No. Arrow's theorem does not depend on "the inputs" being strict rankings (orderings), as opposed to partial rankings. It is common for proofs of Arrow's theorem to only take into account strict orderings, but that's just a simplification and is done without loss of generality. So yes, Arrow's impossibility result still holds if we allow partial and incomplete orderings, which is the point against Mr. Poundstone's claim that "scored" methods bypass Arrow's theorem.
If you still don't get it, try thinking about it this way. If you can prove that white cats are not immortal, you've automatically proven that cats (generally) are not immortal. Similar reasoning is used in Arrow's theorem. Arrow showed that there is no social welfare function that satisfies a set of desirable conditions; he did this by showing that there are no social welfare functions that, when "fed" strict orderings, satisfy those conditions; since there are no social welfare functions that satisfy these conditions, at least when fed strict orderings, there are no social welfare functions that satisfy these conditions regardless of what they are fed.
It isn't a matter of a property being present in a subset that is not present in a more general set. The point is that ranked methods are a subset of scored methods. Ranked methods are "scored" methods that force voters to score candidates in a particular way. Since Arrow's theorem proves that there's no solution for the set of ranked methods, it proves that there's no solution for the more inclusive set of scored methods.
The best analogy is this: if there is no cure for cancer, then there is no cure for everything. If the subset of cases have no solution, then there is no solution for the more general case, since the more general case includes the unsolvable cases! This has nothing to do with vaginas, either.
This guy is dead wrong. He thinks his voting system escapes Arrow's result because it allows "scoring" rather than "ranking." This is utter nonsense. Ranking can be viewed as a particular kind of scoring, i.e., it's possible for everyone to "score" the candidates in such a way that the information on each ballot is equivalent to a "ranked" vote. Since, as this bonehead acknowledges, Arrow's theorem applies to ranking methods, it applies to scoring methods as well, since ranking is a particular kind of scoring. In other words, if your voting system allows "scoring" then it's possible for everyone to simply score the candidates in a way that is equivalent to ranking them. So unless the voting method bars people from scoring the candidates in a "ranked" way (which would be completely absurd), moving to a scoring system cannot avoid Arrow's theorem. This point is common knowledge in social choice theory.
Let me say it again, in a different way: if there is no solution to a particular set of cases, then there is no solution to a broader set of cases that includes that smaller set. This should be obvious.
Here's what the fool wrote/said, in case anyone is curious:
"For decades, there was almost a kind of despair among voting theorists of getting any better system than we had. What's interesting, though, is that the impossibility theorem doesn't apply to systems where you score the candidates rather than rank them. With scoring, you're essentially filling out a report card--if you think there are two candidates who deserve four stars you can give them both four stars--whereas with ranking you have to artificially give one a number one and one a number two. That turns out to be crucial."
Also, some here have criticized some of the conditions of Arrow's theorem, in particular, IIA. Unfortunately, criticisms of IIA are largely misunderstood. Even the philosopher Michael Dummett, who wrote a rather large book on voting theory, gets it wrong. IIA is best understood as the condition that the only information we are going to take into account is that which is present on the ballot; we will not, e.g., ask people whether they hate their "last choice," whether they love their "first choice," where they'd place Stalin or Hitler in the ranking, and so on.
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No. Arrow's theorem does not depend on "the inputs" being strict rankings (orderings), as opposed to partial rankings. It is common for proofs of Arrow's theorem to only take into account strict orderings, but that's just a simplification and is done without loss of generality. So yes, Arrow's impossibility result still holds if we allow partial and incomplete orderings, which is the point against Mr. Poundstone's claim that "scored" methods bypass Arrow's theorem. If you still don't get it, try thinking about it this way. If you can prove that white cats are not immortal, you've automatically proven that cats (generally) are not immortal. Similar reasoning is used in Arrow's theorem. Arrow showed that there is no social welfare function that satisfies a set of desirable conditions; he did this by showing that there are no social welfare functions that, when "fed" strict orderings, satisfy those conditions; since there are no social welfare functions that satisfy these conditions, at least when fed strict orderings, there are no social welfare functions that satisfy these conditions regardless of what they are fed.
It isn't a matter of a property being present in a subset that is not present in a more general set. The point is that ranked methods are a subset of scored methods. Ranked methods are "scored" methods that force voters to score candidates in a particular way. Since Arrow's theorem proves that there's no solution for the set of ranked methods, it proves that there's no solution for the more inclusive set of scored methods. The best analogy is this: if there is no cure for cancer, then there is no cure for everything. If the subset of cases have no solution, then there is no solution for the more general case, since the more general case includes the unsolvable cases! This has nothing to do with vaginas, either.
This guy is dead wrong. He thinks his voting system escapes Arrow's result because it allows "scoring" rather than "ranking." This is utter nonsense. Ranking can be viewed as a particular kind of scoring, i.e., it's possible for everyone to "score" the candidates in such a way that the information on each ballot is equivalent to a "ranked" vote. Since, as this bonehead acknowledges, Arrow's theorem applies to ranking methods, it applies to scoring methods as well, since ranking is a particular kind of scoring. In other words, if your voting system allows "scoring" then it's possible for everyone to simply score the candidates in a way that is equivalent to ranking them. So unless the voting method bars people from scoring the candidates in a "ranked" way (which would be completely absurd), moving to a scoring system cannot avoid Arrow's theorem. This point is common knowledge in social choice theory.
Let me say it again, in a different way: if there is no solution to a particular set of cases, then there is no solution to a broader set of cases that includes that smaller set. This should be obvious.
Here's what the fool wrote/said, in case anyone is curious: "For decades, there was almost a kind of despair among voting theorists of getting any better system than we had. What's interesting, though, is that the impossibility theorem doesn't apply to systems where you score the candidates rather than rank them. With scoring, you're essentially filling out a report card--if you think there are two candidates who deserve four stars you can give them both four stars--whereas with ranking you have to artificially give one a number one and one a number two. That turns out to be crucial."
Also, some here have criticized some of the conditions of Arrow's theorem, in particular, IIA. Unfortunately, criticisms of IIA are largely misunderstood. Even the philosopher Michael Dummett, who wrote a rather large book on voting theory, gets it wrong. IIA is best understood as the condition that the only information we are going to take into account is that which is present on the ballot; we will not, e.g., ask people whether they hate their "last choice," whether they love their "first choice," where they'd place Stalin or Hitler in the ranking, and so on.