[EM] Election-Methods Digest, Vol 106, Issue 2

[Kristofer Munsterhjelm]

On 04/04/2013 02:40 AM, Forest Simmons wrote:
>
>
> On Wed, Apr 3, 2013 at 12:07 AM, Kristofer Munsterhjelm
> <km_elmet at lavabit.com <mailto:km_elmet at lavabit.com>> wrote:
>

>     Perhaps there's some value in making methods that appeal to the
>     right sentiment, even if one has to trade off "objective" qualities
>     (like BR, strategy resistance or criterion compliance) to get there.
>     The trouble is that we can't quantify this, nor how much of
>     sentiment-appeal makes up for deficiencies elsewhere, at least not
>     without performing costly experiments.
>
>
> If I am not mistaken, both methods (Chiastic and this one) are
> strategically equivalent to Approval from a game theoretic point of
> view.  But psychologically they are quite different.  I think that this
> new version is much less likely to elicit approval style responses (at
> the extremes) than ordinary Range voting for example, or even the median
> method with J in the title (I can't think of it at the moment).
>

I found a quite broad reduction for ratings-type methods. I posted it 
when I did, but I'll repeat it.

Say you have a rated ballot set, and candidate C's set of scores is 
represented by the vector s_C (first element in the vector is the first 
voter's rating of the candidate, and so on), then:

- if each candidate gets a meta-score, call it m_C, from some function 
f(s_C),
- the candidate with the highest meta-score wins,
- and f(s_C) is monotone in the sense that increasing a rating in s_C 
never makes f(s_C) evaluate to a lower value than before, and decreasing 
a rating in s_C never makes f(s_C) evaluate to a higher value than before,

- then Approval strategy is optimal.

The reason is that if a voter likes a candidate X, he can never be worse 
off by not giving X a higher score; and if he dislikes X, he can never 
be worse off by not giving X a lower score. Thus the scores migrate to 
the Approval extremes.

And unless I'm missing something, both the chiastic method and your 
method fulfill the properties above.

There's a caveat: the "optimality" might be of a form where it never 
hurts you to go to an extreme, but it doesn't hurt not to either. To 
eliminate that kind of equilibrium, one would have to replace the 
monotonicity property with something stronger.

-

As for the median method, you're probably thinking of Majority 
Judgement. As long as the voters act in the way B&L say they should do, 
and judge candidates to absolute grades rather than comparing the 
candidates to each other, it avoids Approval reduction. B&L use evidence 
from France to argue that enough voters judge to absolute grades that it 
effectively works this way. Maybe your method would also do so, but then 
it would have to be phrased in terms of grades rather than numbers.

I think that Range encourages rating at the extremes, though it doesn't 
seem to always do so. Web sites lend evidence to both sides: IMDB used 
(and uses) Range for the movie ratings, while Youtube switched from 
ratings to Approval-style up-and-down voting. IMDB does some filtering 
on the votes, though, so perhaps that's what is keeping it from reducing 
to approval. In any case, it shouldn't be hard to make a method that's 
more resistant than bare Range in that respect.