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.
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.