Another way to look at the Kelly criterion is to think about betting on a variable number of independent things at once.
If you make a single bet repeatedly, and you use the Kelly criterion, then over time, your log(wealth) is a sum of IID random variables.
So the Law of Large Numbers and Central Limit Theorem hold…
- asymptotically, as time passes
- for log(wealth)
Now imagine that instead, you diversify your wealth across many identical and independent bets. (And you use Kelly to decide how to bet on each one, given the fraction of wealth assigned to it.)
Here, the limit theorems hold…
- asymptotically, as the number of simultaneous bets grows
- for wealth
which is better in both respects. You control the number of bets, so you can just set “n” to a large number immediately rather than having to wait. And the convergence is faster and tighter in term of real money, because the thing that converges doesn’t get magnified by an exp() at the end.
This is regular diversification, which is very familiar. And then, making sequential independent bets turns out to be kind of like “diversifying across time,” because they’re independent. But it’s not as nice as what happens in regular diversification.
In fact, the familiar knee-jerk intuition “never go all-in, bet less than everything!” comes from this distinction, rather than from any result about how to bet on a single random variable if forced to do so.
In the real world, you’re not stuck in an eternal casino with exactly one machine. If you keep some money held back from your bet, it doesn’t just sit there unused. Money you hold back from a bet can be used for things, including other independent bets.
(The Kelly criterion holds money back so it can be used on future rounds of the same bet, which are a special case of “other independent bets.”)
Of course, if you have linear utility (i.e. no risk aversion), you should still go all-in on whichever bet has highest expected return individually. If this were really true, your life would be so simple that most of finance would be irrelevant to it (and vice versa). You’d just put 100% in whichever asset you thought was best at any given time.