I couldn’t access the “Aggregation Procedure for Cardinal Preferences” article. In any case, why isn’t using an aggregate utility function that is a linear combination of everyone’s utility functions (choosing some arbitrary number for each person’s weight) a way to satisfy Arrow’s criteria?
It should also be noted that Arrow’s impossibility theorem doesn’t hold for non-deterministic decision procedures. I would also caution against calling this an “existential risk”, because while decision procedures that violate Arrow’s criteria might be considered imperfect in some sense, they don’t necessarily cause an existential catastrophe. Worldwide range voting would not be the best way of deciding everything, but it most likely wouldn’t be an existential risk.
why isn’t using an aggregate utility function that is a linear combination of everyone’s utility functions (choosing some arbitrary number for each person’s weight) a way to satisfy Arrow’s criteria?
On first inspection, it looks like “linear combination of utility functions” still has issues with strategic voting. If you prefer A to B and B to C, but A isn’t the winner regardless of how you vote, it can be arranged such that you make yourself worse off by expressing a preference for A over B. Any system where you reward people for not voting their preferences can get strange in a hurry.
Let me at least formalize the “linear combination of utility functions” bit. Scale each person’s utility function so that their favorite option is 1, and their least favorite is −1. Add them together, then remove the lowest-scoring option, then re-scale the utility functions to the same range over the new choice set.
Arrow’s Theorem doesn’t say anything about strategic voting. The only reasonable non-strategic voting system I know of is random ballot (pick a random voter; they decide who wins). I’m currently trying to figure out a voting system that is based on finding the Nash equilibrium (which may be mixed) of approval voting, and this system might also be strategy-free.
When I said linear combination of utility functions, I meant that you fix the scaling factors initially and don’t change them. You could make all of them 1, for example. Your voting system (described in the last paragraph) is a combination of range voting and IRV. If everyone range votes so that their favorite gets 1 and everyone else gets −1, then it’s identical to IRV, and shares the same problems such as non-monotonicity. I suspect that you will also get non-monotonicity when votes aren’t “favorite gets 1 and everyone else gets −1”.
EDIT: I should clarify: it’s not 1 for your favorite and −1 for everyone else. It’s 1 for your favorite and close to −1 for everyone else, such that when your favorite is eliminated, it’s 1 for your next favorite and close to −1 for everyone else after rescaling.
I couldn’t access the “Aggregation Procedure for Cardinal Preferences” article. In any case, why isn’t using an aggregate utility function that is a linear combination of everyone’s utility functions (choosing some arbitrary number for each person’s weight) a way to satisfy Arrow’s criteria?
It should also be noted that Arrow’s impossibility theorem doesn’t hold for non-deterministic decision procedures. I would also caution against calling this an “existential risk”, because while decision procedures that violate Arrow’s criteria might be considered imperfect in some sense, they don’t necessarily cause an existential catastrophe. Worldwide range voting would not be the best way of deciding everything, but it most likely wouldn’t be an existential risk.
Here you go: http://dl.dropbox.com/u/85192141/1977-kalai.pdf
On first inspection, it looks like “linear combination of utility functions” still has issues with strategic voting. If you prefer A to B and B to C, but A isn’t the winner regardless of how you vote, it can be arranged such that you make yourself worse off by expressing a preference for A over B. Any system where you reward people for not voting their preferences can get strange in a hurry.
Let me at least formalize the “linear combination of utility functions” bit. Scale each person’s utility function so that their favorite option is 1, and their least favorite is −1. Add them together, then remove the lowest-scoring option, then re-scale the utility functions to the same range over the new choice set.
Arrow’s Theorem doesn’t say anything about strategic voting. The only reasonable non-strategic voting system I know of is random ballot (pick a random voter; they decide who wins). I’m currently trying to figure out a voting system that is based on finding the Nash equilibrium (which may be mixed) of approval voting, and this system might also be strategy-free.
When I said linear combination of utility functions, I meant that you fix the scaling factors initially and don’t change them. You could make all of them 1, for example. Your voting system (described in the last paragraph) is a combination of range voting and IRV. If everyone range votes so that their favorite gets 1 and everyone else gets −1, then it’s identical to IRV, and shares the same problems such as non-monotonicity. I suspect that you will also get non-monotonicity when votes aren’t “favorite gets 1 and everyone else gets −1”.
EDIT: I should clarify: it’s not 1 for your favorite and −1 for everyone else. It’s 1 for your favorite and close to −1 for everyone else, such that when your favorite is eliminated, it’s 1 for your next favorite and close to −1 for everyone else after rescaling.