[EM] Better runoffs

[Dave Ketchum]

On Jul 10, 2012, at 6:51 AM, Kristofer Munsterhjelm wrote:

> When runoffs are subjected to criterion analysis, one usually  
> considers voters to vote in the same order in each round. If they  
> prefer A to B in the first round, and A and B remain in the second  
> round, they'll vote A over B in the second round.

This seems reasonable to me - however much they thought, they decided  
on A vs B for the previous round and have no real need for more  
thinking now.

However, those preferring C or D have only A and B available in the  
top-two runoff round and therefore must change.

Should C and D have lost in the previous round?  Experience with IRV  
demonstrates that those deserving to win can lose due to bad methods  
before runoffs.

Assuming C and D deserved to lose,  their backers need to accept their  
weakness and move on.

Further, C and D could be clones who lost out because the method was  
Plurality, in which clones often lose due to the method.  Plurality  
has primaries to help with this but clones can get nominated via  
multiple parties.
>
DWK
>
> This may not necessarily fit reality. Voters may leave or join  
> depending on whether the second round is "important" or not, and the  
> same for later rounds in exhaustive runoff. But let's consider top- 
> two runoffs and, to begin with, that the voters will stay consistent.
>
> The kind of criterion analysis performed on top-two then says that  
> top-two Plurality runoff is not monotone. Furthermore, it is worse  
> than IRV (i.e. fails participation, consistency, and so on, but also  
> things IRV passes like MDT and mutual majority).
>
> If we want to have a method that does better, what would we need?
>
> Some methods (like Ranked Pairs or Kemeny) pass what is called local  
> IIA. Local IIA says that if you eliminate all candidates but a  
> contiguous subset (according to the output ranking), then the order  
> of those candidates shouldn't change. If you eliminate all  
> candidates but the ones that finished third and fourth and rerun the  
> election, then the candidate that finished third should win. More  
> specifically, for runoff purposes: if you pick the two first  
> candidates to the runoff, and voters are perfectly consistent, then  
> the order doesn't change.
>
> Thus, all that you really need to make a runoff that isn't worse  
> than its base method is that the method passes LIIA. Use Ranked  
> Pairs for both stages and there you go -- if the voters change their  
> minds between rounds, conventional criterion analysis doesn't apply,  
> and if they don't change their minds, you don't lose compliance of  
> any criteria.
>
> However, such runoffs could become quite boring in practice. Say  
> that there are a number of moderates in the first round and people  
> prefer moderates to the rest. After the first round is done, two  
> moderates are retained and run in the second round. What does it  
> matter which moderate wins? The closer they are to being clones, the  
> less interesting the runoff becomes.
>
> More formally, it seems that the whole voting population is not  
> being properly represented. Two candidates represent the middle but  
> nobody represents either side. That might be okay if voters are  
> normally distributed around the candidate, but if they are, you  
> wouldn't need the runoff to begin with.
>
> If that's correct, then it'd be better to have a proportional  
> ordering. That proportional ordering should still put one of the  
> moderates first (assuming he'd be the winner had there been only one  
> round), but also admit one of the side candidates. But here's the  
> tricky part. That proportional ordering method should also pass  
> LIIA, so that all the criterion compliances held by the base method  
> are retained. It's thus necessary that the winner of the base method  
> comes first. Beyond that, however, I have little idea how the method  
> might be constructed, or if it's even possible to have both a  
> proportionality criterion and LIIA.
>
> Finally, if such a method were to be found, one could possibly have  
> more than two candidates in the runoff. The runoff would serve as a  
> way of the method to say "hey, look at these candidates more  
> closely", where their positions could then be compared and voters  
> possibly change their minds. If the method passes LIIA, it doesn't  
> matter how many (or few) candidates you put in the second round -  
> the method acts like the one-round method if all the voters remain  
> perfectly consistent. Practically, also, if there are only two  
> candidates and one is a moderate, the "other" wing not represented  
> might feel cheated out of a chance if only one of the wings are  
> represented. If the centrist and the leftist goes to the second  
> round, the right-wingers may complain that their candidate is not  
> represented, whereas ordinary top-two runoff would have no such  
> problem because both the right-wing and left-wing candidate would be  
> represented at the cost of the centrist.