[EM] hybrid 2-stage and 1-stage Bayesian Regret for comparing multiwinner election systems

[Warren Smith]

So perhaps the best or simplest way to resolve the inherent conflict
(in multiwinner voting systems) between the two goals of (a) good
representation and (b) good winner quality, is... to accept it.

Let the Hybrid Bayesian Regret of a multiwinner voting system be a
LINEAR COMBINATION of 1-stage and 2-stage regret.  Call the weights W1
and W2.

2-stage BR is explained here:
http://www.rangevoting.org/BRmulti.htmli

1-stage regret is simply the sum (over all voters and all winners) of
utilities for the winners for the voters -- subtracted from same thing
with the optimum winners who would have maximized this 1-stage utility
(this subtraction makes it a "regret" rather than a "utility").

You may object "the weights in this linear combination are arbitrary!"

That's a valid objection.  However, I could counter that maybe that is
the way things really are.  This is just yet another "knob" on the
side of the "Bayesian Regret machine" -- others include #candidates,
#winners, utility-generation model, voter-behavior model, etc.

The point is that if we have two candidates, both of whom would vote
the same way on every single issue, those two candidates are
nevertheless NOT entirely equivalent, because members of parliaments
do things other than just voting.
Candidate A who's highly experienced, smart, articulate, unbought,
good at writing legislation, etc is likely to do a better job than
candidate B who lacks those virtues, even if they'd always vote the
same.

To make the argument that W2/W1 is a genuine "knob"...
In hypothetical different parliaments with different internal rules about,
e.g, how legislation gets written, those virtues really might matter
more, or matter less.   In the limit where the MPs really do nothing
other than vote, then A and B
really would be entirely equivalent and we should set W1=0.  In the limit
where MPs never vote and just write legislation which (say) gets voted
on by others,
then W2 should be larger, and maybe W1=0 or anyhow smaller.

Using a hybrid of 2-stage and 1-stage Bayesian Regret, allows us to
make our voting system comparator try to understand/assess this.  As
opposed to ignoring it.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html