[EM] Smith//Score ? normalization

[Kristofer Munsterhjelm]

On 25.01.2022 00:29, Kevin Venzke wrote:
> Hi Kristofer,
> 
> Le lundi 24 janvier 2022, 16:46:06 UTC−6, Kristofer Munsterhjelm <km_elmet at t-online.de> a écrit :
>> That gives me an idea. How about Smith//Lp-cumulative?
>>  
>> That is, first remove everybody who's not part of the Smith set.
>> Renormalize all ballots to have unit p-norm. Then greatest score wins.
>> It probably isn't monotone, but the renormalization should mitigate at
>> least some of the Burr dilemma problems of plain Range.
> 
> I guess that you should rescale, not normalize. What if I rate both A and B
> 10/10 each and they are both in the Smith set? Smith//Score would treat me
> better than that, I guess.

If normalizing is meant in the sense of subtracting the mean, then I
agree, it should better be called rescaling. The "normalization" I'm
thinking of is the one cumulative voting imposes on the voter, or IRNR
does between rounds.

So e.g. if p->infty, you would scale the least preferred Smith set
member to zero and the most preferred to the max of the Range ballot.
(If you have no preference between any candidate in the Smith set, then
what value you set them to doesn't matter as long as each candidate gets
the same rating.)
If p=1, you get ordinary cumulative voting, which tends to have a
bullet-voting incentive. For p=2, there may be strategy resistance to be
had since (IIRC) the optimal zero information ballot for l2-cumulative
voting rates candidates proportional to the utilities.

In any case, the ballot is transformed so that no score is outside of
the range of the Range ballot (e.g. 0-10) and that the p-norm is equal
to the same constant for every voter, unless that voter rates every
candidate equal.

-km