[EM] electing a variable number of seats

[fsimmons at pcc.edu]

Andy,

The choice of epsilon is crucial when the number of seats is small, but it's influence approaches zero as 
the number of seats approached infinity.  

My original version of non-sequential RRV generalized PAV by using a continuous form of the function

f(n) = 1+1/2+...+1/n


The precise generalization is

f(x) = Integral (with respect to t from 0 to 1) of the integrand [(1-t^x)/(1-t)]

A cruder approximation that still works is

f(x) = 1 + 1/2 +...+1/floor(x+0.5) .

If you use either of these versions of f(x) in place of ln(epsilon+x), the problem disappears.

But note that no matter the value of epsilon as x approaches infinity

the value of f(x)/ln(epsilon+x) approaches unity.

Btw, with regard to your question about homogeneous functions, consider that the expression

     p^2/(p+q) 

is homogeneous of degree one, but is decreasing in q, so not allowable as a ballot in the Ultimate 
Lottery.  

Yet on the simplex given by p+q=1, it has the same value as the expression p^2.  My point is that we 
cannot just specify any old function on the simplex p+q=1.

My Best,

Forest

----- Original Message -----
From: Andy Jennings 
Date: Thursday, May 19, 2011 7:25 am
Subject: Re: [EM] electing a variable number of seats
To: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com

> Isn't Jameson right? In the non-sequential version of RRV, if 
> there are
> only two seats to be awarded and C gets niether of them, then 
> the sum of the
> C voter's grades of the elected candidates is zero, which will 
> contribute a
> huge negative value to the sum of the logs.
> 
> But if C is given one of the two seats, even though only one 
> voter out of
> 100 liked C, then the C voter will have a positive sum and all 
> the A voters
> will have a positive sum, so the sum of the logs will be higher.
> 
> I guess you can try to pick a large enough epsilon so that a 
> small group of
> voters doesn't have veto power. Has a good formula been given 
> for choosing
> the appropriate epsilon? If you're considering the limit as 
> epsilon goes to
> zero, then it seems to be vulnerable to one voter bullet voting.
> 
> Andy
> 
> 
> 
> On Wed, May 18, 2011 at 12:15 PM, wrote:
> 
> >
> > Jameson Quinn wrote ...
> >
> > Wait a minute.... so under non-sequential RRV, there is no 
> "leftover Hare
> > quota" of unrepresented voters? If 99 voters vote A100 B99 and 
> one voter
> > votes C100, then C will be in the 2-member parliament? That 
> seems broken.
> >
> > FWS replies:
> >
> > Your question has the same answer regardless of which version 
> of RRV is
> > used
> > (sequential or non):
> >
> > If there are only two seats, A gets the first and B the second.
> > If there are only two seats and repetition is allowed, A gets 
> both of them.
> > If there are 100 seats with repetitions allowed, then A gets 
> 99 of them and
> > C
> > gets one of them.
> >
> > We allow repetition only if A , B, C, etc represent parties 
> (or if the
> > elected
> > body uses a weighted voting system).
> >
> > So the primary interpretation of "A gets two seats" would be 
> two seats come
> > from
> > the party A.
> > ----
> > Election-Methods mailing list - see http://electorama.com/em 
> for list info
> >
>