# Kelly Betting

Arguably there are already sufficiently many articles on the Kelly criterion. Here are a few that I liked:

- Edward Thorp provides a great review of the Kelly criterion (with possible objections),
- This paper proves that a market with Kelly bettors
- has prices that are a wealth-weighted average of traders’ beliefs (no surprise so far),
- learns at the optimal rate and the market price reacts exactly as if updating according to Bayes’ Law (a bit of a surprise?),
- and the market prediction has low worst-case log regret to the best individual participant (quite a surprise, although potentially an artifact of the unrealistic(?) assumption of overly aggressive Kelly traders).

- Proebsting’s “paradox”; not really a paradox, but raises questions about applying it in practice,
- Buck Shlegeris conjectures that any market should eventually be dominated by Kelly bettors,
- A seasoned PredictIt trader presents some arguments why Kelly is actually not so great in practice,
- SimonM on Twitter shows that fractional Kelly is really just (partially) deferring to the market (and then going full-Kelly with the updated probabilities) and more reasons to never go full-Kelly from the same author on LessWrong.
- An unfortunately rather spread-out discussion on LessWrong (see Kelly isn’t (just) about logarithmic utility, Kelly is just about logarithmic utility, and various replies in the comments/other threads); while “X is (just/not) about Y” hardly seems objectively resolvable, the arguments for and against are interesting:
- Kelly optimises not only log-wealth, but also any quantile; it immediately follows that
any other strategy can only beat Kelly at most 1/2 the time. (1/2 is optimal since the other strategy could be Kelly)

- A (sub-)logarithmic utility function can be justified with concerns about ergodicity for repeated bets. (In plain terms: Linear utility functions (e.g. maximising expected wealth) leaves us with having to bet everything on a bet where you double your money with 51% and lose everything with 49%. Doing this many times results in tremendous expected wealth, despite the fact that you go bankrupt almost surely. (Sub-)logarithmic utility functions, however, take care of this problem.)

- Kelly optimises not only log-wealth, but also any quantile; it immediately follows that