[EM] Manipulability stats for more poll methods (fixed footnotes)

[Kristofer Munsterhjelm]

On 2024-05-05 00:25, Michael Ossipoff wrote:
> 
> 
> On Sat, May 4, 2024 at 14:45 Kristofer Munsterhjelm 
> <km_elmet at t-online.de <mailto:km_elmet at t-online.de>> wrote:
> 
> 
>     Yes, that's right. But consider a voter with the following utilities:
> 
>     A: 0.57
>     B: 0.32
>     C: 0.23
>     D: 0.08
> 
>     Normalization to two steps fixes the highest value (0.57) to 1 and the
>     lowest value (0.08) to 0 and rounds off the intermediate values after
>     linearly scaling them. 
> 
> 
> Yes. So far, so good. But…
> 
>     This in essence says that a value is rounded off
>     to 1 if it's greater than or equal to 0.325 (the midpoint between 0.08
>     and 0.57)
> 
> 
> What? You didn’t average the normalized values. You averaged two of the 
> values before normalization. The midrange isn’t usually the same as the 
> mean. You used the midrange as the mean.

Yes, that's my point. What I call "normalize" uses the midrange, because 
it's a generalization of the following approach for mapping utilities to 
a cardinal scale:

Let min be the minimum utility (the utility of your worst candidate), 
and max be the utility of your favorite. Then the normalized ratings 
from 0 to k inclusive are given by

r_i = round( (x_i - min)/(max-min) * k), where x_i is the utility of the 
ith candidate, and r_i is the rating of that candidate.

I thought this was the natural way to create Range ballots from 
utilities. To my knowledge, there's no obvious way to generalize 
above-mean utility thresholding to say, a 0-10 scale.

I'm aware that "normalize" as I've defined it above is different from 
above-mean thresholding, and that the latter is more common in 
treatments of approval voting in particular. For that reason, I use mean 
utility cutoffs for the approval hybrids, and range-based normalization 
for the Range hybrids. My point was just that one shouldn't confuse the 
two and think I used the same approach for approval and Range.

-km