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

Wed Jan 28 12:39:38 PST 2009

```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.

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