Talk:IRV Prime

Hello, Condorcet and Later-no-harm are incompatible - see proof in Woodall. Could you run your method through the example provided there and update the article? Kristomun (talk) 09:12, 31 July 2021 (UTC)

So if you look at Woodall's full paper, he does not say they're incompatible:

 In general, CONDORCET is incompatible with LATER-NO-HELP, LATER-NO-HARM,
 MONO-RAISE-DELETE, MONO-SUB-PLUMP and, in the presence of PLURALITY, MONO-ADD-TOP. 

"in general" --Marcosb (talk) 22:39, 3 August 2021 (UTC)

That's just a figure of speech. Let me quote from the reference I provided:

Theorem 2 says that if an election rule satisfies Condorcet's principle, then it cannot possess any of the seven properties that are crossed in the column headed 2 in Table 1.

(emphasis mine.) The crossed properties are participation, four different monotonicity properties, later-no-help, and later-no-harm. Woodall then proceeds to prove this impossibility; if it were indeed only true some of the time, then one would imagine these proofs would contain the necessary qualifications, but they do not claim any such qualification beyond that the method must pass Condorcet. Kristomun (talk) 09:17, 4 August 2021 (UTC)

There is a problem with the proof, though; it starts with:

 Without loss of generality, suppose a is elected in P. But c becomes the Condorcet winner, and so must be elected by CONDORCET

The theorem begins with the premise that both of those are possible, i.e. that a wins & c can become the Condorcet winner simply by modifying later preferences. But I believe those premise may be unsatisfiable.

The question is: how does a win? And if a wins, is it possible to make c the condorcet winner without putting c above a? (in IRV-prime, it's impossible to fulfill both premises; either a doesn't win, or c cannot be made a condorcet winner)

It's like starting with "suppose the tree exists and the tree doesn't exist" - you can make a lot of faulty theorems by starting with premises that cannot all be satisfied.

By the way, thank you so much Kristomun for the discussion & for your time, very much appreciated!

--Marcosb (talk) 16:41, 4 August 2021 (UTC)

Arrow/IIA

As I understand it, the reference to satisfying Arrow's theorem is meant to imply that the method satisfies IIA. But I don't think that's possible.

In a Condorcet cycle like this:

35: A>B>C
30: B>C>A
25: C>A>B

Who wins in IRV Prime? If it's A, then eliminating B (irrelevant candidate) should make C win by majority rule. If it's B, then eliminating C makes A win; and if it's C, then eliminating A makes B win. I may be missing something, though! :-) Kristomun (talk) 22:22, 31 July 2021 (UTC)

Running through the IRV-Prime steps, first we do classic IRV, which eliminates C & finds winners={A}:

A: 60
B: 30

We now see if any candidate can win against A (we know B can't), i.e. WinnersPrime={C}:

C: 55
A: 35

And as such C is the winner in IRV-Prime; this is a classic case of A=Rock, B=Scissors, C=Paper; if I phrase it to you as "suppose B & C voters were to go up against A: which candidate should they stand behind?" then it becomes clear what IRV prime is trying to do (i.e. higher preferences of a candidate that can win)

--Marcosb (talk) 22:39, 3 August 2021 (UTC)

In that case, if C is the winner and IRV Prime is majority rule in the two-candidate case, eliminating A (an irrelevant candidate) will make B win, which is a violation of IIA. Kristomun (talk) 09:17, 4 August 2021 (UTC)

You're correct, thanks for that example. It does not fulfill IIA: (sorry I'm still learning this stuff)

 if one candidate (X) would win an election, and if a new candidate (Y) were added to the ballot, then either X or Y would win the election.

But if you look at the ballots, this actually makes sense from a social representation perspective:

35: B>C
30: B>C
25: C>B

B best represents the population.

However, when A enters the race, B no longer has the votes to win, so their voters must now compromise. By definition it's still a "spoiler candidate", but it's the opposite effect: in plurality, a spoiler results in a candidate that could win to lose (by vote splitting); in IRV-Prime, it causes votes for a candidate that was going to lose to shift to those voters' next favorite.

--Marcosb (talk) 16:18, 4 August 2021 (UTC)