I added a section called “Deciding how to decide” that (hopefully) deals with this issue appropriately. I also amended the conclusion, and added you as an acknowledgement.
One of the simplest and most intuitive is aggregative utilitarianism, in which we define a utility for each person, add them all together, and make the choice with the larger total utility.
I suggest using the phrase “additive utilitarianism” rather than “aggregative utilitarianism”. It was entirely my fault for saying aggregative utilitarianism in my comment, which was a misnomer; I got it mixed up with aggregative consequentialism. (All flavors of utilitarianism are by definition aggregative because they take into account the utilities from some collection of beings, but not all flavors are additive.)
Note: VincentYu has pointed out in the comments below that VNM utility may be able to deal with the infinites in this problem without taking limits.
Unfortunately, I think that ascribes too much power to VNM utility functions (that term itself is a LessWrongism; elsewhere, they would be called cardinal utility functions or just utility functions). If we had our hands on a VNM utility function, we would be okay (we simply ask it which option it prefers!), but the VNM theorem simply asserts the existence of a utility function given certain basic axioms, and it doesn’t give us the utility function! So, unfortunately, VNM utility also falls flat on its face unless we already know what we prefer. (An important point is that VNM utility functions cannot work with the “utility” described in the problem. It’s an unfortunate historical accident that the word “utility” is overloaded, because VNM utility requires careful handling.)
If we fail to specify the exact type of decision theory we’re using, it is entirely unclear whether taking infinite limits would lead to a self-consistent solution.
If we want to make a decision based on [additive] utility, the infinite problem is ill posed; it has no unique solution unless we take on additional assumptions.
Unfortunately, I think that ascribes too much power to VNM utility functions (that term itself is a LessWrongism; elsewhere, they would be called cardinal utility functions or just utility functions).
I actually don’t recall seeing the usage “VNM utility functions” on less wrong at all, prior to this thread. It may have occurred previously but certainly not with sufficient frequency as to be a ‘lesswrongism’. As you say, the “VNM” is unnecessary in that context since is all the VNM part does is say “it must have a utility function because it adheres to these axioms”.
It is sometimes necessary to explicitly refer to things other than ‘utility functions’ with a ‘VNM’ qualifier. This is largely to pre-empt pedants who, when reading unqualified usage ‘consequentialist’, are not willing to assume that it refers to the only kind of consequentialist that is ever significantly discussed here (those that have utility functions).
VNM utility also falls flat on its face unless we already know what we prefer.
Not quite, but the point stands. The actual requirement is that there is any way to collect any evidence at all about our preferences (or, to be even more general, any way to cause outcomes to be correlated to our preference).
I added a section called “Deciding how to decide” that (hopefully) deals with this issue appropriately. I also amended the conclusion, and added you as an acknowledgement.
I suggest using the phrase “additive utilitarianism” rather than “aggregative utilitarianism”. It was entirely my fault for saying aggregative utilitarianism in my comment, which was a misnomer; I got it mixed up with aggregative consequentialism. (All flavors of utilitarianism are by definition aggregative because they take into account the utilities from some collection of beings, but not all flavors are additive.)
Unfortunately, I think that ascribes too much power to VNM utility functions (that term itself is a LessWrongism; elsewhere, they would be called cardinal utility functions or just utility functions). If we had our hands on a VNM utility function, we would be okay (we simply ask it which option it prefers!), but the VNM theorem simply asserts the existence of a utility function given certain basic axioms, and it doesn’t give us the utility function! So, unfortunately, VNM utility also falls flat on its face unless we already know what we prefer. (An important point is that VNM utility functions cannot work with the “utility” described in the problem. It’s an unfortunate historical accident that the word “utility” is overloaded, because VNM utility requires careful handling.)
Great, I think these are good clarifications!
I actually don’t recall seeing the usage “VNM utility functions” on less wrong at all, prior to this thread. It may have occurred previously but certainly not with sufficient frequency as to be a ‘lesswrongism’. As you say, the “VNM” is unnecessary in that context since is all the VNM part does is say “it must have a utility function because it adheres to these axioms”.
It is sometimes necessary to explicitly refer to things other than ‘utility functions’ with a ‘VNM’ qualifier. This is largely to pre-empt pedants who, when reading unqualified usage ‘consequentialist’, are not willing to assume that it refers to the only kind of consequentialist that is ever significantly discussed here (those that have utility functions).
Not quite, but the point stands. The actual requirement is that there is any way to collect any evidence at all about our preferences (or, to be even more general, any way to cause outcomes to be correlated to our preference).
For the moment, I’m going to strike the comment from the post. I don’t want to ascribe a viewpoint to VincentYu that he doesn’t actually hold.