One other argument I’ve seen for Kelly is that it’s optimal if you start with $a and you want to get to $b as quickly as possible, in the limit of b >> a. (And your utility function is linear in time, i.e. -t.)
You can see why this would lead to Kelly. All good strategies in this game will have somewhat exponential growth of money, so the time taken will be proportional to the logarithm of b/a.
So this is a way in which a logarithmic utility might arise as an instrumental value while optimising for some other goal, albeit not a particularly realistic one.
One other argument I’ve seen for Kelly is that it’s optimal if you start with $a and you want to get to $b as quickly as possible, in the limit of b >> a. (And your utility function is linear in time, i.e. -t.)
You can see why this would lead to Kelly. All good strategies in this game will have somewhat exponential growth of money, so the time taken will be proportional to the logarithm of b/a.
So this is a way in which a logarithmic utility might arise as an instrumental value while optimising for some other goal, albeit not a particularly realistic one.