The Kelly Criterion

November 6, 2025

This is a short summary of the Kelly Criterion, initially developed by J. Kelly Jr. in their 1956 manuscript [1] , which describes the optimal way in which to size bets as to obtain the greatest expected logarithmic returns in a betting game. Plenty of further reading is available on this topic, one useful resource is [2] , which is a short article on the Kelly Criterion from a course taught by T. Ferguson of UCLA in 2009. Ferguson also has a number of other interesting articles on his webpage, which you can find here. There also exist a plethora of other articles from the famous E. Thorp on the matter, see for example [3] . Note, for brevity, the notation used here aims to remain consistent with that of Thorp and Ferguson.

The Game #

Suppose you are asked to play a betting game where you undergo repeated trials against an infinitely wealthy opponent who will wager even-money bets. Suppose further that on each trial, you have a probability of winning of $p > \frac{1}{2}$, and of losing, $q := 1 - p$.

Let $X_{0}$ denote your initial wealth, and $X_{k}$ your wealth after the $k$th trial. Denote by $0 \leqslant B_{k} \leqslant X_{k - 1}$ the bet made on the $k$th trial (we can assume that this is either deterministic or random). If you win the $k$th trial, your wealth increases to $X_{k - 1} + B_{k}$ and $X_{k - 1} - B_{k}$ if you lose, i.e.

Letting $T_{k} = + 1$ if the $k$th trial is won, and $T_{k} = - 1$ otherwise, we can simplify and instead write

where $T_{k}$ is a Bernoulli random variable with success $p$, i.e. $T_{k} \sim \operatorname{Bernoilli} (p)$, and each $T_{k}$ is i.i.d. for $k \in \mathbb{N}$. Recursively expanding, and unravelling, we can further simplify the expression for our wealth after the $k$th trail to

Maximising Wealth After $k$ Trials #

The question we now ask ourselves is, how can we best maximise our wealth, in whatever manner we mean best.

From a naïve point of view, a gambler will aim to determine how to size bets $B_{i}$ to maximise $\mathbb{E} (X_{k})$ given knowledge of the outcome of the previous bets. Indeed, doing so, we see that

Expanding by the law of total expectation, and using the fact that $\mathbb{P} (T_{i} = + 1) = p, \mathbb{P} (T_{i} = - 1) = q$,

Since $p > \frac{1}{2}$, we have $p - q > 0$, therefore to maximise $\mathbb{E} (X_{k})$, we are simply required to maximise $\mathbb{E} (B_{i})$ for each $i = 1, \ldots, k$. As $0 \leqslant \mathbb{E} (B_{i}) \leqslant \max B_{i} = X_{i - 1}$, one can maximise $\mathbb{E} (B_{i})$ by simply (in a deterministic manner) setting $B_{i} = X_{i - 1}$ for each $i = 1, \ldots, k$. That is, the optimal strategy to maximise expected total wealth $\mathbb{E} (X_{k})$ at stage $k$, is to simply wager your whole fortune at each trial.

In doing so, in $k$ trials, you will amass a wealth of

with $\mathbb{E} (X_{k}) = (2 p)^k X_{0}$. Note that the probability of ruin, $1 - p^k$, quickly tends to one as $k \rightarrow \infty$. Although this is a strategy that generates the greatest expected wealth, it is not a sensible strategy, as that wealth is not probable and chance of ruin is high.

Indeed, calculating the variance of your wealth with this strategy, we find that for any $k \in \mathbb{N}$

In particular, as $p > \frac{1}{2}$, $4 p > 2$, and as $p < 1$, $p^k \rightarrow 0$ as $k \rightarrow \infty$, therefore $\mathbb{V} (X_{k})$ grows at least as fast as $X_{0}^2 2^k$ as $k \rightarrow \infty$, which is exponentially fast, i.e. the strategy is not very stable.

One other metric that is interesting to calculate is the Sharpe of this strategy. Define by $R_{i} := \frac{X_{i} - X_{i - 1}}{X_{i - 1}}$ for $i = 1, \ldots, k$, the returns of this strategy after stage $i$. Then, using $B_{i} = X_{i - 1}$,

Since, for each $i = 1, \ldots, k$, $T_{i} \sim \operatorname{Bernoilli} (p)$, and is i.i.d., we have that

and in particular the Sharpe ratio for this strategy is

which is positive for $p > \frac{1}{2}$, approaches infinity as $p \rightarrow + 1$ and zero as $p \rightarrow \frac{1}{2}$. Surprisingly, this strategy has a relatively high Sharpe for $p$ just above $0.5$.

Avoiding Ruin #

To avoid ruin, that is seeing your wealth $X_{k}$ go to zero, we can instead opt for a slightly modified strategy, where we restrict $0 \leqslant B_{k} < X_{k - 1}$, and take a proportional approach and never invest our whole wealth (note that, although this avoids complete ruin, it does not avoid asymptotic ruin, i.e. there exists $k \in \mathbb{N}$ such that $\mathbb{P} (X_{k} < \varepsilon)$ for any $\varepsilon > 0$, for which there are other results). For brevity, write $B_{k} := \pi (p) X_{k - 1}$ for $k \in \mathbb{N}$ for some $\pi : [0, 1] \rightarrow [0, 1)$ yet to be determined. Then, we can write

where $\sharp (T_{i} = + 1, i = 1, \ldots, k)$ denotes the number of times $T_{i} = + 1$ for $i = 1, \ldots, k$, and similarly for $\sharp (T_{i} = - 1)$. The random variable $\sharp (T_{i} = + 1, i = 1, \ldots, k)$ can be understood as the Binomial random variable $Z_{k} \sim \operatorname{Binomial} (k, p)$, with success $p$, allowing us to rewrite

Under this construction, we can see that under an appropriate choice of $\pi$, our wealth could grow exponentially fast, without the chance of absolute ruin, in particular

which in conjunction with the estimator $\frac{Z_{k}}{k} \rightarrow p$ as $p \rightarrow \infty$ for the Binomial random variable $Z_{k}$, we find that for large $k \gg 1$

where $G (p) :=$$p \log (1 + \pi) + (1 - p) \log (1 - \pi)$ denotes the growth rate.

Indeed, which $\pi (p)$ attains maximal growth is known as the Kelly proportion, which can be calculated as $\bar{\pi} (p) := p - q = 2 p - 1$, for $q := 1 - p$.

In particular, the Kelly proportion suggests that for $k \gg 1$, $X_{k}$ would grow at a rate of

which grows exponentially for $p \neq \frac{1}{2}$, and stays at the initial capital $X_{0}$ for $p = \frac{1}{2}$, in which case the Kelly proportion, $\pi$, is $0$, and no capital would be bet. I wish to state that the above strategy does not maximise $\mathbb{E} (X_{k})$, as that would be the strategy of betting your whole wealth at each trial, but instead concerns maximising the asymptotic rate of growth, $G$, of your wealth as $k \rightarrow \infty$.

Kelly Betting System: Maximising Expected Utility #

Instead of aiming to maximise $\mathbb{E} (X_{k})$, consider instead the want to maximise $\mathbb{E} (\log (X_{k}))$ at each $k \in \mathbb{N}$, where $\log (X_{n})$ is often called the utility, see the work of J. von Neumann in [4] for the seminal work on utility theory.

Additionally, assume that $B_{k} := \pi _{k} (p) X_{k - 1}$ with $\pi _{k} : [0, 1] \rightarrow [0, 1]$ which can vary at each step. Then, the expected wealth at trial $k$, given information at trial $k - 1$ can be written as

At each stage $k$, $\mathbb{E} (\log (X_{k}) |X_{k - 1})$ is maximised by the Kelly proportion discussed above

This gives a small justification of the Kelly betting system (i.e. it maximises the expectation of the log utility, whereas for the non log-utility approach, it only maximises the rate of increase of wealth).

This betting system can also be used if the win probabilities change from stage to stage. If there are $n$ stages, and the win probability at stage $k$ is $p_{k}$, then the Kelly betting system at each stage uses the rule of maximising the expected $\log$ of fortune one step ahead, as done above. Thus at stage $k$, you bet the proportion $\pi _{k} (p_{k})$ of your fortune.

For further reading on the matter, one can also review the good and bad properties of the Kelly betting system, [5] , and the application of the Kelly criterion for sports betting and Wall Street, see [3] and references therein.