Interesting. It reminds me of a challenge for denying countable additivity:
God runs a lottery. First, he picks two integers at random (each integer has an equal and 0 probability of being picked, violating countable additivity.) Then he shows one of the two at random to you. You know in advance that there is a 50% chance youâll see the higher one (maybe he flips a coin), but no matter what it is, after you see it youâll be convinced it is the lower one.
Iâm inclined to think that this is a problem with infinities in general, not with unbounded utility functions per se.
Iâm inclined to think that this is a problem with infinities in general, not with unbounded utility functions per se.
I think itâs a problem for the conjunction of allowing some kinds of infinities and doing expected value maximization with unbounded utility functions. I think EV maximization with bounded utility functions isnât vulnerable to âisomorphicâ Dutch books/âmoney pumps or violations of the sure-thing principle. E.g., you could treat the possible outcomes of a lottery as all local parts of a larger single universe to aggregate, but then conditioning on the outcome of the first St. Petersburg lottery and comparing to the second lottery would correspond to comparing a local part of the first universe to the whole of the second universe, but the move from the whole first universe to the local part of the first universe canât happen via conditioning, and the arguments depend on conditioning.
Bounded utility functions have problems that unbounded utility functions donât, but these are in normative ethics and about how to actually assign values (including in infinite universes), not about violating plausible axioms of (normative) rationality/âdecision theory.
After reading the linked comment I think the view that total utilitarianism can be dutch booked is fairly controversial (there is another unaddressed comment I quite agree with), and on a page like this one I think itâs misleading to state as fact in a comment that total utilitarianism can be dutch booked in a similar way that person-affecting views can be dutch booked.
I should have specified EV maximization with an unbounded social welfare function, although the argument applies somewhat more generally; Iâve edited this into my top comment.
Looking at Sliderâs reply to the comment I linked, assuming thatâs the one you meant (or did you have another in mind?):
Slider probably misunderstood Christiano about truncation, because Christiano meant that youâd truncate the second lottery at a point that depends on the outcome of the first lottery. For any actual value outcome X of the original St. Petersburgâs lottery, half St. Pesterburg can be truncated at some point and still have a finite expected value greater than X. (EDIT: However, I guess the sure-thing principle isnât relevant here with conditional truncation, since we arenât comparing only two fixed options anymore.)
I donât understand what Slider meant in the second paragraph, and I think itâs probably missing the point.
The third paragraph misses the point: once the outcome is decided for the first St. Petersburg lottery, it has finite value, and half St. Petersburg still has infinite expected value, which is greater than a finite value.
Yes, I should have thought more about Sliderâs reply before posting, I take back my agreement. Still, I donât find dutch booking convincing in Christianoâs case.
The reason to reject a theory based on dutch booking is that there is no logical choice to commit to, in this case to maximize EV. I donât think this applies to the Paul Christiano case, because the second lottery does not have higher EV than the first. Yes, once you play the first lottery and find out that it has a finite value the second one will have higher EV, but until then the first one has higher EV (in an infinite way) and you should choose it.
But again I think there can be reasonable disagreement about this, I just think equating dutch booking for the person-affecting view and for the total utilitarianism view is misleading. These are substantially different philosophical claims.
Yes, once you play the first lottery and find out that it has a finite value the second one will have higher EV, but until then the first one has higher EV (in an infinite way) and you should choose it.
I think a similar argument can apply to person-affecting views and the OPâs Dutch book argument:
Yes, starting with World 1, once you make trade 1 to get World 2 and find out that trade 2 to World 3 is available, trade 1 will have negative value, but until then trade 1 has positive value and you should choose it.
I agree that you can give different weights to different Dutch book/âmoney pump arguments. I do think that if you commit 100% to complete preferences over all probability distributions over outcomes and invulnerability to Dutch books/âmoney pumps, then expected utility maximization over each individual decision with an unbounded utility function is ruled out.
As you mention, one way to avoid this St. Petersburg Dutch book/âmoney pump is to just commit to sticking with A, if A>B ex ante, and regardless of the actual outcome of A (+ some other conditions, e.g. A and B both have finite value under all outcomes, and A has infinite expected value), but switching to C under certain other conditions.
You may have similar commitment moves for person-affecting views, although you might find them all less satisfying. You could commit to refusing one of the 3 types of trades in the OP, or doing so under specific conditions, or just never completing the last step in any Dutch book, even if youâd know youâd want to. I think those with person-affecting views should usually refuse moves like trade 1, if they think theyâre not too unlikely to make moves like trade 2 after, but this is messier, and depends on your distributions over what options will become available in the future depending on your decisions. The above commitments for St. Petersburg-like lotteries donât depend on what options will be available in the future or your distributions over them.
See this comment by Paul Christiano on LW based on St. Petersburg lotteries (and my reply).
Interesting. It reminds me of a challenge for denying countable additivity:
Iâm inclined to think that this is a problem with infinities in general, not with unbounded utility functions per se.
I think itâs a problem for the conjunction of allowing some kinds of infinities and doing expected value maximization with unbounded utility functions. I think EV maximization with bounded utility functions isnât vulnerable to âisomorphicâ Dutch books/âmoney pumps or violations of the sure-thing principle. E.g., you could treat the possible outcomes of a lottery as all local parts of a larger single universe to aggregate, but then conditioning on the outcome of the first St. Petersburg lottery and comparing to the second lottery would correspond to comparing a local part of the first universe to the whole of the second universe, but the move from the whole first universe to the local part of the first universe canât happen via conditioning, and the arguments depend on conditioning.
Bounded utility functions have problems that unbounded utility functions donât, but these are in normative ethics and about how to actually assign values (including in infinite universes), not about violating plausible axioms of (normative) rationality/âdecision theory.
After reading the linked comment I think the view that total utilitarianism can be dutch booked is fairly controversial
(there is another unaddressed comment I quite agree with), and on a page like this one I think itâs misleading to state as fact in a comment that total utilitarianism can be dutch booked in a similar way that person-affecting views can be dutch booked.I should have specified EV maximization with an unbounded social welfare function, although the argument applies somewhat more generally; Iâve edited this into my top comment.
Looking at Sliderâs reply to the comment I linked, assuming thatâs the one you meant (or did you have another in mind?):
Slider probably misunderstood Christiano about truncation, because Christiano meant that youâd truncate the second lottery at a point that depends on the outcome of the first lottery. For any actual value outcome X of the original St. Petersburgâs lottery, half St. Pesterburg can be truncated at some point and still have a finite expected value greater than X. (EDIT: However, I guess the sure-thing principle isnât relevant here with conditional truncation, since we arenât comparing only two fixed options anymore.)
I donât understand what Slider meant in the second paragraph, and I think itâs probably missing the point.
The third paragraph misses the point: once the outcome is decided for the first St. Petersburg lottery, it has finite value, and half St. Petersburg still has infinite expected value, which is greater than a finite value.
Yes, I should have thought more about Sliderâs reply before posting, I take back my agreement. Still, I donât find dutch booking convincing in Christianoâs case.
The reason to reject a theory based on dutch booking is that there is no logical choice to commit to, in this case to maximize EV. I donât think this applies to the Paul Christiano case, because the second lottery does not have higher EV than the first. Yes, once you play the first lottery and find out that it has a finite value the second one will have higher EV, but until then the first one has higher EV (in an infinite way) and you should choose it.
But again I think there can be reasonable disagreement about this, I just think equating dutch booking for the person-affecting view and for the total utilitarianism view is misleading. These are substantially different philosophical claims.
I think a similar argument can apply to person-affecting views and the OPâs Dutch book argument:
I agree that you can give different weights to different Dutch book/âmoney pump arguments. I do think that if you commit 100% to complete preferences over all probability distributions over outcomes and invulnerability to Dutch books/âmoney pumps, then expected utility maximization over each individual decision with an unbounded utility function is ruled out.
As you mention, one way to avoid this St. Petersburg Dutch book/âmoney pump is to just commit to sticking with A, if A>B ex ante, and regardless of the actual outcome of A (+ some other conditions, e.g. A and B both have finite value under all outcomes, and A has infinite expected value), but switching to C under certain other conditions.
You may have similar commitment moves for person-affecting views, although you might find them all less satisfying. You could commit to refusing one of the 3 types of trades in the OP, or doing so under specific conditions, or just never completing the last step in any Dutch book, even if youâd know youâd want to. I think those with person-affecting views should usually refuse moves like trade 1, if they think theyâre not too unlikely to make moves like trade 2 after, but this is messier, and depends on your distributions over what options will become available in the future depending on your decisions. The above commitments for St. Petersburg-like lotteries donât depend on what options will be available in the future or your distributions over them.