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Cumulative Transferable Voting, round 3

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  • dodecatheon.meadia
    Bruce Gilson has memorialized my last iteration and given the method a snappier name, Cumulative Transferable Voting. ... However, I ran into a problem (see
    Message 1 of 1 , Jun 30, 2010
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      Bruce Gilson has memorialized my last iteration and given the method a
      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.

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