Consider an ordinal election or referendum in which voters rank three alternatives. Let \(v_{ij}\) denote the proportion of voters ranking alternative i first and j second, for \(i, j \in \{a, b, c\}\). (A ballot ranking i first and j second implicitly ranks k last.) Let \(v_{ix}\) denote the proportion of ballots listing \(i \in \{a, b, c \}\) first and no candidate second, which is a relevant possibility if incomplete rankings are permissible.Footnote 16 Then the winner of any election using an anonymous ordinal voting method depends on the vector
$$\begin{aligned} {\mathbf {v}} \equiv (v_{ab}, v_{ac}, v_{ax}, v_{ba}, v_{bc}, v_{bx}, v_{ca}, v_{cb}, v_{cx} ). \end{aligned}$$
Because its components must sum to one, \({\mathbf {v}}\) lives on the eight-dimensional simplex. In principle, that eight-dimensional space could be divided into win regions in the same way the two-dimensional ternary diagram was above. The problem is that visualizing win regions in eight dimensions is essentially impossible.
My solution is to depict win regions as a function of first-preference shares (the proportion of voters ranking each candidate first) given conditional second-preference shares (the proportion of voters ranking each candidate second conditional on ranking a given candidate first). Let \(v_a \equiv v_{ab} + v_{ac} + v_{ax}\) denote a’s first-preference share, with \(v_b\) and \(v_c\) defined equivalently; let \(p_{ab} \equiv \frac{v_{ab}}{v_a}\) indicate the proportion of ballots listing b second conditional on listing a first, with \(p_{ac}\), \(p_{ax}\), \(p_{ba}\), etc. defined similarly and the set of these proportions collectively referred to as conditional second-preference shares. Noting that \(v_c = 1 - v_a - v_b\) and \(p_{ax} = 1 - p_{ab} - p_{ac}\) (and similarly for \(p_{bx}\) and \(p_{cx}\)), we can express each component of \({\mathbf {v}}\) in terms of two first-preference shares (\(v_a\) and \(v_b\)) and six conditional second-preference shares (\(p_{ab}, p_{ac}, p_{ba}, p_{bc}, p_{ca}, p_{cb}\)).Footnote 17 We can then use the ternary diagram to represent the proportion of voters ranking each alternative first \((v_a, v_b, v_c)\), just as we do (implicitly) in the plurality ternary diagram, while dividing the diagram into regions in which each alternative would win, given the conditional second-preference shares and the voting system. I call these regions first-preference win regions.
Note that by definition the winner of any anonymous ordinal voting method depends only on \({\mathbf {v}}\), and note also that (given the conditional second-preference shares) \({\mathbf {v}}\) depends only on \(v_a\), \(v_b\), and \(v_c\). It follows that, again given the conditional second-preference shares, the winner of any anonymous ordinal voting system depends only on \(v_a\), \(v_b\), and \(v_c\) and can therefore be identified on the ternary diagram.
I now apply this approach to prominent ordinal voting rules.
Condorcet methods
A Condorcet method selects as winner a candidate who wins a pairwise contest against every other candidate (i.e. a Condorcet winner). Condorcet methods differ in how to proceed if there is no Condorcet winner. I begin by showing how to identify first-preference regions in which there is a Condorcet winner; I then show how to fill in the remaining cyclic region depending on the specific Condorcet method.
Let a majority tie line for two candidates indicate first-preference shares where, given conditional second-preference shares, the two candidates would tie in a pairwise majority vote. This is true for i and j where
$$\begin{aligned} v_i + v_{ki} = v_j + v_{kj}, \end{aligned}$$
which can be written in terms of first-preference shares and conditional second-preference shares as
$$\begin{aligned} v_i = v_j + v_k (p_{kj} - p_{ki}). \end{aligned}$$
(1)
Equation (1) describes the ij majority tie line.
To construct a majority tie line we identify the two points where the line intersects an edge of the ternary diagram. One intersection point, which we call y, is located where \(v_k = 0\) and \(v_i = v_j = \frac{1}{2}\); this is where half of the voters rank i first and the other half rank j first, producing a majority tie. Point y is labeled for the Deal-No Deal majority tie line on Fig. 3a, b. The location of the other intersection point of the ij-majority tie line, which we call z, depends on the conditional second-preference shares for ballots ranking k first. If \(p_{kj} > p_{ki}\) (i.e. if ballots ranking k first more often rank j second than i second), then z is located where \(v_i = v_k (p_{kj} - p_{ki})\), \(v_j = 0\), and \(v_k = 1-v_i\); otherwise, z is located where \(v_i = 0\), \(v_j = v_k (p_{ki} - p_{kj})\), and \(v_k = 1-v_j\). Figure 3a shows extreme cases for the Deal-No Deal majority tie line: if Remain voters (i.e. those who submit ballots ranking Remain first) always rank Deal second, then the Deal-No Deal majority tie line is the one connecting points y and \(z_1\); if Remain voters always rank No Deal second, then the Deal-No Deal majority tie line is the one connecting point y and \(z_2\).
Figure 3b shows the actual Deal-No Deal majority tie line given the Brexit preferences from the Nov. 2018 poll. Remain voters are much more likely to rank Deal second than No Deal second, such that the Deal-No Deal majority tie line is much closer to the No Deal vertex than the Deal vertex, and Deal beats No Deal in a pairwise majority vote over much more than half of the ternary diagram. The key general point is that the “tilt” of the ij-majority tie line is a measure of how k’s supporters (i.e. voters who rank k first) tend to rank i and j: the more k’s supporters favor i as their second choice, the more the ij-majority tie line tilts away from i’s vertex, indicating that i can tie j in a pairwise vote when receiving a lower first-preference share.
Figure 3c shows all three majority tie lines for the Brexit case. The angle of the majority tie lines reflect Remain voters’ preference for Deal over No Deal (noted immediately above), Deal voters’ preference for No Deal over Remain (indicated by the counter-clockwise tilt of the Remain-No Deal tie line), and No Deal voters’ preference for Deal over Remain (indicated by the counter-clockwise tilt of the Remain-Deal tie line).
Generically, majority tie lines do not intersect at a point; the three majority tie lines divide the ternary into six sub-triangles in which a different transitive majority preference order obtains, plus one sub-triangle (labeled “Cycle” in Fig. 3c) in which there is a majority cycle: for first-preference shares in this triangle (given the pattern of second preferences), a majority prefers Deal over No Deal (because the result is above the Deal-No Deal majority tie line), a majority prefers No deal to Remain (because the result is to the right of the Remain-No deal majority tie line), and a majority prefers Remain to a Deal (because the result is below the Remain-Deal majority tie line).
Figure 3d shows the implied first-preference win regions given the Nov. 2018 pattern of conditional second preferences, with a black dot indicating the first-preference result. (To aid comparison across diagrams, I include faint plurality tie lines and lines where each alternative receives half of first-preference votes; first-preference win region boundaries for Condorcet methods must be within the triangle formed by the latter set of lines.) Deal and Remain essentially tie for first, with the first-preference result lying along the Deal-Remain majority tie line.
In Fig. 3d, I leave the cyclic triangle unfilled, but there are various methods for determining the winner in the event of a cycle among three candidates. On the diagram, the cyclic triangle can be filled in according to the specific method:
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Black’s method (or Condorcet–Borda) decides the winner by Borda count, so the cyclic triangle can be filled in using the method shown in the next section.
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Condorcet-IRV (equivalent in the three-candidate case to what Tideman (2017, p. 232) calls Alternative Smith) decides the winner by instant runoff voting, so the cyclic triangle can be filled in using the IRV method shown further below.
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The Kemeny, Kemeny–Young, or minimax/maximin methodFootnote 18 chooses the candidate with the smallest losing margin. To fill in the cyclic triangle, we must define and draw an additional set of tie lines that identify first-preference results where (given the pattern of second preferences) two losing margins are the same. These tie lines bisect the vertices of the cyclic triangle and meet at a point within it. I give an example for the Brexit poll in Figs. 6 and 7.
Borda count and other positional methods
Given three candidates, a positional method gives each alternative 1 point for each ballot on which it is ranked first and \(s \in [0,1]\) points for every ballot on which it is ranked second; the winner is the candidate with the most points. Special cases include plurality (\(s = 0\)), Borda count (\(s=1/2\)), and anti-plurality (\(s = 1\)).
A positional tie line for two candidates indicates first-preference shares where the two candidates have the same number of points, given s and the conditional second-preference shares. Assuming that an ix ballot (i.e. one that lists i only) awards 1 point to i and 0 to j and k,Footnote 19 candidates i and j tie when
$$\begin{aligned} v_i + s(v_{ji} + v_{ki}) = v_j + s(v_{ij} + v_{kj}), \end{aligned}$$
which can be written in terms of first-preference shares, conditional second-preference shares, and s as
$$\begin{aligned} v_i(1 - s p_{ij}) = v_j(1 - s p_{ji} ) + s v_k (p_{kj} - p_{ki}). \end{aligned}$$
(2)
This describes the ij positional tie line. The first-preference win regions are defined by the intersection of positional tie lines.
To construct the diagram, we begin by identifying the point (y) where the positional tie line intersects the edge connecting the i vertex and the j vertex; at point y, i and j tie while k receives no first-preference support (i.e. \(v_k = 0\)). Substituting \(v_k = 0\) and \(v_j = 1 - v_i\) into Eq. (2) and rearranging, we obtain
$$\begin{aligned} v_i = \frac{1 - sp_{ji}}{2 - s(p_{ij} + p_{ji})}. \end{aligned}$$
(3)
The location of point y depends on how often j’s supporters rank i second and vice versa. If ballots ranking j first never rank i second while ballots ranking i first always rank j second (i.e. \(p_{ji} = 0\) and \(p_{ij} = 1\)), then \(v_i = \frac{1}{2-s}\) at point y; thus for Borda count (\(s = 1/2\)) point y would be located where \(v_i = 2/3\) and \(v_j = 1/3\), which (assuming i is No Deal and j is Deal) is the point labeled \(y_1\) on Fig. 4a. If by contrast ballots ranking j always rank i second while ballots ranking i first never rank j second (i.e. \(p_{ji} = 1\) and \(p_{ij} = 0\)), then \(v_i = \frac{s}{2-s}\) at point y; thus for Borda count (\(s = 1/2\)), point y would be located where \(v_i = 1/3\) and \(v_j = 2/3\), which is the point labeled \(y_2\) on Fig. 4a.
Next we seek to identify point z, where the ij-positional tie intersects another edge of the ternary diagram. The location of this point depends on several conditional second-preference shares, but whether it lies on the ik edge or the ij edge depends on the sign of \(p_{ki} - p_{kj}\): if k’s supporters more often rank j second, then z lies on the edge between the i and k vertices; otherwise, z lies on the edge between the j and k vertices. More specifically, by substituting \(v_j = 0\) and \(v_k = 1 - v_i\) into Eq. (2), we obtain
$$\begin{aligned} v_i = \frac{s (p_{kj} - p_{ki})}{1 - s(p_{ij} - p_{kj} + p_{ki})}, \end{aligned}$$
(4)
which locates z in the case where k’s supporters favor j; by substituting \(v_i = 0\) and \(v_k = 1 - v_j\) into Eq. (2), we obtain
$$\begin{aligned} v_j = \frac{s (p_{ki} - p_{kj})}{1 - s(p_{ji} - p_{ki} + p_{kj})}, \end{aligned}$$
(5)
which locates z in the case where k’s supporters favor i. On Fig. 4a, \(z_1\) indicates the extreme case for the Deal-No Deal tie line (under Borda count) where all Remain supporters rank Deal second, and \(z_2\) indicates the other extreme case where all Remain supporters rank No Deal second.Footnote 20
Broadly, then, the angle of the ij-positional tie line can (like the ij-majority tie line) be thought of as a measure of how k’s supporters feel about i and j: the more k’s supporters favor i as their second choice, the more the ij-positional tie line tilts away from i’s vertex. The connection is not as clear as in the majority tie line case, though, because the location and angle of the line also depends on the second preferences of i’s and j’s supporters.
Figure 4b shows the actual Deal-No Deal tie line for Borda count (given conditional second preferences from the Nov. 2018 Brexit poll). Point y is located near the midpoint of the Deal-No Deal edge, indicating that supporters of Deal and No Deal are about equally likely to list each other’s first choice second; point z is located close to the midpoint of the Remain-No Deal edge, indicating that supporters of Remain are far more likely to list Deal second than No Deal second.
Figure 4c shows all three Borda tie lines. Unlike majority tie lines, positional tie lines always intersect at a point,Footnote 21 dividing the ternary diagram into six subtriangles (each corresponding to an order of finish).
Finally we obtain the first-preference win regions, shown in Fig. 4d. Because Deal attracts most second preferences from voters whose first preference is Remain or No Deal, Deal’s first-preference win region is substantially larger than the other two. The first-preference result is located comfortably inside Deal’s win region, indicating that Deal would win by Borda count.
Instant-runoff voting
In an instant-runoff voting election (e.g. for Australian lower-house seats),Footnote 22 voters submit ranked ballots and the lowest-scoring candidates are successively eliminated until only one remains. In the three-candidate case where elimination is based on first preferences (i.e. plurality votes), this simplifies to a procedure in which the candidate who receives the lowest first-preference share is eliminated and, of the remaining candidates, the winner is the one who is ranked higher on a larger share of ballots.
The IRV ternary diagram is built from a combination of positional tie lines (determining who is eliminated) and majority tie lines (determining which of the remaining candidates is the winner). We first use plurality tie lines to identify regions in which each candidate receives the lowest share of first-preference votes and is eliminated; these elimination regions are shown in Fig. 5a. Next, we superimpose the majority tie lines introduced above. In the region where a given candidate is eliminated, the majority tie line for the other two candidates determines which candidate wins; for example, in the region where Remain is eliminated, the Deal-No Deal majority tie line (singled out in Fig. 3b above) determines which candidate wins. To reflect this, in Fig. 5b I draw each majority tie line with a solid stroke in the region where it determines the winner (and a dashed stroke elsewhere), and I label each portion of the resulting figure according to how the non-eliminated candidates fare. The resulting first-preference win regions are shown in Fig. 5c. The irregular shape of these win regions reflects the combination of positional and majority tie lines that determine a winner in IRV, and will play a role in later illustrations of join-inconsistency and other voting system properties.
To develop intuition about the IRV diagram, consider Fig. 5d, which shows IRV first-preference win regions under a possible shift in conditional second preferences. (We will shortly see that a similar shift actually happened.) Recall that the majority tie line for alternatives i and j reflects the second-preference pattern of voters whose first preference is k. Suppose Deal voters become more favorable to Remain compared to No Deal (i.e. more likely to rank Remain second rather than No Deal); then we would see a clockwise tilt in the Remain-No Deal majority tie line, indicated by the rightward-pointing arrow at the boundary between the Remain-No Deal win regions in Fig. 5d. The other two arrows highlight a hypothetical shift (compared to Fig. 5c) of Remain voters toward Deal (indicated by the counterclockwise movement of the boundary between the Deal and No Deal win regions at the right of the figure) and of No Deal voters toward Deal (indicated by the clockwise movement of the boundary between the Deal and Remain win regions at the left of the figure).
Figure 5c indicates that, given November 2018 Brexit preferences, the IRV winner would be either Remain or Deal. Deal and No Deal are effectively tied for second in first preferences, such that the first-preference result sits at the edge between the region where Deal is eliminated and the region where No Deal is eliminated. If Deal is eliminated, then Remain would defeat No Deal in the “runoff”: the first-preference result is to the left of the Remain-No Deal majority tie line. If No Deal is eliminated, then Remain and Deal would effectively tie in a pairwise majority vote: the first-preference result essentially lies on the Remain-Deal majority tie line. Thus Remain could be helped by a hypothetical shift in first-preference shares that strengthened No Deal, leading to Deal being eliminated and Remain winning. In fact, such a shift in preferences occurred, as we will shortly see.
Other voting rules
We can depict further ordinal voting systems by combining and/or extending elements already introduced. Although I do not depict these below for reasons of space, consider:Footnote 23
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In the Bucklin system, a candidate who wins a majority of top rankings is the winner; if there is no such candidate, the candidate who wins the largest share of first or second rankings is the winner. For the Bucklin diagram, begin by identifying regions where each candidate wins a majority of first preferences (each defined by a line connecting the midpoint of two edges) and fill in the remaining central area using anti-plurality win regions.
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The Coombs system is a variation of IRV in which elimination is based on the number of bottom rankings rather than top rankings. Specifically, a candidate who wins a majority of top rankings is the winner, but if there is no such candidate, the candidate who is ranked last on the most ballots is eliminated and the winner chosen based on rankings of the remaining two. To make the Coombs diagram, start with the regions where each candidate wins a majority of first preferences (as in Bucklin); to fill in the remaining central area, draw anti-plurality tie lines (positional tie lines with \(s = 1\)) to identify elimination regions and use majority tie lines to identify the winner within each elimination region.
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In the Nanson system the winner is determined by successive elimination of candidates who receive a below-average Borda score. For the Nanson diagram we must define a new tie line that identifies first-preference shares where, given second-preference shares, a candidate receives the average Borda score;Footnote 24 these tie lines partition the diagram into regions where either one or two candidates is eliminated. In regions where a single candidate k is eliminated, use the ij-majority tie line to determine which of the remaining candidates is the winner.
An update to the Brexit poll
A new Brexit poll was conducted in June, 2019, producing another set of estimated preference shares.Footnote 25 Table 2 reports the June, 2019, shares alongside the November, 2018, shares reported and illustrated above. Between November and June, first-preference support dropped for Deal and first-preference support for No Deal rose; also, preferences appear to have become more single-peaked, with Deal becoming the second choice for a larger proportion of voters ranking Remain or No Deal first.
Table 2 Brexit preferences, November 2018 and June 2019 Figure 6 displays the ternary diagrams for plurality, Borda, anti-plurality, Kemeny–Young, and IRV for both the November 2018 poll (left column) and the June 2019 update (right column). The plurality diagram shows Remain winning in both polls, with the black dot moving southeast between the two polls to reflect the shift in first-preference support from Deal to No Deal. In the Borda and anti-plurality diagram, the first-preference win region for Deal expands considerably from November 2018 to June 2019 (reflecting the increase in second-preference support for Deal as preferences became more single-peaked), such that even as Deal’s first-preference support drops it wins more comfortably in June 2019 under both voting rules. Whereas Deal and Remain were essentially tied in the November 2018 poll in the Kemeny–Young (Condorcet) system, Deal wins in June 2019 (despite securing less first-preference support) because of the greater second-preference support from No Deal. In IRV, by contrast, the change in preferences was favorable to Remain. Recall that in November 2018 Deal and No Deal were essentially tied for second in first-preference shares; Remain would win if Deal were eliminated, but Remain and Deal would essentially tie if No Deal were eliminated. By June 2019, Deal finishes securely in last place in first-preference shares, after which Remain defeats No Deal in the IRV system. Thus not only do the different systems produce different winners with a given preference profile, but the same change in preferences can be beneficial to a given alternative in one system and harmful in another.
In Fig. 7, we zoom in on the Kemeny–Young diagram for the Nov. 2018 Brexit poll from Fig. 6. Where the majority tie lines form the boundary between two win regions they are drawn with a solid stroke; I use a dashed stroke to show the continuation of these lines where they do not form win region boundaries. Within the triangle defined by these dashed lines, there is a majority cycle where Remain beats Deal, Deal beats No Deal, and No Deal beats Remain. The Kemeny–Young rule says that the winner is the candidate who is defeated by the smallest margin in pairwise comparisons. To partition the majority cycle region according to this rule, we draw a new set of tie lines based on the margin of candidates’ defeats. For example, the Remain-Deal tie line (which bisects the uppermost vertex of the cyclic triangle) indicates first-preference shares where the margin by which Remain is defeated by No Deal is the same as the margin by which Deal is defeated by Remain. Each such tie line bisects an angle of the cyclic triangle, and the three lines meet at a point where the three candidates have the same margin of defeat.
Handling more than three alternatives
So far we have considered cases with only three alternatives, as is common in previous geometrical approaches to voting (e.g. Saari 1994). The diagram can still be useful for many voting methods when there are more candidates. In particular, if there are three candidates a, b, c who would beat any other candidate in pairwise rankings, then any Condorcet method that resolves a cycle through an ordinal method applied to the Smith set can be diagrammed by considering only those three candidates, i.e. by eliminating candidates \(d, e, \ldots\) from all ballots and considering only orderings of a, b, and c. Similarly, if there are three candidates a, b, and c who would be the last candidates standing in IRV or a similar elimination procedure, then we can eliminate all other candidates from the ballots and use the diagram to represent the final rounds of the competition. (Unfortunately the same approach does not work for positional methods: crucial information is lost if we reduce e.g. nine ranks to three.)
Example: 2018 mayoral election in San Francisco
The June 5, 2018, special mayoral election in San Francisco used a version of IRV in which voters could rank up to three candidates. Table 3 reports the number of times each candidate was ranked first, second, and third,Footnote 26 showing that three candidates (London Breed, Mark Leno, and Jane Kim) received the vast majority of first and second preferences. Assuming these three candidates are the last to be eliminated, we can ignore the other candidates and use the diagram to depict the final stages of the counting procedure.
Table 3 Rankings in 2018 San Francisco mayoral election The joint distribution of preferences over the three frontrunners is shown in Table 4. Note that the table has the same format as Table 1 except that a column is added to indicate ballots that rank only one candidate, which arises because voters were not required (indeed, not permitted) to rank all candidates. In fact, around a quarter of all ballots ranked none of the frontrunners and are thus eliminated entirely; of those that ranked at least one frontrunner, around a quarter ranked all three frontrunners, around half ranked ranked exactly two, and around a quarter ranked exactly one.
Table 4 First- and second-preferences within top three candidates in San Francisco mayoral election, June 2018 Using the proportions in Table 4 we can make a diagram exactly as described above. As shown in Fig. 8, the results for Black’s method (Condorcet with Borda to break cycles) and IRV happen to be very similar to the Brexit case. Breed narrowly wins both under Black’s method and IRV.
Example: 2018 election in Maine’s 2nd congressional district
The 2018 congressional election in Maine was the first use of IRV in a U.S. federal election. In Maine’s 2nd district, two candidates (incumbent Bruce Poliquin and Jared Golden) won over 90% of first-preference votes (with Poliquin winning slightly over 46% and Golden winning slightly under 46%). Table 5 shows the preferences over the top three candidates once the fourth-placed candidate had been eliminated. Notably, over half of all ballots ranked only one candidate, including almost a third of ballots ranking Bond first.
Table 5 First- and second-preferences among top three candidates in Maine 2nd district congressional election, Nov 2018 The corresponding diagrams for Black’s method and IRV appear in Fig. 9. In both systems the result was essentially decided by the Golden–Poliquin majority tie line, which depends on Bond’s second-preference shares; given Poliquin’s narrow first-preference advantage, Golden won because voters who ranked Bond first favored Golden as their second choice (by a two-to-one margin).