Correct. This utility function grows fast enough that it is possible for the expected utility after many bets to be dominated by negligible-probability favorable tail events, so you’d want to bet super-Kelly.
E(1−e−money)
If you expect to end up with lots of money at the end, then you’re right; marginal utility of money becomes negigible, so expected utility is greatly effected by neglible-probability unfavorable tail events, and you’d want to bet sub-Kelly. But if you start out with very little money, so that at the end of whatever large number of rounds of betting, you only expect to end up with ∼1 money in most cases if you bet Kelly, then I think the Kelly criterion should be close to optimal.
E(√log(money))
(The thing you actually wrote is the same as log utility, so I substituted what you may have meant). The Kelly criterion should optimize this, and more generally E(log(money)x) for any x>0, if the number of bets is large. At least if x is an integer, then, if log(money) is normally distributed with mean μ and standard deviation σ, then E(log(money)x) is some polynomial in μ and σ that’s homogeneous of degree x. After a large number N of bets, μ scales proportionally to N and σ scales proportionally to √N, so the value of this polynomial approaches its μN term, and maximizing it becomes equivalent to maximizing μ, which the Kelly criterion does. I’m pretty sure you get something similar when x is noninteger.
E(min(money,x))
It depends how much money you could end up with compared to x. If Kelly betting usually gets you more than x at the end, then you’ll bet sub-Kelly to reduce tail risk. If it’s literally impossible to exceed x even if you go all-in every time and always win, then this is linear, and you’ll bet super-Kelly. But if Kelly betting will usually get you less than x but not by too many orders of magnitude at the end after a large number of rounds of betting, then I think it should be near-optimal.
E(log(money+x))
If there’s many rounds of betting, and Kelly betting will get you money≈x as a typical outcome, then I think Kelly betting is near-optimal. But you might be right if money<<x.
Okay, “Kelly is close to optimal for lots of utility functions” seems entirely plausible to me. I do want to note though that this is different from “actually optimal”, which is what I took you to be saying.
(The thing you actually wrote is the same as log utility, so I substituted what you may have meant)
Oops! I actually was just writing things without thinking much and didn’t realize it was the same.
I do want to note though that this is different from “actually optimal”
By “near-optimal”, I meant converges to optimal as the number of rounds of betting approaches infinity, provided initial conditions are adjusted in the limit such that whatever conditions I mentioned remain true in the limit. (e.g. if you want Kelly betting to get you a typical outcome of money≈1 in the end, then when taking the limit as the number N of bets goes to infinity, you better have starting money r−N, where r is the geometric growth rate you get from bets, rather than having a fixed starting money while taking the limit N→∞). This is different from actually optimal because in practice, you get some finite amount of betting opportunities, but I do mean something more precise than just that Kelly betting tends to get decent outcomes.
Thanks for clarifying! Um, but to clarify a bit further, here are three claims one could make about these examples:
As wealth→∞, the utility maximizing bet at given wealth will converge to the Kelly bet at that wealth. I basically buy this.
As n→∞, the expected utility from utility-maximizing bets at timestep n converges to that from Kelly bets at timestep n. I’m unsure about this.
For some finite n, the expected utility at timestep n from utility-maximizing bets is no higher than that from Kelly bets. I think this is false. (In the positive: 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.)
I think you’re saying (2)? But the difference between that and (3) seems important to me. Like, it still seems that to a (non-log-money) utility maximizer, the Kelly bet is strictly worse than the bet which maximizes their utility at any given timestep. So why would they bet Kelly?
Here’s why I’m unsure about 2. Suppose we both have log-money utility, I start with $2 and you start with $1, and we place the same number of bets, always utility-maximizing. After any number of bets, my expected wealth will always be 2x yours, so my expected utility will always be log(2) more than yours. So it seems to me that “starting with more money” leads to “having more log-money in expectation forever”.
Then it similarly seems to me that if I get to place a bet before you enter the game, and from then on our number of bets is equal, my expected utility will be forever higher than yours by the expected utility gain of that one bet.
Or, if we get the same number of bets, but my first bet is utility maximizing and yours is not, but after that we both place the utility-maximizing bet; then I think my expected utility will still be forever higher than yours. And the same for if you make bets that aren’t utility-maximizing, but which converge to the utility-maximizing bet.
And if this is the case for log-money utility, I’d expect it to also be the case for many other utility functions.
...but something about this feels weird, especially with log(money+x), so I’m not sure. I think I’d need to actually work this out.
Here’s a separate thing I’m now unsure about. (Thanks for helping bring it to light!) In my terminology from on Kelly and altruism, making a finite number of suboptimal bets doesn’t change how rank-optimal your strategy is. In Kelly’s terminology from his original paper, I think it won’t change your growth rate.
And I less-confidently think the same is true of “making suboptimal bets all the time, but the bets converge to the optimal bet”.
But if that’s true… what actually makes those bets suboptimal, in those two frameworks? If Kelly’s justification for the Kelly bet is that it maximizes your growth rate, but there are other bet sizes that do the same, why prefer the Kelly bet over them? If my justification for the Kelly bet (when I endorse using it) is that it’s impossible to be more rank-optimal than it, why prefer the Kelly bet over other things that are equally rank-optimal?
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.
Correct. This utility function grows fast enough that it is possible for the expected utility after many bets to be dominated by negligible-probability favorable tail events, so you’d want to bet super-Kelly.
If you expect to end up with lots of money at the end, then you’re right; marginal utility of money becomes negigible, so expected utility is greatly effected by neglible-probability unfavorable tail events, and you’d want to bet sub-Kelly. But if you start out with very little money, so that at the end of whatever large number of rounds of betting, you only expect to end up with ∼1 money in most cases if you bet Kelly, then I think the Kelly criterion should be close to optimal.
(The thing you actually wrote is the same as log utility, so I substituted what you may have meant). The Kelly criterion should optimize this, and more generally E(log(money)x) for any x>0, if the number of bets is large. At least if x is an integer, then, if log(money) is normally distributed with mean μ and standard deviation σ, then E(log(money)x) is some polynomial in μ and σ that’s homogeneous of degree x. After a large number N of bets, μ scales proportionally to N and σ scales proportionally to √N, so the value of this polynomial approaches its μN term, and maximizing it becomes equivalent to maximizing μ, which the Kelly criterion does. I’m pretty sure you get something similar when x is noninteger.
It depends how much money you could end up with compared to x. If Kelly betting usually gets you more than x at the end, then you’ll bet sub-Kelly to reduce tail risk. If it’s literally impossible to exceed x even if you go all-in every time and always win, then this is linear, and you’ll bet super-Kelly. But if Kelly betting will usually get you less than x but not by too many orders of magnitude at the end after a large number of rounds of betting, then I think it should be near-optimal.
If there’s many rounds of betting, and Kelly betting will get you money≈x as a typical outcome, then I think Kelly betting is near-optimal. But you might be right if money<<x.
Okay, “Kelly is close to optimal for lots of utility functions” seems entirely plausible to me. I do want to note though that this is different from “actually optimal”, which is what I took you to be saying.
Oops! I actually was just writing things without thinking much and didn’t realize it was the same.
By “near-optimal”, I meant converges to optimal as the number of rounds of betting approaches infinity, provided initial conditions are adjusted in the limit such that whatever conditions I mentioned remain true in the limit. (e.g. if you want Kelly betting to get you a typical outcome of money≈1 in the end, then when taking the limit as the number N of bets goes to infinity, you better have starting money r−N, where r is the geometric growth rate you get from bets, rather than having a fixed starting money while taking the limit N→∞). This is different from actually optimal because in practice, you get some finite amount of betting opportunities, but I do mean something more precise than just that Kelly betting tends to get decent outcomes.
Thanks for clarifying! Um, but to clarify a bit further, here are three claims one could make about these examples:
As wealth→∞, the utility maximizing bet at given wealth will converge to the Kelly bet at that wealth. I basically buy this.
As n→∞, the expected utility from utility-maximizing bets at timestep n converges to that from Kelly bets at timestep n. I’m unsure about this.
For some finite n, the expected utility at timestep n from utility-maximizing bets is no higher than that from Kelly bets. I think this is false. (In the positive: 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.)
I think you’re saying (2)? But the difference between that and (3) seems important to me. Like, it still seems that to a (non-log-money) utility maximizer, the Kelly bet is strictly worse than the bet which maximizes their utility at any given timestep. So why would they bet Kelly?
Here’s why I’m unsure about 2. Suppose we both have log-money utility, I start with $2 and you start with $1, and we place the same number of bets, always utility-maximizing. After any number of bets, my expected wealth will always be 2x yours, so my expected utility will always be log(2) more than yours. So it seems to me that “starting with more money” leads to “having more log-money in expectation forever”.
Then it similarly seems to me that if I get to place a bet before you enter the game, and from then on our number of bets is equal, my expected utility will be forever higher than yours by the expected utility gain of that one bet.
Or, if we get the same number of bets, but my first bet is utility maximizing and yours is not, but after that we both place the utility-maximizing bet; then I think my expected utility will still be forever higher than yours. And the same for if you make bets that aren’t utility-maximizing, but which converge to the utility-maximizing bet.
And if this is the case for log-money utility, I’d expect it to also be the case for many other utility functions.
...but something about this feels weird, especially with log(money+x), so I’m not sure. I think I’d need to actually work this out.
Here’s a separate thing I’m now unsure about. (Thanks for helping bring it to light!) In my terminology from on Kelly and altruism, making a finite number of suboptimal bets doesn’t change how rank-optimal your strategy is. In Kelly’s terminology from his original paper, I think it won’t change your growth rate.
And I less-confidently think the same is true of “making suboptimal bets all the time, but the bets converge to the optimal bet”.
But if that’s true… what actually makes those bets suboptimal, in those two frameworks? If Kelly’s justification for the Kelly bet is that it maximizes your growth rate, but there are other bet sizes that do the same, why prefer the Kelly bet over them? If my justification for the Kelly bet (when I endorse using it) is that it’s impossible to be more rank-optimal than it, why prefer the Kelly bet over other things that are equally rank-optimal?
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.