The Kelly Fix: Bet to Survive

Lesson 03 handed you the trick: for wealth that multiplies, take the logarithm and the multiplication becomes addition. Compounding in wealth turns into a plain running total in log-wealth, and the quantity you actually live — the time-average growth rate — is just the average of the log returns. That was the diagnosis. This lesson is the cure. If the number that governs you is the expected log return, then don’t merely measure it — maximise it. Turn the crank on that idea and out drops a single, beautiful bet size that grows your money fastest over time and, as a free bonus, flatly refuses to ever wipe you out. It has a name: the Kelly criterion.

The move: optimise the growth rate itself

Analogy. Lesson 03 gave you a compass — the log — that always points at your fate rather than the crowd’s. It would be strange to own a compass and then walk in a random direction. The obvious next step is to walk the way the compass points: pick the bet size that pushes the time-average growth rate as high as it will go.

Precise definition. For a repeated multiplicative bet, your time-average growth rate is the expected logarithm of the per-round wealth factor, $g = E[\ln(W_{\text{after}}/W_{\text{before}})]$ — the geometric growth rate from lesson 03. Choosing your bet size to maximise $E[\ln W]$ is the Kelly criterion. That is the whole idea in one line: maximise expected log-wealth.

Origin. The rule is named for John L. Kelly Jr., a physicist at Bell Labs who published it in 1956 while thinking about how much of a gambler’s bankroll to wager given a noisy signal about the outcome. He was foreshadowed by more than two centuries: Daniel Bernoulli proposed logarithmic utility back in 1738 to resolve the St. Petersburg paradox, arguing that a ducat means less to the rich than to the poor. Kelly’s information-theory colleague Claude Shannon championed the rule, and the mathematician Ed Thorp took it out of the seminar room and onto casino floors and Wall Street, using it to size bets in blackjack and then in his hedge fund.

The two roads, one destination. Here is the quietly stunning part. Bernoulli and Kelly arrived at “maximise expected log-wealth” from utility and information — psychology and signals. Ole Peters’ ergodicity argument (this whole course) arrives at the same log objective from physics — from the plain fact that you live one path through time and multiplicative growth compounds. Two completely different starting points; one identical answer. When two roads meet like that, you should suspect you’ve hit something real.

When to use it

Reach for Kelly whenever you face a repeated, multiplicative bet on your own evolving bankroll and you want it to grow as fast as it sustainably can. It is the bridge from “the time average is what matters” (a diagnosis) to “so stake exactly this much” (a prescription).

The Kelly bet, worked

Precise formula. For the cleanest case — a simple win/lose bet — Kelly gives an explicit fraction of your bankroll to stake:

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

Term by term:

• $b$ = the net odds: profit per unit staked if you win. A bet that returns your stake plus the same again (double-or-nothing) has $b = 1$; a bet that pays 2-to-1 has $b = 2$.
• $p$ = the probability of winning.
• $q = 1 - p$ = the probability of losing.

Read the numerator out loud and it tells a story: $bp$ is your expected gain per unit staked, $q$ is your expected loss, so $bp - q$ is your edge. Divide the edge by the odds $b$ and you get the fraction of your bankroll to put at risk. No edge ($bp = q$) means $f^* = 0$ — Kelly tells you not to bet at all. Negative edge means $f^*$ goes negative, i.e. you should be on the other side of the bet.

Fully worked example. Take a double-or-nothing bet — win and you gain your stake ($b = 1$), lose and you lose your stake — with a win probability of $p = 0.6$ (so $q = 0.4$). Then:

$f^* = \frac{(1)(0.6) - 0.4}{1} = \frac{0.6 - 0.4}{1} = \frac{0.2}{1} = 0.2$

So you stake 20% of your bankroll each round. On a $1,000 bankroll that is a $200 bet — not the $1,000 an “it’s +EV, bet it all” instinct would scream for, and not the $5 a nervous gambler might risk. Twenty percent is the exact fraction that makes this bankroll grow fastest over the long run.

Pitfall. The formula assumes you know $p$ and $b$ exactly. In real life you estimate them, and a hopeful overestimate of your edge feeds straight into an oversized $f^*$ — which, as the next section shows, is the dangerous direction to be wrong in.

When to use it

The $f^* = (bp - q)/b$ form is your go-to for any binary “win this much / lose your stake” wager: sports bets, a single trade with a defined stop, a hand of advantage-play blackjack. For messier payoffs, the underlying rule (“maximise expected log-wealth”) still holds; you just solve it numerically instead of with the tidy formula.

The growth curve is a hump

Analogy. Think of bet size like the dose of a drug. Too little does nothing. The right dose heals. Too much poisons you — and past a point, more dose is strictly worse and more toxic. Growth-versus-bet-fraction has exactly this shape: a hump.

Precise definition. Plot time-average growth $g$ against the fraction $f$ you stake each round. At $f = 0$ you never bet, so $g = 0$. As $f$ rises, $g$ climbs to a single positive peak — that peak sits at the Kelly fraction $f^*$. Push $f$ past the peak and $g$ falls back down, crosses zero, and goes negative: you are now guaranteed to shrink over time. There is one best bet size, and both under- and over-shooting cost you growth.

Worked example — the flagship coin. Take this course’s coin: heads pays +50%, tails costs −40%, a fair 50/50 flip, staking a fraction $f$ of your bankroll each round. The per-round wealth factors are $(1 + 0.5f)$ on a win and $(1 - 0.4f)$ on a loss, and the time-average growth is $g = 0.5\ln(1 + 0.5f) + 0.5\ln(1 - 0.4f)$. Evaluate it across bet sizes:

Read it as a journey. Betting nothing gets you nowhere. Around a quarter of your bankroll you hit the positive peak — the fastest sustainable growth this coin allows. By half your bankroll you’ve given the peak back and you’re merely treading water. And staking the whole thing — the naive “+5% expected value, bet it all!” move from earlier lessons — plants you firmly in negative territory at about −5.1% per round, grinding toward zero.

The overbetting cliff. Notice the asymmetry. Betting more than Kelly is not a bolder version of the same good idea — it is strictly worse: you accept lower growth and wilder swings at the same time. You are paying extra risk to reduce your return. That is the overbetting cliff, and it is the single most important shape in this course.

When to use it

Keep the hump in your mind’s eye every time you size a position. It reframes the question from “is this bet good?” (yes/no) to “how much of this good bet maximises my growth?” — and warns you that erring high isn’t merely aggressive, it’s self-defeating.

Do the experiment yourself. Run the default coin at full stake first: the ensemble average balloons while the typical path is ground toward zero — the non-ergodic ruin you now know by heart. Then drag the bet fraction slider down and re-run. Somewhere around a 20–25% stake the time-average growth chip flips from negative to positive and the typical path finally climbs — you’ve found the Kelly region. Now push the bet fraction back up toward the maximum and watch the growth go negative again. That U-turn is the far side of the hump: overbetting.

Kelly respects ruin — for free

Analogy. Imagine a hiker whose map assigns an infinitely bad score to a single cliff-edge square. No route planner, however greedy for shortcuts, will ever plot a path through that square — not because you bolted on a “don’t fall off cliffs” rule, but because the maths makes the cliff infinitely repulsive on its own.

Precise definition. The Kelly objective is expected log-wealth, and $\ln(0) = -\infty$. Wiping out — reaching zero — therefore carries an infinite penalty in the very quantity Kelly is maximising. So a Kelly bettor will never stake the whole bankroll on anything that could zero them out, because even a tiny probability of hitting $-\infty$ obliterates the expected value. Ruin-aversion is baked into the mathematics, not bolted on afterward.

Worked example. Consider a bet that could, on a loss, take your entire bankroll. Staking $f = 1$ means one loss sends your wealth to $0 and your log-wealth to $-\infty$. Averaged in, that single branch drags the expected log-wealth to $-\infty$ no matter how juicy the winning branch is. Kelly’s maximiser therefore recoils from $f = 1$ automatically and settles on some interior fraction — for the +60% coin above, 20%. You never had to say “and please don’t go bankrupt”; the logarithm said it for you.

Pitfall. This free protection assumes the log objective. Swap it for maximising expected wealth (the plain ensemble average — the arithmetic mean) and the guardrail vanishes: expected-wealth maximisation happily tells you to bet everything on any +EV gamble, straight off the cliff. The safety is a property of the logarithm specifically.

When to use it

Lean on this whenever a bet carries genuine ruin risk. Sizing by expected log-wealth means you never have to hand-tune a “max loss” rule — the objective already treats bankruptcy as the infinitely bad outcome it is.

Half-Kelly and the real world

Analogy. Full Kelly is like driving at the exact speed that gets you there fastest on a perfect road. Half-Kelly is easing off the accelerator: you arrive almost as quickly, with far less white-knuckling, and you don’t crash when the road turns out to be bumpier than the map promised.

Precise definition. Half-Kelly means staking half the Kelly fraction — $f^*/2$ — every round. The trade is remarkably kind: because growth is a smooth hump that is flat near its top, halving the bet keeps roughly three-quarters of the maximum growth rate while cutting the variance of your bankroll by about half. You give up a sliver of speed for a large reduction in gut-wrenching swings.

Worked example. In the double-or-nothing bet above, full Kelly stakes 20%. Half-Kelly stakes 10%. Because the peak of the growth hump is broad and flat, dropping from 20% to 10% costs you only a modest slice of long-run growth — but the ride is dramatically smoother, and, crucially, if your true win probability was actually 0.55 rather than the 0.6 you estimated, the 10% bet is comfortably below the real Kelly fraction (still on the safe, growing side of the hump) instead of above it.

Why practitioners fractionalise. Real edges are estimated, not known, and estimation errors bias you toward overbetting: an optimistic $p$ inflates $f^*$, nudging you toward the cliff where growth turns negative. Betting a fraction of Kelly is insurance against your own overconfidence — it keeps you on the gently-sloping left side of the hump, where being a bit wrong costs you a little growth, rather than the steep right side, where being a bit wrong costs you everything.

Pitfall. Estimating the edge too high is the classic failure mode. Overstate $p$ or $b$, compute $f^*$ from the inflated numbers, and you think you’re betting Kelly while you’re actually overbetting — sitting past the peak on the negative-growth slope, convinced you’re optimising.

When to use it

Default to some fraction of full Kelly — a half, or even a quarter — any time your inputs are uncertain (which is almost always). It’s the standard practitioner hedge: nearly all the growth, a fraction of the volatility, and a wide margin of safety against a wrong edge estimate.

Where Kelly bites

Kelly is a scalpel, and scalpels cut the surgeon if mishandled. Know its assumptions before you trust its number.

• It assumes you know the odds. The formula takes $p$ and $b$ as given. Feed it wrong inputs and it confidently returns a wrong — usually too large — bet size. Garbage in, overbetting out.
• Wrong inputs cause overbetting specifically. Because $f^*$ rises with your estimated edge, the natural human error (optimism) pushes you toward the cliff, not away from it. The error is asymmetric and it hurts.
• It maximises long-run growth, not comfort. Even bet perfectly, a full-Kelly path is volatile — deep, prolonged drawdowns are entirely normal on the way to fastest growth. Kelly optimises where you end up, not how smooth the trip feels.
• It is about growth, not your personal risk tolerance. Kelly answers “what size grows my money fastest?” — not “what size lets me sleep?” If full Kelly’s swings would make you bail at the bottom (locking in the loss), the growth-optimal size is the wrong size for you. That’s precisely why half-Kelly exists.

Pitfall. Treating Kelly as a guarantee of a smooth ride is the deepest misunderstanding. It guarantees the fastest sustainable growth, and fast growth and calm nerves are different things. Size down to the volatility you can actually live through — because a bettor who quits at the trough never collects the long run Kelly is optimising for.

When to use it

Use Kelly to convert a decision: it turns “should I take this bet?” into “how much should I stake?” — the far more useful question. Position sizing for any repeated multiplicative bet — trades, wagers, capital allocations — is exactly its home turf. Just feed it honest inputs, and fractionalise to match the uncertainty in those inputs and the volatility you can stomach.