[EM] Question about the Plurality Criterion

[Kristofer Munsterhjelm]

On 06/24/2013 04:10 PM, Benjamin Grant wrote:
> As I have had it explained to me, the Plurality Criterion is: "If there
> are two candidates X and Y so that X has more first place votes than Y
> has any place votes, then Y shouldn't win".
>
> Which I think means that if X has, for example, 100 votes, then B would
> have to appear on less than 100 ballots and still **win** for this
> criterion to be failed, yes?
>
> I cannot imagine a (halfway desirable) voting system that would fail the
> Plurality Criterion – can anyone tell me the simplest one that would?
> Apart from a lame one like “least votes win”, I mean?

That depends on what you put into a candidate not being ranked on the 
ballot. If you think that voters mean that all the candidates they rank 
are better than those they don't rank, then Plurality obviously makes 
sense. On the other hand, if not ranking a candidate simply means the 
voter has no opinion, then the Plurality criterion is no longer as obvious.

A very simple system that fails the Plurality criterion for this reason 
is mean (average) Range. In this system, you take the mean rating of 
each candidate, and greatest mean wins; but in this particular variant, 
if you don't rate candidate X, you don't change his mean in any way.

So you could have a candidate A that's ranked with a mean of 8.5 by 1 
million voters, and a candidate B that's ranked with a mean of 9 by 500 
000 voters (and otherwise not ranked). Say more than 500k of the ballots 
list A first. Then B is barred from winning by the Plurality criterion. 
Yet by the logic of mean Range, B should win because, according to that 
logic, the voters who didn't list B were just saying they didn't *know* 
what rating B should have and instead left the task of determining B's 
mean to the others who did rate B.

Now, pure mean Range has a problem in that candidates who are only known 
by a few fanatics could get an illegitimate win, so some sort of soft 
quorum (like IMDB does for its movies) is probably better. I just use 
mean Range as an example of a system that isn't obviously insane yet 
fails the Plurality criterion (or one particular way the Plurality 
criterion might be extended to rated ballots).