-bet Kelly overbets

Two independent bets: per-bet says 10%+10%; joint≈9.9% each\text{Two independent bets: per-bet says } 10\% + 10\%;\ \text{joint} \approx 9.9\%\text{ each}
Two perfectly correlated: joint is 5% each, not 10%+10%\text{Two perfectly correlated: joint is } 5\%\text{ each, not } 10\% + 10\%
Independent bets barely need the correction. Two bets that always land together are really one bet at double stake, so Kelly splits the single-bet fraction between them. That’s why same-game bets punish naive sizing most.

Where the criterion comes from

Not folklore: seventy years of information theory with a paper trail. The formula is Kelly’s.[1] The proof that maximizing expected log wealth asymptotically outgrows any essentially different rule is Breiman’s.[2] Fractional Kelly in practice is Thorp’s first-hand account.[3] The MacLean–Thorp–Ziemba volume collects the whole literature.[4]

  1. Kelly, J. L., Jr. (1956). “A New Interpretation of Information Rate.” Bell System Technical Journal 35(4), 917–926. – the criterion itself: the stake that maximizes expected log wealth.
  2. Breiman, L. (1961). “Optimal Gambling Systems for Favorable Games.” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 65–78. University of California Press. – the optimality proof behind “provably optimal.”
  3. Thorp, E. O. (2006). “The Kelly Criterion in Blackjack Sports Betting, and the Stock Market.” In S. A. Zenios & W. T. Ziemba (eds.), Handbook of Asset and Liability Management, Vol. 1, 387–428. North-Holland. – fractional Kelly in practice, and why overbetting is the deadlier error.
  4. MacLean, L. C., Thorp, E. O. & Ziemba, W. T., eds. (2011). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific. – the collected literature, including the growth–security tradeoff behind betting a half or a quarter.

Every formula here lives on the Formula Sheet for quick reference.

A season, stress-tested

The scene up top uses a fat 55% edge so the shapes read. This season is the stress test: a thin 51% edge, a 2% Kelly stake, 1,000 bets, and every strategy gets the same wins and losses. Martingale doubles after each loss until one streak eats the bankroll. The flat bettor never resizes, so a cold run digs a hole the fixed stake cannot climb out of. Kelly resizes every bet, which is the whole trick. Replay it: the shapes survive the luck changing.

Check your understanding

Frequently asked questions

What is the Kelly criterion?

A staking formula that sizes a bet to maximize the long-run growth rate of a bankroll, given a win probability and payout.

What is the Kelly criterion formula?

For decimal profit b per dollar and win probability p, the Kelly fraction is f* = (b · p − q) / b, where q = 1 − p.

Why use fractional Kelly instead of full Kelly?

Edge estimates are noisy, and overbetting is far more damaging than underbetting. Half or a quarter of the Kelly stake keeps most of the growth while cutting volatility and estimation risk.

How do you size multiple simultaneous bets with Kelly?

Per-bet Kelly overallocates simultaneous bets, especially correlated ones. Optimize the stakes jointly to maximize expected log wealth.