Kelly with several bets on at once: why you cannot add up the stakes

The Kelly formula sizes one bet that settles before the next begins. When several bets are open at once, the right stakes are smaller than the single-bet figure. Ten independent 55% even-money bets come to about 9.0% each, not 10%, and the total at risk is close to 90% of the bankroll, which is why people cap exposure.

The Kelly criterion calculator gives the single-bet stake. This page covers what to do when there are many.

Why the single-bet number overstates

Staking a share f of bankroll on each of n open bets means some of the bets lose together. If every bet is at even money and the stakes are 10% each, ten losses at once cost 100% of the bankroll. Kelly maximises the expected log of the final bankroll across every combination of results, so the bad combinations pull the growth-maximising stake down as n rises.

It grows with the number of bets

Computing the growth-maximising equal stake for n independent bets, each a 55% chance at even money, gives:

• 1 bet: 10.0% (total 10%)
• 2 bets: 9.9% each (total 19.8%)
• 5 bets: 9.6% each (total 48%)
• 10 bets: 9.0% each (total 90%)

The per-bet stake falls slowly, so the total commitment climbs fast. A card of ten such bets would ask for nearly the whole bankroll, with some of the money in play for days. That is the arithmetic of the method working as designed, and it is the reason most people refuse to follow it that far.

Three things the table assumes

• Independence. Two bets on one game, two legs sharing one result or two props on the same player move together. Their combined risk is higher than the table says, and the stakes should be lower again. The page on correlation explains why.
• Correct probabilities. Each of the ten estimates carries the error described in the estimate article. Ten wrong estimates staked at 9% each compound the problem.
• Money returns in time. If bets settle at different times, the stake on a later one can be sized on the bankroll left after the earlier ones. Open exposure is what the table limits.

Practical limits people use

The usual response is a ceiling on total exposure. Some bettors cap each stake at 1% to 2% of bankroll and the open total at 5% to 10%, whatever the formula says. Others take the Kelly stake for each bet, scale every one of them down by the same factor until the total fits under the ceiling, and keep the ratios between them. A ceiling is a risk preference, not a mathematical result, and it is fine to set it by what size of drawdown you can sit through.

Working it for your own card

For two or three bets, take each single-bet Kelly stake, halve it, and check that the sum of stakes is under your ceiling. For a larger card the stakes become small enough that the unit sizing in the unit article does the same job with less effort.

Start with the single bet. The Kelly criterion calculator returns the single-bet figure for each play; then scale to your ceiling.