A blog for better streets and public spaces in Portland, Maine.

Monday, October 10, 2011

Ranked Choice Game Theory

My most recent column for the Portland Daily Sun analyzes some of the strategic ramifications of the city's new Ranked Choice Voting election for mayor this fall. Read it here.

Unfortunately, the city's other daily paper chose to publish a more error-prone and confused op-ed column a day later. I've talked to a few people over the weekend who expressed some confusion over the topic. The author of the Press Herald's column, an overseas statistician, does make some good points, but he also makes a number of misleading ones. So I'm going to use this space to break it down and add in some corrections and clarifications.

To start with, though, here's the City Charter's 3-step explanation of the Ranked Choice Election:

(1) The ballot shall give voters the option of ranking candidates in order of preference.

(2) If a candidate receives a majority (defined as fifty percent [50%] of the votes plus one or more votes) of first preferences, that candidate is elected.

(3) If no candidate receives a majority of first preferences, an instant runoff re-tabulation shall be performed... In each round, each voter's ballot shall count as a single vote for whichever candidate the voter has ranked highest and has not been eliminated in a prior round. The candidate with the fewest votes after each round in which no candidate receives a majority of votes shall be eliminated and the votes re-tabulated until one candidate receives a majority. [source]
Step 3 is the most important piece. If there's a 15-way election, it's very unlikely that step 2 will come to pass - the front runners might get 15% or 20% of the votes, tops. That means that several rounds of instant runoff re-tabulations will happen. The last-place candidate will be eliminated from the running, and ballots that had picked him or her as their #1 choice will be re-distributed to other candidates according to those ballots' #2 choices.

In each round, only the top choice out of the remaining candidates gets counted from each ballot, and the elimination and redistribution process continues until a candidate emerges with more than 50% of the ballots' top choices. OK, so let's look at Jack O'Brien's Press Herald piece:
The key variable in the balloting process is the "break," or the count that decides the election.

If one candidate obtains a majority in the first count, the break is one.

If it takes five counts -- so that all votes up to the fifth preference are tallied -- then the break is five.
The last sentence is the column's first false statement, and it's a big one. Having five instant runoff rounds does not mean that all votes up to the fifth preference are tallied: a ballot's #5 choice will only be counted if the same ballot's #1 choice, #2 choice, #3 choice, and #4 choice have all been eliminated from consideration in the previous four rounds.

In fact, the statistical probability of someone actually choosing the 5 least popular candidates from a field of 15 is pretty slight. For an election with a break n, it's actually very unlikely that election officials will get around to tallying your nth-choice votes.


Further, the break divides the candidates into the "head" -- those candidates who survived until the break -- and the "tail," those who are eliminated before the break. Only the votes given to candidates in the head ultimately matter.

Votes given to the tail disappear by the time the runoff process has worked itself out. Finally, the break is important because it determines how important depth of support is to the outcome -- it doesn't matter how many sixth-rank votes you have if the break is at five.

The "break" is a useful concept, as are the "head" and "tail." It's absolutely true that 6th-choice votes won't matter at all if the break happens in round five. But, as I mentioned before, it's extremely unlikely that your 5th choice votes will count in that scenario, either.

This might deserve a simulated election before I go further. Taking O'Brien's lead, I'm going to show you an Ranked-Choice election scenario where the "break" is eight (i.e., there are 7 rounds of instant runoff reallocations, and seven candidates get eliminated), and where only 1st- and 2nd-choice votes influence the final outcome.

Here's a first-round result for our hypothetical election. Only the top choices for each ballot get counted, and because all the candidates are in the running at the outset, everyone's top choice is the same as their 1st choice:

Here, eight cellar candidates all roughly get 5% of the vote. The front-runner, Brennan, takes 16%. Obviously there's nobody with 50% of top-choice votes, so no winner in the first round. Bragdon, with roughly 4% of the vote, gets eliminated.

For simplicity's sake let's suppose that everyone who voted for Bragdon picked Brennan as their second choice. All of Bragdon's ballots then get re-allocated to Brennan (because, with Bragdon out, Brennan is the best choice of the remaining candidates for those voters).

Round 2 of the count then looks like this (in the graphs below, black bars will indicate first-choice votes, and red bars will indicate second-choice votes reallocated from other candidates):

Now Brennan's up to a shade over 20%. The next-last place candidate, Jodie Lapchick, now gets eliminated. I promised to finish this in seven rounds and there's a long way to go, so let's suppose that Lapchick's voters also all picked Brennan as their second choice candidate. Round three:

Brennan's up to 25% - halfway there. You get the idea, now: Haadow will get eliminated next, then Carmona, then Bryant. Let's continue to assume that the vast majority of these ballots pick Brennan as their second choice, and skip ahead to round 8, where 7 candidates have been eliminated:

So, there's a hypothetical, but entirely possible, instant runoff election where the break is eight, but only 1st- and 2nd-choice votes get counted before the winner passes the crucial 50% mark.

It does not matter in this scenario whom anyone picked as their 3rd or 4th or 5th choices (or whether those choices got marked at all). It only mattered whom the eliminated candidates' 1st-choice voters picked for their second choice.

Continuing on in O'Brien's column, let's confront his other big fallacy:

The most obvious, and most disconcerting, difference between the old process and the new is how RCV lends itself to tactical voting, where voters deliberately limit their range of preferences to advance a favorite candidate or candidates.

Suppose a resident votes just for her first and second choices, leaving the rest of the ballot blank. That means the voter is actively withholding support for all other candidates, and this makes her vote more valuable than someone who votes for five, 10 or all 15 candidates.

This is not true. And shame on the Press Herald for printing this, because if voters make the mistake of following this strategy, it will neuter their ability to make their values known in a crowded field of candidates.

Refer again to the simulated election I mentioned above. Suppose that Bragdon's (or Lapchick's, or Vail's) voters had all read O'Brien's column and decided to withhold support for any other candidates, based on the idea that their vote would be more "valuable" that way. What would happen once Bragdon was eliminated from the running? The ballot counters, not seeing any other choices marked, would set those ballots aside - effectively treating them as blank ballots.

A blank ballot is not valuable to anyone. Employing this strategy is effectively the same as turning your back from the election out of spite if your top choices get eliminated from the running. We have enough good candidates in this race that nobody should be doing that.

I've heard rumors from various candidates' canvassing efforts that some voters might be employing this strategy - i.e., that some voters seem less willing to mark other candidates as second choices on their ballots, based on the mistaken idea that their ballots will be more "powerful" that way.

Let's run another simulated election to see why that might be hazardous. This time assume a wider spread among 1st-choice rankings. In this scenario, the lower-tier candidates might get 2% or 3% of the vote, and the front-runners get closer to 15% to 20%. Here's the initial ranking:

Again, Bragdon gets eliminated first, but this time, let's assume that Bragdon's voters choose a variety of candidates for their second choices, mostly to Eder, Miller, and Bryant:

And then a third round, when Lapchick's ballots get redistributed to her voters' follow-up choices:

... and so on. By round six, some of the candidates (Bryant and Carmona) who had previously received second-choice votes from others have themselves been eliminated. Those ballots will then get their 3rd choices get counted (3rd choice votes are denoted by the purple bars below), assuming they've marked 3rd choices on their ballots:

We've eliminated more than a third of the candidates and the front-runners still haven't broken the 20% mark. This is pretty realistic for what we can expect in the actual election. Let's skip ahead to round 11:

Here, the efforts of Rathband, Miller, and Marshall to appeal to voters who had picked others for their first choices have paid off. Our hypothetical election is suddenly a 5-way race with Rathband in the lead. But here's the interesting part: Brennan, the initial front-runner, has fallen behind, having failed to pick up many second- or third-choice votes from the eliminated candidates. In fact, Brennan is now in last place. Which means he's eliminated.

Now, if all the people who had picked Brennan as their top choice had picked one of the remaining candidates in any of their 2nd, 3rd, or 11th-choice spots, they could make a huge difference in the rest of the race - this is a block of voters that represents 20% of the electorate, after all.

But if a substantial block of those voters hadn't picked anyone besides Mike Brennan, then those ballots will be taken out to sit on the sidelines for the remaining rounds. Of the remaining candidates in this scenario, Marshall would be eliminated next. The substantial bulk of his voters' follow-up choices are likely to go to Rathband and Miller, putting both of them well on the way to the crucial 50% threshold, and victory.

Now, it's true that the one-vote strategy doesn't explicitly do Brennan as a candidate any harm in this scenario - he got eliminated regardless of what his voters marked down as 2nd and 3rd choices. But a one-vote strategy also wouldn't do Brennan (or his voters) any good.

If you're voting for a candidate because you'd like them to implement specific policies, or to instill specific values in government - and if you'd like other candidates and voters to also care about those policies and values - then it makes no sense to choose only one candidate on the ballot, and sit out the rest of the election in the (not unlikely) event that that candidate gets eliminated in a runoff round.

With this crowded field of candidates, it's a strong possibility that your first-choice candidate - even if he or she is considered a "front runner" - will be eliminated at some point in the process. Whoever eventually wins should be more receptive to implementing your first-choice candidate's ideas if he or she knows that your candidates' runner-up votes provided a substantial boost to putting them past the 50% threshold. Wouldn't you, as a voter, rather see a winner who at least partially shares your first-choice candidate's values, even if your first choice doesn't get to be mayor? Or would you rather just skulk on the sidelines while the runoff process plays out?

One last fallacy in O'Brien's article - it's a false conclusion he draws from the false premise (debunked above) that your 8th-choice vote counts as soon as the runoff reaches the 8th round:

Another conceivably powerful strategy could be named "Eight for Haadoow," after the candidate Hamza Haadoow.

Haadoow likely has enough support among the Somali community to be in the head if the break is fairly high (say, seven or eight). Lacking broad exposure to the city at large, he could succeed by relying on his home constituency to allow him to survive the early rounds and by trying to garner large amounts of weak support by trying to get other voters, who might only be slightly familiar with him but otherwise positively disposed, to rank him as No. 8.

This kind of strategy could also favor someone like Chris Vail, the Portland firefighter and Peaks Island native, who can draw from these two small-but-significant constituencies.

Marketing yourself as the "eighth best" is not - I repeat, IS NOT - a winning strategy. Remember that election officials will only count the top choice on your ballot out of the remaining candidates in each round. Suppose I vote for Mavodones, Brennan, Rathband, and Miller in my top 4 spots, and put Haadow or Vail down at #8. It's extremely unlikely that those four candidates in my top 4 will all be eliminated before Haadow and Vail, and thus it's extremely unlikely that my #8 choice will be counted.

As above, this strategy would only work for Haadow or Vail if they both manage to reach the eighth round AND if the ballots that had ranked them in the #8 spot had all picked losers in spots #1 through #7. Which is highly improbable.

These candidates would be better off spending their time to try to get as many first-choice votes as they can to survive early-round elimination. And come to think of it, that's the strategy that every one of the fifteen candidates should be pursuing.


Rob Richie said...

Great analysis -- and spot on. Professor O'Brien in fact recognizes this too, as he thought Portland was using another form of ranked choice voting for which his observations would have been accurate.

The real "bad guy" here is the Portland Press Herald editor who approved this. It was just shockingly irresponsible to not check in with anyone else about the accuracy of column. It would be directly analogous to printing a commentary by a supporter of Rick Perry in which he/she said as a matter of fact that Mitt Romney had been born in another country and thereby could not be president.

Fortunately, the Herald ran a corrective commentary today.

Anonymous said...

It took you 9 pages to explain this Ranked Voting. Unbelievable.

Only the elite would believe this is an easy to understand.

Jack O'Brien said...

It's clearly long overdue, but I just recently came across this post and felt I should say 'Thanks!'

What you've written is exactly correct - the rules as I spelled them out were clearly wrong and I appreciate you being so thoughtful in responding, and in turn, help to clarify the issue I muddled.

While I appreciate the consideration that Rob is trying to show, the fact is that I wrote the piece and I'm responsible for it. And I'm really appreciative of people who took me to task for my inaccuracies.

Thanks again,

Jack O'Brien