Theory AWhy the biggest bet isn't the best one
Every flip is a good bet
A dollar you stake comes back as $2 60% of the time and as nothing 40% of the time, so on average it returns $1.20. A bet that pays 20% on average sounds like one to make as big as possible. But stake everything and the first loss ends the game: the chance of surviving even ten all-in flips is 0.6%.
A win and a loss don't cancel out
Every gain and loss is a percentage of whatever you have at the time. Bet 50%: a win takes $100 to $150, and a loss then takes 50% of that larger balance, leaving $75. Losing first and winning second lands in the same place. Each win paired with a loss multiplies your money by 1 − stake², here 0.75, so the bigger the stake, the more every pair costs.
Where a typical run ends up
Over 100 flips you should expect about 60 wins and 40 losses. Order doesn't matter, so a typical run ends at $100 × (1 + stake)60 × (1 − stake)40. At a 50% stake that is $3.
Small stakes grow slowly, and big stakes get eaten by the win-loss penalty. Growth peaks at 20%, where a typical run ends near $749. Past 39% a typical run loses money, even though you still win 60% of the flips.
The Kelly criterion: bet your edge
The peak isn't at 20% by coincidence. In 1956 John Kelly, a researcher at Bell Labs, showed that the stake with the fastest long-run growth on an even-money bet is your edge: the chance you win minus the chance you lose, 60% − 40% = 20%. When a win pays b dollars for every dollar risked and p is your chance of winning, the rule becomes
stake = p − (1 − p) / b
A few bets run through it:
| Bet | Win chance p | Pays b | Kelly stake |
|---|---|---|---|
| This game | 60% | 1 to 1 | 0.6 − 0.4 / 1 = 20% |
| A generous coin flip | 50% | 2 to 1 | 0.5 − 0.5 / 2 = 25% |
| A long shot | 25% | 5 to 1 | 0.25 − 0.75 / 5 = 10% |
| Red at roulette | 47% | 1 to 1 | 0.47 − 0.53 / 1 = −6%, don't bet |
A bigger payout raises the stake even at worse odds. A long shot with a real edge still gets a small stake, because you lose most of the time. When the answer is zero or below, as at roulette, the edge belongs to the house and the Kelly bet is nothing.
Same flips, different stakes
A sample run of 100 flips with exactly 60 wins, replayed at fixed stakes. Play 20 flips and this switches to yours. The luck is identical in every panel; only the stake changes.
In real life
Real edges are never known this precisely, and they shrink as you use them, because other people notice and the opportunity closes. Betting too little only slows you down, while betting too much can lose money, so many professional bettors and investors stake a fraction of Kelly. Half Kelly here (10%) keeps about 75% of the growth rate with much smaller swings.
Questions about the Kelly criterion
What is the Kelly criterion?
The Kelly criterion is a formula for how much of your money to put on a bet that favors you, so your money grows as fast as possible over many bets. John Kelly published it at Bell Labs in 1956. For a bet you win with probability p that pays b dollars per dollar risked, the Kelly stake is p − (1 − p) / b of your current balance.
What is the best bet size on a coin that lands heads 60% of the time?
20% of your current balance on every flip: the 60% chance of winning minus the 40% chance of losing. Over 100 flips with 60 wins, that turns $100 into about $749.
What happens if you bet more than the Kelly amount?
Your money grows more slowly and swings harder. On a 60/40 coin, any stake above about 39%, roughly twice Kelly, shrinks a typical run even though you still win most flips. Betting everything loses it all on the first loss.
What is half Kelly?
Half Kelly means betting half the Kelly stake, 10% on a 60/40 coin. It keeps about 75% of the growth rate with much smaller swings, and it protects you if you have overestimated your edge, which is why many professional bettors and investors bet a fraction of Kelly.
What is gambler's ruin?
Gambler's ruin is going broke. In a fair or losing game, a gambler who keeps playing is certain to go broke eventually. Even when every bet favors you, betting a fixed dollar amount or too large a share of your money leaves a real chance that a losing streak wipes you out. Betting a fixed fraction of your current balance shrinks each bet as you lose, so you never quite reach zero.
Can you use the Kelly criterion for investing?
Yes, as a guide to position sizing, but real investments don't come with known odds. Estimates of an edge are usually too optimistic, so investors who use Kelly tend to bet a fraction of it. For an investment with continuous returns, the Kelly fraction is roughly its expected excess return divided by the variance of its returns.
How do people do when they play this game?
Poorly. In a 2016 experiment by Victor Haghani and Richard Dewey, 61 people, mostly finance students and young professionals, bet real money on a coin that landed heads 60% of the time, starting with $25 and with winnings capped at $250. The average payout was $91, and many bet far more than 20% per flip, some of them everything. Read the paper