Assuming a roughly 50-50 split the inverse square-root rule is right. Now my issue is why you incorporate that factor in scenario 2, but not scenario 3. I honestly thought I was just rephrasing the problem, but you seem to see it differently? I should clarify that this isn’t you unconditionally receiving a speck if you’re willing to, but only if half the remainder are also so willing.
The point of voting, for me, is not an attempt to induce scope insensitivity by personalizing the decision, but to incorporate the preferences of the vast majority (3^^^^3 out of 3^^^^3 + 1) of participants about the situation they find themselves in, into your calculation of what to do. The Torture vs. Specks problem in its standard form asks for you to decide on behalf of 3^^^^3 people what should happen to them; voting is a procedure by which they can decide.
[Edit: On second thought, I retract my assertion that scenario 1) and 2) have roughly the same stakes. That in scenario 1) huge numbers of people who prefer not to be dust-specked can get dust-specked, and in scenario 2) no one who prefers not to be dust-specked is dust-specked, makes much more of a difference than a simple doubling of the number of specks.]
By the way, the problem as stated involves 3^^^3, not 3^^^^3, people, but this can’t possibly matter so nevermind.
There are actually two differences between 2 and 3. The first is that in 2 my chance of affecting the torture is negligible, whereas in 3 it is quite high. The second difference is that in 2 I have the power to save huge numbers of others from dust specks, and it is this difference which is important to me, since when I have that power it dwarfs the other factors so much as to be the only deciding factor in my decision. In your ‘rephrasing’ of it you conveniently ignore the fact that I can still do this, so I assumed I no longer could, which made the two scenarios very different.
I think also, as a general principle, any argument of the type you are formulating which does not pay attention to the specific utilities of torture and dust-specks, instead just playing around with who makes the decision, can also be used to justify killing 3^^^^3 people to save one person from being killed in a slightly more painful manner.
Assuming a roughly 50-50 split the inverse square-root rule is right. Now my issue is why you incorporate that factor in scenario 2, but not scenario 3. I honestly thought I was just rephrasing the problem, but you seem to see it differently? I should clarify that this isn’t you unconditionally receiving a speck if you’re willing to, but only if half the remainder are also so willing.
The point of voting, for me, is not an attempt to induce scope insensitivity by personalizing the decision, but to incorporate the preferences of the vast majority (3^^^^3 out of 3^^^^3 + 1) of participants about the situation they find themselves in, into your calculation of what to do. The Torture vs. Specks problem in its standard form asks for you to decide on behalf of 3^^^^3 people what should happen to them; voting is a procedure by which they can decide.
[Edit: On second thought, I retract my assertion that scenario 1) and 2) have roughly the same stakes. That in scenario 1) huge numbers of people who prefer not to be dust-specked can get dust-specked, and in scenario 2) no one who prefers not to be dust-specked is dust-specked, makes much more of a difference than a simple doubling of the number of specks.]
By the way, the problem as stated involves 3^^^3, not 3^^^^3, people, but this can’t possibly matter so nevermind.
There are actually two differences between 2 and 3. The first is that in 2 my chance of affecting the torture is negligible, whereas in 3 it is quite high. The second difference is that in 2 I have the power to save huge numbers of others from dust specks, and it is this difference which is important to me, since when I have that power it dwarfs the other factors so much as to be the only deciding factor in my decision. In your ‘rephrasing’ of it you conveniently ignore the fact that I can still do this, so I assumed I no longer could, which made the two scenarios very different.
I think also, as a general principle, any argument of the type you are formulating which does not pay attention to the specific utilities of torture and dust-specks, instead just playing around with who makes the decision, can also be used to justify killing 3^^^^3 people to save one person from being killed in a slightly more painful manner.