2012-10-23

Candidate strategies with Instant Runoff Voting

In a recent email to her supporters, San Francisco District 5 supervisor candidate Hope Johnson outlines a strategy for the upcoming Instant Runoff Voting election.
Johnson and Resignato are launching The People's Ticket to give voters a better choice than the big-money downtown and progressive machine candidates who are constantly at odds with each other.  They are asking D5 constituents to vote for them in either the first and second or first and third positions, in any order, on November 6’s ranked choice ballot.
A friend asks, "Does this make sense, from a strategic voting perspective?" The answer is, it's complicated. Say you have these hypothetical preferences:
35% X > Johnson > Resignato
32% Johnson > X > Resignato
33% Resignato > X > Johnson
Johnson has only 32% of the first-place rankings, so she's the first eliminated, which causes 32% of the votes to be transferred to X in the next round. X then trounces Resignato 67% to 33%.

But say Johnson and Resignato successfully convince their supporters to adopt the strategy. Then it would look like this:
35% X > Johnson > Resignato
32% Johnson > Resignato > X
33% Resignato > Johnson > X
Johnson is still eliminated first (indeed, this strategy cannot possibly affect the election prior to the elimination of the first of the two colluders). But now Resignato trounces X 65% to 35%. So the strategy worked.

But things aren't so simple! The strategy helped one of the candidates, but it also caused 32% of voters to get their 3rd choice instead of their 2nd. So from their point of view, it was the opposite of strategic. While the Johnson supporters certainly want the Resignato supporters to help Johnson, they don't want to help Resignato, particularly within the privacy of the voting booth. And vice versa. This is the classic prisoners' dilemma.

History has disproved the notion that voters care more about loyalty to their favorite candidates than about getting what's best for themselves. Consider that exit polls in the 2000 U.S. presidential election showed that 90% of Nader-favoring voters claimed to have voted for someone other than Nader. Obviously a Nader supporter who preferred Gore to Bush was maximizing his expected value by voting for Gore instead of Nader. Likewise, we expect that Johnson and Resignato would have a hard time convincing people to vote against their own best interests.

There's also some interesting game theory related to which partner you should pick, if engaging in this strategy. Notice that if X were to team up with Johnson in the last example, that would lift X one ranking up (above Resignato) on 32% of the ballots. The same deal with Resignato would create a one-rank lift on 33% of ballots, but above Johnson. Which strategy makes sense? In this case, it doesn't matter, because X loses head-to-head against any rival. But in general, the strategy is to team up with someone whom you think will be ranked higher than your greatest rival, by a lot of voters. But you also want to pick someone whom you believe will be eliminated before you, otherwise you don't benefit from the strategy.

I'll leave the more complicated analysis to my math Ph.D. associates at The Center for Election Science.

6 comments:

Nathan Wisman said...

Wow. Some of the outcomes you mention seem so counter-intuitive, and it seems like that last thing you want your voting system to be is counter-intuitive. How does IRV rank in voter satisfaction compared to other systems? It seems like strange outcomes like these would really put a dent in satisfaction...especially when a large group shows up to the polls and executes a strategy like this one...only to see it fail massively.

Clay Shentrup said...

Nathan,

Measuring voter satisfaction is a complex business, largely because we can't read voters' minds and precisely quantify how satisfied a particular outcome made them. And election data is, from a statistical point of view, not very abundant.

So the best method we have is computer election simulations, in which we can precisely read (and even rewrite) voters' preferences. We can even average the performance of various systems over a large number of randomized elections, to get incredible statistical confidence in the results. What we're left with is a metric called "Bayesian regret". You can read more about it here:
ScoreVoting.net/BayRegsFig.html
ScoreVoting.net/BayRegDum.html

As you can see in the graph on the first page, IRV does pretty poorly. It still comes out way ahead of ordinary vote-for-one Plurality Voting, but it's important to remember that San Francisco didn't switch to IRV from Plurality. We had Top-Two Runoff, which actually has comparable performance to IRV, and is a lot simpler. Here's an even more detailed analysis which includes figures for TTR.
ScoreVoting.net/StratHonMix.html

Thanks for your comment.

AuntTom said...

I don't like the effects of instant runoff voting. Candidates win with far less than 50% of the voters. Not good! I think there should be real run-offs when no one candidate gets more than 50%. If we've got to settle for representative democracy, then let's have it be at least sort of representative! The district supervisor idea was meant to get us free from the downtown interests but chopping up the city into districts that only small numbers of citizens get to vote for is not a good solution. And the effect is even worse with instant runoff voting.

Clay Shentrup said...

AuntTom,

To be fair, there's a legitimate question of what it even means for a candidate to win with "50% of the voters". Consider these hypothetical voter preferences for example:

% of voters - their ranking
35% W > Z > X > Y
32% X > Z > Y > W
17% Y > X > Z > W
16% Z > Y > X > W

To be clear, the first row means that 35% of the voters prefer W, and least prefer Y.

With ordinary Plurality Voting, W would win with 35% of the vote (assuming everyone votes sincerely for their favorite candidate).

A traditional top-two runoff (TTR) would pit W and X in a second round head-to-head "runoff" election. Unless voters radically changed their minds prior to the election, X would win with a landslide.

But wait! A 51% majority of the voters prefer Z to X. And Z would win a majority in a head-to-head election with every other candidate. Z is the Condorcet (beats-all-by-majority) winner, who would be elected using any Condorcet-compliant voting method (e.g. the Schulze method).

And to add even more weirdness, IRV elects Y!

So four different voting systems (three of which are specifically intended to elect a "majority winner") produce four different outcomes.

The moral here is to ignore the mathematically impossible task of ensuring a majority winner, and instead try to elect the candidate who makes the most voters the most satisfied. This leads us to look at Bayesian regret figures, which show that we get a big benefit by using Score Voting or Approval Voting instead of archaic systems like Plurality Voting or Instant Runoff Voting.

Anonymous said...

I've been voting regularly since 1964 and I haven't a clue as to what you are describing/suggesting up there.

Clay Shentrup said...

@Anonymous,

I'd be happy to clarify if you have a specific question. We use Instant Runoff Voting in San Francisco. Do you vote in San Francisco?