[EM] Arrow/Gibbard and impossibility (Re: Scientific American and the "Perfect Electoral System")

[Kristofer Munsterhjelm]

On 2023-11-13 19:35, Richard, the VoteFair guy wrote:
> Just because it's impossible to get zero IIA failure rates doesn't mean 
> "we have to let go of" it, in the sense of not trying to reduce IIA 
> failures.
> 
> Although all methods fail IIA, measuring HOW OFTEN those failures occur 
> is insightful.  Some methods have much higher failure rates than others.
> 
> As I've said before, I believe reducing failure rates is more important 
> than regarding fairness criteria as pass/fail (yes/no) flags that are 
> worth counting simplistically.

Suppose that a majoritarian method fails Condorcet. Then there exist 
elections where the CW is X, but the method elects Y. Then eliminating 
every candidate but Y and X makes X beat Y, so these elections have IIA 
failure.

Suppose that there's a Condorcet cycle. Then every majoritarian ranked 
election method fails IIA: suppose without loss of generality that the 
method elects A, and that B beats A pairwise. Eliminating every 
candidate but A and B leads B to win, hence an IIA failure.

Suppose a majoritarian ranked method passes Condorcet and the election 
has a Condorcet winner. Then the method passes IIA for that election, 
because eliminating any set of non-winning candidates still leaves that 
candidate a Condorcet winner.

So if we want to minimize IIA failure, and IIA failure exists whenever 
we can remove a set of candidates who did not win and thus change the 
winner, we would want to pass Condorcet.

-km