It's rank, all right – Part 6 (of 6, I promise)

I've finally finished learning (to the extent I can) the in's and out's of the "rank choice" voting system that Portland is about to use to elect the City Council under the new charter. And with that, I can run out to their conclusion the hypothetical election results that I started counting last week.

Here's the full tale of the vote-counting. For those of you who started this with me in my previous post, I've marked with asterisks below where the new stuff starts. If you remember everything previously covered (God bless you), you can skip down to those asterisks to pick up where we left off. Everybody else, pull up a chair as we embark on the odyssey from the start.

Let's say in a race for three City Council seats, there are seven candidates. (In the real races there are three or four times that many contenders, but I think I can figure everything out with only seven.) The first-choice votes are counted, and they are as follows:

First choice tally.

Because there were 1,000 first-choice votes cast, the number needed to win a seat is 251 – 25 percent plus one vote. That target never changes throughout the painful process we are about to undertake: 251 votes is needed to win a seat. That number was chosen by the charter children because no more than three people can get to 251. This is quite different from how a winner is chosen in the single-winner mayor's race, which I've explained here.

And as you can see, CA got more than that, and wins one of the three seats, right off the bat. But the other two seats are still in play.

The next thing that happens is that because CA had "surplus" votes – more than the 251 it took to win – the second-choice votes of CA fans (that is, the 260 voters who chose CA first) are now going to count in favor of CB, CC, CD, CE, CF, and CG. But each of those second-choice transfers is going to be scaled back by applying a fraction that shows how much greater than the 251-vote threshold CA's first-choice total was.

The magic fraction that is going to tell us how much strength CA fans' second-choice votes should be given at this point is 9/260. That's because CA's first-choice vote total was only 9 votes greater than the 251 threshold, and CA got 260 votes altogether.

And so now we're going to take each CA fan's second-choice vote, multiply it by 9/260, and hand that fraction of a vote over to whoever that CA fan ranked in second place. If when that addition is done, any of the remaining six "active" candidates gets to 251, they win a seat.

But it turns out, on our hypothetical second-choice ranking, none of candidates CB through CG make it to victory. Here's the tally after the CA "surplus" business is taken into account:

Result after allocation of CA surplus.

At this point, I believe the official terminology is that Round 1 is over. You might note that not all the CA fans ranked a second choice. Only 235 out of 260 did. That's to be expected.

Okay then, now it's on to the next step, which is to eliminate the lowest vote-getter, CG. Sorry, CG fans, you lose. But you still count. We are now going to take out the ballots of all of the people who chose CG first (the CG fans), and see what their second choice was. If that was CA, we'll go on to their third choice. And once we have identified this next choice of each CG fan, we will add it to the total of CB, CC, CD, CE, or CF.

But we're not done yet with this round. The other thing we have to do is get out the ballots of the 17 CA fans whose second choice was CG. We are going to identify the next choice (if any) on each of those ballots and add them, too, to the totals of CB, CC, CD, CE, and CF – but as before, only at their surplus fraction, which on these facts is small because CA came in only slightly above the threshold.

After we do this, we ask whether any of CB, CC, CD, CE, or CF have reached 251. If so, they win the second of the three council seats (or the second and third). But on our facts, it turns out that the votes coming out of the CG elimination don't push anyone over the top. CB and CC are still close to, but not actually, elected:

Result after elimination of CG and transfer of CG's votes.

I think the official jargon is that that's the end of Round 2. Notice, all the candidates got to keep the votes they picked up in the CA surplus. They will get to keep them through all the rounds until the candidate is either elected or eliminated. And the same goes for the votes they just picked up from the CG fans.

Okay, it's on to the next step, which is to eliminate CF. Sorry, CF fans, your favorite candidate, CF, will have to stay at their day job, or in their tent on the sidewalk, as the case may be. We are now going to take out the ballots of all of the people who chose CF first (the CF fans), and see what their second choice was. If that was CA or CG, we'll go on to their next choice. And once we have identified these next choices (if any) of the CF fans, we will add them to the totals of CB, CC, CD, and CE.

And we're not done yet with this round. The other thing we have to do is get out the ballots of the 8 CA fans whose second choice was CF. We are going to identify the next choice (if any) on each of those ballots and add them, too, to the totals of CB, CC, CD, and CE – but as before, only at their surplus fraction, which again, is small because CA got only slightly above the threshold.

No CG fans' votes were transferred to CF in the last round, and so we don't need to worry about that at this stage.

Okay, let's say we've done everything just described, and we get this (click on the image for a bigger version):

Result after elimination of CF and transfer of CF's votes.

Nobody's reached 251, and so we've got to do another round. I believe what we just did gets us to the end of Round 3. 

* * * * * * * * * *

In Round 3, we're going to be eliminating CE, and transferring CE's votes to the three remaining contestants for the two remaining seats: CB, CC, and CD. But by "CE's votes," I mean all the votes that counted for CE – not just the first-choice votes of the CE fans, but also the votes CE picked up from CA, CF, and CG.

With the votes CE got from CF and CG, we simply pull those ballots out again, see what the voter's next choice was after CE, but only votes for CB, CC, and CD count. If the next choice is CA, CF, or CG, we take the next choice, if any. Each vote counts as 1.

With the votes CE got from the CA surplus, we pull out the ballots again, and take the next choice among the remaining three candidates, but as always, we multiply those transferred votes by CA's surplus fraction, 9/261, which sucks a lot of the life out of them. 

Okay then, let's do all that. It gives us this (click on the image for a bigger version):

Result after elimination of CE and transfer of CE's votes.

Wow, important development there. The CE fans' second choices were overwhelmingly for CD, and that pushed CD over the top. CD is elected! But neither CB or CC has reached 251, and so we need to move on to the next step.

Before we go there, though, notice that two of the three seats have been filled, and we still haven't looked at any of the second or later choices of the CB fans or the CC fans. The CE fans just pushed CD ahead of the CB fans' and CC fans' first choice. What if all the CB fans had CC as their second choice, and all the CC fans had CB as their second choice? Too bad; they're irrelevant.

All right, on to the next step, which in this scenario is going to be the last. It's time to transfer surplus votes from CD to CB and CC. But just as we did with CA's surplus way back when, the later-choice votes of the CD fans are going to be multiplied by a fraction, which in this case is going to be teeny-weeny. CD is only 1.0895 votes over the threshold, and so CD's surplus fraction is only .0072. For this reason, CD's surplus is unlikely to make much of a difference.

But you never know; here CB and CC are close. And so let's go ahead and perform the CD surplus transfer. 

Now, here's where it gets even messier than ever. The votes that CD got from the CA surplus have to be multiplied by two fractions: first by CA's surplus fraction, then by CD's surplus fraction. The other votes for CD, in the most recent round, are multiplied only by CD's surplus fraction.

So what are we looking at? We start with the CD fans' next choice as between CB and CC; those get multiplied by .0072 before being added to the CB and CC totals. Then we look at the next choice of all the other voters whose votes counted for CD in the round in which CD won – except those CD got from the CA fans. These, too get multiplied by CD's surplus fraction, .0072 before being added to the CB and CC totals.

We then pull out the ballots of the CA fans that got transferred to CD along the way. We look for those voters' next choice as between CB and CC. But those votes get multiplied by two fractions before being added to the CB and CC totals: CD's fraction, .0072, and CA's fraction, .0346. It turns out that those doubly surplus votes count at a value of only .0002 of a vote each. 

My head is swimming, but because those fractions get so minuscule, I'm confident that I'm close enough for government work. Here's what I get for the final showdown between CB and CC (and it's so long a process, you'll have to click on the image to see the whole thing):

Result after allocation of CD surplus.

The key column is all the way over on the right:

Yikes! There's nobody else to eliminate, and neither CB nor CC has made it to the election threshold of 251. Do we have a runoff? 

No. In this case, CC, as the lower vote-getter, is defeated. CB wins the third seat with less than 24 percent of the votes cast.

None of the second or later choices of the CB fans or CC fans counted for anything. They were not even looked at.

* * * * * * * * * *

There is so much not to like in this "system" that I'm going to collect my thoughts and put my critique in a separate post. But that's how you get to the final result in a seven-person race. In the real Portland, some of the races have 30 candidates in them, but the counting will likely just be a much longer version of this ordeal. For a computer, it's no sweat. But if a recount is required, Katy bar the door.

Readers, if you see mistakes on my part in this description of the process, I'm all ears. But I've done my best to get it right.