Great post, I find it really valuable to engage in this type of meta-modeling, i.e., deriving when and why models are appropriate.
I think you’re making a mistake in Section 2 though. You argue that a mode optimizer can be pretty terrible (agreed). Then, you argue that any other quantile optimizer can also be pretty terrible (also agreed). However, Kelly doesn’t only optimize the mode, or 2% quantile, or whatever other quantile: it maximizes all those quantiles simultaneously! So, is there any distribution for which Kelly itself fails to optimize between meaningfully different states (as in your 2%-quantile with 10% bad outcome example)? I don’t think such a distribution exists.
(Note: maybe I’m misunderstanding what johnswentworth said here, but if solving for any x%-quantile maximizer always yields Kelly, then Kelly maximizes for all quantiles, correct?)
Yep, I actually note this in footnote 3. I didn’t change section 2 because I still think that if each of these is individually bad, it’s pretty questionable to use them as justification for Kelly.
Note that if a strategy Sb is better or equal in every quantile, and strictly better in some, compared to some Sw, then expected utility maximization will prefer Sb to Sw, no matter what the utility function is (so long as more money is considered better, ie utility is monotonic).
So all expected utility maximizers would endorse an all-quantile-optimizing strategy, if one existed. This isn’t a controversial property from the EU perspective!
But it’s easy to construct bets which prove that maximizing one quantile is not always consistent with maximizing another; there are trade-offs, so there’s not generally a strategy which maximizes all quantiles.
So it’s critically important that Kelly is only approximately doing this, in the limit. If Kelly had this property precisely, then all expected utility maximizers would use the Kelly strategy.
In particular, at a fixed finite time, there’s a quantile for the all-win sequence. However, since this quantile becomes smaller and smaller, it vanishes in the limit. At finite time, the expected-money-maximizer is optimizing this extreme quantile, but the Kelly strategy is making trade-offs which are suboptimal for that quantile.
(Note: maybe I’m misunderstanding what johnswentworth said here, but if solving for any x%-quantile maximizer always yields Kelly, then Kelly maximizes for all quantiles, correct?)
That’s my belief too, but I haven’t verified it. It’s clear from the usual derivation that it’s approximately mode-maximizing. And I think I can see why it’s approximately median-maximizing by staring at the wikipedia page for log-normal long enough and crossing my eyes just right [satire].
Great post, I find it really valuable to engage in this type of meta-modeling, i.e., deriving when and why models are appropriate.
I think you’re making a mistake in Section 2 though. You argue that a mode optimizer can be pretty terrible (agreed). Then, you argue that any other quantile optimizer can also be pretty terrible (also agreed). However, Kelly doesn’t only optimize the mode, or 2% quantile, or whatever other quantile: it maximizes all those quantiles simultaneously! So, is there any distribution for which Kelly itself fails to optimize between meaningfully different states (as in your 2%-quantile with 10% bad outcome example)? I don’t think such a distribution exists.
(Note: maybe I’m misunderstanding what johnswentworth said here, but if solving for any x%-quantile maximizer always yields Kelly, then Kelly maximizes for all quantiles, correct?)
Yep, I actually note this in footnote 3. I didn’t change section 2 because I still think that if each of these is individually bad, it’s pretty questionable to use them as justification for Kelly.
Note that if a strategy Sb is better or equal in every quantile, and strictly better in some, compared to some Sw, then expected utility maximization will prefer Sb to Sw, no matter what the utility function is (so long as more money is considered better, ie utility is monotonic).
So all expected utility maximizers would endorse an all-quantile-optimizing strategy, if one existed. This isn’t a controversial property from the EU perspective!
But it’s easy to construct bets which prove that maximizing one quantile is not always consistent with maximizing another; there are trade-offs, so there’s not generally a strategy which maximizes all quantiles.
So it’s critically important that Kelly is only approximately doing this, in the limit. If Kelly had this property precisely, then all expected utility maximizers would use the Kelly strategy.
In particular, at a fixed finite time, there’s a quantile for the all-win sequence. However, since this quantile becomes smaller and smaller, it vanishes in the limit. At finite time, the expected-money-maximizer is optimizing this extreme quantile, but the Kelly strategy is making trade-offs which are suboptimal for that quantile.
That’s my belief too, but I haven’t verified it. It’s clear from the usual derivation that it’s approximately mode-maximizing. And I think I can see why it’s approximately median-maximizing by staring at the wikipedia page for log-normal long enough and crossing my eyes just right [satire].
That clears up my confusion, thanks!