One Electoral Method is Inferior to Others When Voters Are Deeply Divided

The fourth in a series of essays on the philosophy of democracy and its implications.

In the previous essay in this series, I described a hypothetical election between three candidates—Blue, Red, and Gray—to illustrate the challenge of electing a single winner to represent the entirety of an electorate that is divided along two dimensions of policy contestation. Blue, a progressive Democrat, and Red, a MAGA populist, disagree vehemently on all the salient hot-button issues that characterize contemporary culture wars, including immigration, but both are willing to add to the national debt in order to fund popular government spending programs. Gray, by contrast, is a fiscal hawk, whose main campaign message is a dire warning about the danger of excessive deficit spending. On the cultural war issues, Gray is much closer to Red than Blue, as Gray comes from the previously dominant but now entirely eclipsed fiscally conservative wing of the MAGA-remade Republican party.

The hypothetical election was conducted using the Convergence Voting electoral method, meaning that the ballots that voters received asked them to identify their preferred candidate for each pair: Blue versus Red, Blue versus Gray, and Red versus Gray.  Because voters differed in their opinions along the two dimensions of policy conflict, the outcome of the election was that each candidate was preferred by a majority of voters against one of the two opponents (and thus each candidate was also disfavored by a majority of voters compared to the other opponent). Specifically:

Red over Blue: 51-49

Blue over Gray: 52-48

Gray over Red: 57-43

The previous essay discussed who should win the election given these outcomes.  It advocated electing Blue as the candidate whose margin of defeat is the smallest—2 points, rather than 4 or 14—and thus is the candidate who comes closest to being majority-preferred to each opponent. The essay reasoned that this electoral rule, called Maximum Convergence Voting, is the way most consonant with fundamental democratic values to choose a single winner because it puts into office the candidate with the broadest support among the whole electorate.  Or, to express the same point a bit differently, Maximum Convergence Voting elects the candidate whom the fewest voters wish to displace with one of the competitors. In this way, Maximum Convergence Voting has a solid claim to being the fairest—most democratic—method for electing a candidate to an office, like the presidency or a governorship, that can be held only by one individual at a time and yet must represent the entire polity despite deep divisions within the electorate along multiple dimensions of public policy.

The previous essay contrasted Maximum Convergence Voting with another possible way to break the 1-1 tie among all three candidates in terms of their single head-to-head victory and single head-to-head defeat against their two opponents. Called a Borda count after its progenitor, this other method would elect the candidate whose average (or combined) margin of victory (and defeat) is the largest. In this hypothetical election, this other method would declare Gray the winner—because Gray’s 14-point win and 4-point loss combine to yield the highest average: 5 points, compared to 1 for Blue and -6 for Red (a negative average for Red).

This Borda method has its contemporary defenders, and a case can certainly be made for it being the best way to choose among these three candidates given their 1-1 tie. By electing the candidate with the highest average margin of victory, it maximizes the sum of all preferences among the different candidates across all voters. Still, the previous essay contended that Maximum Convergence Voting better accords with the purposes of democratic elections for single-winner offices like president or governor, because the goal is not to maximize the net utility of all voters (as it would be in the context of other types of collective choices) but instead to install into office a single individual to make decisions on behalf of the public as a whole. For this kind of election, picking the candidate with the broadest range of support is a better and fairer way to treat all voters as equal citizens on this important matter of democratic representation, rather than picking the candidate who maximizes the sum of all preferences.

Whatever position one takes on the debate between Maximum Convergence Voting and Borda, both are far superior to a third alternative. There is an altogether different way to settle the kind of three-way electoral competition illustrated by the hypothetical involving Blue, Red, and Gray.  Instead of giving voters ballots that ask them to indicate their preferences between each pair of candidates, this third alternative would give voters ballots that enable them to rank the candidates in order of preference. To be clear up front, there is nothing inherently wrong with using this kind of ranked-choice ballot. Both the Maximum Convergence Voting and Borda methods can be administered with ranked-choice ballots instead of the direct head-to-head comparisons described earlier (by computing each voter’s head-to-head preferences for each pair of candidates from their relative rankings on each ballot), although in the context of a three-candidate election the ballot in the form of expressing direct head-to-head comparisons is more straightforward for voters to cast, easier for election officials to administer, and, crucially, much more transparent and accessible in terms of immediately understanding the state of the race as the returns are reported after the polls close.

What distinguishes the third method from both Maximum Convergence Voting and Borda is how it treats the rankings on the ranked-choice ballots. Magnifying the significance of first-choice rankings on each ballot, rather than giving equal weight to each preference expressed by each ranking on a ballot, this third alternative eliminates the candidate with the fewest first-choice votes and electing whichever of the two remaining candidates is preferred by more voters. This method, as those familiar with it will immediately recognize, is what is often called Instant Runoff Voting or the Hare method, after the British lawyer Thomas Hare who first proposed it. I prefer to call it Lowest Plurality Runoff because that is a more precise term and, as we shall shortly see, there are other instant runoff methods using ranked-choice ballots that yield a different outcome from Hare’s invention and, in fact, yield the same outcome (at least in some scenarios) as Maximum Convergence Voting.

To illustrate the important difference between Hare’s method and others, including Maximum Convergence Voting and Borda, using the same three-candidate hypothetical we have been considering, we need to know the ranked-choice ballots that correspond to the direct head-to-head preferences we have already seen. Here they are:

4: Blue>Red>Gray

36: Blue>Gray>Red

9: Gray>Blue>Red

12: Gray>Red>Blue

27: Red>Gray>Blue

12: Red>Blue>Gray

The above notation, hopefully, is clear: the first line indicates that 4% of ballots ranked Blue first, Red second, and Gray third. If anyone wishes to check that these ranked-choice ballots produce the same head-to-head results as stated above, they can copy-and-paste this set of rankings into Rob LeGrand’s extraordinarily helpful ranked-ballot calculator and click on the Borda method to produce the table showing:

Red over Blue: 51-49

Blue over Gray: 52-48

Gray over Red: 57-43

(Note: LeGrand’s calculator reports Borda scores as a candidate’s combined, not average, win-loss margin, but mathematically these two are equivalent because each candidate has the same number of head-to-head matchups: in this case, two).

Given these ranked ballots, we can quickly see that Red—not Blue or Gray—wins the election if Hare’s method is used. Gray has the fewest first-choice votes and thus is eliminated, leaving Red to defeat Blue by the slimmest of margins: 51-49. (One can replicate this result using the Hare function of LeGrand’s calculator.) This result differs from the outcome under either Maximum Convergence Voting or Borda.  As we have seen, Blue wins the election if Maximum Convergence Voting is used to determine the result given this set of voter preferences. (The Simpson function of LeGrand’s calculator verifies this, as Paul B. Simpson was an early proponent of what I call Maximum Convergence Voting.) Gray wins if Borda is used.

Hare’s method yields a result that is not only different from Maximum Convergence Voting or Borda; it is a distinctly inferior result. Red is the weakest candidate by either measure. Red has the lowest Borda score, one that is negative, averaging a 6-point margin of defeat (not victory). In contrast, both Gray and Blue have positive Borda scores.

Using the methodology of Maximum Convergence Voting, we see that Red has both the smallest margin of victory, 2 points, and the largest margin of defeat, 14 points. The triangular visualizations employed in the previous essay clearly show Red on the bottom of the three candidates, in terms of their relative support within the electorate, once the degree of their equivalence is removed from consideration. In sum, Red is clearly the least deserving of the three candidates to represent the entirety of the electorate given the totality of its preferences, yet Red is the one that Hare’s method would put into this singular office (like the presidency or a governorship).

Hare’s method is an outlier in this respect even compared to other instant runoff methods. Total Vote Runoff(also called Baldwin, as in LeGrand’s calculator) eliminates the candidate with the lowest Borda score (because a candidate’s Borda score can be understood as measuring the total number of votes a candidate receives against each opponent) and, in a three-candidate race, simply elects whichever remaining candidate is preferred by more voters. If Total Vote Runoff is used for this election, Blue wins—not Red. Indeed, Red is immediately eliminated for having the lowest Borda score, and then Blue prevails over Gray: 52-48.

Bottom-Two Runoff is another instant runoff method. It operates by comparing the two candidates with the lowest first-choice pluralities and eliminates whichever of the two is preferred by fewer voters (in other words, whichever of these two loses the head-to-head comparison between them). In a three-candidate election, this method elects whichever of the two remaining candidates is preferred by more voters. If Bottom-Two Runoff is used for our hypothetical election, Blue again wins: the two candidates with the lowest first-choice pluralities are Gray, with 21, and Red, with 39; as between these two, more voters prefer Gray over Red, 57-43, and thus Red is eliminated; so (again), Blue prevails, because more voters prefer Blue to Gray, 52-48.

Indeed, Blue would win if the conventional method of the electing the candidate with the highest plurality is used: Blue’s 40% plurality beats Red’s 39%. This is not to say that electing the candidate with the largest plurality is a good electoral method. It most definitively is not. But this does show that the Hare method is quite the outlier in electing Red, when many other methods agree upon electing Blue, and even Borda would elect Gray.

What is most troublesome about Hare electing Red in this example is that Red is the most polarizing candidate. If you force voters to choose between Blue and Red by eliminating Gray—and thus collapse the electoral choice into a single dimension of policy conflict when in fact voters diverge in their preferences along two different dimensions—you end up with Red. Yet Red is the most divisive candidate in the race, the one that more voters would prefer to replace with an opponent: Gray, 57 to 43. Red is also the candidate with the lowest overall level of support, as measured by Red’s lowest Borda score.

If the goal is to elect a single candidate to represent all voters in this one office, whether president, governor or some other single-member seat, you can’t do any worse in this three-candidate race than electing Red. That is not a good outcome for democracy—which is why Hare is not a good method, especially when an election is being contested among multiple candidates in multiple policy dimensions.

Thus, if we want to choose a method that serves the entire electorate well, we should return to the choice between Maximum Convergence Voting and Borda. We could also consider other methods, like Total Vote Runoff; in a three-candidate election, it has the virtue of eliminating the weakest candidate overall (the one with the lowest Borda score), and then electing whichever remaining candidate is preferred by a majority of voters. But that method is more complicated than either Maximum Convergence Voting or Borda, and simplicity is a very strong virtue in the context of democratic elections for the entire polity.

For reasons I have summarized here, and elaborated in the previous essay, I think Maximum Convergence Voting is the better method given the goals of democracy. But if we are endeavoring to undertake electoral reform to make election results more representative of what voters overall want, and to avoid exacerbating polarization unnecessarily, let’s not choose an electoral method that causes the least representative—and most polarizing—candidate to win.