My own anti-preference function seems to have a form something like this:
U(N,I,T) = kI(1 - e^(-NT/a))
where a and k are constants with appropriate units.
Relevant “intuitions” not listed before:
1) For the purposes of this thought experiment, who suffers a pain doesn’t matter. Therefore:
1a) Transferring an instant of pain from one person to another, without changing the (subjective) intensity of the pain, doesn’t change the “badness” of the situation. Two people suffering torture for 25 years simultaneously equals one person suffering 25 years of torture and then a second 25 years of torture. Therefore:
1b) U is a function of N*T and I; U(N,I,T) = Ū(NT,I) for some Ū.
2) Pains that are sufficiently more intense are qualitatively different from pains that are less intense, but insufficiently more intense pains are not. Therefore:
2a) For a sufficiently large difference in intensity, no amount of less intense pains are worse than a single more intense pain (of sufficient aggregated duration). Therefore:
2b) Ū(NT,I) approaches a finite limit as NT approaches infinity and I is held constant, because there must be a NT,I1,I2 such that Ū(NT,I1) > lim(NT->+â) Ū(NT,I2) and I2 > 0.
2c) For insufficiently large differences in intensity, there is an amount of less intense pains that are worse than a more intense pain.
3) My mathematical intuition suggests to me that an equation of the form U(N,I,T) = kI(1 - e^(-NT/a)) has the properties I want.
There’s a catch here. N and T are fairly well-defined, measurable terms. Intensity of pain doesn’t have a well-defined scale, though; the term “I” has to be some function mapping subjective feelings (or, more precisely, brain states) to some subset of real numbers. Depending on how you do this mapping and how you choose the constant a, you can get many different preference orderings.
In the case of nearly infinite dust specks vs. 50 years of torture, the anti-preference function gives U(SPECKS) = kI(speck) and U(TORTURE)= kI(torture)(1-e^(-50 personyears/a)). Using the time-tested technique of “making stuff up”, I assign I(speck) = 0.001 (as stipulated, one speck is a very small pain), I(torture)=1 (as torture is understood to be a very large pain), and a = 100 personyears (approximately, a human lifespan). This gives U(SPECKS) = 0.001 and U(TORTURE) = 0.393.
U(SPECKS) < U(TORTURE), and therefore, I prefer SPECKS. It’s based on actual math! ;)
My own anti-preference function seems to have a form something like this:
U(N,I,T) = kI(1 - e^(-NT/a))
where a and k are constants with appropriate units.
Relevant “intuitions” not listed before:
1) For the purposes of this thought experiment, who suffers a pain doesn’t matter. Therefore:
1a) Transferring an instant of pain from one person to another, without changing the (subjective) intensity of the pain, doesn’t change the “badness” of the situation. Two people suffering torture for 25 years simultaneously equals one person suffering 25 years of torture and then a second 25 years of torture. Therefore:
1b) U is a function of N*T and I; U(N,I,T) = Ū(NT,I) for some Ū.
2) Pains that are sufficiently more intense are qualitatively different from pains that are less intense, but insufficiently more intense pains are not. Therefore:
2a) For a sufficiently large difference in intensity, no amount of less intense pains are worse than a single more intense pain (of sufficient aggregated duration). Therefore:
2b) Ū(NT,I) approaches a finite limit as NT approaches infinity and I is held constant, because there must be a NT,I1,I2 such that Ū(NT,I1) > lim(NT->+â) Ū(NT,I2) and I2 > 0.
2c) For insufficiently large differences in intensity, there is an amount of less intense pains that are worse than a more intense pain.
3) My mathematical intuition suggests to me that an equation of the form U(N,I,T) = kI(1 - e^(-NT/a)) has the properties I want.
There’s a catch here. N and T are fairly well-defined, measurable terms. Intensity of pain doesn’t have a well-defined scale, though; the term “I” has to be some function mapping subjective feelings (or, more precisely, brain states) to some subset of real numbers. Depending on how you do this mapping and how you choose the constant a, you can get many different preference orderings.
In the case of nearly infinite dust specks vs. 50 years of torture, the anti-preference function gives U(SPECKS) = kI(speck) and U(TORTURE)= kI(torture)(1-e^(-50 personyears/a)). Using the time-tested technique of “making stuff up”, I assign I(speck) = 0.001 (as stipulated, one speck is a very small pain), I(torture)=1 (as torture is understood to be a very large pain), and a = 100 personyears (approximately, a human lifespan). This gives U(SPECKS) = 0.001 and U(TORTURE) = 0.393.
U(SPECKS) < U(TORTURE), and therefore, I prefer SPECKS. It’s based on actual math! ;)