[EM] Anyone got a good analysis on limitations of approval and range voting?

[Kristofer Munsterhjelm]

robert bristow-johnson wrote:
> 
> On Nov 25, 2009, at 3:26 PM, Kristofer Munsterhjelm wrote:
> 
>> robert bristow-johnson wrote:
>>
>>> my understanding is that the later-no-harm result happens only if the 
>>> case of a Condorcet cycle (the prevalence of which i am dubious 
>>> about).  where there is a Condorcet winner and that person is 
>>> elected, is there still possible later harm?
>>
>> As far as I remember, Condorcet and LNHarm has the property that 
>> LNHarm isn't, by itself, violated as long as there is a CW, but the 
>> transition from CW to no CW (or vice versa) makes it inevitable that 
>> there will be a LNHarm-violating discontinuity *somewhere*.
> 
> the degree of inevitability is an issue.  if "inevitable" is measured as 
> a binary value, the i s'pose it's inevitable.  if "inevitable" is 
> measured as a probability of a cycle occurring per election-year, then i 
> think it's a small number.  if cycles are rare, the mean percentage of 
> elections that have Condercet cycles is small.  when we somehow figure 
> out a merit metric for an election system, a low-likelihood of a 
> pathology that has low cost (say, if a cycle happens you elect using IRV 
> rules, how bad can that be?) should contribute (negatively) negligibly 
> to the merit metric.

I just want to be precise here: when I talk about inevitability, I talk 
about mathematical inevitability. That is, (if I'm right,) for any 
possible completion method, there will be some ballot set that, when 
some of the ballots are modified to add later candidates to otherwise 
truncated ballots, the candidates that are ranked higher are harmed by that.

The elections don't have to be realistic - what's sufficient in the term 
of criterion compliance is whether it's possible to contrive an example.

Whether or not that has any real bearing on the issue of public 
elections using the method is another question, but that's an argument 
one may field against any criterion compliance/failure. For instance, 
some of those who favor IRV say that monotonicity failures can't be 
manipulated and so are irrelevant.

In any case, I think we agree, but I wanted to clear that up. I'll 
repeat that I don't think LNHarm compliance is that important, although 
others disagree, of course.


A final note is this: while the above may make it seem like I think 
criterion compliance is pointless (after all, the failure could be 
hidden in an obscure election scenario that will never happen), that's 
not quite true. If a method passes Condorcet, you at once know it will 
always pick the CW if one exists; you don't have to sit down and reason 
whether the failure is acceptable by whatever metric you are using. 
Thus, if we can pass criteria without having to pay too much, we should; 
a method passing a criterion is a guarantee for that method that it 
won't ever misbehave in the way drawn up by the criterion, and such 
absolute guarantees are nice things to have. If we can't have them, 
*then* we can start talking about whether or not it matters, but in the 
ideal world, we would have a method that pass the criteria that are 
worth passing.