[EM] electing a variable number of seats

[fsimmons at pcc.edu]

Raph Frank  wrote ...
________________________________________
On Fri, Feb 18, 2011 at 7:54 AM, Juho Laatu <juho4880 at yahoo.co.uk> wrote:
> If you want to keep this property, the approach proposed by Michael Rouse could determine
> the number of board members. If most votes go to few candidates, then there would be 5 members
> (with different weight). If the votes are more distributed, then all candidates (up to 9 candidates)
> that get support over some agreed limit would be elected. Alternatively one could use the number
> of unrepresented votes as the criterion on how many members to elect. This approach would
> improve proportionality and keep the size of the board small at the same time.

You could still use PR-STV to give a proportional result.

There is a formula which defines the "effective number of parties".
It is also used in economics to define how many firms there are in a
market.

The formula is

1/sum((vote share squared))

So, if the first choice totals were

A: 20%
B: 30%
C: 15%
D: 12%
E:  18%
F: 5%

The result gives:

1/(0.2*0.2 + 0.3*0.3 + 0.15*0.15 + 0.12*0.12 + 0.18*0.18 + 0.05*0.05) = 4.96

This says that there are around 5 groups in the vote, which is about right.

If the voters were less concentrated, you get a larger number

A: 8%
B: 12%
C: 7%
D: 14%
E: 6%
F: 10%
G: 9%
H: 11%
I: 4%
K: 19%

would give 8.56

The rule could be that you use that formula using the first choice
votes and round to the nearest whole number between 5 and 9.

Also, a property of the formula is that if the votes are exactly
evenly distributed, then the number will be equal to the number of
candidates.  For example, if there were 8 candidates and each got
exactly 1/8 of the vote, then the number of seats would be equal to 8.

You can then use standard PR-STV with that number as the seats target.

Forest replies:

Or, once you have decided on the number n of seats. you could complete the election with Asset 
Voting: the candidates get together and negotiate the redistribution of votes until n candidates have the 
quota of 100%/(n+1) or more of the votes. If (n+1) candidates tie with exactly this quota, then elect all of 
the tied candidates.

This way you wouldn't have to have each voter rank the candidates.