[EM] PR with weighted rep. power

[Kristofer Munsterhjelm]

In a post in September, Toby Pereira said the following about the 
"Random Ballots" method I had implemented in quadelect:

> This is very interesting. Thank you for doing this analysis. Random
> ballots, as I understand your implementation, I think would not be
> strategyproof. If there are c candidates to be elected, then every
> voter ranks their top c. And then you sequentially pick c ballots at
> random, electing the top-ranked unelected candidate on each ballot.
> Is that correct? In that case, I think if a voter is fairly confident
> that their favourite candidate will be elected anyway, it makes sense
> for them to put other candidates above them that might not get
> elected anyway. If my second favourite candidate is not very popular
> among other voters, and my favourite is, I am likely to put my second
> favourite top.

And I responded that that too is interesting, because that looks like 
vote management (Hylland free riding), and I hadn't expected vote 
management to be that robust a strategic feature.

But a few days ago I had a thought. Suppose we change the method to the 
following:

- Pick c random ballots. For simplicity, disallow equal-rank.
- Let each candidate's voting power be the fraction of these ballots on 
which he is the favorite.
- Elect every candidate with positive voting power. (Note that there may 
be fewer than c distinct winners.)
These candidates are the winners, and their votes have the given weight 
in the assembly.

This turns the method into a sort of randomized party list. Is it 
strategyproof?

The voting stage seems to be, at least, because in effect each candidate 
has an infinite number of clones, and if you prefer A to B, you would 
rather want to give an assembly seat to A (or a clone of A) than to B. 
Note that the number of seats need to be variable, otherwise there would 
be an incentive to vote for separate candidates to push opposition 
candidates out of the assembly.

If there's an incentive to strategy for this method, it would be in how 
voter weights add up to majorities (power indices being nonlinear). But 
I'm not sure there's an incentive there either.

So if it is strategyproof, we have a weighted variable seat method that 
is proportional without vote management/free riding; and we also have 
results that say that Droop proportionality is incompatible with 
complete immunity to free riding. The natural question then is: how 
close to DPC can we go and have immunity to free riding? Is it possible 
by just having a variable size assembly, or do we need some amount of 
weighted representative power too?

I don't know what the answers would be, but they're interesting thoughts.

-km