The basic rough argument for Kelly betting goes something like this.
First, assume we’re making a sequence of T independent bets, one-after-another, with multiplicative returns (similar to e.g. financial markets). We choose how much money to put on which bets at each timestep.
Returns multiply, so log returns add. And they’re independent at each timestep, so the total log return over T timesteps is a sum of T independent random variables. “Sum of T independent random variables” makes us want to invoke the Central Limit Theorem, so let's assume whatever other conditions we need in order to do that. (There are multiple options for the other conditions.) So: total log return will be normally distributed for large T, with mean equal to the sum of expected log return at each timestep.
Then the key question is: for any given utility function, will it be dominated by the typical/modal/median return, or will it be dominated by the tails? For instance, the utility function u(W) = W is dominated by the upper tail: agents maximizing that utility function will happily accept a probability-approaching-1 of zero wealth, in exchange for an exponentially tiny chance of exponentially huge returns. On the other end [...]
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First published:
November 20th, 2025
Source:
https://www.lesswrong.com/posts/e5bCcXutaN8Zc2m7w/what-is-the-basin-of-convergence-for-kelly-betting
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