The Kelly Criterion
The Kelly criterion is a famous formula for answering one question: when you have a genuine edge, how much of your money should you put on the line each time? It's used by serious investors and professional bettors to size positions for the fastest long-run growth without blowing up. But its single most important lesson is also the one most people miss — Kelly only works when the odds are in your favor, and in a casino they never are.
The formula
Kelly tells you what fraction of your bankroll to wager:
f* = (bp − q) / b
- f* — the fraction of your bankroll to bet
- b — the net odds (profit per $1 staked if you win)
- p — your probability of winning
- q — your probability of losing, which is 1 − p
Read the top of the fraction carefully: bp − q is your edge. It's how much your weighted winnings beat your weighted losses. If that number is positive, you have an advantage and Kelly tells you how hard to press it. If it's zero or negative, the whole formula collapses to a single instruction: bet nothing.
A worked example
Suppose a coin is biased in your favor: it lands heads 60% of the time, and a friend foolishly pays you even money — win $1 for every $1 you stake. Here b = 1, p = 0.6, q = 0.4.
f* = (1 × 0.6 − 0.4) / 1 = 0.20
Kelly says bet 20% of your bankroll on each flip. With a $1,000 bankroll, that's $200 on the first toss, then 20% of whatever you have next, and so on. Betting this fraction maximizes how fast your money compounds over many repetitions. Bet much less and you grow slower than you could; bet much more and you court disaster.
Key idea: the numerator bp − q is your edge. No positive edge, no Kelly bet. The formula can't conjure an advantage that isn't there — it can only size one that already exists.
Why casino games make Kelly say zero
Now run the same math on a real casino bet. Take an even-money color bet on a 37-pocket roulette wheel. You win 18 times out of 37, so p ≈ 0.486 and q ≈ 0.514, with b = 1.
f* = (1 × 0.486 − 0.514) / 1 = −0.028
The answer is negative. A negative Kelly fraction means the only edge belongs to the other side of the table — the right "bet" for you is zero, or, if you could, to take the house's seat. Every game in a casino, including Crash, Plinko, Slots, Blackjack and Roulette, carries a built-in house edge, so a Kelly calculation on any of them lands on the same verdict: there is no stake size that turns a losing game into a growth machine.
| Situation | Edge (bp − q) | Kelly says |
|---|---|---|
| Biased coin in your favor | Positive (+0.20) | Bet 20% of bankroll |
| Fair 50/50 bet, even money | Zero | Bet nothing |
| Any casino game | Negative | Bet nothing |
Over-betting and the road to ruin
Even when you do have an edge, betting more than Kelly — called over-betting or going "over-Kelly" — is dangerous. Push past the optimal fraction and your long-run growth doesn't just flatten, it reverses: the swings get so violent that a bad streak can wipe you out before the edge pays off. Double the Kelly fraction in the coin example and your expected growth rate falls all the way back to zero despite the genuine advantage. Bet more than that and you're statistically marching toward ruin even though every individual bet is in your favor.
Fractional Kelly
Because full Kelly is volatile and real-world edges are usually estimates rather than certainties, most practitioners use fractional Kelly — betting half or a quarter of what the formula recommends. Half-Kelly gives up only about a quarter of the growth rate while cutting the wild swings roughly in half. It's the cautious version: it accepts slightly slower compounding in exchange for a far smoother, safer ride and a buffer against having overestimated your edge.
The reason disciplined investors size positions with Kelly is the same reason it's useless as a casino trick: it's a tool for pressing a real advantage, not for inventing one. On Riskr the money is fake, so you can test any sizing scheme you like — and watch why no fraction beats a negative-edge game over the long run.