Yes, potential actions are discrete and outcomes are arbitrarily distributed.
It seems like this paper or this paper might be relevant to your interests. (PM me your email if you don't have access to them.)
No, I mean that the Kelly criterion says that allocation to a bet should be proportional to expected value over payoff. If I hold expected value constant and integrate over payoff the integral diverges. Intuitively I would expect to see a finite integral, reflecting that Kelly restricts how much risk I should be willing to take.
Kelly tells you how much risk you should be willing to take for a particular b; integrating over b is not meaningful, since it's integrating over multiple bets. (Note that f is E/b, if E is the expected value, and 1/x diverges. Since p is capped by 1, then E is capped by b, and the maximum risk you should take is betting everything, if p=1 i.e. it's a sure thing.)
If you put a probability p(b) on any particular payout, you might get something meaningful out of integrating p(b)E/b, but it's not clear to me that's the right way to do things.
Interesting. I should try this later.
It won't work out very prettily, but it is instructive. Basically, that tells you how much your bet should have differed from Delta, given what happened. You can then figure out what would have been optimal for that sequence, then do a weighted sum over sequences. (If your utility function isn't scale invariant, and only log is, then you need information on how long the game runs; if you're allowed to change the fraction of your wealth that you put up each time, then it's an entirely different problem.)
The Kelly criterion is the optimal way to allocate one's bankroll over a lifetime to a series of bets assuming the actor's utility increases logarithmically with the amount of money won. Most importantly the criterion gives motivation to decide between investments with identical expected value but different risk of default. It essentially stipulates that the proportion of one's bankroll invested in a class of bets should be proportional to the expected value divided by the payoff in case it pans out.
Now, nothing in the formalism restricts the rule to bets or money for that matter, but is applicable to any situation an actor as assumed above faces uncertainty and possible payoff in utility. Aside from the obvious application to investments, e.g. bonds, this is also applicable to the purchase of insurance or cryonic services.
Buying an insurance can obviously be modeled as bet in the Kelly sense. A simple generalisation of the Kelly criterion leads to a formula that allows to incorporate losses.
An open question, to me at least, is if it possible to generalise the Kelly criterion to arbitrary probability distributions. Also, how can it be that integration over all payoffs for constant expected value evaluates as infinity?
Finally, how would a similar criterion look like for other forms of utility functions?
I did not put this question in the open thread because I think the Kelly criterion deserves more of a discussion and is immediately relevant to this site's interests.