[EM] "Independence of cycles" and a possible new method.

[Kristofer Munsterhjelm]

On 12/12/21 11:04 AM, Toby Pereira wrote:
> If you look at the example here: 
> https://www.rangevoting.org/TobyCondParadox.html 
> <https://www.rangevoting.org/TobyCondParadox.html> having a tie cycle in 
> a Condorcet election can change the result. The ballots are essentially:
> 
> 
> 1 voter: A>B>C
> 2 voters: B>A>C
> 
> Plus
> 
> 2 voters: A>B>C
> 2 voters: B>C>A
> 2 voters: C>A>B
> 
> A is the Condorcet winner in this election, beating both B and C by 5 
> votes to 4. However, in the top section of ballots, B is the Condorcet 
> winner, and the bottom section is a three-way tie as it is a perfect 
> cycle. Adding in tie cycles to change the result could be seen as a 
> failure of "independence of cycles" (or perhaps there is a better name). 
> This is also a specific case of failure of consistency. Or maybe a weak 
> failure because B doesn't win both the sub-elections. B would have an 
> outright win from the top ballots and tie from the bottom ballots, but 
> loses to A when all are combined.
> 
> This example, by the way, is a simplified version of something that 
> Donald Saari used to promote the Borda Count.
> 
> The problem is that in an election with more than three candidates, you 
> couldn't simply remove the cycles and calculate the result. Ballots and 
> candidates would potentially be involved in many intertwined cycles, so 
> there would be no straightforward way of doing it.
> 
> But what you can do is compare every possible triplet of candidates 
> (like Condorcet methods compare pairs). For each triplet, all tie cycles 
> are removed and you look at the head-to-heads.

You could do this, but as I understand the example, the method would no 
longer be a Condorcet method. You could also define the irrelevance of 
cycles criterion, perhaps something like:

Removing a constant number of voters who together form an exact tied 
Condorcet cycle should not modify the output.

Though I'm not sure what the implications would be - or if it's possible 
to pass by any method that fails IIA.

As for using triplets instead of pairs, I think Stensholt suggested that 
doing so might be a way to generalize his BPW method, which is only 
defined for three candiates. Similarly, it might be a way of 
generalizing my fpA-fpC, though I'm again unsure how to do so and 
preserve the desired properties of DMTBR and monotonicity.

-km