[OscarCunningham]

I guess one thing you can say about the Kelly criterion without mentioning utility is that if Alice uses the Kelly criterion and Bob uses some other strategy (which is still of the form "bet some fixed proportion of your money each turn") then the probability that Alice has more money than Bob tends to 1 as the number of turns increases.

Of course in the cases where Bob has more money he might have much more money, so this fact isn't very relevant to them unless they have appropriate utility functions.

Another thing that occurs to me is that your utility function is changed by the opportunities you expect to encounter. If your utility function for money would normally be U_0, and you are about to be allowed to make a bunch of bets, then your current utility function, U_-1, is equal to the expectation of U_0 under the probability distribution that results from you betting optimally starting with however much money you have.

Maybe there's a family of utility functions for which if U_0 is in that family then U_-1 is approximately logarithmic? Then that would be a good justification for using the Kelly criterion if you have a long string of bets ahead of you. On the other hand I just checked the HARA (https://en.wikipedia.org/wiki/Hyperbolic_absolute_risk_avers...) family of utility functions, and they're all stable under the process I described. So there are certainly a lot of functions that don't become logarithmic.

[bluecalm]

Thanks for the interesting comment and a link. I will be using your the first paragraph from your post in the future :)