Potential for odd election outcomes with ranked choice voting system, says mathematician

November 7, 2011
Keith Devlin, Stanford mathematician with the Human-Sciences and Technologies Advanced Research Institute, lecturing. Credit: Stanford University

"Instant runoff" voting – which San Franciscans will use next week to choose their new mayor, county sheriff and district attorney – requires voters to rank their three top choices in each race, instead of simply voting for their first choice.

Advocates say instant runoff does a better than job of producing a winner who truly represents most of the voters' preference; opponents say the opposite. It all boils down to what you think of the math, which raises an obvious question: What does a mathematician think?

We posed the question to Stanford mathematician (and NPR Math Guy) Keith Devlin.

"There is no perfect voting method," said Devlin. "A famous 1950 result by the Nobel Prize–winning mathematician Kenneth Arrow, who spent much of his career at Stanford, tells us that there is no voting system that meets all basic democratic requirements.

"One advantage of the system we use in our national elections, the 'first past the post' method, is that it is easy to understand. Everyone votes for one candidate and the one with the most votes wins. But ease of understanding is about the only thing in its favor, and all the experts agree it is just about the worst possible system.

"For one thing, in a field with at least three candidates, split votes between two top candidates can result in a third person being elected who is disliked by a large majority of the electorate. Another problem is more social than numerical, in that it forces candidates into opposing camps, slugging it out in a verbal prizefight.

"That may have a lot to do with the current dysfunction in Washington," Devlin suggests, "where no one is willing to compromise or make deals."

The method being used in San Francisco is a variant of the "instant runoff" process, also known as "ranked choice" voting. Here, voters select their top three choices out of the field of candidates and rank them in order of preference. In the mayoral race this year, that means ranking three out of 16 contenders.

Tallying the votes

All the first choice votes are counted and then an iterative tallying process begins.

First, the contender with the lowest number of first-choice votes is dropped from the competition. Each voter who had ranked that candidate as their number one choice then has their given to whichever candidate they selected as their second choice. The votes are re-tallied and, as before, the contender with the lowest vote total is eliminated.

This process continues for as many rounds as needed until one candidate has over 50 percent of the votes tallied in a round, at which point he or she is declared the winner.

Costs vs. benefits

Advocates point out that this method can save costs. With the one-vote approach previously used in San Francisco, if no candidate received over 50 percent of the votes in the , a special runoff election had to be held, entailing the expense of printing new ballots and paying poll workers. With the instant runoff approach, a second election is not needed.

Opponents of the method point out that voters whose choices are repeatedly eliminated effectively get to vote several times, and moreover the process gives equal value to a person's third-place ranking of a candidate and someone else's top-choice vote.

But there are other problems, Devlin points out. "For example, with ranked choice voting, you can get a winner who is the first choice of only a relatively small minority of the voters.

"Undesirable outcomes such as this can arise," he explains, "because the candidates are eliminated and their votes reassigned one after another, and the order in which that happens can make a huge difference.

"A shift of a large block of votes in an early round can eliminate a candidate who would have gone on to win had she survived until a later round and then picked up more votes to boost her tally."

In math, if it can happen, it will

Devlin's example is not a just theoretical possibility. In the 2010 race for supervisor in San Francisco's District 10, the eventual winner received just 11.8 percent of the first-place votes, ultimately edging out the candidate who had the most first-place votes, according to reports in the San Francisco Chronicle.

There is also Jean Quan's win over Don Perata last year for mayor of Oakland. Quan had only 24.4 percent of first-choice votes; Perata had 33.7 percent, the Chronicle reported.

The known vagaries of the voting method have resulted in some candidates trying new approaches to getting votes. For example, mayoral candidate Michela Alioto-Pier sent out a mailing urging voters to "please consider at least making Michela your number 2 choice for Mayor."

A mailing from the San Francisco Republican Party suggested two mayoral "candidates to avoid."

Devlin thinks that as candidates become more aware of the election math and the possibilities it opens up, more of these tactics are likely to be seen. "Particularly in the era of social media, where it is possible for large numbers of voters to coordinate their actions," he adds.

Is this a misuse of election math? Devlin does not think so. "Arrow's theorem tells us that the only mathematically ideal system is a dictatorship, and no American wants that."

It isn't rocket science

"So voting is not like physics or engineering, where we have to do what the math tells us. Rather, it is one of those cases where we can make the math work for us – to use it to achieve our own ends as a society. The voters will make the selection, but the math we choose can shape the kind of government we get. Do we want politics to be about partisanship and fighting, where half the electorate will always end up as losers and we just keep seesawing between the two, or do we encourage cooperation and compromise, where no one gets everything but everyone gets something?" Devlin says.

Devlin says that ranked choice voting now in use in San Francisco almost certainly encourages coalition building and reduces negative campaigning. "The question is, do you think that is a good thing? I have my opinion, but there I am being a citizen, not a ."

Explore further: Alternative Vote could lock out extremist candidates, says UK researcher

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not rated yet Nov 07, 2011
""There is no perfect voting method," said Devlin."

The reason for this is that voters need to understand the issues to make correct choices, the reasoning behind who, what and why you are voting for one and not the other, and this involves studying, weeding through sound bites, coming to an intimate understanding of issues and the repercussions involved.

Very few people do this, voting for politicians for most people is simply "cult of personality" or "I voted for him because he has a nice smile".
So with that in mind you can understand why:

""There is no perfect voting method," said Devlin."

Voting isn't "broken" the people making the decisions are.
not rated yet Nov 07, 2011
@f33dback Um, no, that's not it. The reason there is no perfect system (even in an ideal world of perfectly informed voters) has to do with the Arrow Impossibility Theorem. It's interesting stuff, look it up.
5 / 5 (1) Nov 07, 2011
The democratic superiority of these kinds of systems is that each voter can participate in the final decision between the last two candidates, even if only with what is a lesser-of-two-evils choice, while still being able to record what kind of candidate they really prefer. This both enhances the legitimacy of the result and gives fringe opinions an appropriate venue to be tested.

This is not to say that top-remaining-candidate ranked voting in single-seat constituencies is a great method (in particular, it often inappropriately eliminates the centrist in a close three-way race). Methods that take some account of lower-ranked candidates do better, and those that use multi-seat constituencies can do better still. With some adjustments, this style of election could work very well.

Arrow's analysis shows that it is not possible to be completely fair to the candidates. But they are not the point. We should be more concerned with making elections play a positive role in our society.
3 / 5 (2) Nov 07, 2011
Democrat was anathema to the Founding Fathers. Democrat is anathema to liberty and freedom.
5 / 5 (1) Nov 07, 2011
"For example, with ranked choice voting, you can get a winner who is the first choice of only a relatively small minority of the voters.

Yes, but your 'second choice candidate' can win *ONLY IF* your first choice *COULD NOT POSSIBLY* win under *any* electoral system.

In such a 'strange' scenario, under the current system, a candidate which was the *LAST* choice of the majority would likely win. That's screwy.

Instant run-off (aka STV) means more democratic representation. If more people like Person A than Person B, you're going to have Person A elected before Person B. ALWAYS.

That is not true in First Past the Post. In FPP, you often have to vote strategically (vote for Candidate A, even though you really want to vote for Candidate B, because you don't want Candidate C to win.

Under FPP, you *ensure* divisive politics. Under more representational voting, you eliminate the "you're either with us or against us" mentality that is actually true to a large degree under FPP.

not rated yet Nov 07, 2011
John Cleese provides an excellent primmer on Ranked voting (aka Proportional Representation or Single Transferrable Voting):


It's both entertaining and informative. (Note: It applies to voting in members of the British Parliament, but it applies equally to any election for any body).
not rated yet Nov 07, 2011
@temple, I think the way you wrote your opinion was confusing. However, after reading it through several times, I believe I agree with what you're trying to say.
not rated yet Nov 07, 2011
@f33dback Um, no, that's not it. The reason there is no perfect system (even in an ideal world of perfectly informed voters) has to do with the Arrow Impossibility Theorem. It's interesting stuff, look it up.

You're wrong, human stupidity is exactly the reason why there is no perfect (read useful) voting system, because stupidity can't be compensated for in this system, there is no such thing as "perfect" when dealing with humans, but there is useful.
So if you want to throw theorems out there remember what you're dealing with, human stupidity.
But hey Einstein said it better.
"Two things are infinite: the universe and human stupidity; and I'm not sure about the the universe."

If people would take the time to educate themselves the current system would be fine, all this article really talks about is watering down an already damaged (by human stupidity) system, no other "theorems" needed.
1 / 5 (1) Nov 07, 2011
These mathematician need a law class. The US isn't a democracy it's a Constitutional Republic. The Founders considered a democracy the most tyrannical form of government.

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