Then it’s not quite as clear-cut. Maybe most of the time A is only slightly better than B, but the tiny possibility of A being worse than usual involves A being enormously worse than usual, or the tiny possibility of B being better than usual involves B being enormously better than usual. But this is still a pretty big hint that A is better in expectation.A is better unless your expected utility calculations are radically altered by negligible-probability tail events.
But this is exactly the situation of comparing Kelly betting to max-EV betting! When you’re betting your whole stack each timestep, what you’re doing is squeezing all of the value into that tiny possibility that B is better than usual, such that it outstrips A all on its own. If you assume that this is a bad idea and we shouldn’t do it, then sure, something like Kelly betting pops out, but I find it unsatisfying.
I took the main point of the post to be that there are fairly general conditions (on the utility function and on the bets you are offered) in which you should place each bet like your utility is linear, and fairly general conditions in which you should place each bet like your utility is logarithmic. In particular, the conditions are much weaker than your utility actually being linear, or than your utility actually being logarithmic, respectively, and I think this is a cool point. I don’t see the post as saying anything beyond what’s implied by this about Kelly betting vs max-linear-EV betting in general.
I basically endorse what kh said. I do think it’s wrong to think you can fit enormous amounts of expected value or disvalue into arbitrarily tiny probabilities.
Yes, I would agree with this. If we suppose our utility function is bounded, then when given unlimited access to a gamble in our favor, we should basically be asking “Having been handed this enormous prize, how do I maximize the probability that I max out on utility?”
Hm, but that actually doesn’t give back any specific criterion, since basically any strategy that never bets your whole stack will win. What happens if you try to minimize the expected time until you hit the maximum?
Optimal play will definitely diverge from the Kelly criterion when your stack is close to the maximum. But in the limit of a large maximum I think you recover the Kelly criterion, for basically the reason you give in this post.
“Having been handed this enormous prize, how do I maximize the probability that I max out on utility?” Hm, but that actually doesn’t give back any specific criterion, since basically any strategy that never bets your whole stack will win.
That’s not quite true. If you bet more than double Kelly, your wealth decreases. But yes, Kelly betting isn’t unique in growing your wealth to infinity in the limit as number of bets increases.
If the number of bets is very large, but due to some combination of low starting wealth relative to the utility bound and slow growth rate, it is not possible to get close to maximum utility, then Kelly betting should be optimal.
But this is exactly the situation of comparing Kelly betting to max-EV betting! When you’re betting your whole stack each timestep, what you’re doing is squeezing all of the value into that tiny possibility that B is better than usual, such that it outstrips A all on its own. If you assume that this is a bad idea and we shouldn’t do it, then sure, something like Kelly betting pops out, but I find it unsatisfying.
I took the main point of the post to be that there are fairly general conditions (on the utility function and on the bets you are offered) in which you should place each bet like your utility is linear, and fairly general conditions in which you should place each bet like your utility is logarithmic. In particular, the conditions are much weaker than your utility actually being linear, or than your utility actually being logarithmic, respectively, and I think this is a cool point. I don’t see the post as saying anything beyond what’s implied by this about Kelly betting vs max-linear-EV betting in general.
I basically endorse what kh said. I do think it’s wrong to think you can fit enormous amounts of expected value or disvalue into arbitrarily tiny probabilities.
Yes, I would agree with this. If we suppose our utility function is bounded, then when given unlimited access to a gamble in our favor, we should basically be asking “Having been handed this enormous prize, how do I maximize the probability that I max out on utility?”
Hm, but that actually doesn’t give back any specific criterion, since basically any strategy that never bets your whole stack will win. What happens if you try to minimize the expected time until you hit the maximum?
Optimal play will definitely diverge from the Kelly criterion when your stack is close to the maximum. But in the limit of a large maximum I think you recover the Kelly criterion, for basically the reason you give in this post.
That’s not quite true. If you bet more than double Kelly, your wealth decreases. But yes, Kelly betting isn’t unique in growing your wealth to infinity in the limit as number of bets increases.
If the number of bets is very large, but due to some combination of low starting wealth relative to the utility bound and slow growth rate, it is not possible to get close to maximum utility, then Kelly betting should be optimal.