Imagine a universe that lasts forever, with zero uncertainty, constant equally good opportunities to turn wealth into utility, and constant high investment returns (say, 20% per time period).
In this scenario you could (mathematically) save your wealth for an infinite number of periods and then donate it, generating infinite utility.
It sounds paradoxical but infinities generally are, and the paradox only exists if you think there’s a sufficient chance that the next period will exist and have opportunities to turn wealth into utility relative to the interest rate—that is, you ‘expect’ an infinitely long lasting universe.
A less counterintuitive approach with the same result would be to save everything with that 20% return and also donate some amount that’s less than 20% of the principal each period. This way each period the principal continues to grow, while each year you give away some amount between 0-20% (non-inclusive) and generate a finite amount of utility. After an infinite number of time periods you have accumulated an infinite principal and also generated infinite utility—just as high an expected value as the ‘save it all for an infinite number of time periods and then donate it’ approach suggested above!
I think you and Ben have picked up the part of the problem I wasn’t focusing on. I’m less concerned about the totalist version: I think you can spin a version of the story where you should donate the end of time, and that’s just the best thing you can do.
My point was that, given you accept the totalist philanthropist’s paradox, there’s an additional weirdness for person-affecting views. That’s the bit I found interesting.
Although, I suppose there’s a reframing this that makes the puzzle more curious. Totalists get a paradox where they recognise they should donate at the end of time, and that feels odd. Person-affecting views might think they dodge this problem by denying the value of the far future, but they get another kind of paradox for them.
In this scenario you could (mathematically) save your wealth for an infinite number of periods and then donate it, generating infinite utility.
How is there anything (i.e. “and then”) after an infinite amount of periods (taking altogether an infinite amount of time)? Are you introducing hyperreals or nonstandard analysis? Are you claiming this is just a possibility (from our ignorance about the nature of time) or a fact, conditional on the universe lasting forever?
I think it’s extremely unlikely that time works this way, but if you’re an EU maximizer and assign some positive probability to this possibility, then, sure, you can get an infinite return in EU. Most likely you’ll get nothing. It’s a lot like Pascal’s wager.
I’m almost certain time doesn’t work this way in our universe! But for the paradox to exist we have to imagine a universe where an infinite amount of time really can pass. I’m not an expert in these expected value paradoxes for different kinds of infinity—might be worth asking Amanda Askell who is.
Either way, the mixed strategy of saving and donating some gives us a way out.
It’s worth pointing out that if time just advances forever, so that your current time is just “T seconds after the starting point”, then it is simultaneously true that:
time is infinite
every instant has a finite past (and an infinite future)
The second point in particular means that even though time is infinite, you still can’t wait an infinite amount of time and then do something. I think that’s what MichaelStJules was getting at.
Your mixed strategy has its own paradox, though – suppose you decide that one strategy is better than another if it “eventually” does more total good – that is, there’s a point in time after which “total amount of good done so far” exceeds that of the other strategy for the rest of eternity. You have to do something like this because it doesn’t usually make sense to ask which strategy achieved the most good “after infinite time” because infinite time never elapses.
Anyway, suppose you have that metric of “eventual winner”. Then your strategy can always be improved by reducing the fraction you donate, because the exponential growth of the investment will eventually outpace the linear reduction in donations. But as soon as you reduce the fraction to zero, you no longer get any gains at all. So you have the odd situation where no fraction is optimal – for any strategy, there is always a better one.
In a context of infinite possible outcomes and infinite possible choice pathways, this actually isn’t that surprising. You might as well be surprised that there’s no largest number. And perhaps that applies just as well to the original philanthropist’s paradox – if you permit yourself an infinite time horizon to invest over, it’s just not surprising that there’s no optimal moment to “cash in”.
As soon as you start actually encoding your beliefs that the time horizon is in fact not infinite, I’m willing to bet you start getting some concrete moments to start paying your fund out, and some reasonable justifications for why those moments were better than any other. To the extent that the conclusion “you should wait until near the end of civilization to donate” is still a counterintuitive one, I claim it’s just because of our (correct) intuition that investing is not always better than donating right now, even in the long run. That’s the argument that Ben Todd and Sanjay made.
Imagine a universe that lasts forever, with zero uncertainty, constant equally good opportunities to turn wealth into utility, and constant high investment returns (say, 20% per time period).
In this scenario you could (mathematically) save your wealth for an infinite number of periods and then donate it, generating infinite utility.
It sounds paradoxical but infinities generally are, and the paradox only exists if you think there’s a sufficient chance that the next period will exist and have opportunities to turn wealth into utility relative to the interest rate—that is, you ‘expect’ an infinitely long lasting universe.
A less counterintuitive approach with the same result would be to save everything with that 20% return and also donate some amount that’s less than 20% of the principal each period. This way each period the principal continues to grow, while each year you give away some amount between 0-20% (non-inclusive) and generate a finite amount of utility. After an infinite number of time periods you have accumulated an infinite principal and also generated infinite utility—just as high an expected value as the ‘save it all for an infinite number of time periods and then donate it’ approach suggested above!
Infinities are weird. :)
I think you and Ben have picked up the part of the problem I wasn’t focusing on. I’m less concerned about the totalist version: I think you can spin a version of the story where you should donate the end of time, and that’s just the best thing you can do.
My point was that, given you accept the totalist philanthropist’s paradox, there’s an additional weirdness for person-affecting views. That’s the bit I found interesting.
Although, I suppose there’s a reframing this that makes the puzzle more curious. Totalists get a paradox where they recognise they should donate at the end of time, and that feels odd. Person-affecting views might think they dodge this problem by denying the value of the far future, but they get another kind of paradox for them.
Yeah not saying anything in contradiction to you, just adding my own two cents on the thing.
How is there anything (i.e. “and then”) after an infinite amount of periods (taking altogether an infinite amount of time)? Are you introducing hyperreals or nonstandard analysis? Are you claiming this is just a possibility (from our ignorance about the nature of time) or a fact, conditional on the universe lasting forever?
I think it’s extremely unlikely that time works this way, but if you’re an EU maximizer and assign some positive probability to this possibility, then, sure, you can get an infinite return in EU. Most likely you’ll get nothing. It’s a lot like Pascal’s wager.
I’m almost certain time doesn’t work this way in our universe! But for the paradox to exist we have to imagine a universe where an infinite amount of time really can pass. I’m not an expert in these expected value paradoxes for different kinds of infinity—might be worth asking Amanda Askell who is.
Either way, the mixed strategy of saving and donating some gives us a way out.
It’s worth pointing out that if time just advances forever, so that your current time is just “T seconds after the starting point”, then it is simultaneously true that:
time is infinite
every instant has a finite past (and an infinite future)
The second point in particular means that even though time is infinite, you still can’t wait an infinite amount of time and then do something. I think that’s what MichaelStJules was getting at.
Your mixed strategy has its own paradox, though – suppose you decide that one strategy is better than another if it “eventually” does more total good – that is, there’s a point in time after which “total amount of good done so far” exceeds that of the other strategy for the rest of eternity. You have to do something like this because it doesn’t usually make sense to ask which strategy achieved the most good “after infinite time” because infinite time never elapses.
Anyway, suppose you have that metric of “eventual winner”. Then your strategy can always be improved by reducing the fraction you donate, because the exponential growth of the investment will eventually outpace the linear reduction in donations. But as soon as you reduce the fraction to zero, you no longer get any gains at all. So you have the odd situation where no fraction is optimal – for any strategy, there is always a better one.
In a context of infinite possible outcomes and infinite possible choice pathways, this actually isn’t that surprising. You might as well be surprised that there’s no largest number. And perhaps that applies just as well to the original philanthropist’s paradox – if you permit yourself an infinite time horizon to invest over, it’s just not surprising that there’s no optimal moment to “cash in”.
As soon as you start actually encoding your beliefs that the time horizon is in fact not infinite, I’m willing to bet you start getting some concrete moments to start paying your fund out, and some reasonable justifications for why those moments were better than any other. To the extent that the conclusion “you should wait until near the end of civilization to donate” is still a counterintuitive one, I claim it’s just because of our (correct) intuition that investing is not always better than donating right now, even in the long run. That’s the argument that Ben Todd and Sanjay made.