[EM] Strategies for RRV/RSV and BR for multi-member constituencies

[Kristofer Munsterhjelm]

Raph Frank wrote:
> http://www.rangevoting.org/RRV.html
> 
> I wonder what would be reasonable strategies for RRV.

For RRV, and probably for any sort of multiwinner method that works 
according to the "elect and punish" cycle (I think that's Warren's term, 
but it's fairly descriptive), the method is susceptible to vote 
management. RRV would be more so since the votes are so fine grained. If 
a party knows its rough support, it could ask the voters to deweight 
their ratings so that its candidates squeak into first place. Of course, 
there's a risk in vote management: if the party tells the voters to 
deweight by too much, or the voters do it anyway, the party can lose a lot.

Perhaps one could make a CWP version of Schulze STV somehow. In the 
comparison of A1..An vs A2..An,B1, count towards the first if the voter 
prefer all of A* to B1. Perhaps one could count towards it according to 
the margin of the lowest rating of one of the As against B1. So if the 
candidates are A1, A2, A3 vs A2, A3, B1, and a voter has a ballot A2: 
10, A3: 5, A1: 3, B1: 1, this voter would give the former group two 
points above the latter (since 2-1 = 1). The WV equivalent would give 
the former group three points. I don't know if this is a good idea; 
intuitive ones can be bad, but can be good too.

> Anyway, testing strategies is hard for multi-winner elections as there
> is no simple rule like BR.  Previously, there has been some
> suggestions on how to handle PR-elections including virtual
> parliaments and multi-dimensional issue spaces.
> 
> An easy option is to just average the utility of all the winners.
> However, this doesn't take into account the benefit of diversity of
> candidates elected, which is one of the big points of PR.
> 
> A method that elects candidates with utilities of
> 
> 0,10,20,30,40,50
> 
> will count as the same as one that elects candidates with utilities of
> 
> 30,30,30,30,30
> 
> However, the first is more likely to be a PR result.
> 
> One option would be to take the median candidate for each voter as his
> utlity for that result.  This somewhat simulates the concept that a
> legislature follows the median member.
> 
> OTOH, since the legislature is likely to be much larger than the
> district in question, maybe each elected candidate can be considered
> as independent.  If one member of your party gets elected from your
> district, then that increase the probability of that party getting
> into government by X%.  It is likely that if 2 members of the party
> get elected, it will raise the probability by 2 times X% (plus a small
> extra amount).  Thus, the utility of each member of the party getting
> elected adds linearly.  Any non-linearility should be small especially
> if the number of seats to be elected is kept low.
> 
> Thus, maybe just summing/averaging the utilities of all the winners is
> the correct option (even though I don't really like it).
> 
> The only exception is if your party has no seats at all.  In that
> case, the first member being elected would be worth alot.  However,
> that just means that you would have high utility for that candidate
> and anyway, your party would only be able to run 1 candidate in your
> district, if it was small.

What we want with a PR method is to make an assembly that is a mirror of 
the people, or of the people's wishes. For simplicity, I'll assume 
voters are selfish and want candidates like themselves, so the assembly 
is a mirror of the people, but the argument holds even through the 
indirect layer of the votes. Usually (disregarding party influence), the 
voters want people like themselves but better (if they only want people 
like themselves, we could just pick the assembly by lot).

Averaging simply destroys too much information, so we can't determine 
whether the assembly is populated by candidates like the people, a 
gaggle of centrists, or half extreme left and half extreme right.

Since we want the assembly to mirror the people, the most obvious thing 
to check would be whether this is the case. Two methods seem intuitively 
easy. The first is to give each voter a point in some multidimensional 
issue space and then check if the distribution of those points in the 
assembly is like that among the people. The second is to construct the 
assembly and then let it vote on a number of decisions, where there is 
some rule as to how the people would have voted.

The former can be done in either a binary fashion (as my multiwinner 
election program does), or in a continuous fashion: construct a curve 
(or surface) of the distribution of points in voting space, then 
determine the similarity of the curve/surface for the people and for the 
assembly. You may have to use a kernel density estimator or something 
similar to get a good estimate of the distribution, especially for the 
assembly, and I don't know how to get the optimal bandwidth parameters 
in multidimensional space for a KDE, thus I haven't implemented that in 
my program.

The "construct the assembly and then let it vote" approach might be 
reducible to ordinary Bayesian Regret. The idea would be this: 
single-winner BR assigns utilities to all candidates and voters. Call 
the candidates' utilities their "defined utilities". For a single winner 
method, if the candidate is the optimal, he'll vote the way the people 
votes most of the time. In the long run, with the correct function for 
evaluating utility of a decision that goes "your way", the utility 
gained by electing this candidate would average out to the candidate's 
defined utility, and the measurement would thus be the expected value, 
from which one can get the Bayesian regret.