I think the Kelly betting criterion always gives “sensible” results. By which I mean: there’s no hyper-st-petersburg-lottery for which maximizing expected log wealth means investing infinity times your current wealth, even if diverges; the Kelly criterion should always give you a finite fraction of your wealth (maybe >1) you ought to bet.
(Sorry if this isn’t a novel idea, just noticed this and needed to put it down somewhere)
Sketch proof for a toy model, I think this generalizes.
Assume we are deciding what fraction, , of our wealth to wager on a bet that will return dollars, where is a random variable that takes values with probability . The fraction of our wealth that we don’t wager is unaffected.
We assume , and to ensure the question is interesting, at least one and at least one (otherwise one should obviously invest as much/little (respectively) as one can).
Our expected log wealth (as a multiple of what we stated with), having invested is
It is very easy to get this to diverge, e.g for all positive integers .
The Kelly criterion says we should look for maxima of. Formally, we have
and we want to solve .
The first observation to make is that f’(q) converges for almost all values of : even if grows rapidly with , the summands above will tend to , whose sum must converge if the are probabilities.
The exceptions are the simple poles at each , and a possible pole at if diverges—for this argument, we will assume it does, otherwise everything converges and we can do this the normal way.
The second is that is negative everywhere, except at the poles mentioned above, so any stationary point of is a local maximum.
Finally, consider the smallest [1]. There is an associated pole . is negative to the left of this pole, and positive to the right (as is negative everywhere); this is also true for the pole at . As there are no poles between and
I think the Kelly betting criterion always gives “sensible” results. By which I mean: there’s no hyper-st-petersburg-lottery for which maximizing expected log wealth means investing infinity times your current wealth, even if
(Sorry if this isn’t a novel idea, just noticed this and needed to put it down somewhere)
Sketch proof for a toy model, I think this generalizes.
Assume we are deciding what fraction,
We assume
Our expected log wealth (as a multiple of what we stated with), having invested
It is very easy to get this to diverge, e.g
The Kelly criterion says we should look for maxima of
and we want to solve
The first observation to make is that f’(q) converges for almost all values of
The exceptions are the simple poles at each
The second is that
Finally, consider the smallest
I’m not sure what would happen if there were no smallest x_i < 1