# 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 theorems, possible objections and many applications),
- 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 fixed quantile (as the number of bets goes to infinity, by the CLT); 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)

- Note that, in general, for finitely many bets Kelly only minimises the median. This is because betting more than Kelly increases the variance of the logarithm of our wealth, while decreasing its mean. In some cases betting a bit more than Kelly thus increases >50 quantiles (e.g. what you get in the luckiest 10% of all outcomes).
Maximises asymptotic long run growth

Given a target wealth, is the strategy which achieves that wealth fastest

Asymptotically outperforms any other strategy (ie \(\mathbb E[X/X_\text{Kelly}] \leq 1\))

- 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 fixed quantile (as the number of bets goes to infinity, by the CLT); it immediately follows that
- In finance, Kelly is essentially the same as optimising the Sharpe ratio.