[EM] a response to Kristofer Munsterhjelm re: Fuzzy Options.

[Kristofer Munsterhjelm]

I'll reply to your other post, but I think this one is the easiest to 
reply to, so I'll do so first.

David L Wetzell wrote:
> 
> On Thu, Nov 3, 2011 at 9:14 AM, Kristofer Munsterhjelm 
> <km_elmet at lavabit.com <mailto:km_elmet at lavabit.com>> wrote:
> 
>     David L Wetzell wrote:
> 
>         And I don't think the Condorcet criterion is /that important/,
>         as I think in political elections, our options are inherently
>         fuzzy options and so all of our rankings are prone to be ad hoc.  
> 
> 
>     If opinions are fuzzy, that means that the voters' true distribution
>     within political space would differ somewhat from the distribution
>     you would infer by looking at the votes alone.
> 
> it could also mean that political spaces are at best somewhat useful 
> constructs and that the "true" distribution is something that's 
> constantly being manipulated, not taken as given.   This understanding 
> makes me tend to be more middle-brow than most people on this list.

Well, yes, but people seem to vote at least in a somewhat spatial 
manner. Consider, for instance, Tideman's voter models and SVD/PCA fits 
like http://politics.beasts.org/ . While the results might have been 
colored by the polarizing effects of Plurality, I think it shows we can 
use political space as a way of visualizing voting behavior. If nothing 
else, it at least gets the point across in an intuitive manner - one can 
generalize later.

SVD/PCA has also been applied to legislature records, where one would 
think the Plurality polarizing effect to be diminished. See, for 
instance, http://www.govtrack.us/congress/spectrum.xpd .

>     KM:In terms of the 2D Yee diagrams, this means that if the voters
>     are centered on a certain pixel, their votes might behave as if it
>     was centered on one of the neighboring pixels (since each pixel in a
>     2D Yee diagram gives who would win if the population were normally
>     distributed around that point in political space and preferred
>     candidates closer to them). So in a Condorcet method, this might
>     sometimes lead to the wrong candidate being elected. It would do so
>     in the case where the true distribution is on one side of the
>     divider between two Voronoi cells, and the distribution inferred
>     from the votes alone is at the other side.
> 
> dlw:Yup, getting the condorcet winner all of the time isn't the 
> slam-dunk it's purported to be. 
> 
> 
>     KM:However, fuzzy opinions can cause greater problems with IRV.
>     Because IRV is sensitive to the order of eliminations, it doesn't
>     just have the clean cell transitions of Condorcet; it can also have
>     disconnected regions near the edges or in the middle of one of the
>     regions. In essence, these are the same as the "island of other
>     candidates" artifacts, but in two dimensions rather than one.
> 
> 
> dlw: I'm afraid you lost me there.   

Okay, I'm going to go a bit more thoroughly through it. For the sake of 
the model, I'm going to assume a 2D opinion space with a normal 
distribution around a central point that depends on the society in 
question. These assumptions can be altered, but the difference between 
methods like most Condorcet methods and methods like IRV will persist, 
to my knowledge. I'll eplain what the difference is, but first, to get 
some things out of the way:

If you have a spatial model with an attendant distribution for each 
point, then you have win regions. Inside each win region, the candidate 
assigned with that win region will win. For instance, if the center of 
opinion is solidly within the left-mainstream region, then the Democrat 
candidate will win. Similarly, if the center of opinion is solidly 
within the right-mainstream region, then the Republican candidate will win.

Look at the "Equilateral" diagram on http://zesty.ca/voting/sim/ . There 
you can see that if the center of opinion is closer to, say, the red 
candidate than to the other two, then the red candidate wins, and it is 
the same with the green and blue candidates.

Now, what effect does vote fuzziness have? The effect is that the 
election outcome is as if the voters were centered on a particular 
pixel, when in fact they were centered on a pixel some distance away. 
The distance between the two arise from noise.

This is not a problem if the voters are solidly inside a given win 
region, because both points (the voters' true center as well as the 
center you'd infer from the votes alone) lie within the same win region. 
So vote fuzziness degrades the outcome only when the voters' true center 
is close to the border between two (or more) win regions, because then 
the noise can push the inferred center to the other side of the border, 
which could lead to another candidate being elected.

So how often could this happen? Well, that depends on how many points of 
the voting space have another region close to it. If the regions are 
compact or the borders between straight, most of the outcomes will be 
solidly within a region and therefore will give the right result even if 
vote fuzziness adds noise. However, if the borders are long, zigzags all 
over the place, or have disconnected regions inside some other 
candidate's win region, these all increase the probability that the true 
center will be in one candidate's region, but the inferred center (from 
the votes, which is the only thing the method knows about) will be in 
another candidate's region.

Finally, looking at Yee diagrams such as "Nonmonotonicity", "Square", 
"Shattered", and "Disjoint" on Ka-Ping Yee's page (whose link I've given 
above), or the IRV pictures on http://rangevoting.org/IEVS/Pictures.html 
shows that elimination methods in general and IRV in particular can have 
such zigzags and disjoint regions whereas Condorcet methods usually 
don't. Hence, IRV is more affected by vote fuzziness than are Condorcet 
methods, which was what I wanted to show.

The difference then, as I said I would explain, is that IRV has longer 
borders and less compact win regions, which makes wrong-winner outcomes 
more likely if there is vote fuzziness. If vote fuzziness impacts 
Condorcet methods, it impacts IRV more.

If you take a step back, it's not that hard to understand. In IRV, a 
change in one of the elimination rounds can lead a different candidate 
to be eliminated, which in turn can lead a different candidate to be 
eliminated in the next turn, and a different candidate to be eliminated 
as a consequence of this, and so on. A single vote can change the 
outcome radically. IRV can be sensitive to initial conditions. On the 
other hand, if you look at something like Borda (not a good method, but 
it shows the concept clearly enough), a single vote will only alter the 
point sum of each candidate by a bounded amount. This might be enough to 
hand the victory to another candidate, but it is more well-behaved: the 
initial difference doesn't amplify into greater differences that amplify 
into greater differences in turn.

>     KM:It may be the case that voters are not centrally distributed in
>     political opinion-space, but I think the observation can be
>     generalized. If I'm right, why put up with a method that, by
>     sensitivity to the elimination order, amplifies the fuzziness of the
>     votes?
> 
> 
> dlw: Well, 1. IRV3 doesn't let folks rank all of the options and so it 
> hopefully has more quality control on which options are ranked.

IRV with a few candidates will do better than IRV with many candidates, 
but with the number of candidates held the same, other methods will 
still be more well behaved. Also, you could consider the 3-rank limit 
(if only the number of rankings rather than candidates are limited) 
another form of noise. The noise is, in essence, the same as if every 
voter didn't know their later preferences and so truncated after three. 
(I'll note, though, that it's not unambiguously noisy: IRV restricted to 
one rank would simply be Plurality, and Plurality doesn't have the 
zigzag and disjointness of IRV).

> 2. by not always giving us the "center", it does permit learning about 
> the different viewpoints.  Remember, since I'm middle-brow, I don't put 
> as much significance on optimizing within the distribution of political 
> opinion space. 

You can get that with other methods, too. I think a good comparison 
would be to PR. In most countries that make use of proportional 
representation, the method also includes a threshold whose purpose is to 
keep sliver parties from gaining kingmaker power - e.g. a far right 
party only joining a coalition if the coalition agrees to kick out all 
the immigrants.

In a similar manner, if the "always center" or "weak CW with no core 
support" objections are important and could break Condorcet, well, just 
institute a threshold. Say that a candidate who gets less than n% of the 
votes (or who is ranked on less than n% of the ballots) will not be 
counted.

If you want to be really sophisticated, you could probably use a 
cloneproof method for the threshold to avoid perverse incentives, but 
that's getting a bit too complex - all I'm saying is that if a weak 
center is a problem, there are ways around it.

> 3. It introduces some uncertainty in the circulation of the elites, 
> which can give alternative viewpoints a chance to get a better hearing. 
>  When a new third party gains ground, it'll get a serious hearing and 
> hopefully the de facto center will be moved. 

All the good methods do that, and they do that by providing competition. 
On a level playing field, minor parties can pull major parties in their 
direction (but not unduly so) because the method is responsive - and in 
a predictable way - to shifts in votes and sentiment.

Plurality does very badly here because strategic Plurality isn't 
responsive. IRV is better, but it can be unpredictable, as the 
simulations show. Yes, it's possible to weaken the effects of the 
unpredictable nature by making alliances (as you've suggested could be 
done in Burlington), but that imposes a further burden on the parties. 
Instead of requiring the parties to know when to cooperate and when to 
go at it alone, just have the voting method deal with it by shifting the 
center smoothly in the direction given by the people's preferences.