[EM] ASCII maps

[Kevin Venzke]

Hi Kristofer,

--- En date de : Lun 21.2.11, Kristofer Munsterhjelm <km-elmet at broadpark.no> a écrit :
> Then it tries to
> come up with a nice map
> > that minimizes inaccuracy.
> 
> You could try using synthetic coordinate algorithms for
> mapping the distances to 2D. I did that for competing
> entries in a programming game, once, using the centralized
> Vivaldi algorithm as described in
> pdos.csail.mit.edu/papers/vivaldi:sigcomm/paper.pdf.
> 
> Another option would be to use principal components
> analysis, but I know less about that.

Oh dear. That looks very complicated. I just used the number of scenarios
where the methods agreed. So, the maximum distance between two methods is
27. When the methods are plotted I find the Euclidean distance, subtract
from the desired distance, and square it. Add all that up and that's the
measure of a poor fit.

To improve the fit it tries rerolling a few candidates, or tweaking their
positions.

> > Roughly left-to-right there seems to be a "all
> preferences" to "first
> > preferences" emphasis spectrum.
> 
> To test the idea that left-to-right is "all preferences" to
> "first preferences", try including Borda... or
> antiplurality. They should be to the left if that's correct,
> because they don't privilege first preference very much.
> Perhaps Coombs would be down by IRV but significantly to the
> left.

Ok, we will start to hit some difficulties.
1. The scenarios evaluated all assume no majority favorite. That will
probably skew how some of these are measured.
2. I can't prevent truncation. Not a big problem, I just assume that
truncation means split the preference between the unranked.
3. I can't easily do methods like Borda or Margins because you need to
know the actual faction sizes. So you have to plot not just a method,
but the relative faction sizes assumed.

That said, I have some results for these. Unless I made terrible errors,
there are 23 ways to map Borda based on the faction size ratios. There
are 3 ways to do Antiplurality. And 5 for Coombs.

I did a number of Borda ("b") maps. This one is typical, in that DAC
is closer to WV than to Bucklin, and IRV is not on the WV side of DSC.
The latter quality isn't reliable. I suspect that either the large
quantity of b's is creating too much work to get an accurate map, or else
this actually is accurate given the methods being plotted.

It seems clear that Borda is not appearing on the "first preference
emphasis" half.

............................................................
..............b.........................IR..................
............b...b.b.........................................
...............b.....b..b...................................
................b..b.......b.....................FP.........
.......................b...............DS...................
...........Bu...b.....b.....................................
.................b.......b.b................................
............bb.....bb.b.....................................
............................................................
............b....b..........................................
...................WV....DA.................................
............................................................
............................................................
............................................................

I put Antiplurality "a" and Coombs "c" on the same map. Much simpler.
The pentagon shape and order are preserved. Both methods are on the left
side, and Coombs is closer to IRV.

............................................................
..................DA........................................
.............................................FP.............
.........Bu...........................DS....................
........a...................................................
..........a.................................................
..........a.................................................
............................................................
.................WV.........................................
..................c.........................................
....................c..c.............IR.....................
.....................c..c...................................
............................................................
............................................................
............................................................

> As for finding something between DSC and DAC, you could try
> DHSC. This meets neither LNHarm nor LNHelp but might be
> "balanced" between the two. DHSC simply consists of creating
> both the DAC and DSC structures, then adding them up and
> running the DAC/DSC algorithm (intersect sorted sets until
> there's only one left, skipping intersections that would
> turn the set empty) on the result.

I'm pretty confident DHSC would do the trick, but I think that one would
have to be a cloud of points again. Hopefully not as bad as Borda.

I think what I'm going to do is add a criterion to the generator/fixer
that tries to make methods that are somewhat close to both DAC and DSC,
and see if anything looks interesting.

Kevin