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

[Toby Pereira]

 As Kristofer says, minimising IIA failure (at least with ranked ballots) just means using Condorcet, and I think they would all be equal in that respect. However, even if there is a Condorcet winner, one could still argue, at least sometimes, that there has been an IIA failure. Say there are three main candidates - A, B and C. Polls suggest that it's close and that there will be no Condorcet winner, but that A will likely win under the particular method used. C then drops out before the election causing B to win. One might call this an IIA failure.
The other question is whether C dropping out was a good thing. We went from probably not having a Condorcet winner to having one as a result of C dropping out. Arguably C standing gives us more information overall about voter preferences. And as A would have won if C had stood, A is arguably the most likely "best" winner.
So while it might be nicer to have a clear Condorcet winner, in situations where it's close enough for there to be a possible cycle if certain candidates stand, arguably it's better if they do stand, causing the cycle and giving us more information.
Toby
    On Monday, 13 November 2023 at 20:55:48 GMT, Kristofer Munsterhjelm <km_elmet at t-online.de> wrote:  
 
 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
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