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As a person with a math background, I think the way the 'dictator voter' is defined in Arrow's theorem is rather silly, and, thus, I never got why it's considered to be the proof that this sort of 'democracy' is impossible - I stumbled upon the theorem about 10 years ago and had the same issue with this part that I have now.
Also, I find another case more interesting, one where it is proven that a system of voting can't allow for all three:
Anonimity - every voter's input is weighed the same way as the input of every other voter.
Unanimity - if every voter votes for a particular thing A, then A is the elected result of the vote.
Continuity - the mapping f(x_1, x_2, x_3,..., x_n), where x_i is the vote of the ith voter, from the topological space of vote-vectors to the topological space of vote results, is continuous.
I first encountered an examination of this case here:
https://www.youtube.com/watch?v=v5ev-RAg7Xs
However, even in this case there are a couple of issues:
Specifically for the case of the video, it assumes that preference spaces are (homeomorphic to) standard S^1 and not any other spaces, which I find a faulty assumption, and I'm not familiar with any general proofs that axioms 1, 2, and 3 can't be satisfied at the same time.
I don't even think that we should expect continuity, or that continuity is important.
I feel like the preference space assumption was reasonable? Effectively asking "how important is x_i" for every issue i and then normalizing the result. Works at the limits, too, if something is considered infinitely important.
It does depend on how one asks about the preferences. Given a different question one might get a non-complete or non-transitive preference function. Also I think that if there were dependent preferences (e.g. more roads, but only if work-from-home isn't available) then that wouldn't be continuous? Cause the preference for one would jump with the sign change of the other. Continuity might even be harmful.
Honestly I just hate it because it's a rather unmathematical approach to say "voting is the problem" and not "our definition of fair voting is flawed."
I feel like the preference space assumption was reasonable?
There are other reasonable assumptions that can be made about the preference space. Like, for example, we could assume that it should be a space with discrete topology of some relevant cardinality.
Works at the limits, too, if something is considered infinitely important
Not sure what you mean by that, as each vote x_i is generally not a function or a similar structure, so it can't have a limit in any relevant sense.
Honestly I just hate it because it's a rather unmathematical approach to say "voting is the problem" and not "our definition of fair voting is flawed."
I largely agree. The way these people define what a fair voting system is is rather silly, and Arrow's theorem and the like are definitely not worthy of any sort of prizes. Arrow's theorem in particular is extremely silly, IMO, in the way that it defines a voter as a 'dictator', despite the fact that it can just as well be any other voter, and despite the fact that our supposed 'dictator' changes depending on the order in which we go over the profiles. Everybody is a dictator.
Like, for example, we could assume that it should be a space with discrete topology of some relevant cardinality. [...] Not sure what you mean by that, as each vote xi is generally not a function or a similar structure
Yeah that was badly written, sorry. I was taking the xi's as well-defined preference-based utility functions, so "i is xi important". That's not even continuous unless one could say "how much of our resources will be spent on i," which is a simplification itself. Maybe instead of issues having functions ki describing all possible choices regarding an issue? By limit I meant someone saying "i is infinitely important."
Anyway, I think it's possible to build a reasonable, continuous, preference model, depending on what the set of topics/issues looks like. Whether the properties required of the set of issues would be reasonable... I think not. I think one would end up with something maybe not discrete but certainly not continuous. Hence the second paragraph in my previous comment.
Arrow’s theorem
I've never heard of this. Just off the first sentence on Wikipedia, I'd question the existence of independent alternatives. It looks like non-dictatorship is defined to be ordering invariant?
I've never heard of this. Just off the first sentence on Wikipedia, I'd question the existence of independent alternatives. It looks like non-dictatorship is defined to be ordering invariant?
The issue is that, in the dictator voter proof of Arrow's theorem, they prove that for every ordering of voting profiles between cases where the result of the vote is A and where the result of the vote is B there has to be a first profile where the result is not A. The profiles differ by just one vote, so they declare the relevant voter a dictator. The problem is that who this 'dictator' is depends on the order in which we change the votes. As such, we can literally argue that everybody is a dictator.
I do not think that this 'non-dictatorship' rule is a reasonable requirement for democratic systems.
As a person with a math background, I think the way the 'dictator voter' is defined in Arrow's theorem is rather silly, and, thus, I never got why it's considered to be the proof that this sort of 'democracy' is impossible - I stumbled upon the theorem about 10 years ago and had the same issue with this part that I have now.
Also, I find another case more interesting, one where it is proven that a system of voting can't allow for all three:
Anonimity - every voter's input is weighed the same way as the input of every other voter.
Unanimity - if every voter votes for a particular thing A, then A is the elected result of the vote.
Continuity - the mapping f(x_1, x_2, x_3,..., x_n), where x_i is the vote of the ith voter, from the topological space of vote-vectors to the topological space of vote results, is continuous.
I first encountered an examination of this case here:
https://www.youtube.com/watch?v=v5ev-RAg7Xs
However, even in this case there are a couple of issues:
Specifically for the case of the video, it assumes that preference spaces are (homeomorphic to) standard S^1 and not any other spaces, which I find a faulty assumption, and I'm not familiar with any general proofs that axioms 1, 2, and 3 can't be satisfied at the same time.
I don't even think that we should expect continuity, or that continuity is important.
I feel like the preference space assumption was reasonable? Effectively asking "how important is x_i" for every issue i and then normalizing the result. Works at the limits, too, if something is considered infinitely important.
It does depend on how one asks about the preferences. Given a different question one might get a non-complete or non-transitive preference function. Also I think that if there were dependent preferences (e.g. more roads, but only if work-from-home isn't available) then that wouldn't be continuous? Cause the preference for one would jump with the sign change of the other. Continuity might even be harmful.
Honestly I just hate it because it's a rather unmathematical approach to say "voting is the problem" and not "our definition of fair voting is flawed."
There are other reasonable assumptions that can be made about the preference space. Like, for example, we could assume that it should be a space with discrete topology of some relevant cardinality.
Not sure what you mean by that, as each vote x_i is generally not a function or a similar structure, so it can't have a limit in any relevant sense.
I largely agree. The way these people define what a fair voting system is is rather silly, and Arrow's theorem and the like are definitely not worthy of any sort of prizes. Arrow's theorem in particular is extremely silly, IMO, in the way that it defines a voter as a 'dictator', despite the fact that it can just as well be any other voter, and despite the fact that our supposed 'dictator' changes depending on the order in which we go over the profiles. Everybody is a dictator.
Yeah that was badly written, sorry. I was taking the xi's as well-defined preference-based utility functions, so "i is xi important". That's not even continuous unless one could say "how much of our resources will be spent on i," which is a simplification itself. Maybe instead of issues having functions ki describing all possible choices regarding an issue? By limit I meant someone saying "i is infinitely important."
Anyway, I think it's possible to build a reasonable, continuous, preference model, depending on what the set of topics/issues looks like. Whether the properties required of the set of issues would be reasonable... I think not. I think one would end up with something maybe not discrete but certainly not continuous. Hence the second paragraph in my previous comment.
I've never heard of this. Just off the first sentence on Wikipedia, I'd question the existence of independent alternatives. It looks like non-dictatorship is defined to be ordering invariant?
The issue is that, in the dictator voter proof of Arrow's theorem, they prove that for every ordering of voting profiles between cases where the result of the vote is A and where the result of the vote is B there has to be a first profile where the result is not A. The profiles differ by just one vote, so they declare the relevant voter a dictator. The problem is that who this 'dictator' is depends on the order in which we change the votes. As such, we can literally argue that everybody is a dictator.
I do not think that this 'non-dictatorship' rule is a reasonable requirement for democratic systems.
I found a YouTube link in your comment. Here are links to the same video on alternative frontends that protect your privacy: