Kelly Criterion for Bonus Sizing — how much bankroll to deploy per offer

Why bet-sizing is the missing lever

Bonus hunters spend hours on offer selection and almost no time on stake selection. That is backwards. Two players with the same edge on the same bonus can end the month at wildly different bankrolls — because the one who over-stakes gets ruined by variance before the edge shows up.

The Kelly criterion is the answer to “how much do I risk per positive-EV opportunity?” It was derived at Bell Labs by John Kelly (1956) and later canonised by Ed Thorp for blackjack. It also works — with adjustments — for casino bonus play, and it is the sizing rule the Portfolio Builder uses under the hood.

The one formula

For a bet with probability p of winning b units per unit staked (and probability 1 − p of losing the stake), the growth-optimal fraction of bankroll to risk is:

f* = (bp − q) / b        where q = 1 − p

Equivalently, when EV and variance are the natural inputs (which they are for bonuses):

f* ≈ EV / Var

That is the version to remember. The optimal fraction of bankroll to deploy on a bonus is its expected value divided by its variance.

Reading it in bonus terms

For a match bonus you get:

• EV — the Monte Carlo expected value of the offer at your intended stake level. Published in every CasiMath datasheet as a euro figure (e.g., EV = +€12.40).
• Var — the variance of the wagered-through outcome. A high-volatility slot at a large wagering multiple produces enormous variance; low-volatility slots at low wagering produce modest variance.

Divide EV by Var and you get the fraction of your total bankroll that the offer optimally consumes.

Worked example. A €100 deposit-match bonus with EV = +€8 and outcome variance σ² = €6,400 (a very ordinary high-vol bonus).

f* = 8 / 6,400 = 0.00125 = 0.125% of bankroll

If your total gambling bankroll is €10,000, the offer is sized for you: €10,000 × 0.125% = €12.50 of deposit. If your bankroll is €800, the offer is oversized by an order of magnitude — even though its EV is technically positive.

That is Kelly’s brutal message: positive EV does not mean playable at any size.

The half-Kelly rule (and why we default to it)

Full Kelly is the growth-optimal fraction if your EV and Var estimates are perfectly correct. They never are. The consequence of over-estimating EV is severe — full-Kelly on a mismeasured edge produces long-run losses even when the raw math looks positive.

The industry-standard adjustment is half-Kelly: stake 0.5 × f* instead of the full fraction. You give up ≈25% of the growth rate in exchange for cutting drawdown variance roughly in half. For casino bonuses — where EV estimation is honest but not precise, and where the number of parallel edges is small — half-Kelly is the correct default.

CasiMath’s Portfolio Builder ships with 0.5× as the shipped sizing recommendation. If you know your model error is small, you can lift it toward 0.7-0.8. If you are new to bonus hunting, lift the fraction down to 0.25.

Where Kelly breaks for casino play

Three real-world constraints that make casino bonuses a modified-Kelly problem, not a raw one:

1. Minimum deposits. Kelly says “deploy €12 on this offer” but the casino minimum deposit is €20. If the minimum is more than 2× your Kelly fraction, skip the offer — you are over-deployed by rule, not choice. This is the single most-common way small bankrolls get destroyed on individually-positive bonuses.
2. Wagering time-locks. Kelly assumes independent, resolvable bets. Bonus wagering locks the stake for the duration of the play-through — you cannot re-deploy that capital elsewhere while the bonus is active. A Kelly-optimal offer with a 30-day expiry ties up bankroll differently than one with 24 hours.
3. Correlated offers. Two bonuses from the same operator, or two variants of the same offer at different casinos, are not independent — the same catalogue, same variance, sometimes the same regulator risk. Sum their Kelly fractions carefully.

The CasiMath sizing table (rule-of-thumb version)

Not everyone runs Monte Carlo before every deposit. This lookup captures 80% of the answer:

Two guardrails on top of the table:

• Never deposit more than 2% of total bankroll on any single offer, no matter how good the EV. Concentration risk beats math.
• Never chase minimum-deposit forcing. If min-dep > 2% of bankroll, the offer is not for you. Wait for one that is.

How this ties into the rest of the framework

• The bankroll management guide is your session floor — Kelly sits one level up and tells you how to allocate across offers.
• The volatility and variance guide explains why the Var term dominates the formula. Doubling volatility halves your Kelly fraction.
• The wagering requirements guide is where the time-lock term comes from — high wagering does not just reduce EV, it also blocks re-deployment.
• The Portfolio Builder applies half-Kelly automatically across your active bonus stack and warns when concentration hits the 2% ceiling.

The CasiMath move

We publish EV, variance, and clearance on every offer. Kelly turns those three numbers into a stake size that grows the bankroll instead of grinding it. Positive EV without correct sizing is a slow ruin dressed as a strategy — and the correct sizing is almost always smaller than the casino’s minimum deposit is designed to make you feel.