[EM] Kristofer, April 3, '12, Approval vs Condorcet

[Kristofer Munsterhjelm]

On 05/15/2012 10:11 AM, Michael Ossipoff wrote:

> Or dither if you want to, but don't blame it on the voting system.

I think you have misunderstood some key concepts I have been using. Let 
me try to make them more clear.

When I talked about "dithering", I did not mean it in the sense of 
"hesitating, worrying, being uncertain" (although there can be elements 
of that in Approval strategy). I was instead making an analogy, which I 
think was quite appropriate, to the "dithering" of the graphics world. 
This is why I also referred to grayscale and black-and-white.

As applied to graphics, dithering is a process where you reduce the 
number of shades (or colors) in a picture, and emulate the presence of 
more shades (or colors) by placing dots of the colors you do have 
available in the proportion required. This is a process that, like 
deciding upon how to vote in Approval, makes use of global information 
to make a local decision - in the graphics case, whether the pixel at 
any particular coordinate should be white or black. See 
https://en.wikipedia.org/wiki/Floyd-Steinberg_dithering for an example.

My point about the dithering analogy is that the method imposes a 
constraint and then the software/voters have to calculate how to make a 
good approximation within the constraint. When I say "I'd rather not 
have to do that", I mean that I would like to just submit my full 
information - the "whole picture" - rather than having to deal with how 
to render or dither it into a format compatible with the constraints of 
the method.

Furthermore, since the analogy shows that one's moving quite a bit from 
the "front end" (the actual voting method) into the "back end" (which 
may involve polls), it seems that if one's very concerned with strategy, 
as you are, one should also consider strategy in the back-end. For 
instance, if I'm a Bush voter in a three-way race, I might report that 
I'm voting Nader alone. This will make it seem that Nader has sufficient 
support to win, so the Nader-voters vote for Nader alone. But they've 
been had, because now Gore lacks the Nader+Gore votes to get past Bush, 
and Bush wins. Of course I would have to turn a significant fraction of 
the Bush voters in this direction, but your Condorcet examples also 
require the coordination of a significant fraction of the voters for a 
strategic end. (And if you say that the Nader voters would be more 
cautious when they see Bush's support, then Approval+back-end doesn't 
really pass IIA, does it? Nor is the strategy as easy as frontrunner 
plus anymore.)

The above could be formalized by making a DSV version of Approval. If 
that DSV method is ranked, then one of two things will happen: Either it 
fails criteria like FBC or monotonicity, which means that 
Approval+back-end might not be as good as you think, or it doesn't, 
which means DSV Approval would be a good ranked method on its own. If 
Approval is good even when people play the strategic game, why not have 
the method itself play the game for them?

Okay. The second concept I think you misunderstood was "half-empty" vs 
"half-full" and "noise". I'll first explain the first distinction, and 
then the second.

I'll use another analogy here for Condorcet vs Approval. Approval is 
like an analog TV signal. If you add uncertainty, it degrades a little. 
Add more noise, and it degrades more until, if the voters have 
absolutely no idea what they're doing and vote randomly, you have snow.

On the other hand, Condorcet is like a digital signal. You keep adding 
uncertainty (or strategy) and nothing happens until a break point, then 
it starts to get weird, and a little more will get it past the point and 
everything goes black.

Range is like Approval in this respect, and MJ is like Condorcet*. When 
I refer to your "half-empty" view, I refer to that you seem to think 
that voters (alone, i.e. not coordinated by party central) will add 
strategy into Condorcet even when they're on the side of the break point 
where nothing happens -- because they fear that if they don't, then when 
it is driven to the other side and gets seriously weird, the weirdness 
will favor the other side**. Similarly, in MJ, you think that it'll 
reduce to Approval because the voters will think that, even though 
voting strategically probably won't make a difference, the method might 
be pushed over to the other side and then they need all the help they 
can get.

Plurality is not all that good on the right side of the break point and 
in any event breaks very easily, so people strategize there. But I think 
that in sufficiently robust methods (like Condorcet or MJ), it's not 
going to happen; or if it does because of voter irrationality, the 
voters can be taught through feedback that they don't need to assume the 
worst. The benefit, am I right, is that when the voters generally stay 
on the right side of the break point, they can enjoy the expressivity of 
Condorcet and not have to think about dithering (quantizing) their ballots.

So if strategy isn't what I concern myself about, then what is? It is 
the idea that voters, through incomplete information, get a bad result 
-- and that's "noise".

I'll give an example first of all. Say you have a three-way 1D election 
with the Greens to the left, Democrats to the right, and Republicans a 
little further to the right. The median voter is somewhat to the left of 
the midpoint between the Greens and the Democrats. Then, for each voter 
in an Approval election, there is some probability that he gets the 
"Nader only or Nader+Gore" decision wrong. Say that he "gets it wrong" 
if he votes differently than he would after n-> inf iterations of 
complete information of others' votes in the last round (i.e. the 
equilibrium which would probably consist of voting Nader alone).
Then if enough people get it wrong, the winner will be Gore instead of 
Nader, and since each voter has some probability of getting it wrong, 
then there is also some probability of enough people getting it wrong to 
swing the election to Gore rather than Nader. If the election is 
three-way and close, that probability could be significant***; and if it 
swings the wrong way, then I think the voters who voted the wrong way 
would get very annoyed. At least I would. I would go "so this method 
burdens me with a quantization decision and if I happen to decide 
wrongly, possibly through no fault of my own, it punishes me? Is this a 
joke?". Enough voters feel like this, and there would be a backlash. 
Approval would in that respect then be like IRV: appearing sensible when 
you have two + minor parties, but in a three-way race, the problems surface.

In contrast, in Condorcet the voters just rank. Black's theorem handles 
the rest, and I really doubt the voters would truncate in great enough 
numbers when the only real contestants are Bush, Gore, and Nader. 
Because of the robust nature of Condorcet (the "digital signal" analogy 
way near the start), even when people make mistakes about whether they 
prefer Gore to Nader or vice versa, the method generally arrives at the 
reight outcome.
You could even be more rigorous by considering methods to be 
maximum-likelihood estimators in the presence of noise. If the noise is 
of the form that "people get X vs Y wrong with fixed probability", then 
Kemeny is the MLE, for instance - the one most likely to get at what the 
voters had intended if their expression wasn't noisy.

--

* The comparison is more apt than one would think at first glance, 
because there is a relation between average and Borda on one hand and 
median and Condorcet on the other. See 
http://lists.electorama.com/pipermail/election-methods-electorama.com/2011-January/026999.html 
for more information.

** You seem to be more inclined to look at ballots in a strategic manner 
in general, I think. For instance, when you said that you would only use 
a few ranks in (some Bucklin variant I don't remember), that struck me 
as quite strange. Are not ranked ballots expressions of preferences? No 
- not if they're primarily *strategic*. My initial puzzlement was at how 
you directly jumped to the instrumental point of view. Of course, I 
could be wrong in my inference because I don't know with absolute 
certainty how you really think (as you don't with me). We only have what 
we write to go on.

*** This could be tested through simulations.