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

[fsimmons at pcc.edu]

From: Andy Jennings 
> On Mon, Jul 18, 2011 at 6:00 PM, 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-
> determinednumber 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.

Good points!

> 
> 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.


I would like to point out that if the election were single winner, this RRV result of 50%A+50%B would be 
the closest thing to a consensus lottery.  In other words, your method of using RRV with repetition is a 
great way of generating proportional lotteries.  So it seems that good lotteries are easier to generate 
than good PR results.

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

If I am thinking straight, PAV would give the same max score of  

20*(1+1/2)+40

to three slates of three candidates {A, B, C}, {A, B, D}, and {A, B, E}.

If higher resolution range values were available, non-sequential PAV would probably favor one of these 
three.

This reminds me of "envy free" versus merely fair division.  The {C, D, E} slate would be "envy free" as 
well as fair, even though {A, B, C} would give more total satisfaction, while still being fair in the sense 
that each voter got at least the satisfaction guaranteed by PR.  In the case of {A, B, C} the D and E 
voters might well envy the extra satisfaction of the C voters.

> 
> Andy
>