Hmm. I seem to be flinching away from both answers, and I think I know why. It’s because I’m unable to decide whether utility really does multiply (after all, one could advocate the utility function “The minimum happiness within the population”, instead of the sum).
So I’m happy to make the factual claims that “Sum-utility ⇒ pick ‘torture’” and “Min-utility ⇒ pick ‘specks’”; I just can’t see any procedure for choosing between sum and min. So I’ll formulate a test to see which I believe: I’ll gradually reduce the severity of the non-speck option. So, specks versus someone getting tortured for 25 years, I’m still unsure. Specks versus someone getting slapped in the face, I choose slap over specks. Therefore I’m not following min-utility, so I’m willing to accept that really I’m following sum-utility, so in the original problem I pick Torture. I don’t like this, because my brain wants to be scope-insensitive and refuses to understand 3^^^3, but when I made one of the outcomes not flinch-worthy that outcome got picked, and I’m pretty sure that my reasons for picking Slap ought to scale up to Torture, so there it is.
I started this post not knowing what answer I would reach despite having spent several minutes on the question. I think I’ve now been trying to resolve this for over half an hour, and I still feel uncomfortable. My mind has just now come up with a third alternative, which is that utilities should perhaps be rated with hyperreals, so that 3^^^3 1 is still less than 1 H (for an infinite hyperinteger H), in which case we could pick Specks without discarding sum-utility. But I probably wouldn’t have thought of that while I was locked up and couldn’t choose an answer. I am now feeling comfortable, which suggests that this is what I actually believe about utilities. Of course, now there is an experiment I could do to try and falsify this: try to construct a chain of things starting at a dust speck and ending at torture, where each link in the chain is only a finite amount worse than the one before. I know I can get from speck to slap, because I chose Slap over Specks. I also think a hefty kick up the arse is only finitely worse than a slap. I next try to get to a broken arm, but I’m unwilling to do that (at least, in a single step), so I need to find something intermediate. In fact I think I should try and find something I can jump down to from a broken arm, because a broken anything seems scary in a new way. A deep-bruised hand? Yes, I think that relates finitely to a broken arm. I also think that finitely many kicks up the arse are worse than a deep-bruised hand. Given that my instinctive feeling about the relation of kick to arm was very similar to my feeling about the relation of speck to torture, I conclude that in fact my scale of utilities is constrained to the finite.
The point to this post (if there is one) is that a useful method seems to be to vary the parameters of the problem until you can get an answer, and then look to see whether that illuminates the original problem. (Come to think of it, ISTR that’s one of Polyà′s How To Solve It tips.) But to evaluate this method, I need to see whether it can also produce the opposite result. So, I need to vary the parameters in ways that favour Specks. If I reduce the 3^^^3 to something smaller, I eventually pick Specks because I get a number I think I can comprehend—but that number has to be so much smaller than 3^^^3 that I don’t think it’s relevant to the original problem. If I make the ‘torture’ option involve something worse than torture, I still pick it—I can’t think of anything that’s sufficiently worse than torture that doing that to someone could make me pick Specks when I didn’t pick Specks against torture.
So the method does constrain, and I pick Torture. There, finally finished this post.
Of course, you can also use the chain of negative-utility cases to make a direct argument for specks vs. torture.
Say you prefer 1 slap to N1 specks. Then you prefer 1 kick to N2 slaps, 1 bruise to N3 kicks, 1 broken arm to N4 bruises, and so on, up until the last step where you prefer years of torture to Nk of something.
It follows that the specks vs. torture point comes at N1 x N2 x N3 x …. x Nk. This is pretty much always going to be less than 3^^^3 -- if the steps were truly small, the factors are all going to be less than a trillion or so, and there’s probably going to be less than a trillion steps, and (1 trillion)^(1 trillion) is still insignificant compared to 3^^^3.
Hmm. I seem to be flinching away from both answers, and I think I know why. It’s because I’m unable to decide whether utility really does multiply (after all, one could advocate the utility function “The minimum happiness within the population”, instead of the sum).
So I’m happy to make the factual claims that “Sum-utility ⇒ pick ‘torture’” and “Min-utility ⇒ pick ‘specks’”; I just can’t see any procedure for choosing between sum and min. So I’ll formulate a test to see which I believe: I’ll gradually reduce the severity of the non-speck option. So, specks versus someone getting tortured for 25 years, I’m still unsure. Specks versus someone getting slapped in the face, I choose slap over specks. Therefore I’m not following min-utility, so I’m willing to accept that really I’m following sum-utility, so in the original problem I pick Torture. I don’t like this, because my brain wants to be scope-insensitive and refuses to understand 3^^^3, but when I made one of the outcomes not flinch-worthy that outcome got picked, and I’m pretty sure that my reasons for picking Slap ought to scale up to Torture, so there it is.
I started this post not knowing what answer I would reach despite having spent several minutes on the question. I think I’ve now been trying to resolve this for over half an hour, and I still feel uncomfortable. My mind has just now come up with a third alternative, which is that utilities should perhaps be rated with hyperreals, so that 3^^^3 1 is still less than 1 H (for an infinite hyperinteger H), in which case we could pick Specks without discarding sum-utility. But I probably wouldn’t have thought of that while I was locked up and couldn’t choose an answer. I am now feeling comfortable, which suggests that this is what I actually believe about utilities. Of course, now there is an experiment I could do to try and falsify this: try to construct a chain of things starting at a dust speck and ending at torture, where each link in the chain is only a finite amount worse than the one before. I know I can get from speck to slap, because I chose Slap over Specks. I also think a hefty kick up the arse is only finitely worse than a slap. I next try to get to a broken arm, but I’m unwilling to do that (at least, in a single step), so I need to find something intermediate. In fact I think I should try and find something I can jump down to from a broken arm, because a broken anything seems scary in a new way. A deep-bruised hand? Yes, I think that relates finitely to a broken arm. I also think that finitely many kicks up the arse are worse than a deep-bruised hand. Given that my instinctive feeling about the relation of kick to arm was very similar to my feeling about the relation of speck to torture, I conclude that in fact my scale of utilities is constrained to the finite.
The point to this post (if there is one) is that a useful method seems to be to vary the parameters of the problem until you can get an answer, and then look to see whether that illuminates the original problem. (Come to think of it, ISTR that’s one of Polyà′s How To Solve It tips.) But to evaluate this method, I need to see whether it can also produce the opposite result. So, I need to vary the parameters in ways that favour Specks. If I reduce the 3^^^3 to something smaller, I eventually pick Specks because I get a number I think I can comprehend—but that number has to be so much smaller than 3^^^3 that I don’t think it’s relevant to the original problem. If I make the ‘torture’ option involve something worse than torture, I still pick it—I can’t think of anything that’s sufficiently worse than torture that doing that to someone could make me pick Specks when I didn’t pick Specks against torture.
So the method does constrain, and I pick Torture. There, finally finished this post.
Of course, you can also use the chain of negative-utility cases to make a direct argument for specks vs. torture.
Say you prefer 1 slap to N1 specks. Then you prefer 1 kick to N2 slaps, 1 bruise to N3 kicks, 1 broken arm to N4 bruises, and so on, up until the last step where you prefer years of torture to Nk of something.
It follows that the specks vs. torture point comes at N1 x N2 x N3 x …. x Nk. This is pretty much always going to be less than 3^^^3 -- if the steps were truly small, the factors are all going to be less than a trillion or so, and there’s probably going to be less than a trillion steps, and (1 trillion)^(1 trillion) is still insignificant compared to 3^^^3.
Of course. Except that I think you mean trillion^trillion, not trillion*trillion.
Er. Right. Fixed. And it’s a testament to the magnitude of 3^^^3 that I need to change absolutely nothing else.