What if the Voters Want What Voters Actually Want?
In our first look into this topic, we used a simple model of voters and candidates to find some mixed results for Ranked Choice voting. It performed better than the Plurality system that dominates US elections, but it was outclassed by its systems and didn’t always benefit from more candidates joining in on an election. We should reflect a bit on why Ranked Choice voting looked mediocre at best and the ways in which we need to be more clever before we can pass judgment on Ranked Choice voting’s performance.
The model in Part 1 was extremely simple — voters and candidates had random ideologies and voters voted solely based on which candidates were more ideologically similar to themselves. This means that candidates on the ideological extremes tend to be perceived less favorably by voters overall than more moderate candidates. There is reason to believe this can lead to “worst case scenario” Ranked Choice elections, where a relatively unpopular candidate wins an election, whereas Borda Count is better suited to avoiding such results.
Ranked Choice voting works by counting up all the first place votes for candidates, eliminating the candidate with the fewest first place votes, then reallocating votes for the eliminated candidate to the next active candidates in those voters’ rankings. Imagine we are running an election with candidates who can be neatly placed along a Left/Right spectrum, where voters prefer candidates based on how close they are to them along this spectrum.
If our voters are evenly distributed along the line, Candidate F is most likely to get the fewest first place votes and thus get eliminated first. Candidate F’s voters are then moved to their next favorite choice, Candidate E. This means Candidate E is likely to avoid elimination longer, greatly increasing her chance to win. So our setup in Part 1 — where voters and candidates are uniformly distributed in an ideological space, and voters vote for candidates based on how ideologically similar they are — can be particularly troublesome for Ranked Choice voting; while the most extreme candidates might be generally disadvantaged, this can turn into an advantage for their slightly less extreme competitors. With a decent number of candidates, voters will be more evenly distributed among the candidates and the early elimination of fringe candidates can facilitate consolidation around their slightly less fringe ideological neighbors, potentially leading to a snowball effect where a less popular candidate wins an election. In Borda Count elections, candidates are simply assigned points based on where they are ranked, so there is no way to get a snowball rolling. Taking these concerns into account, we’ll look at worst-case elections along with average election results.
Does the model of voting behavior where voters prefer candidates based on how ideologically similar they are to them unfairly disadvantage Ranked Choice voting? The voters in our model should vote how actual voters vote, and actual voters, at least in the US, don’t vote based solely on which candidate is most ideologically similar to themselves. They also prefer candidates who would push the polity closer to their own preferred position. “A Unified Theory of Voting” by Samuel Merrill III and Bernard Grofman discusses this idea in some mathematical detail, and we’ll use their account of this idea moving forward.
Consider a hypothetical voter Alice who is choosing between two candidates, Kirsten and John. They can be placed neatly along a left/right spectrum, where the current status quo of the polity is represented in orange:
Alice is a fairly centrist person, and so is John. But Alice would prefer to move the polity left, and John would prefer to move it right. Kirsten, however, would move the country towards Alice’s preferred status. So Alice might prefer to vote for Kirsten, even though Kirsten’s views differ more from Alice’s view than John’s do. However, Alice might see a candidate like Leon as being too extreme — perhaps he would move the country too far or at least do things Alice finds repulsive. In fact, this seems to be a better account of how voters prefer candidates than simply ideological similarity, according to Merrill and Grofman. Voters seem to value candidates according to a formula like
where V is the position of a voter, C is the position of the candidate and β weights how much voters care about how far a candidate would push the polity in their preferred direction versus how ideologically similar a candidate is to them. When β is 1, the second term, which describes ideological distance, is the only factor that matters, as in Part 1. Distance matters less and direction matters more as β shrinks until voters only care about how far a candidate would move the country in their preferred direction when β is 0. (Merrill and Grofman propose a slightly more complicated formula, but for now, we’ll use this version.) Let’s look at how our voting rules perform as β varies.
First, let’s look at the average utilities voters assign to election winners as β changes. This time, I ran 100 simulations of elections with 100 voters each, because larger simulations are starting to take a long time. Where Blue is Borda Count, Red is Ranked Choice, Orange is Plurality Runoff, Green is Plurality, and Gray is a Random candidate, we can graph the number of candidates in an election (horizontal axis) against the average utility voters assign to the election winner (vertical axis):
Since 100 elections of each voting rule for each β is a fairly small sample size, we shouldn’t read to much into any particular data point. We also shouldn’t compare the numerical values of the utilities across different values of β, as that is an apples-to-oranges comparison. Nonetheless, some overall trends are certainly clear. Borda Count consistently outperforms all other voting rules, and consistently performs better in elections with many candidates. (We’ll discuss the nice mathematical reason why this is the case some day in the future.) If we aren’t using Borda Count, elections with many candidates and voters that care about both ideological distance and direction tend to end up with results that are about as good as randomly picking candidates. And Ranked Choice doesn’t do much to distinguish itself from Plurality and Plurality runoff; making voters care less about how ideologically similar candidates are to them actually makes Ranked Choice perform worse, as caring more about which direction candidates would move the polity pushes more voters towards more extreme candidates and helps get the snowball running.
We can also compare the worst-case scenario outcomes of our voting rules. We might care more about reducing the harm of worst-case outcomes of elections than making them perform best on average, where worst-case outcomes are the election results in our simulations with the smallest average utility voters assign to the winner. So out of our 100 simulations of each voting rule (and for each number of candidates and each value of β), we’re only concerned about the worst one. According to this metric, the higher this minimum average utility is, the better the voting rule has performed at avoiding a worst-case scenario. Again, Borda Count comes out looking very good, Plurality Runoff tends to be second best, and Plurality and Ranked Choice follow. The sample size means our results are somewhat noisy, but we’re mostly trying to get an idea of how these voting rules before along our path of making these simulations more closely reflect reality.
Again, Borda Count looks strong and Ranked Choice struggles to distinguish itself from Plurality Runoff and Plurality.
In our first two parts of this investigation, the results of our simulations have been very positive for Borda Count and fairly negative for Ranked Choice, which continues to perform especially poorly in elections with larger numbers of candidates. But so far, our voters have been simple, honest folk who vote purely based on how much they like each candidate. In future installments, we will look at what happens when our voters vote with their heads along with their hearts — do voters who take into account the viability of candidates and vote strategically give us different results for how well these voting systems perform?