[EM] The use of ranked ballots for proportional representation.

[Abd ul-Rahman Lomax]

This is an exercise in examining the rationale behind the use of 
ranked ballots for proportional representation. It is elementary for 
some readers, I know. But I think it may be worth going through the process.

If there are V voters and S seats to be elected, it would be 
convenient if the voters nicely arrayed themselves such that each of 
S candidates is the first choice of V/S voters. Unfortunately, it 
does not work that way. One serious way that it does not work is that 
a candidate may be the first choice of *more* than V/S voters.

If a candidate is elected with these excess votes, what happens to 
the excess votes? Are they wasted? Or are they used in some other way?

In Asset Voting, excess votes may be recast by the candidate 
receiving them. However, we are here considering a method which uses 
only ranked ballots to distribute excess votes.

It is clear that any candidate receiving the quota (V/S) of 
first-place votes is properly elected. There can be no more than S 
such candidates.

Now, if we simply recounted the ballots with the immediate 
first-place winners eliminated, an inequity would arise. Those whose 
first-place votes were used to elect the initial winners would be, as 
it were, voting again, while those whose first-place choices were not 
so selected would only be voting once.

Now, if somehow we knew which votes were the ones that brought an 
initial first-place winner to the quota, we could then discard those 
ballots and all the votes that came from them. But the election 
outcome could depend on which votes were first and which came later; 
it seems undesirable that an election outcome would shift depending 
on what seems irrelevant: voting counting sequence.

If, when a candidate is elected, in this first round, all ballots 
electing that candidate remain, but are devalued proportionally, we 
have the initial counting phases of STV, with fractional transfer. 
That is, if a candidate is elected with a total first-place vote 
count of T votes, when Q = V/S is the quota, the total set of ballots 
with that first-place choice is now treated as if it were (T - Q) 
ballots. The votes in each position are multiplied by (T - Q)/T.

If, as a simple example, twice the quota of voters voted for a 
candidate, that candidate would be elected. From all those ballots, 
the second-place vote totals would remain in the next round, divided by two.

Now, a point may come in this process where there are one or more 
remaining candidates to be elected, and none of them have the quota 
in the reduced ballots. It is clear that all the candidates elected 
prior to this point are legitimate winners: they all represent, as 
well as possible, a quota of the voters. It is at this point that 
Condorcet methods may become relevant, for the selection of the 
remaining winners. And further than this I'm not going tonight....