Which Voting System Could be Best for Our Polarized Politics?

Part 3 of Is Ranked Choice Voting the Hero We Need?

Top Image from ABC (the Australian one)

In previous posts (Part 1 and Part 2) on the topic of modelling elections, we’ve looked at how well a few different voting systems satisfied our voters’ desires for winning candidates in some simple scenarios. Now let’s look at how these voting systems perform in a highly polarized political climate.

With good reason, alternative voting systems have been proposed as a possible way to reduce polarization in the US. Plurality-based elections dominate American politics, reinforcing (though not guaranteeing) a two-party system by turning third parties into potential spoilers for the major parties most similar to them. But besides contributing to the existence of a polarized two-party system, we might also worry that Plurality is a particularly bad voting system given that a polarized political climate already exists. Being able to win by having the largest faction of devoted fans is a boon for bombastic partisans and a doom for centrists who are broadly acceptable but lack rabid followings. Swinging from partisan leader to opposite partisan leader after each election means that a lot of time and energy is spent undoing what the last leaders did, and the political situation always leave some large proportion of voters entirely dissatisfied. How much could changing our voting systems allow us to avoid these problems? Let’s look at how some different voting systems (Plurality, Plurality Runoff, Ranked Choice, and Borda Count) play out in a polarized political environment.

Major Takeaways:

• In most situations and according to most measures of success, Borda Count outperforms Plurality, Plurality Runoff, and Ranked Choice.
• Plurality is uniquely bad at selecting candidates who are broadly satisfactory to all voters and yet is still not particularly good at selecting candidates who are narrowly satisfactory to partisan voters.
• Ranked Choice and Plurality Runoff are fairly similar across most results, with Ranked Choice tending to do slightly better.
• Results can vary quite significantly depending on what voters want from a winning candidate and what we want our elections to produce.

What is a partisan electorate?

What does a politically polarized environment look like? In ideological space, it might look something like the image on the right. Rather than being evenly distributed among all possible ideologies (as I assumed in previous posts on voting), voters are concentrated into two separate peaks on opposite sides of ideological space. So most of my simulated voters are in either a left-leaning or right-leaning block, with fewer voters in between or at the extreme ideological fringes.

Mathematically, I do this by having all voters and candidates belong to one of two normal distributions (imagine a two-dimensional Bell curve). These are separated along the x-axis, so that one peaks at (0,-.5) and the other peaks at (0,5). We can think of these as representing the two major partisan alignments, akin to Democrats/Democrat-leaners and Republicans/Republican-leaners in the US context. The highest concentrations of voters and candidates are around the center of these two political alignments, and there are relatively few voters in the ideological center and at the ideological fringes. Voters are polarized on one dimension (the x-dimension), but voters are not polarized on the other (the y-dimension). The x-dimension represents the issues that are polarized and the y-dimension represents the issues that are not and on which there is a bipartisan consensus. The standard deviation of each of these partisan distributions is .2, meaning that most voters clearly fall into one of the partisan groups and relatively few could be considered “swing voters” in the sense of being roughly ambivalent about the two partisan alignments.

We’ll make the two political parties be roughly the same size by having each voter and candidate be equally likely to be in either partisan distribution.

What makes an election successful?

What do we want our voting systems to reward in such a politically polarized environment? Do we want to elect leaders who best balance the concerns of both parties or do we want leaders who agree with whatever bipartisan consensus there is, without concern for where they stand on polarized issues? Do we instead want leaders who clearly represent the center of their political parties? I’ll use three different measures of success to track these differing concerns.

Average Utility: This measures how satisfied about election winners voters are on average. This is how closely election winners matched what voters were looking for in candidates when the voters were deciding how to rank the candidates on their ballots. The higher the Average Utility, the better the voting system did at selecting a candidate that fit what the voters, on average, wanted from a candidate.

Relevance: This measures how well election winners agree with the bipartisan consensus, where it exists, regardless of where they sit on partisan issues. We might care about this if our political system is designed to make passing laws difficult without large majorities, so change is unlikely to occur on partisan issues but much more likely to occur on issues for which there is a broad bipartisan consensus. The higher the Average Relevance score, the better the voting system did at choosing candidates who fit the bipartisan consensus (i.e. candidates that are positioned closer to the x-axis).

Party Fit: This measures how close election winners are to the peak of the partisan distribution to which they belong. This tells us how well election winners represent their partisan voting bloc. We might want election winners to clearly represent one of our two partisan alignments. The higher the Average Party Fit score, the better the voting system did at selecting candidates that are good fits for their partisan alignment (i.e. candidates that are closer to the center of their corresponding partisan distribution).

The voting systems we’ll consider

Plurality: Each voter votes for one candidate. The candidate with the most votes wins.

Plurality Runoff: Each voter votes for one candidate. The two candidates with the most votes advance to a final round, where voters vote between them. Whichever candidate gets more votes in this one-on-one election wins. (California, Louisiana, and some other US jurisdictions use versions of this system instead of party primaries.)

Ranked Choice: Voters rank all candidates. The first place rankings of all candidates are tallied, and the last place candidate is eliminated. Votes for the eliminated candidate are then reassigned to the next active candidate on each voter’s ballot. The new first place rankings of all candidates are tallied, and the last place candidate among all candidates who have not been previously eliminated is now eliminated. Votes are reassigned as before. Repeat this process until one candidate remains. The final remaining candidate wins. (Maine, New York City, and several other US jurisdictions use this system for some elections.)

Borda Count: Voters rank all candidates. Candidates receive points from each voter based on where they are ranked. For instance, the last place candidate on my ballot receives zero points from me, the second-to-last place candidate on my ballot receives one point from me, the third-to-last candidate on my ballot receives two points from me, and so on until my top ranked candidate gets the most points from me. The candidate with the most total points from all ballots wins. (Borda Count can be scored in other ways, but I’ll use this simple version.)

Simulating elections

I ran 1000 simulation with 1000 voters for each voting system and value of ß, where ß tells us what voters are looking for in candidates. (For a more in-depth discussion of ß and what it means, see my post on the topic.) ß=1 means that voters only care about how ideologically similar candidates are to them. ß=0 means that voters only care about how much candidates would push the status quo toward (or away from) their preferred direction. These voters prefer the most extreme candidates on their side of the political divide. ß=.5 means that voters are balanced between caring about these two kinds of concerns.

Average Utility

All graphs plot the number of candidates (x-axis) versus the Average Utility (y-axis).

When voters only care about how ideologically similar candidates are to them (ß=1), Borda Count is the runaway best performer, Plurality Runoff and Ranked Choice get similar results, and Plurality is notably worse than the other three. Borda uniquely improves the more candidates join the race.

When voters are divided between caring about how ideologically similar to them and how much candidates will push the status quo in their preferred direction (ß=.5), Borda Count blows the other voting systems out of the water. With more than four candidates, non-Borda systems begin to perform worse than randomly choosing candidates.

When voters only care about the direction candidates will move politics (ß=0), Borda, Ranked Choice, and Plurality Runoff are essentially indistinguishable, whereas Plurality is uniquely bad.

Relevance

All graphs plot the number of candidates (x-axis) versus the Average Relevance (y-axis).

With ß=1, all voting systems tend to do better at selecting candidates who agree with the bipartisan consensus the more candidates there are. Ranked Choice tends to do slightly better than the other voting systems.

With ß=.5, Borda Count again distinguishes itself from the other voting systems, with the others being quite similar until the number of candidates gets very large and Plurality Runoff begins to look uniquely bad.

When ß=0, Plurality gets a rare strong result while Borda count looks uniquely bad. Perhaps now is a good time to emphasize that ß=0 voters are really weird — for instance, a slightly left-of-center ß=0 voter prefers candidates that are as far left as possible. These voters are the most extreme partisans imaginable.

Party Fit

All graphs plot the number of candidates (x-axis) versus the Average Party Fit (y-axis).

With ß=1, Ranked Choice is consistently the best at selecting candidates who clearly represent their side in the partisan divide. Borda Count, in contrast, does very poorly according to this metric.

The ß=.5 case looks completely different from the ß=1 case. Borda Count is far and away the best voting system for selecting candidates that clearly represent their partisan side, and increasing the number of candidates increases the gap between Borda and the other systems.

When ß=0, all voting systems tend to be worse at choosing clear partisan winners than randomly selecting candidates. Again, the weirdness of ß=0 voters, who prefer the most extreme candidates possible, can be blamed for this result.

What we learn about alternative voting systems

If we want to make our voters as happy with election results as possible (by maximizing Average Utility), Borda Count looks like the best option. It consistently produces higher average utility results than the other voting systems. Following it are Plurality Runoff and Ranked Choice, which give very similar results on this metric, and finally Plurality, which is consistently the worst of the four voting systems.

If we are most concerned with ensuring that our election winners agree with whatever bipartisan consensus exists or with having elections that pick candidates whose views represent the center of one side in the partisan divide, which voting system we should use will depend on what we think our voters care most about in candidates. If voters care mostly about how ideologically similar candidates are to them (B=1), Ranked Choice narrowly edges out the other voting systems as the best choice, but Borda Count is the runaway best choice when voters are equally concerned about choosing candidates who match their ideological positions and who will push politics as much as possible in their preferred direction. (When ß=.75, Borda Count and Ranked Choice perform remarkably similar on these metrics, with the other two voting systems following behind.)

We also saw that having purely directional voters (ß=0) who prefer candidates who will move politics in their preferred directions as much as possible can lead to some bizarre results. Thankfully, even in a polarized political climate, this simple model of voting behavior is not that realistic.

In my opinion, Borda Count comes out of these tests looking like the strongest of the four voting systems. But I still have a lot more planned to make my election model more realistic and to add more voting systems to contend with the others.

Looking Ahead

In these simulations, I’ve assumed that voters in Ranked Choice and Borda Count completely fill out their ballots by ranking every candidate, no matter how many there are. This is both unrealistic for elections with a lot of candidates and masks a lot of questions for Borda Count in particular, as the way that Borda Count handles incomplete ballots might make a big difference in its performance. Adding in incomplete ballots also gives me a way to model another voting system — Approval Voting. My guess is that, as I continue to make my election models more nuanced and realistic, Approval Voting will give Borda Count strong competition for Best Overall Voting System, but time will tell.