Indeed, but “single-winner” has a technical meaning here that’s rather more restrictive than that. Unless each voter could choose their vote aribitrarily from among the set of those overall outcomes, it’s not single-winner.
can you give the meaning of “single winner” and the reason you think not directly voting for the single winner will remove the problems?
in the US presidential elections, no voters can directly vote for his desired overall outcome. we have a primary system. are you saying that the primary system makes arrow’s theorem’s problems go away for us?
This appears to be very confused. Arrow’s theorem is a theorem, a logical necessity, not a causal influence. It does not go around causing problems, that can be then be warded off by modifying your system to avoid its preconditions. It’s just a fact that your voting system can’t satisfy all its desiderata simultaneously. If you’re not talking about a single-winner voting system it’s simply an inapplicable statement. Perhaps the system meets all the desiderata, perhaps it doesn’t. Arrow’s theorem simply has nothing to say about it. But if you take a system that meets the preconditions, and simply modify it to not do so, then there’s no reason to expect that it suddenly will start meeting the desiderata. For instance, though I think it’s safe to say that US presidential elections are single winner, even if they somehow didn’t count as such they’d fail to satisfy IIA. My point is just don’t bring up theorems where they don’t apply.
If you want a formal statement of Arrow’s theorem… well it can take several forms, so take a look at the definitions I made in this comment about their equivalence. Then with that set of definitions, any voting system satisfying IIA and unanimity (even in just the “weak” form where if everyone votes for the same linear order, the overall winner is the winner of that linear order) with a finite set of voters must be a dictatorship. (There’s a version for infinite sets of voters too but that’s not really relevant here.)
Your claims implied Arrow’s Theorem doesn’t apply to the US election system. Or pretty much any other. You also defined “single-winner” so that the single US president who wins the election doesn’t qualify.
OK, you’re right I didn’t actually contradict you. But don’t you think that’s a mistake? I think it does apply to real life voting systems that people use.
Perhaps you misunderstand what is meant when people say a theorem “applies”. It means that the preconditions are met (and that therefore the conclusions hold), not simply that the conclusions hold. The conclusions of a theorem can hold without the theorem being at all applicable.
If you meant “Arrow’s theorem is not applicable to any voting system on Earth” why did you object to my statements about PR? The PR issue is irrelevant, is it not?
For one thing, all elections have a single overall outcome that wins.
Indeed, but “single-winner” has a technical meaning here that’s rather more restrictive than that. Unless each voter could choose their vote aribitrarily from among the set of those overall outcomes, it’s not single-winner.
can you give the meaning of “single winner” and the reason you think not directly voting for the single winner will remove the problems?
in the US presidential elections, no voters can directly vote for his desired overall outcome. we have a primary system. are you saying that the primary system makes arrow’s theorem’s problems go away for us?
This appears to be very confused. Arrow’s theorem is a theorem, a logical necessity, not a causal influence. It does not go around causing problems, that can be then be warded off by modifying your system to avoid its preconditions. It’s just a fact that your voting system can’t satisfy all its desiderata simultaneously. If you’re not talking about a single-winner voting system it’s simply an inapplicable statement. Perhaps the system meets all the desiderata, perhaps it doesn’t. Arrow’s theorem simply has nothing to say about it. But if you take a system that meets the preconditions, and simply modify it to not do so, then there’s no reason to expect that it suddenly will start meeting the desiderata. For instance, though I think it’s safe to say that US presidential elections are single winner, even if they somehow didn’t count as such they’d fail to satisfy IIA. My point is just don’t bring up theorems where they don’t apply.
If you want a formal statement of Arrow’s theorem… well it can take several forms, so take a look at the definitions I made in this comment about their equivalence. Then with that set of definitions, any voting system satisfying IIA and unanimity (even in just the “weak” form where if everyone votes for the same linear order, the overall winner is the winner of that linear order) with a finite set of voters must be a dictatorship. (There’s a version for infinite sets of voters too but that’s not really relevant here.)
You’re very condescending and you’re arguing with my way of speaking while refusing to actually provide a substantive argument.
it does apply. you’re treating yourself as an authority, and me a beginner. and just assuming that knowing nothing about me.
you made a mistake which i explained in my second paragraph. your response: to ignore it, and say you were right anyway and i should just read more.
I am, in fact, inferring that based on what you wrote.
Then I must ask that you point out this mistake more explicitly, because nothing in that second paragraph contradicts anything I said.
Your claims implied Arrow’s Theorem doesn’t apply to the US election system. Or pretty much any other. You also defined “single-winner” so that the single US president who wins the election doesn’t qualify.
OK, you’re right I didn’t actually contradict you. But don’t you think that’s a mistake? I think it does apply to real life voting systems that people use.
Perhaps you misunderstand what is meant when people say a theorem “applies”. It means that the preconditions are met (and that therefore the conclusions hold), not simply that the conclusions hold. The conclusions of a theorem can hold without the theorem being at all applicable.
If you meant “Arrow’s theorem is not applicable to any voting system on Earth” why did you object to my statements about PR? The PR issue is irrelevant, is it not?