I think the interesting question is what to do when you expect many more, but only finitely many rounds. It seems like Kelly should somehow gradually transition, until it recommends normal utility maximization in the case of only a single round happening ever. Log utility doesn’t do this. I’m not sure I have anything that does though, so maybe it’s unfair to ask it from you, but still it seems like a core part of the idea, that the Kelly strategy comes from the compounding, is lost.
Ah, I see, interesting.
This is the sort of argument you want to be very suspicious of if youre confused, as I suspect we are. For example, you can now just apply all the arguments that made Kelly seem compelling again, but this time with respect to the new, logarithmic utility function. Do they actually seem less compelling now? A little bit, yes, because I think we really are sublinear in money, and the intuitions related to that went away. But no matter what the utility function, we can always construct bets that are compounding in utility, and then bettors which are Kelly with respect to that utility function will come to dominate the market. So if you do this reverse-inference of utility, the utility function of Kelly bettors will seems to change based on the bets offered.
Yeah, I agree with this.
(all the classical foundations imply bounded utilities, including VNM. This doesn’t seem to be well known here)
Yeah. I’m generally OK with dropping continuity-type axioms, though, in which case you can have hyperreal/surreal utility to deal with expectations which would otherwise be problematic (the divergent sums which unbounded utility allows). So while I agree that boundedness should be thought of as part of the classical notion of real-valued utility, this doesn’t seem like a huge deal to me.
OTOH, logical uncertainty / radical probabilism introduce new reasons to require boundedness for expectations. What is the expectation of the self-referential quantity “one greater than your expectation for this value”? This seems problematic even with hyperreals/surreals. And we could embed such a quantity into a decision problem.
I’m generally OK with dropping continuity-type axioms, though, in which case you can have hyperreal/surreal utility to deal with expectations which would otherwise be problematic (the divergent sums which unbounded utility allows).
Have you worked this out somewhere? I’d be interested to see it but I think there are some divergences it can’t adress. There is for one the Pasadena paradox, which is also a divergent sum but one which doesn’t stably lead anywhere, not even to infinity. The second is an apparently circular dominance relation: Imagine you are linear in monetary consumption. You start with 1$ which you can either spend or leave in the bank, which doubles it every year even after accounting for your time preference/uncertainty/other finite discounting. Now for every n, leaving it in the bank for n+1 years dominates leaving it for n years, but leaving it in the bank forever gets 0 utility. Note that if we replace money with energy here, this could actually happen in universes not too different from ours.
What is the expectation of the self-referential quantity “one greater than your expectation for this value”?
What is the expectation of the self-referential quantity “one greater than your expectation for this value, except when that would go over the maximum, in which case it’s one lower than expectation instead”? Insofar as there is an answer it would have to be “one less than maximum”, but that would seem to require uncertainty about what your expectations are.
Have you worked this out somewhere? I’d be interested to see it but I think there are some divergences it can’t adress.
It’s a bit of a mess due to some formatting changes porting to LW 2.0, but here it is.
I’ve gotten the impression over the years that there are a lot of different ways to arrive at the same conclusion, although I unfortunately don’t have all my sources lined up in one place.
I think if you just drop continuity from VNM you get this kind of picture, because the VNM continuity assumption corresponds to the Archimedian assumption for the reals.
I think there’s a variant of Cox’s theorem which similarly yields hyperreal/surreal probabilities (infinitesimals, not infinities, in that case).
If you want to condition on probability zero events, you might do so by rejecting the ratio formula for conditional probabilities, and instead giving a basic axiomatization of conditional probability in its own right. It turns out that, at least under one such axiom system, this is equivalent to allowing infinitesimal probability and keeping the ratio definition of conditional probability.
(Sorry for not having the sources at the ready.)
There is for one the Pasadena paradox, which is also a divergent sum but one which doesn’t stably lead anywhere, not even to infinity.
Here’s how it works. I have to assign expectations to gambles. I have some consistency requirements in how I do this; for example, if you modify a gamble g by making a probability p outcome have v less value, then I must think the new gamble g′ is worth p⋅v less. However, how I assign value to divergent sums is subjective—it cannot be determined precisely from how I assign value to each of the elements of the sum, because I’m not trying to assume anything like countable additivity.
In a case like the St Petersburg Lottery, I believe I’m required to have some infinite expectation. But it’s up to me what it is, since there’s no one way to assign expectations in infinite hyperreal/surreal sums.
In a case like the Pasadena paradox, though, I’m thinking I’ll be subjectively allowed to assign any expectation whatsoever—so long as all my other infinite-sum expectations are consistent with the assignment.
You start with 1$ which you can either spend or leave in the bank, which doubles it every year even after accounting for your time preference/uncertainty/other finite discounting. Now for every n, leaving it in the bank for n+1 years dominates leaving it for n years, but leaving it in the bank forever gets 0 utility. Note that if we replace money with energy here, this could actually happen in universes not too different from ours.
Perhaps you can try to problematize this example for me given what I’ve written above—not sure if I’ve already addressed your essential worry here or not.
What is the expectation of the self-referential quantity “one greater than your expectation for this value, except when that would go over the maximum, in which case it’s one lower than expectation instead”? Insofar as there is an answer it would have to be “one less than maximum”, but that would seem to require uncertainty about what your expectations are.
Yes, uncertainty about your own expectations is where this takes us. But that seems quite reasonable, especially because we only need a very small amount of uncertainty, as is illustrated in this example.
However, how I assign value to divergent sums is subjective—it cannot be determined precisely from how I assign value to each of the elements of the sum, because I’m not trying to assume anything like countable additivity.
This implies that you believe in the existence of countably infinite bets but not countably infinite dutch booking processes. Thats seems like a strange/unphysical position to be in—if that were the best treatment of infinity possible, I think infinity is better abandoned. Im not even sure the framework in your linked post can really be said to contain infinte bets: the only way a bet ever gets evaluated is in a bookie strategy, and no single bookie strategy can be guaranteed to fully evaluate an infinte bet. Is there a single bookie strategy that differentiates the St. Petersburg bet from any finite bet? Because if no, then the agent at least cant distinguish them, which is very close to not existing at all here.
In a case like the St Petersburg Lottery, I believe I’m required to have some infinite expectation.
Why? I haven’t found any finite dutch books against not doing so.
Perhaps you can try to problematize this example for me given what I’ve written above—not sure if I’ve already addressed your essential worry here or not.
I dont think you have. That example doesn’t involve any uncertainty or infinite sums. The problem is that for any finite n, waiting n+1 is better than waiting n, but waiting indefinitely is worse than any. Formally, the problem is that I have a complete and transitive preference between actions, but no unique best action, just a series that keeps getting better.
Note that you talk about something related in your linked post:
I’m representing preferences on sets only so that I can argue that this reduces to binary preference.
But the proof for that reduction only goes one way: for any preference relation on sets, theres a binary one. My problem is that the inverse does not hold.
Ah, I see, interesting.
Yeah, I agree with this.
Yeah. I’m generally OK with dropping continuity-type axioms, though, in which case you can have hyperreal/surreal utility to deal with expectations which would otherwise be problematic (the divergent sums which unbounded utility allows). So while I agree that boundedness should be thought of as part of the classical notion of real-valued utility, this doesn’t seem like a huge deal to me.
OTOH, logical uncertainty / radical probabilism introduce new reasons to require boundedness for expectations. What is the expectation of the self-referential quantity “one greater than your expectation for this value”? This seems problematic even with hyperreals/surreals. And we could embed such a quantity into a decision problem.
Have you worked this out somewhere? I’d be interested to see it but I think there are some divergences it can’t adress. There is for one the Pasadena paradox, which is also a divergent sum but one which doesn’t stably lead anywhere, not even to infinity. The second is an apparently circular dominance relation: Imagine you are linear in monetary consumption. You start with 1$ which you can either spend or leave in the bank, which doubles it every year even after accounting for your time preference/uncertainty/other finite discounting. Now for every n, leaving it in the bank for n+1 years dominates leaving it for n years, but leaving it in the bank forever gets 0 utility. Note that if we replace money with energy here, this could actually happen in universes not too different from ours.
What is the expectation of the self-referential quantity “one greater than your expectation for this value, except when that would go over the maximum, in which case it’s one lower than expectation instead”? Insofar as there is an answer it would have to be “one less than maximum”, but that would seem to require uncertainty about what your expectations are.
It’s a bit of a mess due to some formatting changes porting to LW 2.0, but here it is.
I’ve gotten the impression over the years that there are a lot of different ways to arrive at the same conclusion, although I unfortunately don’t have all my sources lined up in one place.
I think if you just drop continuity from VNM you get this kind of picture, because the VNM continuity assumption corresponds to the Archimedian assumption for the reals.
I think there’s a variant of Cox’s theorem which similarly yields hyperreal/surreal probabilities (infinitesimals, not infinities, in that case).
If you want to condition on probability zero events, you might do so by rejecting the ratio formula for conditional probabilities, and instead giving a basic axiomatization of conditional probability in its own right. It turns out that, at least under one such axiom system, this is equivalent to allowing infinitesimal probability and keeping the ratio definition of conditional probability.
(Sorry for not having the sources at the ready.)
Here’s how it works. I have to assign expectations to gambles. I have some consistency requirements in how I do this; for example, if you modify a gamble g by making a probability p outcome have v less value, then I must think the new gamble g′ is worth p⋅v less. However, how I assign value to divergent sums is subjective—it cannot be determined precisely from how I assign value to each of the elements of the sum, because I’m not trying to assume anything like countable additivity.
In a case like the St Petersburg Lottery, I believe I’m required to have some infinite expectation. But it’s up to me what it is, since there’s no one way to assign expectations in infinite hyperreal/surreal sums.
In a case like the Pasadena paradox, though, I’m thinking I’ll be subjectively allowed to assign any expectation whatsoever—so long as all my other infinite-sum expectations are consistent with the assignment.
Perhaps you can try to problematize this example for me given what I’ve written above—not sure if I’ve already addressed your essential worry here or not.
Yes, uncertainty about your own expectations is where this takes us. But that seems quite reasonable, especially because we only need a very small amount of uncertainty, as is illustrated in this example.
This implies that you believe in the existence of countably infinite bets but not countably infinite dutch booking processes. Thats seems like a strange/unphysical position to be in—if that were the best treatment of infinity possible, I think infinity is better abandoned. Im not even sure the framework in your linked post can really be said to contain infinte bets: the only way a bet ever gets evaluated is in a bookie strategy, and no single bookie strategy can be guaranteed to fully evaluate an infinte bet. Is there a single bookie strategy that differentiates the St. Petersburg bet from any finite bet? Because if no, then the agent at least cant distinguish them, which is very close to not existing at all here.
Why? I haven’t found any finite dutch books against not doing so.
I dont think you have. That example doesn’t involve any uncertainty or infinite sums. The problem is that for any finite n, waiting n+1 is better than waiting n, but waiting indefinitely is worse than any. Formally, the problem is that I have a complete and transitive preference between actions, but no unique best action, just a series that keeps getting better.
Note that you talk about something related in your linked post:
But the proof for that reduction only goes one way: for any preference relation on sets, theres a binary one. My problem is that the inverse does not hold.