snappier name, Cumulative Transferable Voting.

On 24 Jun 2010 18:42:41 -0700, Bruce R. Gilson wrote:

> ...

>

> http://important-information.net78.net/vote/multiple/ctv/

>

However, I ran into a problem (see link on that page) which made me

realize that a summable Cumulative Transferable Vote would be neither

feasible nor satisfactory to the voter.

So I've come up with a new variation. It's a non-summable version and

has rounds similar to those in STV.

In a multi-winner election, a voter would like to just vote for

candidates and not have to worry about whether they were giving too

much weight to one candidate, or whether they have used all their

votes. They would like their vote for popular candidates to be

reduced to exactly what they need to be elected (and no more) and let

the rest of their vote be distributed among their other choices

proportionally. If some of their choices are eliminated, their

remaining votes should be redistributed as if they hadn't wasted them

on the losing candidates.

Voters use a range ballot and cast scores for any number of candidates

using a range of 0 through 10. The total weight of a ballot is 1, so

the more candidates you vote for, the more your individual vote is

spread out among all your choices.

For example, say I vote A:10, B:10, C:6, D:4, E:1, F:1

The sum of all scores is 10 + 10 + 6 + 4 + 1 + 1 = 32

In general, let's denote k non-normalized scores by alpha_i, for i

in the range 0 to k-1, and the sum of all alphas as S.

The normalized vote distribution for my example ballot is

A:10/32, B:10/32, C:6/32, D:4/32, E:1/32, F:1/32

In general, let's denote these k normalized scores by beta_i, for i in

the range 0 to k-1.

So far, so good. It's like score voting but with fractional votes.

Here's the trick:

Let's say that when all the votes are tallied, A has more votes than

required for the quota, therefore we should multiply the normalized A

votes on all ballots by a coefficient gamma, equal to the quota

divided by A-total, to reduce A's original total to exactly the amount

necessary to meet the quota.

Then we should proportionally increase the normalized votes given to

the other candidates.

However, if we decrease A's NON-normalized vote by a certain amount X,

the ballot's score sum is reduced by that same amount, which would

automatically increase the normalized votes for the other

(non-eliminated) candidates in exactly the distribution that the voter

would desire. So let's solve for that value:

What is the amount X such that

(alpha_j - x) / (S - x) = gamma * (alpha_j / S)

to reduce the score of candidate j? Solving for x, we get

x = alpha_j * (1 - gamma) / (1 - (gamma * alpha_j)/S)

Substituting alpha_j/S = beta_j,

x = alpha_j * (1 - gamma) / (1 - gamma*beta_j)

This formula is very convenient, because all we need to do for a

quota-reduction or elimination round recount is, for each ballot,

(a) reduce the original non-normalized score of one candidate by x

(b) recalculate the ballot's normalized scores

(c) tally the ballot's new scores into the current round's totals

If candidate j is eliminated completely, that means gamma is zero, and

x is equal to alpha_j, just as you'd expect.

Continue as one does for STV: reduce scores of candidates who have

been seated and redistributed, until none meet the quota, then

eliminate candidates and redistribute until another candidate meets

the quota or the number of non-eliminated candidates equals the number

of seats not yet filled.

A caveat and a corollary:

1) Vote transfer can be applied ONLY on non-bullet-vote ballots, which

means we need to store information about bullet voting and

calculate gamma a little more carefully in practice.

2) On a ballot with redistribution, you can recalculate the new

normalized scores easily as

beta_i := beta_i * S / (S - x)

for all candidates i other than candidate j whose score is

reduced by gamma.

Ted

--

Frango ut patefaciam -- I break so that I may reveal