The Kelly's Criterion, famous for its various application in sports betting and asset management, is detailed in a paper that J. L. Kelly published in 1956 while working under the Bell lab. In a situation where a gambler places repeated bets on an event with success probability of $p\%$, Kelly proves that the optimal bet size each time is $(2p-1)\%$ of the gambler’s total capital. This is due to the fact that as the bets continues, the logarithm of gambler’s wealth is concave with respective to the bet size.

## Quick Proof

Let $p$ denote the true probability of an event which the gambler bets on. $V_0$ and $V_N$ denotes the initial capital and capital after $N$ bettings, and $B$ denotes the percentage bet size relative to the capital. Among the $N$ bets, $W$ denotes the number of successes. Therefore:

Taking the logarithm of both sides and then take derivative w.r.t. $B$:

Taking the second derivative and we can see that $\log{V_N}$ is concave w.r.t. $B$:

Setting the partial derivative to zero we can then solve for $B'$ that maximizes $\log{V_N}$:

As $N$ goes to infinity, the optimal bet size becomes $2p-1$:

## No-Limit Hold’em Application

Suppose we have a bankroll of $2000$ and wants to play $1/2$ No-Limit Hold’em. A standard buy-in size is 100BB which amounts to 200. Suppose we have an inherent edge in this game, how can we vary[1] the buy-in to maximize our long-term profit based on the Kelly’s Criterion?

Let’s say we adopt a play style where we play very tight and always all-in pre-flop. We continues until someone calls our all-in and will exit the game no matter the outcome. For now, we will ignore the blinds we are losing by waiting for a hand, since we earn some blinds when players fold to our all-ins. This game now becomes very similar to the gambler situation above, that we either double our buy-in or lose it. Since we assumed that we had an inherent edge[2] $p>50\%$, we should be always betting a buy-in of $(2p-1)\%$ of our bankroll. But how do we know our edge?

We know that sample mean is an unbiased estimator of population mean. We may use our average historical win percentage[3] as an estimate. Suppose we played 10 sessions first all with standard buy-in and made 200 in profits. Then our edge can be calculated:

Based on Kelly’s Criterion, in the $11$th session, we want to bet $10\%$ of our total bankroll of $2200$, which is a buy-in of $220$.

Now suppose we win the $11$th session, then:

Our next optimal buy-in will become $2420 * 18\% = 435.6$.

## Limitations

In the scenario above, we can observe large swings in the estimated win rate and updated optimal buy-in amount after a single session. This is because the high variance of the aggressive, all-in-only play style we adopted.

In normal poker plays, the variance will be much lower and as the play history grows, the updates will become incremental. The question now becomes whether Kelly’s Criterion still applies if there are more than $2$ outcomes. In fact, Kelly had made a general case for multiple outcome scenarios in his paper. I will continue this exploration in a future post.

[1]: We are also making the assumption that varying buy-in will not change our inherent edge/win rate. This assumption can be supported by implementing a floor of 100BB and a cap to our buy-in, no matter what Kelly’s Criterion suggests.

[2]: In reality, this play style is difficult to earn an edge, as players will only call our all-in with an even tighter range, causing our win rate to drop below 50%.

[3] Note that this only works if our winning is i.i.d. However, as we sit at a table longer, players will be more familiar with our strategy and therefore negatively affect our win rate.