Yeah, I was still being sloppy about what I meant by near-optimal, sorry. I mean the optimal bet size will converge to the Kelly bet size, not that the expected utility from Kelly betting and the expected utility from optimal betting converge to each other. You could argue that the latter is more important, since getting high expected utility in the end is the whole point. But on the other hand, when trying to decide on a bet size in practice, there’s a limit to the precision with which it is possible to measure your edge, so the difference between optimal bet and Kelly bet could be small compared to errors in your ability to determine the Kelly bet size, in which case thinking about how optimal betting differs from Kelly betting might not be useful compared to trying to better estimate the Kelly bet.
Even in the limit as the number of rounds goes to infinity, by the time you get to the last round of betting (or last few rounds), you’ve left the n→∞ limit, since you have some amount of wealth and some small number of rounds of betting ahead of you, and it doesn’t matter how you got there, so the arguments for Kelly betting don’t apply. So I suspect that Kelly betting until near the end, when you start slightly adjusting away from Kelly betting based on some crude heuristics, and then doing an explicit expected value calculation for the last couple rounds, might be a good strategy to get close to optimal expected utility.
Incidentally, I think it’s also possible to take a limit where Kelly betting gets you optimal utility in the end by making the favorability of the bets go to zero simultaneously with the number of rounds going to infinity, so that improving your strategy on a single bet no longer makes a difference.
I think that for all finite n, the expected utility at timestep n from utility-maximizing bets is higher than that from Kelly bets. I think this is the case even if the difference converges to 0, which I’m not sure it does.
Why specifically higher? You must be making some assumptions on the utility function that you haven’t mentioned.
You could argue that the latter is more important, since getting high expected utility in the end is the whole point. But on the other hand, when trying to decide on a bet size in practice, there’s a limit to the precision with which it is possible to measure your edge, so the difference between optimal bet and Kelly bet could be small compared to errors in your ability to determine the Kelly bet size, in which case thinking about how optimal betting differs from Kelly betting might not be useful compared to trying to better estimate the Kelly bet.
So like, this seems plausible to me, but… yeah, I really do want to distinguish between
This maximizes expected utility
This doesn’t maximize expected utility, but here are some heuristics that suggest maybe that doesn’t matter so much in practice
If it doesn’t seem important to you to distinguish these, then that’s a different kind of conversation than us disagreeing about the math, but here are some reasons I want to distingish them:
I think lots of people are confused about Kelly, and speaking precisely seems more likely to help than hurt.
I think “get the exact answer in spherical cow cases” is good practice, even if spherical cow cases never come up. “Here’s the exact answer in the simple case, and here are some considerations that mean it won’t be right in practice” seems better than “here’s an approximate answer in the simple case, and here are some considerations that mean it won’t be right in practice”.
Sometimes it’s not worth figuring out the exact answer, but like. I haven’t yet tried to calculate the utility-maximizing bet for those other utility functions. I haven’t checked how much Kelly loses relative to them under what conditions. Have you? It seems like this is something we should at least try to calculate before going “eh, Kelly is probably fine”.
I’ve spent parts of this conversation confused about whether we disagree about the math or not. If you had reliably been making the distinction I want to make, I think that would have helped. If I had reliably not made that distinction, I think we just wouldn’t have talked about the math and we still wouldn’t know if we agreed or not. That seems like a worse outcome to me.
Why specifically higher? You must be making some assumptions on the utility function that you haven’t mentioned.
Well, we’ve established the utility-maximizing bet gives different expected utility from the Kelly bet, right? So it must give higher expected utility or it wouldn’t be utility-maximizing.
Yeah, I wasn’t trying to claim that the Kelly bet size optimizes a nonlogarithmic utility function exactly, just that, when the number of rounds of betting left is very large, the Kelly bet size sacrifices a very small amount of utility relative to optimal betting under some reasonable assumptions about the utility function. I don’t know of any precise mathematical statement that we seem to disagree on.
Well, we’ve established the utility-maximizing bet gives different expected utility from the Kelly bet, right? So it must give higher expected utility or it wouldn’t be utility-maximizing.
Right, sorry. I can’t read, apparently, because I thought you had said the utility-maximizing bet size would be higher than the Kelly bet size, even though you did not.
Yeah, I was still being sloppy about what I meant by near-optimal, sorry. I mean the optimal bet size will converge to the Kelly bet size, not that the expected utility from Kelly betting and the expected utility from optimal betting converge to each other. You could argue that the latter is more important, since getting high expected utility in the end is the whole point. But on the other hand, when trying to decide on a bet size in practice, there’s a limit to the precision with which it is possible to measure your edge, so the difference between optimal bet and Kelly bet could be small compared to errors in your ability to determine the Kelly bet size, in which case thinking about how optimal betting differs from Kelly betting might not be useful compared to trying to better estimate the Kelly bet.
Even in the limit as the number of rounds goes to infinity, by the time you get to the last round of betting (or last few rounds), you’ve left the n→∞ limit, since you have some amount of wealth and some small number of rounds of betting ahead of you, and it doesn’t matter how you got there, so the arguments for Kelly betting don’t apply. So I suspect that Kelly betting until near the end, when you start slightly adjusting away from Kelly betting based on some crude heuristics, and then doing an explicit expected value calculation for the last couple rounds, might be a good strategy to get close to optimal expected utility.
Incidentally, I think it’s also possible to take a limit where Kelly betting gets you optimal utility in the end by making the favorability of the bets go to zero simultaneously with the number of rounds going to infinity, so that improving your strategy on a single bet no longer makes a difference.
Why specifically higher? You must be making some assumptions on the utility function that you haven’t mentioned.
So like, this seems plausible to me, but… yeah, I really do want to distinguish between
This maximizes expected utility
This doesn’t maximize expected utility, but here are some heuristics that suggest maybe that doesn’t matter so much in practice
If it doesn’t seem important to you to distinguish these, then that’s a different kind of conversation than us disagreeing about the math, but here are some reasons I want to distingish them:
I think lots of people are confused about Kelly, and speaking precisely seems more likely to help than hurt.
I think “get the exact answer in spherical cow cases” is good practice, even if spherical cow cases never come up. “Here’s the exact answer in the simple case, and here are some considerations that mean it won’t be right in practice” seems better than “here’s an approximate answer in the simple case, and here are some considerations that mean it won’t be right in practice”.
Sometimes it’s not worth figuring out the exact answer, but like. I haven’t yet tried to calculate the utility-maximizing bet for those other utility functions. I haven’t checked how much Kelly loses relative to them under what conditions. Have you? It seems like this is something we should at least try to calculate before going “eh, Kelly is probably fine”.
I’ve spent parts of this conversation confused about whether we disagree about the math or not. If you had reliably been making the distinction I want to make, I think that would have helped. If I had reliably not made that distinction, I think we just wouldn’t have talked about the math and we still wouldn’t know if we agreed or not. That seems like a worse outcome to me.
Well, we’ve established the utility-maximizing bet gives different expected utility from the Kelly bet, right? So it must give higher expected utility or it wouldn’t be utility-maximizing.
Yeah, I wasn’t trying to claim that the Kelly bet size optimizes a nonlogarithmic utility function exactly, just that, when the number of rounds of betting left is very large, the Kelly bet size sacrifices a very small amount of utility relative to optimal betting under some reasonable assumptions about the utility function. I don’t know of any precise mathematical statement that we seem to disagree on.
Right, sorry. I can’t read, apparently, because I thought you had said the utility-maximizing bet size would be higher than the Kelly bet size, even though you did not.