[EM] A quick and dirty variant of STAR

[Kristofer Munsterhjelm]

I've been trying, on and off, to find ways to "properly" generalize STAR 
so that it's cloneproof and also respects intensity of preference. But 
this is really hard. It's possible to get a mostly unambiguous 
definition of honest rated ballots (up to an affine transformation) by 
basing the ratings on lotteries.

But incommensurability is a pain: how do you count two voters' rated 
ballots equal if one of them thinks that only totalitarian dictators 
deserve a 0/10 and only perfect candidates deserve a 10/10, while the 
other normalizes however slight his preferences are?

Not having any good answer, I can only resort to normalizing every 
ballot so every voter gets the same power over the contest that counts, 
whatever it is. (Or all of them at once, if some kind of Condorcet 
analog can be found, but I haven't had any luck yet.)

So here's a quick and dirty method template based on this idea and my 
response to Chris Benham:

1. Take the input rated ballots and clone every candidate three times.
2. Use some multiwinner method to pick three finalists.
3. If they're all clones of the same candidate, that candidate wins. If 
they're two candidates and a clone, then the pairwise winner wins.
4. Otherwise, remove every candidate but the three and normalize every 
voter's ballot to span the whole scale. (Other normalization methods 
could be used, e.g. l-2 norm normalization.)
5. Select the Range winner according to the normalized ballots.

This might be cloneproof, but I'm not sure if it's robust enough; it's 
possible that adding a near-clones could affect the behavior of the 
multiwinner method. That's part of why it's quick and dirty.

The other part is that in close three-candidate rated elections, steps 
1-4 do nothing. Then step 5 becomes simply ordinary Range. And these 
elections would then exhibit the same kind of Burr dilemma problem that 
causes Range to degrade into Approval, placing the burden of manual DSV 
back on honest voters. And that's just what we'd want to avoid!

So one way to avoid that is to use some other norm than max-norm when 
normalizing. But then it has kind of weird behavior, because if the 
finalists are one candidate and two clones of another candidate, then 
the candidate whose clone got into the finalist set will be punished by 
the lp normalization. Thus the two candidate case is no longer "whoever 
beats the other pairwise wins", but there's an odd nonmonotonicity/free 
riding analog where getting too much support is bad, because it makes 
the multiwinner method also pick a clone of you, which hurts you in the 
final.

So that's another reason it's quick and dirty. But I thought I'd throw 
it out there anyway!

-km