[...] - Independence of irrelevant alternatives: If we prefer A to B as a group, and we all assign the same scores to A and B when C is introduced, then the total score of A must still be higher than that of B, even after C is brought in. Hence, our decision between A and B does not depend on C.
Wrong : this assumes we assign scores to alternatives in a way that doesn’t depend of he others. This is wrong for two reasons:
Tactical voting, as taw pointed out
We value thing as compared to alternatives : if A is good and B is so-so, you’ll rate B differently if there was a third alternative C that’s downright awful.
You didn’t “beat” arrow’s theorem by finding an alternate way to express preferences, but by making the fourth constraint less strict. Once you release that constraint, I’m sure you can find an algorithm that uses preference ordering instead of preference rating that fulfills the constraints just as well (i.e. not very well).
(Still, I agree with the general point, that we should be wary of over-interpreting what mathematical models tell us)
That’s not what independence of irrelevant alternatives means; it means that, given the same relative rankings, the voting algorithm will always make the same decision, with or without the alternative. It doesn’t mean that the voters will make the same decisions.
The independence of irrelevant alternatives (“pairwise independence”) can also apply to individuals, it is just that Arrow’s theorem doesn’t directly require it.
However, a better statement of the independence of irrelevant alternatives for Arrow is “if all individuals have preferences satisfying pairwise independence, then the group’s decision function should also satisfy it”.
So for group IIA to matter we should have individual IIA
Wrong : this assumes we assign scores to alternatives in a way that doesn’t depend of he others. This is wrong for two reasons:
Tactical voting, as taw pointed out
We value thing as compared to alternatives : if A is good and B is so-so, you’ll rate B differently if there was a third alternative C that’s downright awful.
You didn’t “beat” arrow’s theorem by finding an alternate way to express preferences, but by making the fourth constraint less strict. Once you release that constraint, I’m sure you can find an algorithm that uses preference ordering instead of preference rating that fulfills the constraints just as well (i.e. not very well).
(Still, I agree with the general point, that we should be wary of over-interpreting what mathematical models tell us)
That’s not what independence of irrelevant alternatives means; it means that, given the same relative rankings, the voting algorithm will always make the same decision, with or without the alternative. It doesn’t mean that the voters will make the same decisions.
The independence of irrelevant alternatives (“pairwise independence”) can also apply to individuals, it is just that Arrow’s theorem doesn’t directly require it. However, a better statement of the independence of irrelevant alternatives for Arrow is “if all individuals have preferences satisfying pairwise independence, then the group’s decision function should also satisfy it”. So for group IIA to matter we should have individual IIA