[EM] Question on RCV/IRV multi-seat method used in Minneapolis

[Raph Frank]

On Tue, Sep 23, 2008 at 5:59 PM, Kathy Dopp <kathy.dopp at gmail.com> wrote:
> Ralph,
>
> Thank you.
>
> Wow. I just realized - What an absolute arithmetic complicated mess or
> arbitrarily unfair system this will be whenever the third choice votes
> of voters must be transferred in this same split-vote manner.

It should be OK as long as the random selection is actually reasonably random.

For example, say you vote

A>B>C

A gets 11000 votes (1000 surplus)
B gets 9500

your vote then has a 1/11 chance of being passed on to your second
choice.  However, even if your ballot isn't picked, other ballots with
the same 2nd choice will be picked, so that the proportions are
preserved.

Assuming that B gets 600 of those ballots and is thus elected with a
100 surplus.  Only the 600 ballots that were transferred to B are
considered, the 9500 that B held are ignored.  Your ballot would have
a 100 in 600 chance of being picked (assuming it was picked in the
first stage).

If fractional votes are used, your vote would be at full strength for
the first count, 1/11 for the 2nd count and 1/66 for the 3rd count, so
even if there is a little noise in the 3rd count, it isn't as
important.

Ofc, some (including me) would argue that all 9500 + 600 ballots
should be considered for the 2nd transfer, in which case, your ballot
would have approx 0.9% of a vote.

In any case, the random selection after splitting into piles greatly
reduces the effect of the randomness.

> Has anyone described the mathematical formulas for transferring excess
> votes above the threshold amounts by using a mathematical system that
> accurately reflects all voters rankings or has this only been done by
> using the arbitrary unfair (inaccurate) random selection of ballots
> method when it comes to using voters' third or lower rankings in these
> multi-seat elections?

Yes, Meek's method calculates it exactly (or at least to, say, 1
millionth of a vote).

Each candidate is given a 'keep value'.  For each ballot a candidate
keeps that proportion of the vote and passes the rest to the next
highest.

All eliminated candidates have a keep value of 0 and all elected
candidates have a keep value between 0 and 1.  All unelected but still
uneliminited candidates have a keep value of 1.

So, for example, if A was elected and C was eliminated, the keep values might be

A: 0.7
B: 1
C: 0

If you voted A>C>B>D, then your vote would be split

A is first and keeps 0.7 of a vote, and 0.3 passed to C.
C keeps none (as his keep value is 0) and passes the 0.3 to B
B is next and keeps all of the 0.3 (as keep 1) and thus passes nothing to D
D (and lower) get nothing as B keep everything remaining

There is a process for working out the keep values that has a PR-STV
effect.  The keep values are set so that each elected candidate ends
up with exactly a quota.