> Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above in the general case.
Nobel Laureate Amartya Sen[0] has demonstrated that, while there is no system that satisfies all four characteristics in the general case, there are systems that either satisfy all four conditions either probabilistically or satisfy all four conditions subject to some very weak assumptions.
The example I've heard him use is of the 2000 election in Florida, with Bush, Gore, and Nader (let's ignore Buchanan for simplicity). While technically there are 3! = 6 possible ways to rank the candidates, in practice, the ranking (Nader, Bush, Gore) is much less likely than (Nader, Gore, Bush) or (Gore, Nader, Bush). If we introduce one minor assumption about the relative frequencies of the rankings, we can prove that instant-runoff voting[1] does always satisfy all four of Arrow's criteria[2].
To use an analogy from computer science, the halting problem is undecidable in the general case, but that doesn't prevent static analysis tools from spotting many infinite loops; it just means it can't spot all infinite loops with 100% accuracy.
[2] A different example: instead of making assumptions about the relative frequencies, we could make assumptions about the number of axes that candidates may have and the way they cluster around them. This realistically depicts both two-party and multiparty elections in most parts of the world, since political positions are not uniformly distributed along n dimensions.
You're right that it was a smaller group, of course, but it wasn't an empty one.
Jello Biafra's endorsement of Nader plus Gore's attack on Twisted Sister for the Parents Music Resource Council means N-B-G was actually the order of preference for anyone where "the right to rock out" was their single voting issue.
In college I was part of a club that had an elaborate election procedure for officers. I'm pretty sure it violates Arrow's Theorem, but it was also nonterminating!
Oh, god. I don't remember the details, but it involved crossing off the bottom third of candidates every round until there was only one left. But somehow, this didn't always get rid of everyone, and you could get stuck in a state where there was no bottom third.
> Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above in the general case.
Nobel Laureate Amartya Sen[0] has demonstrated that, while there is no system that satisfies all four characteristics in the general case, there are systems that either satisfy all four conditions either probabilistically or satisfy all four conditions subject to some very weak assumptions.
The example I've heard him use is of the 2000 election in Florida, with Bush, Gore, and Nader (let's ignore Buchanan for simplicity). While technically there are 3! = 6 possible ways to rank the candidates, in practice, the ranking (Nader, Bush, Gore) is much less likely than (Nader, Gore, Bush) or (Gore, Nader, Bush). If we introduce one minor assumption about the relative frequencies of the rankings, we can prove that instant-runoff voting[1] does always satisfy all four of Arrow's criteria[2].
To use an analogy from computer science, the halting problem is undecidable in the general case, but that doesn't prevent static analysis tools from spotting many infinite loops; it just means it can't spot all infinite loops with 100% accuracy.
[0] https://en.wikipedia.org/wiki/Amartya_Sen
[1] https://en.wikipedia.org/wiki/Instant-runoff_voting
[2] A different example: instead of making assumptions about the relative frequencies, we could make assumptions about the number of axes that candidates may have and the way they cluster around them. This realistically depicts both two-party and multiparty elections in most parts of the world, since political positions are not uniformly distributed along n dimensions.