[Election-Methods] Approval based equal ranks in PR-STV and Meek's method

[raphfrk at netscape.net]

 Continuing on from the previous post.

The first issue is deciding how to share votes if you have more than one elected candidate at the same rank.

For example, assume the vote is

A=B>C

and w(a) = 0.6 and w(b) = 0.3

How many votes should each candidate get.

One option is

A = 0.6
B = 0.3
C = 0.1 (remainder)

This has problems if w(a)+w(b) was greater than 1.? Also, if the vote was

A>B>C

A = 0.6
B = 0.12
C = 0.28

or

B>A>C

A = 0.42
B = 0.3
C = 0.28

then the vote given to C would be larger (0.28 in both cases)

This doesn't seem fair.? Both of these three voters should only have to 'spend' the same amount in total to support A and B, rather than A=B requiring 0.9, but A>B or B>A only requiring 0.72.? 

A fairer way to do it would be

A=B>C

C = 0.28 (so as to match above)

Then A and B should get a share proportional to their keep values

A gets 0.72*(0.6)/(0.6+0.3) = 0.48
B gets 0.72*(0.3)/(0.6+0.3) = 0.24


More generally, assuming;

A and B are elected
w(x) = keep factor for candidate x

In, A>B>C, C gets (1-w(a))*(1-w(b))

Thus, in, A=B>C, C should still get (1-w(a))*(1-w(b))

The remainder after passing this amount, to the next rank or hopefuls at this rank is shared between all elected at this rank, in proportion to their keep values.

w(x) = keep factor for candidate x
V(x) = vote fraction received by x
F(x) = fraction received by elected candidate x
E(rank) = excess after all elected candidates have received vote share
P(rank) = vote share passed to next rank

E(rank) = P(rank-1)*(1-w(a))*(1-w(b))*...*(1-w(z))? -> including all elected candidates at that rank

The remainder is then shared between all the elected.

F(a) = w(a) / (w(a)+w(b)+w(c)+.....)? -> including all elected candidates at that rank

V(a) = F(a)*(P(rank-1)-E(rank)) -> for all elected candidates at that rank

Each hopeful at that rank gets E(rank)

If there are no hopefuls then P(rank) = E(rank),
Otherwise, it is zero.

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This treats A=B>C and A>B>C the same as far as votes for candidate C are concerned.

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This doesn't solve the problem with de-electing candidates.? I was thinking of giving each candidate 2 vote totals.

The first vote total is as above and is used to determine who gets eliminated.

However, a second total is also calculated.? This is a worst-case scenario situation for each candidate.? 

It calculates the lowest possible vote total for each candidate assuming,
the candidate was elected
all current candidates were elected
all remaining seats were filled (by some other hopeful candidates)

It is this total that must be brought above the quota before the candidate can be deemed elected.

Doing a full search on all possible combinations of hopefuls to elect, could take a while.? However, it could be calculated on a ballot by ballot basis and then it would be very quick.? The only rank where it would matter would be the last rank to be considered (i.e. the highest rank with at least one hopeful).

This would result in a total which is lower than the actual value.? However, if this value was above the quota, then it would be certain that that candidate was able to meet the quota no matter which other candidates were deemed elected.? This value would also be used when updating? the keep values so as to ensure it stays above the quota for all candidates, once they have been deemed elected.

Also, as more and more candidates were elected and eliminated, there will be fewer ranks which are a mix of hopeful and elected.? This would bring the elected total and the elimination total closer together.

Since it only affects elimination order, this still maintains proportionality for solid-coalitions.? Also, a coalition with 1 hopeful left and more than a quota of supporters and 1 candidate left will be guaranteed to have that candidate above a quota as he will be the first choice of all those voters.


 


Raphfrk
--------------------
Interesting site
"what if anyone could modify the laws"

www.wikocracy.com

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