The Kelly criterion is an elegant, but often misunderstood, result in decision theory. To begin with, suppose you have some amount of some resource, which you would like to increase. (For example, the resource might be monetary wealth.) You are given the opportunity to make a series of identical bets. You determine some fraction $f$ of your wealth to wager; then, in each bet, you gain a fraction $f$ with probability $p$ , and lose a fraction $f$ with probability $(1 - p)$ .^[1]

In other words, suppose $W_{n}$ is your wealth after $n$ bets. We will define $Z_{n} = log W_{n}$ , and we will suppose for simplicity that $Z_{0} = 0$ . Then $Z_{n} = \sum_{t = 1}^{n} R$ , where $R$ is a random variable defined as:

R = {\begin{matrix} log (1 + f) & with probability p log (1 - f) & with probability (1 - p) \end{matrix}

Now suppose that, for some reason, we want to maximize $E [Z_{n}]$ . By linearity of expectation, $E [Z_{n}] = \sum_{t = 1}^{n} E [R]$ . Hence, we should simply maximize $E [R]$ . This amounts to solving:

\begin{matrix} 0 & = & \frac{\partial}{\partial f} E [R] 0 & = & \frac{\partial}{\partial f} [p log (1 + f) + (1 - p) log (1 - f)] 0 & = & (1 - f) p - (1 - p) (1 + f) f & = & 2 p - 1 \end{matrix}

This, $f = 2 p - 1$ , is known as the Kelly bet. For example, it says that if you have a 60-40 edge, then you should bet $f = 2 (0.6) - 1 = 0.2$ , i.e., bet $20 %$ of your current wealth on each bet.

That all seems pretty reasonable. But why do we want to maximize $E [Z_{n}]$ ? If we were to simply maximize expected wealth, i.e., $E [W_{n}]$ , then a straightforward calculation shows that we should not bet Kelly -- in fact, we should bet $f = 1$ ("YOLO"), wagering the entire bankroll on every bet. This seems extremely counterintuitive, since, after $n$ bets, our wealth would then be:

W_{n} = {\begin{matrix} 0 & with probability 1 - p^{n} 2^{n} & with probability p^{n} \end{matrix}

In other words, as $n$ grows large, we would almost surely go bankrupt! Nevertheless, this would be the way to maximize $E [W_{n}]$ . Kelly, whatever its merits, does not maximize $E [W_{n}]$ -- not even in the long run. Especially not in the long run.

We now come to the perennial debate: why does Kelly seem "obviously right", and YOLO "obviously wrong"? There are many answers usually offered to this question.

First, what we believe to be the correct answer:

Utility is not linear in wealth. As originally observed by Bernoulli, utility tends to be approximately logarithmic in wealth. If utility happens to be exactly logarithmic in wealth, then the Kelly bet is optimal. F

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