The Power of None

Abstract

Borda with None (BwN) adds the binding option of “None of these candidates,” N, to the Borda Count (BC), differing from a method of Dodgson in the scoring of ties. The method incorporates the benefits of approval voting, which allows a simple yes/no vote on each candidate and avoids the disadvantages of adding N as a binding outcome to plurality voting. We prove that BwN uniquely satisfies five rational properties which can be viewed as potential axioms that provide a theoretical basis for the method. It allows individual voters to express a personal veto over all or part of the slate which could, with sufficient numbers, become a veto by the electorate. The power of none, then, is offering a voice to otherwise disaffected voters. We introduce a metric similar to one used in the Bak–Sneppen Evolutionary model, candidate fitness, a number between 0 and 1, which measures a candidate’s ranking relative to N. We illustrate its evolution with a hypothetical example. We examine courses of action if N is ranked in the first place. Using BwN to accommodate partial voting is proposed. We explore the general applicability of BwN. Finally, we address the practical considerations for introducing BwN.

If I must choose the lesser of two evils I will choose neither.

—Karl Kraus

“. . .I do not feel compelled to vote . . . if I think that none of the parties or candidates deserves to be supported.”

—Blais (2000, p. 143)

In the United States, voter turnout as a percent of the voting age population is among the lowest among developed countries (DeSilver, 2017). While the reasons people choose not to vote are numerous (Blais, 2000), many cite not liking the candidates or issues as the primary explanation. For example, this was the primary reason given by 12.9% of the 15,167 nonvoters polled after the 2008 U.S. presidential election (U.S. Census Bureau, 2012, Table 12), by 12.7% of the 19,141 nonvoters polled after the 2012 U.S. presidential election (U.S. Census Bureau, 2013, Table 10), and by 24.8% of the 18,933 nonvoters polled after the 2016 U.S. presidential election (U.S. Census Bureau, 2017, Table 10). The latter was the highest percent of any explanation given for not voting. Unfortunately, the voices of these nonvoters are not heard. Although a two-party structure is intended to present clear choices to the electorate, the numbers voluntarily choosing to relinquish their franchise suggest otherwise.

This article offers a voting method whose theoretical foundation rests on five rational properties that it uniquely satisfies and that might encourage those who sit out, particularly because they are unhappy with the choices, to participate. While not specifically mentioning the Borda Count (BC; Borda, 1781/1995), the method that is the basis of our proposed procedure, Dahl (2003) observed that the prevailing “first past the post” electoral system has undesirable consequences. The principal consequence is that the electoral system discourages participation by under representing minority interests. For instance, a party could gain complete control of a legislature by winning by one vote in every district. With additional parties, third and smaller factions are so under represented that their influence is vanishing. This favors a political landscape dominated by two parties. We note that Dahl’s observation suggests a causal link from under representation (which throws away voter preferences) and consequent voter disaffection or nonparticipation. The information preserving property of BC is interesting as a remedy.

It is our working hypothesis that voter disaffection is a direct result of two incentives that discourage voter participation. The first is that “first past the post” elections nullify the preferences of minority interest voters, effectively disenfranchising them. The second is that “first past the post” can allow a candidate to be elected that is unacceptable to the majority of electors.

What is needed to encourage the fielding of more acceptable slates and greater voter participation is a departure from plurality voting and the ability for the each voter to reject an unsatisfactory slate of candidates in part or in total by being able to exercise a personal veto. With sufficient numbers, it would result in a veto by the electorate.

The idea of modifying an existing voting method to allow individual voters the ability to express dissatisfaction with one or more candidates has been explored by other researchers. For example, Alcantud and Laruelle (2014) extended approval voting (AV), a vote-for-k-candidate rule, to “Dis&approval voting” which limits each voter to awarding each candidate 1 for approval, 0 for nonapproval, or a third option, –1, for disapproval. Saari (2001a, 2008) critiqued AV which he noted is an example of a “multiple voting system” in which each voter in essence selects one of many possible positional rules. Other problems arise because there is no ranking of the candidates required so voters are limited in what they can express and winners are thus selected using incomplete information.

This article describes a positional voting method that might not only entice nonvoters to go to the polls but also offers all who do so a more complete way of expressing their preferences. In addition, it preserves that information through to the decision of the electoral process. This scheme is based on a blending of two ideas that have been around for quite a while, the BC which is the only nondictatorial positional voting method that satisfies four rational properties, with one additional option, “None of these candidates,” denoted by N, hereafter referred to as BwN. C. L. Dodgson (1873/1995; also known as Lewis Carroll) recommended an almost identical method differing only in the scoring of ties. BwN will be shown to be the unique method that answers the question “What is the most rational way to aggregate a set of preferences that includes N as an option?”

Encouraging greater civility during a campaign is another potential benefit of BwN that is a direct result of requiring rank-ordering of candidates. Surveys of voters in cities that use ranked choice voting (RCV) conducted for FairVote (website) indicate they perceived that campaigns were less negative than in cities not using RCV. The reason given is “it is riskier for Candidate A to offend Candidate B’s supporters by attacking or besmirching Candidate B, since the Candidate A may lose second- or third- rankings from Candidate B’s supporters in the process” (FairVote, n.d.).

BwN allows the definition of a candidate fitness metric or simply, fitness, based on the candidate’s ranking vis-à-vis N. The fitness, a number in the interval [0,1], is similar to that used in the Bak–Sneppen Evolutionary model (BSE model; Bak & Sneppen, 1993). As defined in a later section, a candidate’s fitness is below 0.5 if it is ranked below N. We illustrate how the evolution of a candidate’s fitness can be tracked over the course of a campaign.

While encouraging greater voter turnout is one potential benefit of BwN, it must be balanced against the possibility that N actually wins and the implications that would carry. We consider options for when this might occur.

The BC is indifferent to what or whom is being ranked be it by voters or according to criteria. In other words, BC is a context-free method of aggregating preferences. Thus, BwN is applicable with the obvious substitution of “None of these alternatives,” when ranking alternatives that are not candidates for office. We will discuss the general application of BwN in a later section.

The next section gives a brief overview of the BC and introduces N as an option. The use of BwN in an election for mayor of a fictitious town is demonstrated in section three. The fitness of a candidate is defined in the fourth section, and an example is presented that tracks it in a hypothetical campaign. Options to consider if N is ranked first in an election using BwN are looked at in section five. The sixth section discusses how BwN can be used to incorporate partial rankings. Using BwN as a decision-making methodology more generally is discussed in the seventh section. The eighth section considers practical issues that might arise when implementing BwN. Conclusions comprise section nine.

About the Borda Count

The BC, named for Jean-Charles de Borda, an eighteenth century French mathematician, is a type of positional voting scheme (Borda, 1781/1995). Positional voting requires voters to rank order their preferences and assigns a specific number of points to each candidate based on rank. When all votes are tallied, a societal ranking results with the top ranked candidate having the largest number of points declared the winner. In the case of the BC, if there are n candidates, the first-ranked candidate receives a Borda Score of n – 1, the second, n – 2, and so on, with the candidate ranked last getting a Borda Score of 0. If the candidates are tied, each receives the average of the total points associated with the positions the group occupies. This is the sole difference with Dodgson’s method which assigns the highest value to all candidates who are tied. For example, if there are seven candidates and a voter ranks those that occupy the fourth through sixth positions as tied, the Borda Score of each is (3 + 2 + 1) / 3 = 2, whereas Dodgson assigns each the highest value, 3.

Saari (1998, 2000a, 2000b, 2001a, 2001b, 2008) has proven that the BC is unique among all positional voting methods in that it satisfies four properties:

• Pareto condition: if all voters prefer A to B then the societal outcome of the method also has A ≻ B where “≻” means “is preferred to”

• Complete transitivity outcome: if A ≻ B and B ≻ C then A ≻ C; complete means that each voter will rank each pair

• Unrestricted Domain: each voter can select any strictly transitive ranking of alternatives

• Intensity of Binary Independence (IBI) or Intensity form of Independence of Irrelevant Alternatives (IIIA): “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” (Saari, 2001b, pp. 189-190).

IBI (IIIA) replaces Independence of Irrelevant Alternatives (IIA), which Saari demonstrated negates complete transitivity outcomes, and in doing so eliminates Arrow’s dictator (Arrow, 1963). The measure of intensity for the BC is the number of alternatives between the two being compared unless one or both are tied with other alternatives. In search of the most defensible aggregation methods, Arrow was the first to select a set of rational properties to serve as a theoretical foundation but ended up with an unsatisfying result. By substituting IBI for IIA, Saari used the complete information and established the uniqueness of the BC in satisfying the new set of four properties.

In comparing an alternative that is not tied with one that is tied with another, the intensity number is measured from the “center of mass” of the tied alternatives, in other words, the average of the rankings. For the seven candidate race above, the “center of mass” of the three tied alternatives that occupy positions 4 through 6 is (4 + 5 + 6)/3 = 5.

A unique feature of the BC is that the difference between the Borda Scores of contiguous alternatives is constant, except in cases of ties. But even in the case of ties, the total number of points each voter assigns is the same, n (n – 1) / 2, for n candidates. To ensure that “the ‘intensity number’ is always one less than the difference between BC Scores” (Saari, 2001b, p. 192), it must be measured from the “center of mass” that the tied alternatives occupy. One exception is when comparing alternatives that are tied. In this case, the intensity number is 0 as is the difference between Borda Scores. In Dodgson’s method when there are ties, the total number of points assigned by voters can vary resulting in disproportionate influences on the societal ranking though the standard measure of intensity remains valid when comparing an untied alternative with one that is tied with another. The BC is also unique among all positional voting methods in that “all possible inconsistencies in Borda rankings over subsets of candidates are strictly due to Condorcet terms” (Saari, 2008, p. 157). ¹ What this translates to is that the BC best expresses voters’ preferences.

Because it is exactly Borda with N specified as an alternative, BwN uniquely satisfies the same four properties. That BwN distributes the same number of points evenly for each voter eliminates the possibility of an individual having a disproportionate influence on the outcome. So in the case of ties, BwN satisfies the anonymity condition (Saari, 2001b, p. 193; 2008, p. 166) while Dodgson does not.

Summarizing:

• Theorem (Saari, 2001b, p. 190): “The Borda Count is a nondictatorial procedure that always has transitive outcomes and that satisfies Unrestricted Domain, IBI, and Pareto.”

That the BC uniquely satisfies this Theorem follows because “only the BC differential . . . uses the intensity information” (Saari, 2001b, p. 192). The difference in Borda Scores between contiguously ranked alternatives that are not tied is a constant which can be normalized to 1, the same as the measure of intensity of preference.

• Corollary: BwN is the only nondictatorial procedure that satisfies complete transitivity outcome, Unrestricted Domain, IBI, Pareto, and anonymity.

So the uniqueness of BwN rests on the theoretical foundation based on the four properties selected by Saari to prove the uniqueness of BC plus a fifth, anonymity.

Kenton Elects a Mayor

Consider the election for mayor of Kenton, a tiny mythical town with 35 eligible voters. Alice, Brent, and Carla decide to run. Fourteen of the eligible voters cannot stand any of the three and refuse to go to the polls. The remaining 21 voters have the following preferences, which define a profile:

• Six rank the candidates as follows: Alice ≻ Brent ≻ Carla but note that all 6 find Carla unacceptable

• Five rank the candidates as follows: Brent ≻ Alice ≻ Carla

• Ten rank the candidates as follows: Carla ≻ Brent ≻ Alice

Assigning a Borda Score of 2 for each first place vote, 1 for each second place vote, and 0 for each third place vote then summing for each candidate, we see that Brent prevails with a Borda Score of 26, Carla comes in second with 20, and Alice finishes third with 17. Using plurality voting that assigns 1 point to the top ranking candidate and zero to all the others, Carla would win with her 10 first place votes over Alice with 6 and Brent with 5. What’s different? The BC uses all of the information each voter provides, that is, it takes into account the full range of preferences, versus plurality, which only counts the top choice.

While the BC is unique among all positional voting schemes, the outcome is still only as good as the selection of candidates and the participation of as many eligible voters as possible. In the election for mayor of Kenton, 14 eligible voters stayed away from the polls because Alice, Brent, and Carla were all unacceptable to them. How might we attract them to the polls and, at the same time, offer a richer selection of alternatives to those who feel they must select the lesser (or least) evil? We do so by allowing for one additional choice, N.

This idea has been around for well over a century at least and has been adopted to a greater or lesser extent in several countries (Wikipedia, n.d.). In 1975, Nevada became the first, and remains the only, state to enact a “None of these candidates” option, but it is nonbinding (Nevada Revised Statutes, 2013). ² Recently, organizations such as “Voters for None of the Above, n.d.” (website) have formed to advocate the mandatory inclusion of a binding “None of the Above” option. Because it utilizes the entire set of preferences of each voter to determine the outcome of an election and has been proven to uniquely satisfy rational properties, the BC is the most natural method to incorporate N.

Let’s look at how BwN would work in the election for mayor of Kenton. The profile is now expanded to include the 14:

• Fourteen (who would otherwise sit out) rank the candidates as follows: N ≻ Alice ≈ Brent ≈ Carla (≈ denotes a tie) ³

• Six (who can’t stand Carla) rank the candidates as follows: Alice ≻ Brent ≻ N ≻ Carla

• Five rank the candidates as follows: Brent ≻ Alice ≻ Carla ≻ N

• Ten rank the candidates as follows: Carla ≻ Brent ≻ Alice ≻ N

To determine the Borda winner with four options, assign a Borda Score of 3 to the first choice, 2 to the second, 1 to the third, and 0 to the fourth, then sum for each choice. Because Alice, Brent, and Carla are tied for second place, each gets (2 + 1 + 0) / 3 = 1 from each of the 14 new voters who put N in first place. BwN ranks Brent in first place with a Borda Score of 61, Alice in second with 52, Carla in third with 49, and N finishes last with 48.

Because the BC satisfies IBI (IIIA) but not IIA, the relative ordering of Alice and Carla is reversed when N is introduced as a candidate. This occurs because for six of the voters, the insertion of N increases the strength of preference of Alice and Brent over Carla by 1 each. There are also 14 additional voters who strongly prefer N over the other three choices. (The intensity measured from the “center of mass” or average ranking of the three tied candidates, (2 + 3 + 4) / 3 = 3, is 1). Note that by IBI (IIIA), if N is ranked in last place by all the voters, the resulting societal ranking of the other candidates would remain unaltered. Also, by the Pareto Condition, the societal ranking would have N in last place. So in a campaign in which all the candidates are minimally acceptable to all the voters, BwN gives the same result as the BC.

Adding N as an option in the current plurality voting method could result in N more frequently coming in first than it would with BwN. For the profile above, N is ranked first with 14 votes, followed by Carla with 10, Alice with six, and Brent with five. Allowing N in a plurality voting system would increase the chances of a minority overruling a majority that is satisfied with the slate offered.

A Candidate Fitness Metric

The BSE model (Bak & Sneppen, 1993) is a simple tool for studying the evolution of species. In the most basic form, the BSE model places each species on a line with periodic boundary conditions and assigns each a random barrier (a measure of stability) or fitness equally distributed between 0 and 1. At each time step, the species with the lowest fitness mutates, that is, is assigned a new fitness between 0 and 1 randomly as are its two neighboring species, reflecting their interdependence.

Conceptually, each alternative, x_i, i = 1 to n, which could be a slate of candidates representing a single party in a parliamentary system or an individual candidate, presented to BwN represents a species in the BSE model. The fitness of alternative x_i at time t, f(x_i), is calculated as follows. For each voter, let ρ(x_i, N;t) be the difference between the Borda Scores of x_i and N at time, t. Note that ρ(x_i, N;t) = 0 if there is a tie, > 0 if x_i is preferred to N, and <0 if N is preferred to x_i. If all possible ties are permitted, the number of rankings of n alternatives is given by:

$$\mathbf{P}(n)=\sum _{k=1}^{n}k!S\left(n,k\right)={\displaystyle \sum }_{k=1}^n{\displaystyle \sum }_{i=0}^k{\left(-1\right)}^i\left(\begin{array}{c}k\\ i\end{array}\right){\left(k-i\right)}^n,$$

where S(n, k) is the Stirling number of the second kind which denotes the number of ways n candidates can be placed in k indistinguishable bins. Here, the bins represent the number of tied alternatives in a given ranking. But the rankings need to be distinguished, because, for instance, A ≈ B ≻ C ≈ D is clearly not the same as C ≈ D ≻ A ≈ B. So to ensure the proper count, we multiply the S(n, k) by k! to obtain the number of ways to order rankings with k ties. For example, Ρ(3) = 13 and Ρ(4) = 75. Let V be the total number of voters and V_j be the number of voters selecting ranking j of the alternatives, then V = ${\sum }_{j=1}^{P\left(n\right)}{V}_j$ . The fitness of candidate x_i at time t, f(x_i; t) ≡ $([({{\displaystyle \sum }}_{j=1}^{{\rm P}\left(n\right)}{\rho }_j(x_i,N;t))/(V(n+1))]+1)/2$ . Note that 0 ≤ f(x_i; t) ≤ 1. If 0 ≤ f(x_i; t) < 0.5, N is ranked higher than x_i; if f(x_i; t) = 0.5, N and x_i are tied; and if 0.5 < f(x_i; t) ≤ 1, x_i is ranked higher than N. To illustrate using the profile above for the election of mayor of Kenton, at election time, t_E, f(Alice; t_E) = 0.519, f(Brent; t_E) = 0.562, and f(Carla; t_E) = 0.505.

The fitness of a candidate can be calculated at any point in the election cycle whenever a rank ordering of all candidates including N is available. Assuming polls adopt BwN, this can allow tracking of each candidate’s fitness over time and before any election. If a candidate’s fitness ≤ 0.5, either the candidate can drop out of the race or redirect campaign efforts in the hope of raising fitness above 0.5. To illustrate how this might play out, we consider a simple example of a seven candidate race. Table 1 displays the fitness of each candidate at the beginning of the campaign at Iteration 0. For each subsequent iteration, or new poll, the fitness of each of the two candidates with the smallest fitness ≤ 0.5 is replaced by a uniformly distributed random number between 0 and 1. In this particular example, each candidate’s fitness exceeds 0.5 after the 10th iteration so the slate is completely acceptable and N would be ranked last if the election were held then. A more realistic example would be based on a dynamic model in which the number of candidates can change along with the fitness of each.

Table 1.

Hypothetical Evolution of Candidate Fitness in a Seven Person Race.

Candidate
Iteration	1	2	3	4	5	6	7
0	0.2153	0.7355	0.1894	0.3510	0.2687	0.4751	0.6055
1	0.1089	0.7355	0.0839	0.3510	0.2687	0.4751	0.6055
2	0.0932	0.7355	0.0354	0.3510	0.2687	0.4751	0.6055
3	0.3385	0.7355	0.0274	0.3510	0.2687	0.4751	0.6055
4	0.3385	0.7355	0.3364	0.3510	0.7011	0.4751	0.6055
5	0.0114	0.7355	0.5451	0.3510	0.7011	0.4751	0.6055
6	0.7153	0.7355	0.5451	0.9788	0.7011	0.4751	0.6055
7	0.7153	0.7355	0.5451	0.9788	0.7011	0.4590	0.6055
8	0.7153	0.7355	0.5451	0.9788	0.7011	0.2003	0.6055
9	0.7153	0.7355	0.5451	0.9788	0.7011	0.3032	0.6055
10	0.7153	0.7355	0.5451	0.9788	0.7011	0.5977	0.6055

Options If “None of These Candidates” Is Ranked First

It is likely that a primary reason permitting N as a binding option has not been accepted is the problem of what to do if N wins an election. From a welfare standpoint, denying the electorate the ability to reject some or all of a slate of candidates incurs a moral hazard. Without N, one can present unacceptable candidates that more or less fit the aims of those who wish to game the system to the detriment of the constituency. Introducing N as a candidate presents the electorate the opportunity to actively defeat efforts to game the system and so is inherently more democratic.

In a BwN voting system, N would win with lower frequency than an unacceptable candidate in the current plurality voting system. Nevertheless, because it might, we examine some possible dispositions in the event N prevails.

The most obvious are

having the office remain vacant, having the office filled by appointment, re-opening nominations or holding another election (in a body operating under parliamentary procedure), or it may have no effect whatsoever, as in India and the US state of Nevada, where the next highest total wins regardless. (Wikipedia, n.d.)

Each has its issues. For example, it may not be practical to leave the office open for any length of time necessitating extending the term of the incumbent or appointing someone until a new election results in a clear win. Should nominations be reopened, candidates who were beaten by N should be excluded from participating. Additional elections are time and resource consuming and are no guarantee that a new slate will yield a winner. While it might be tempting to select the second-ranked candidate when his or her fitness ≤ 0.5, this is not recommended because it undermines a key benefit of BwN, namely giving voters individual vetoes so that when ranked first, N becomes binding. It might be made more palatable, however, if compensation is reduced and a new election is scheduled as soon as practical in which the new incumbent cannot run.

If during a party caucus or primary election to select delegates to a convention, no candidate achieves a fitness above N, a slate of delegates not bound to any of the candidates can be sent. In this case, the N option would stand for uncommitted.

The selection of the president of the United States is more complicated because it is made by electors who are chosen by voters in each state. The electors are generally, though not always, committed either by state law or party pledge to vote for the candidate of the party they represent. If the voters in a state rank N in first place over all electors, the state could be represented in the Electoral College by an unpledged slate comprised of independent electors. In this case, the N option would stand for unpledged.

While each of these options has shortcomings, the expectation is that, should BwN be adopted with N binding, more agreeable slates and candidates will emerge so that the issue arises infrequently.

BwN and Partial Voting

BwN offers an alternative approach to dealing with partial rankings. As stated above, if there are n candidates and a voter ranks all of them, the BC assigns the highest ranked candidate n – 1 down to 0 for the lowest ranked. Note that when all voters submit a ranking of all the candidates this is equivalent to assigning n to the first, n – 1 to the second, and so on with the candidate ranked last getting 1. But if a voter only ranks m candidates where 1 ≤ m < n, how should Borda Scores be assigned? Emerson (2013) explores various proposed solutions and endorses what Saari (2008, p. 197, footnote 6) recommends, assigning m to the top-ranked candidate and deducting 1 for the next highest on down to the lowest rank who would receive 1. Emerson refers to this as the modified BC (MBC).

While MBC satisfies all four properties of BC, it does not give all voters equal weight in the outcome. In other words, it does not satisfy anonymity. BwN can be applied as follows to ensure that partial rankings are given equal weight without handing any candidates an unequal advantage. When a voter ranks m candidates, N followed by the candidates that were not ranked are appended and assumed to be tied. For example, in a seven candidate race, suppose that a voter chooses to rank only 3 as follows C1 ≻ C2 ≻ C3. It would be tallied with the other rankings as C1 ≻ C2 ≻ C3 ≻ N ≻ C4 ≈ C5 ≈ C6 ≈ C7. This is similar to the “BC averaged” considered in Emerson (2013, Table 2) but with a clear demarcation between acceptable and unacceptable candidates.

In the end, the choice of how to deal with partial ballots comes down to whether it is more important to punish voters who submit incomplete rankings, perhaps to give their candidates a disproportionate advantage, by diminishing their influence or to ensure that each voter has an equal impact.

BwN for General Applications

While well suited to address many of shortcomings of our current electoral system, BwN is also a decision methodology appropriate for many other applications. In some instances, the objective is no longer maximum participation because an important decision is typically made by a small dedicated group who must remain involved. The primary goal instead is an efficient consideration of and selection from a sufficiently broad range of alternatives.

One example is the ranking of wines by a panel of judges. This can be done either to obtain a top to bottom consensus, such as in the Judgment of Paris tasting in 1976 (Hulkower, 2009), or to award medals, which is a binning rather than a ranking. In the former case, N can be added to the list of wines being considered. Its position in the consensus ranking would then separate acceptable from unacceptable beverages. In the latter case, Hulkower (2012) offered an application of BwN:

Each judge rank orders each of the wines and inserts three markers, N_G, N_S, and N_B. Wines ranked above N_G are the judge’s selections for Gold medals. Similarly, those ranked between N_G and N_S are the judge’s choice for a Silver medal and those between N_S and N_B, for a Bronze medal. Those ranked below N_B would go without any medal (Hulkower, 2012, p. 4).

The BC can be an appropriate method for combining rankings by disparate criteria as part of trade studies (Hulkower & Neatrour, 2016). Especially during the earliest phases when the widest range of alternatives are under consideration, implementing BwN by inserting N as an option for subjective criteria can help narrow the choices going forward.

Similarly, customer preference surveys that ask for rankings of features would benefit by adding N. This would encourage respondents to rank all alternatives and enable them to segregate those that are desirable from those that are not. Using BwN then ensures that complete information is incorporated in the aggregated ranking, something that is not done if limits are placed on the number of features to be ranked. If N is ranked first, the product or service fails to gain approval and is sent back to the drawing board.

Practical Considerations

Adopting BwN for elections would likely face several challenges. These include resistance to ranking all the candidates instead of choosing just one, susceptibility to strategic voting, and determining courses of action should N prevail.

While we have not identified any evidence of resistance to ranking, the issue remains open. As noted above, several cities use RCV which requires partial ranking of candidates. Online surveys regularly employ drag-and-drop software for ranking preferences (see, for example, https://help.surveymonkey.com/articles/en_US/kb/How-do-I-create-a-Ranking-type-question). This can be modified for elections and to accommodate ties, if permitted. Certainly, introducing BwN should be done after voters are educated about the method and its benefits and over a period of time to allow them to get comfortable with ranking. An obvious place to start is in school elections and in cities that are already using RCV. If some voters chose to continue to vote for just their top choice, then their vote can be considered partial and handled as described in the section “BwN and Partial Voting.”

Gibbard (1973) and Satterthwaite (1975) independently proved that when n ≥ 3, all voting procedures are susceptible to manipulation. In other words, all voting methods admit the possibility of strategic voting where an elector ranks candidates insincerely, that is, different from what his or her true preferences are. In particular, one can readily construct examples of strategic voting for the BC and BwN from a given profile. For instance, suppose the 10 voters who preferred Carla in the election for mayor of Kenton decided to vote Carla ≻ N ≻ Brent ≈ Alice instead of Carla ≻ Brent ≻ Alice ≻ N to give their candidate a boost over the other two. The new consensus ranking would be N ≻ Carla ≻ Alice ≻ Brent which would trigger the need to consider one of the alternatives listed in the “Options if ‘None of these candidates’ is ranked first.” But such a strategy would be counterproductive for the 10 voters because their true preference had N in last place and the outcome may eliminate the chance for their second choice, Brent, to assume office.

Saari (1990) has demonstrated that when n = 3, the BC is the least susceptible to manipulation on a micro scale, that is, by individual voters as opposed to on a macro scale by a large group. For n > 3, this is not the case, but he “shows that the BC always fares fairly well” (Saari, 1990, p. 21). Saari advises “While manipulative behavior is an important concern, it must never become the single deciding factor in the choice of a system” (Saari, 1994, p. 12). BwN inherits from Borda the unique ability to yield a consensus ranking that best reflects voter preferences and uses all of the information in the ranking instead of part, making a strong case for its adoption. In contrast, plurality voting is always more susceptible to manipulation while disregarding all but voters’ first preferences and, if N is permitted, as illustrated above, possibly increasing the likelihood it would win.

In a previous section, we addressed what might be done if N is ranked first. We also illustrated when a strategic vote by a bloc caused N to win. Clearly, the interplay between the temptation to vote strategically and the possibility that N might come out on top must be considered. But until BwN is actually implemented, the most desirable ways of dealing with N winning an election remain uncertain. In any case, we recommend a slow roll out in cases where the stakes are lower if N wins so that the benefits and pitfalls of the method can be tested in real-world elections.

Conclusion

BwN inherits the unique properties of the BC and thus shares its theoretical foundation. Because it also satisfies the anonymity condition, BwN overcomes the problem of disproportionality that plagues Dodgson’s method due to how it scores tied alternatives. It therefore provides the answer to the question “What is the most rational way to aggregate a set of preferences that includes N as an option?” It also can be used to incorporate partial rankings more fairly without any candidate gaining an unfair advantage.

Giving the electorate a veto over an objectionable slate of candidates as an alternative to simply sitting out an election is the power of none. As such, it counteracts the pernicious effects and social costs of foisting unacceptable slates of candidates on an electorate which becomes increasingly alienated. The fitness of a candidate defined via BwN is a metric that can track the candidate’s attractiveness over the course of a campaign. There is also evidence that the ranking of candidates required by BwN would encourage greater civility. For all the reasons the BC is mathematically superior to other positional voting schemes, BwN will ensure that its power is not exercised by a minority of voters and will truly reflect the opinion of an electorate that includes the participation of the previously disaffected.

BwN extends the concept of checks and balances that is a key strength of American democracy by giving veto power to the electorate who necessarily rely on political parties and other intermediaries for the selection of slates of candidates and on electors to choose a chief executive. Broader participation of the electorate with the veto power of none should encourage a selection of candidates with better electability and the rejection of those with extreme positions or too narrow a focus. Then N would not prevail.