Summable PAD voting

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  • Number of ballots which "prefer" each candidate X, denoted P(X).
  • For each pair of candidates X and Y, total of ballot fractions which count as preferring X "first" and which also prefer Y; denoted PP(X,Y). (See below for example).
  • Of the ballots counted as first "prefering" each candidate X, number of candidates which "approve" each candidate Y, denoted PA(X,Y). Ballots which prefer no candidate should be included in these tallies as PA(?,Y).

Say that on my ballot I preferred X, Y, and Z; approved of A and B; marked "don't know" for C and D; and disapproved E and F. Say that 2/3 of XYZ pre-approved C, while a different 2/3 of them pre-disapproved D; so that I count as approving C and disapproving D. Thus, my ballot would add to the following tallies:

  • P(X), P(Y), P(Z): +1
  • PA(X,A), PA(X,B), PA(X,C): +1/3 (since only 1/3 of my ballot is counted for X "first".)
  • PP(X,Y), PP(X,Z): +1/3 ("first for X, but also prefers Y and Z")
  • PA(Y,A), PA(Y,B), PA(Y,C), PA(Z,A), PA(Z,B), PA(Z,C), PP(Y,X), PP(Y,Z), PP(Z,Y), PP(Z,X): +1/3 (as above, but with Y or Z first)

Note that if a ballot only prefers one candidate, it only adds 1 or 0 to each tally, with no fractions.

Given such tallies, it is easy to approximate the above algorithm. For instance, if P(X) is over one quota Q, and PP(Y,X) is zero for all Y?X, then when electing X you should multiply each of P(X), PP(X,Y) for every Y, and PA(X,Y) for every Y, by the ratio (P(X)-Q)/P(X); this has the effect of reducing P(X) by Q, almost as if you'd eliminated Q physical ballots which prefer X.