Democracy might be mathematically impossible – here’s why. Head to https://brilliant.org/veritasium to start your free 30-day trial and get 20% off an annual premium subscription.
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.
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.