You mean, the potential actions are discrete but the potential outcomes for those actions are continuous, with a probability measure over those outcomes, or that there is a non-discrete set of possible actions, or something else?
Yes, potential actions are discrete and outcomes are arbitrarily distributed.
I'm not sure I'm understanding this correctly. Are you asking how the St. Petersburg Paradox works?
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.
Before you take the derivative with respect to Delta, apply the desired utility function, and then take the derivative.
Interesting. I should try this later.
(Note that linear utility functions behave the same as logarithmic utility functions, and Wikipedia's treatment assumes a linear utility function, not a logarithmic one.)
The Kelly criterion is the natural result when assuming a logarithmic utility function. For a linear utility function it arises if the actor maximizes expected growth rate.
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 ...
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.