Robust Kelly

The Kelly criterion is optimal if you know the true probability $p$. But you never know it: you have an estimate $\hat{p}$, with some error. Robust Kelly covers the methods that build this uncertainty into the stake calculation.

The problem: an asymmetric error

The expected growth rate of the bankroll for a fraction $f$ is:

This curve is asymmetric around its maximum: staking too much costs far more than staking too little.

Approach 1: shrinkage towards the market

Combine your estimate with the market's fair probability:

$w$ (between 0 and 1) reflects how much you trust your model relative to the market. It should be estimated from your track record (calibration), not picked by gut feeling.

Approach 2: Kelly on the lower bound

Estimate the uncertainty in $\hat{p}$ (a standard deviation $\sigma_p$, derived for example from your model's calibration), and apply Kelly to a cautious bound:

Approach 3: distributionally robust Kelly

A more general formulation: define an uncertainty set $\mathcal{P}$ (all plausible probabilities) and choose the stake that maximises growth in the worst case:

For a single bet with $\mathcal{P} = [p_{\min}, p_{\max}]$, the worst case is always $p_{\min}$: you're back to approach 2. The formulation becomes truly useful for multiple simultaneous or correlated bets, where the worst case isn't obvious and is solved by numerical optimisation (convex programming).

What about the Bayesian approach?

If you replace $p$ with a posterior distribution and maximise expected growth, the optimal stake for a single bet uses only the mean of that distribution. Robustness therefore comes from that mean itself being pulled towards the market (the prior), which amounts to approach 1. Academic work (Baker & McHale, 2013) shows that when there's estimation uncertainty, reducing the Kelly stake improves actual growth.

The link with fractional Kelly

½ Kelly or ¼ Kelly is a simple, implicit form of robust Kelly: it scales all stakes down by the same factor, without modelling the uncertainty. The approaches above go further by cutting more on the bets whose estimate is the most uncertain (for example very high EVs, often caused by errors).

Practical recommendations

1. Be wary of very high EVs: they're often the most wrong. Shrinkage naturally brings them back down to earth.
2. Calibrate $w$ or $\sigma_p$ on your actual track record (see the module on calibration).
3. Combine: shrinkage towards the market then a Kelly fraction (for example ½).
4. Always cap the individual stake (for example 2 to 5% of the bankroll) and total exposure.