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Moving Beyond Plurality and Ranked Choice Voting: The Case for Borda
Neal D. Hulkower

Nine Democrats and 7 Republicans are competing to fill the seat in the newly created 6th congressional district where I happen to live in Oregon. Following the primaries of May 17th, 2022, the field will narrow and in November voters will be asked to choose from a slate that will include one Republican. one Democrat, and likely some smaller party candidates by selecting one as is required by plurality voting, by far the most commonly used method in the United States and in many countries around the world. Given the unequaled importance of elections in a functioning democracy, does this winner take all method serve us well or does it contribute to the polarization and alienation of the electorate? Is there a better method of voting that will allow voters to more completely express themselves and yield an outcome that best expresses the preferences of the people? 

In his book Chaotic Elections! A Mathematician Looks at Voting, Prof. Donald Saari of the University of California, Irvine, draws the analogy between ways of determining class ranking and voting methods. He notes: “In this comparison, ‘students’ represent ‘candidates’ and the instructor of each course represents a ‘voter.’” (p. 20) He points out that plurality voting in which only a voter’s first choice is counted is tantamount to deciding class rankings “solely on the number of A’s a student receives.” (p. 20). The absurdity of considering only the number of the highest grade while ignoring the distribution of other grades to select a valedictorian is so clear that no school uses it. Instead, a student’s class ranking is determined by his or her grade point which is calculated from the numbers of B’s, C’s, D’s, and F’s, as well as A’s. There is one positional voting method that asks voters to rank each candidate from first to last. That is the electoral analog to using grade point average to place a student in a class. However, the current favorite of electoral reformers now used in several places in the United States, Ranked Choice Voting (RCV), is not it. 

The idea that a candidate who wins over 50% of the votes must be declared the winner is so ingrained in our political psyche that when RCV is proposed as an alternative to plurality voting, it too must somehow preserve majority rule. In an Op-Ed piece in the Boston Globe on 20 September 2020, Jeff Jacoby quotes Evan Falchuk who headed the group that unsuccessfully advocated the adoption of RCV in Massachusetts: “Democracy is supposed to be majority rules. We should have a system where the majority wins.” 

A growing number of cities and states, including New York City, Maine, and Alaska, adopted RCV. While this move toward positional voting is laudable, it does not eliminate many problems. Most seriously, RCV immediately devolves into an altered form of plurality voting and thus inherits many of its shortcomings. One fundamental problem with plurality voting is that, like determining class ranking by only considering the number of A’s, it ignores the complete information a voter who ranks all the candidates provides. If a candidate does not receive over 50% of the first-place votes, a convoluted process of eliminating the candidate with the lowest number of first-place votes and reallocating his or her votes to second-place candidates kicks in. This continues until a candidate emerges with at least 50% of the reassigned votes, so multiple passes may be required before a winner emerges. A voter who only ranked a candidate who is eliminated becomes disenfranchised. Using the analogy to determine class ranking, RCV inflates the grades of some students after others are expelled from school to arrive at a winner. Since the “majority” is artificially created by cannibalizing the individual rankings, RCV can be viewed as the bastard child of majority rule and positional voting. 

There is, however, a positional voting method that uses all the information voters provide, disenfranchises no one, uniquely satisfies several rational properties, and selects a winner in a single pass. Named for Jean-Charles de Borda, an eighteenth-century French mathematician, the Borda Count is the exact analog to determining class ranking by grade point average, the universally recognized fairest method for doing so. With the Borda Count, the top- ranked candidate of each voter gets a Borda Score equal to one less than the total number of candidates, the second-ranked gets two less, and so on down to the candidate ranked last who gets 0. So, for 4 candidates, the Borda Scores are 3, 2, 1, 0 for the highest through lowest ranked candidates. Candidates that are tied each get the average of the total of the scores for the positions they occupy. The societal outcome is the ranking of candidates based on the totals of each candidate’s Borda Scores. The Borda Winner is the candidate with the largest total. The total Borda Score for each candidate represents his or her strength of preference which is a more complete representation of the voters’ assessment than the percent of first place votes. 

But what if some or all the candidates are unacceptable to a voter? Certainly, one may refuse to vote, but then his or her voice is not heard. Recently, John Neatrour and I published a paper in the peer-reviewed open source online journal, SAGE Open, that offers a remedy. “The Power of None” introduces Borda with None (BwN), a marriage of two ideas, the Borda Count and None of these candidates (N) as a binding option. It is almost the same as a method recommended by C. L. Dodgson (aka Lewis Carroll) in 1873 (see pages 279-286 in Classics of Social Choice for a description of Carroll’s method) differing only in the way ties are scored. 

Let’s see how BwN would work for a hypothetical election involving four candidates: Alice, Brent, Carla, and Derek, compared to plurality voting and RCV. Adding N, voters have 5 alternatives to rank. The table below shows the rankings of 6400 voters. Notice that for 2200 voters, all four candidates are ranked equally below first placed N, so each gets the average of the rankings: (2+3+4+5)/4=3.5. 

With plurality voting, N wins with 2200 first place votes or 34.38% of the vote, followed by Derek with 1200 (18.75%), Alice with 1100 (17.19%), Carla with 1000 (15.63%), and Brent with 900 (14.06%). So even though the other 4200 voters find all or some of the slate acceptable, a minority would overrule their preferences. This is equivalent to picking N as the valedictorian since it has the plurality of A’s despite the fact that it has 1500 D’s and 2700 F’s. 

Converting the rankings to Borda Scores by subtracting the rank from 5, then multiplying the scores by the number of voters and summing the results for each candidate gives the following societal outcome: Brent is the Borda Winner with a total Borda Score of 15600 followed by Alice (14900), Carla (13600), N (10300) and Derek (9600). Using the Borda Count without N yields the same ranking of the four candidates, but this will not always be the case. 

In this example, the Borda Winner, Brent, differs from the plurality winner, N. This, too, will not always be the case. For instance, if we use Borda without None, Brent is both the Borda and plurality winner. But here, Brent’s total Borda Score or strength of preference, based on the complete rankings of all the candidates by all the voters, is over 50% greater than N’s, so it better reflects the will of the voters. 

Since no candidate ranked in first place with over 50% of the vote, the RCV method requires that we first eliminate Brent, the Borda Winner, who had the fewest first place votes, and allocate them to Carla who was ranked second by the 900 voters who ranked Brent first. Continuing, we next must eliminate Alice and give her 1100 first place votes to Carla. Since this still doesn’t give any of the remaining candidates over 50%, we eliminate Derek and give his 1200 first place votes to Carla. This leaves Carla, who ranked third using BwN and fourth in plurality voting, the RCV winner with 4200 votes (65.62%) and N with 2200 votes (34.38%). So, Carla benefited from having her grades inflated to A’s from B’s and C’s when 3 of her classmates were expelled. Unlike BwN which produces a societal outcome using all information provided by voters, RCV alters voter preferences in order to attain its desired target of “majority rule.” If voters provide a complete ranking of candidates, why not use Borda or BwN which only requires a single tally of the ballots? 

Since the three methods yielded three different societal outcomes, how can we decide how to decide the winner of an election? Clearly, the analogy to obtaining the fairest and most representative class ranking has not been sufficient to do away with plurality voting or make RCV less attractive. Taking a mathematical approach, we identify four properties that a method should satisfy. The first two, Universal Domain and Complete Transitive Outcome, describe permissible votes and rational outcomes. Universal Domain means that all possible transitive rankings are permitted. A transitive ranking is one that doesn’t chase its tail. For example, if Abby is ranked higher than Bruce who is ranked higher than Carl, then neither Abby nor Bruce is ranked lower than Carl, and Bruce is not ranked above Abby. (An example of a nontransitive outcome is when Team A beats Team B which beats Team C which beats Team A.) If all voters’ rankings are transitive, the method that yields a transitive outcome satisfies Complete Transitive Outcome. 

The two other properties address the consistency of the method, requiring that the societal outcome follows reasonably from the collection of voter preferences called a profile. The Pareto condition states that if each of the voters prefers Abby to Bruce then the societal outcome ranks Abby ahead of Bruce. Intensity of Binary Independence (IBI) also known as Intensity form of Independence of Irrelevant Alternatives (IIIA) addresses what to expect of the societal outcomes of two profiles that are similar but not identical. 

Saari explains that a voting method “which ranks alternatives satisfies … IBI if society’s relative ranking of any two alternatives is determined only by each voter’s relative ranking of the pair and the intensity of that ranking. That is, for any pair of alternatives…, if each voter’s relative ranking and intensity ranking is the same for two profiles p1 and p2, then society’s ranking of this pair is the same for both profiles” (pp. 189-190). Let’s illustrate what this means with an example. Consider two profiles, Profile 1 which comprises the rankings for 5 candidates, Erica, Frank, Greta, Hal, and Izzy, and Profile 2 which contains the rankings of the remaining 4 candidates if Izzy pulls out after the voting is complete so that none of the relative rankings change. The tables below compare the two. 

Notice that in both profiles 10 voters prefer Erica to Frank with no one in between. In other words, the intensity of the preference measured by the number of intervening candidates is 0. Another 10 prefer Erica to Frank with an intensity of 1. Also, 15 voters prefer Frank to Erica with an intensity of 0. In Profile 1, 25 voters prefer Greta to Hal with an intensity of 0 and 10 prefer Hal to Greta with an intensity of 2. In Profile 2, 25 voters prefer Greta to Hal with an intensity of 0 but now the remaining 10 prefer Hal to Greta with an intensity of 1. Using the Borda Count, the societal outcome for Profile 1 is Erica (with a Borda Score of 80), Hal (75), Greta (70), Frank (65) and Izzy (60) while Profile 2 yields Erica (60), Greta (55), Hal (50) and Frank (45). Since the relative ranking and intensity of Erica and Frank for each of the voters is the same in both profiles, the relative ranking of this pair in the societal outcome for each is the same. On the other hand, since the relative ranking and intensity of Greta and Hal are not the same for each voter in both profiles, it is not guaranteed that the relative rankings will be the same. In this case, they are not. With the intensity of preference of Hal over Greta weakened in Profile 2 compared to Profile 1, the relative rankings are reversed, and Greta moves ahead of Hal. If this were an election to choose the top three candidates to fill three city council seats, the two profiles would yield different outcomes. While in this example the first-place winner, Erica, is the same for both profiles, this may not always be the case. 

Saari has proved that the Borda Count uniquely satisfies the four rational properties. What this means is that the Borda Count, unlike plurality voting, RCV, or any other positional voting method, best expresses voters’ preferences. Furthermore, BwN is constructed to ensure that no voter has a disproportionate influence on the outcome, especially in the case of ties, something that is not required of Borda per se. Thus, BwN uniquely satisfies a fifth property called anonymity, in addition to the four properties that Borda does. Our paper shows how BwN can be used to ensure that the impact on the societal outcome is the same for a voter who ranks a few but not all of the candidates as a voter who ranks all. 

So, what happens if N does come in first? Our paper considers various options that depend on what’s at stake in an election. Though each has issues, we believe that giving individual voters the ability to veto all or part of a slate outweighs the potential problems. We also expect that if BwN is adopted, the acceptability of candidates and slates will increase because of the power of none wielded by the electorate.

As a society, we look beyond the number of “A’s” a student has and determine class ranking on the basis of all grades. While allowing positional voting, RCV seems to have become a favorite among those seeking an alternative to plurality voting but are still more focused on preserving “majority rule” at all costs. In contrast, adopting Borda or BwN will ensure that each voter’s complete ranking is used without reallocation to select candidates who are the most acceptable to the electorate thus enhancing the chance for greater voter satisfaction and participation. Unfortunately, neither will be an option in Oregon or anywhere else, for that matter, anytime soon.

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