[EM] Correspondences between PR and lottery methods (was Centrist vs. non-Centrists, etc.)

[Andy Jennings]

On Mon, Jul 18, 2011 at 6:00 PM, <fsimmons at pcc.edu> wrote:

> Andy and I were thinking mostly of Party Lists via RRV.  His question was
> that if we used RRV, either
> sequential or not, would we get the same result as the Ultimate Lottery
> Maximization.  I was able to
> show to our satisfaction, that at least in the non-sequential RRV version,
> the results would be the
> same.  It seems like the initial differences between sequential and
> non-sequential RRV would disappear
> in the limit as the number of candidates to be seated approached infinity.
>
> Would that imply P=NP?    In other words, sequential RRV might be an
> efficient method of
> approximating a solution (for large n) of non-sequential RRV (which is
> undoubtedly NP hard).  What
> would be analogous in the Traveling Salesman Problem?  Don't hold your
> breath, but it would be
> interesting to sort out the analogy, if possible.
>


I am still hopeful that sequential RRV with a large number of seats, leaving
each candidate in as if they were their own party, would be a good and
tractable way to choose legislators and give them each a different amount of
"voting power".  I'm hoping it would be possible to calculate the
proportions in the limit as n goes to infinity.

But sequential RRV is completely ignorant about how many seats need to be
filled, so it's not really going to find the globally optimum N-winner
representative body like ULM and non-sequential RRV aim to do.  This
"infinite sequential RRV" might be good  when there is no pre-determined
number of seats to fill but instead we want the method to choose the number
of winners.  For real elections, however, I suspect that it will give some
voting power to every candidate, so maybe it's not that good for choosing a
representative body.

Here's an example, on the other hand, where this method chooses too few
winners:
10 voters approve A and C
10 voters approve A and D
10 voters approve A and E
10 voters approve B and C
10 voters approve B and D
10 voters approve B and E

If you're choosing two winners, I think the obvious winners are A and B.
 But if you want to choose three winners, I think the obvious choice is C,
D, and E.  Only a method that knows how many winners you're going to choose
can make the correct decision here.  In this case, RRV will choose A and B.
 If A and B are "left in" (pretending they are parties even if they are
candidates) then RRV will continue to alternate between A and B.  In the
limit, it will give half of the voting power to A and half to B.  This is
just not helpful if you wanted to choose three winners.

ULM and non-sequential RRV evaluate each possible combination of winners and
can do the right thing in the three winner case.

Andy
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