What is my utility function if I want to minimize the mean time of doubling my bankroll, provided I am making +EV bets?
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Recall that Kelly maximizes the expected rate of bankroll growth.
Hence, if we approximate outcomes on the continuum (which would be essentially subsumed by limiting ourselves to "sufficiently small" odds and edge combinations), the desire to minimize mean doubling (or tripling, or n-tupling) time at every decision point would simply correspond to log utility.Comment -
Um, close, but not quite. Imagine I have a bankroll of 184 units. I have a positive ev bet that Kelly suggests I bet 16.5 to win 15. My starting bankroll was 100 units. If doubling your bankroll is significantly better than almost doubling your bankroll, then the bet size should justifiably be increased.
BTW, I think it should be noted all three of us are anal ballbusters, which is quite the image.Comment -
On the other hand, the further you are away from your goal, the closer kelly becomes to being the absolute truth.Comment -
The above solution specifically refers to "the desire to minimize mean doubling ... time at every decision point". And even with that condition set we still include the caveat that "we approximate outcomes on the continuum".
If on the other hand, the OP were referring to reaching a stated fixed goal (which could as well be defined, without loss of generality as double the bankroll at an arbitrary fixed point) then the solution would require an approximation utilizing the Brownian motion equations (which also requires the above continuity assumption). For an instructive example of how one might go about this, please see this post.Comment -
Yes, that is what I have in mind. I am sorry about confusion. Also, I want to set an important condition that tripling the bankroll is only negligibly better than doubling it.
How do I derive the utility function from this? I do not think I am up to the task. If you can solve this I would greatly appreciate this....then the solution would require an approximation utilizing the Brownian motion equations (which also requires the above continuity assumption). For an instructive example of how one might go about this, please see this post.Comment -
I think you need to more fully define these preferences.
Initially, you were implying that the sole goal of the market operator was to reach his fixed bogey via a Markov process as quickly as possible.
But now, however, it would also appear that there's some additional (even if minimal) utility to be gained from overshooting the bogey.
This would also beg the question as to whether there would be more utility to be gained from embarking on a "slow" but low risk strategy at a point sufficiently close to the bogey, than a "faster", but higher risk strategy fairly far from the bogey.
Putting it another way, is the partial derivative of utility with respect to current bankroll (holding expected time to hit bogey constant) nonzero?
Basically, the interplay between the state variables needs to be made more explicit.Comment -
Lets say that this additional utility approaches zero and can be ignored.
Yes, I think so. This should help with keeping the bet sizes more in line. Obviously, the real life implementaion of this does not allow any serious stakes increases.This would also beg the question as to whether there would be more utility to be gained from embarking on a "slow" but low risk strategy at a point sufficiently close to the bogey, than a "faster", but higher risk strategy fairly far from the bogey.Comment
