Kelly Growth for Poker Cash Games with Continuous Returns
Summary
The document frames a bankroll-sizing problem for poker cash games, where returns over a fixed number of hands are treated as approximately normally distributed rather than as discrete tournament payouts. It gives an example with a stated mean and standard deviation over one hundred hands, then describes rescaling those quantities to a per-hand basis. It also reports formulas for a Kelly bankroll and for the bankroll threshold at which a player should move between two stakes.
The central question is how to calculate expected logarithmic growth at arbitrary bankroll levels when a session can lose multiple buy-ins, rather than being limited to losing the entire initial stake. The author says a numerical approximation by dividing the return distribution into many discrete outcomes was inefficient and sensitive to resolution and hand-count choice. The document does not include the answer to the posed problem; the accepted response only says the author later found one. Its formulas and assumptions are presented without derivation or validation, and poker results do not directly establish a trading strategy.
Key ideas
- Cash game results can be modeled as continuous, approximately normal returns over a chosen hand count.
- The author rescales mean and volatility from a block of hands to a single hand.
- The document presents formulas for a Kelly bankroll and a threshold between two stakes.
- A log growth calculation must account for outcomes that can lose multiple buy-ins.
- The numerical method described is inefficient and the document omits the eventual solution.
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Full text
# Kelly Criterion for cash game poker (normally distributed returns)
# Kelly Criterion for cash game poker (normally distributed returns)
I'm trying to apply the Kelly Criterion to poker. Poker players have been stuck using outdated bankroll management techniques for decades, and I want to change that.
My goal is to graph the log growth of playing poker with respect to the size of your bankroll, given some edge or return distribution.
For most poker formats (MTTs, Spins, HUSnG, etc) the outcomes of each buy-in are discrete, so it's quite easy to calculate and graph the log growth for any bankroll.
For example, here's one I made for Lottery Spin & Go's which have a discrete payout structure. You can even graph multiple stakes with different returns to see when you should move up or down.
However, I'm struggling to create this graph for Cash games which have continuous, normally(ish) distributed returns.
There are many quick formulas for finding the kelly bet for normal returns. But I want to be able to graph the log growth for arbitrary bankrolls and stakes as I did in the above picture.
Here's an example. A player wins, on average, \$10 per hundred hands. Their standard deviation after 100 hands is \$100.
- µ = \$10,
- σ = \$100
Note that we can transform the mean and std dev to per one hand instead of per hundred. After a single hand, µ = \$0.10 and σ = \$10. The more hands you play, the smaller the gap between your expected returns and standard devation.
The kelly bankroll is calculated as: $\frac{µ^2 + σ^2}{µ}$ = \$1010
We can also calculate the optimal bankroll between multiple stakes. According to Mathematics of Poker:
Critical bankroll between two stakes = $\frac{σ_1^2 - σ_2^2}{2µ_1 -2µ_2} + 0.5(µ_1+µ_2)$ In other words you'd move up to the higher when your bankroll exceeded that amount.
Ok, so here's the problem. How do I calculate log growth for an arbitrary bankroll? I don't care about the risk-free return. All the formulas I've found assume you can only lose 100% of your investment, but that's not the case here. You can lose many buy-ins over the course of 100 hands. It's not clear what 'f' is.
My bad solution was to split the probability distribution into thousands of discrete chunks and solve it that way. It works but it's incredibly inefficient and requires a lot of resolution to approach the theoretical answer. Moreover, it doesn't retain the same maximum if I change the number of hands.
I'm sure there must be a better approach. Any help is appreciated!
## Answer by Tom Boshoff (score 3)
https://quant.stackexchange.com/a/73754
Well it took me a while but I found the answer.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.