APV description

APV may be described as follows (refer to Subsection Notation reference for descriptions of the mathematical notation):

Voters complete their ballots by assigning to each candidate a nonnegative number representing the voter’s self-assessed level of support for or rating of the candidate relative to the other candidates. If a voter fails to assign a value to a candidate, the value should default to zero so that voters do not have to assign a value to every candidate on the ballot. Conceptually, a voter’s completed ballot may represent any nonnegative function on the set of candidates. (See an example of a possible APV ballot system in Subsection Discussion that permits the allocation of support to candidates individually and by party.)

Voters have control over the complexity of their voting task by their choice of a numerical rating system, with binary approval voting being the simplest. Note that APV encourages the participation of political parties in elections by permitting voters to give the same, or nearly the same, level of support to candidates who represent the same political party.

APV stores and processes votes as probability mass functions (pmfs) on sets of candidates. A pmf on a set of candidates is nonnegative, sums to one over the set, and represents each vote’s allocation of support among the candidates in the set. APV automatically converts votes into their corresponding pmfs using function pmf (see Subsection Notation reference for a description of function pmf).

APV returns the set of elected candidates as a function of the following inputs:

1. The number, n > 0, of open positions to be filled by the election.
2. A vector, c, of unique positive integers identifying the candidates from whom the n open positions may be filled. (|c| must be at least n. If |c| equals n, APV returns c as the set of elected candidates.)
3. A matrix, V, of the votes cast in the election, where, for each vote (row) index, x, and candidate (column) index, y, V(x, y) is the nonnegative number (relative level of support or rating) that the x th voter specified for the y th candidate on the voter’s ballot. The elements of c must be column indices of V.
4. A row vector, w, of positive numbers (weights) such that |w| equals the number of rows that V has and, for each row index, x, of V, w(x) is the weight given to the x th vote when computing each candidate’s weighted-vote total.

APV uses a threshold value for candidate weighted-vote totals to identify elected candidates. The threshold is t₀ = ‖w‖₁/(n + 1), where ‖w‖₁ is the sum of the weight values in vector w.

APV selects candidates to fill the open positions as follows:

1. 1. While the number, |c|, of candidates represented in c exceeds the number, n, of open positions, do the following:
   1. 1.1. Set F equal to the restriction, V(:, c), of the votes-cast matrix, V, to the candidates represented in c. Note that each column of F represents the same candidate as the corresponding element of c.
   2. 1.2. Subtract the smallest value in each row of F from every value in the row.
   3. 1.3. Convert the rows of F to their corresponding pmf vectors by setting F equal to pmf(F).
   4. 1.4. Set f equal to the row vector w * F. Each element of f is the weighted-vote total of the candidate represented by the corresponding element of c.
   5. 1.5. While the weighted-vote total of any candidate exceeds the threshold [i.e., while any(f > t₀)], do the following:
      1. 1.5.1. A weighted-vote total is deficient if it is less than t₀. Every candidate whose weighted-vote total is not deficient is identified as an elected candidate. Set nd equal to find(f ≥ t₀), which contains the indices of f at which f is not deficient. Note that 1 ≤ |nd| ≤ n if no rounding errors have occurred.
      2. 1.5.2. If |nd| > n, then rounding errors have occurred at the limit of the computer’s numerical precision. Correct them by setting f(y) equal to t₀ for every index, y, of f such that f(y) > t₀. Then continue from Instruction 1.6.
      3. 1.5.3. If |nd| = n, then return c(nd) as the vector whose elements identify the n election winners and exit APV.
      4. 1.5.4. The portion of a candidate’s weighted-vote total above the threshold, t₀, does not contribute to the candidate’s election and is, therefore, wasted unless redistributed to candidates whose weighted-vote totals are deficient to help fill the remaining n − |nd| > 0 unfilled positions. A weighted-vote total is excessive if it exceeds t₀. Remove the excess weighted-vote totals from the candidates with excessive weighted-vote totals as follows: for each index, y, of f such that f(y) > t₀, set F(:, y) equal to F(:, y) * t₀/f(y).
      5. 1.5.5. Transfer the excess weighted-vote totals to the candidates with deficient weighted-vote totals as follows: Set d equal to find(f < t₀) which contains the indices of f at which f is deficient. For each row index, x, of F, set F(x, d) equal to Note that every row of F represents a pmf on the set of candidates represented by c, i.e., F equals pmf(F).
      6. 1.5.6. Update f as follows: For each index, y, of f such that f(y) > t₀, set f(y) equal to t₀. And set f(d) equal to w * F(:, d). Note that f equals w * F.
   6. 1.6. No excess candidate weighted-vote totals remain. After all excess candidate weighted-vote totals have been reallocated, a candidate is identified as defeated if the candidate cannot be elected without eliminating a candidate with a larger weighted-vote total. Identify and remove defeated candidates from c as follows:
      1. 1.6.1. Let T be the largest element of f such that Note that no candidate in c(f < T) can be elected without eliminating one or more candidates in c(f ≥ T) because ‖f(f < T)‖₁ < T ≤ t₀ and |c(f ≥ T)| > n.
      2. 1.6.2. If any(f < T), then remove the candidates in c(f < T) from c by setting c(f < T) equal to [](null). (Note that APV takes advantage of opportunities to eliminate multiple candidates simultaneously.)
      3. 1.6.3. Otherwise, T is the smallest element of f, i.e., c equals c(f ≥ T). At least two elements of f equal T because, if not, the smallest element of f(f > T) satisfies the conditions of Instruction 1.6.1 above, contradicting T’s status as the largest such element of f. Set Tindices equal to find(f ≤ T), which contains the indices of f at which f equals T. Then randomly select a candidate from c(Tindices) and remove it from c.
2. 2. Return c as the vector whose elements identify the election winners and exit APV.

This completes the APV algorithm. Note that the algorithm has two possible exit points: at Instructions 1.5.3 and 2. Also note that the algorithm does not make any approximations: all computations are exact to the limit of the computer’s numerical precision.

The next subsection, Notation reference, describes the mathematical notation used above. The Octave scripts listed in Subsection Octave scripts provide further algorithm details and optimizations [13]. Subsection Discussion discusses APV and provides an example of a possible APV ballot system that allows the voter to allocate support to candidates both individually and by party.

Notation reference

Subsection APV description uses the following notation:

• Nonnegative functions on a set of candidates are represented by nonnegative row vectors whose column indices represent the candidates and whose value at each column index is the value of the function at the candidate represented by the column index.
• For every finite vector, v, |v| denotes the length (number of elements) of v.
• For every finite nonnegative vector, v, denotes the 1-norm of v, i.e., the sum of the vector’s elements.
• * denotes the matrix-multiplication operator.
• For every nonempty finite nonnegative row vector, v, pmf(v) denotes the pmf vector corresponding to v. Note that where e is the row vector whose length is |v| and every element’s value is one.
• In general, for every nonempty finite nonnegative matrix, M, pmf(M) denotes the matrix with the same number of rows as M such that, for each row, r, of M, the corresponding row of pmf(M) is pmf(r).
• For every vector, a, and finite vector, b, of a’s indices, a(b) denotes the vector of length |b| and of the same shape (row or column) as a, whose jth element, a(b)(j) is
• For every matrix, M, finite vector, r, of M’s row indices, and finite vector, c, of M’s column indices,
  • M(r,c) denotes the |r| by |c| matrix whose element M(r,c)(x,y) with row index x and column index y is
  • M(r,:) denotes M(r,c) when |c| is a column index of M and c(y) = y for every column index, y, of M.
  • M(:, c) denotes M(r,c) when |r| is a row index of M and r(x) = x for every row index, x, of M.
• For every logical vector, b,
  • find(b) denotes the vector obtained from a copy, a, of b by setting every element, a(j), of a whose value is “true” equal to j and deleting every element of a whose value is “false.”
  • any(b)’s value is “true” if b has an element equal to “true,” and is “false” otherwise.
• For every finite vector, a, and logical vector, b, such that |b| = |a|, a(b) denotes

Octave scripts

APV was coded, tested, and refined, in the Octave scripting language [13]. An Octave script is a list of computer instructions in the Octave scripting language. The script is in a text file with a.m extension. An Octave interpreter reads script files and executes the computer instructions contained in them. Octave scripts that implement APV are given in this subsection along with the scripts used to run the examples of APV’s election performance given in Subsection Apportioned-voting examples of the next section. The Octave interpreter is freely available for download and use from https://www.octave.org/download.html and https://ftp.gnu.org/gnu/octave/. [13] is a detailed reference to the Octave scripting language and interpreter and is available from https://www.gnu.org/software/octave/doc/v9.2.0/ and the Documentation submenu of the Octave interpreter’s Help menu. The Octave and MATLAB^® scripting languages are closely related and most scripts are easily portable [13].

The following subsections list the Octave scripts used to implement the APV algorithm given in Subsection APV description and obtain the results of the examples of APV’s election performance given in Subsection Apportioned-voting examples [13]. Function apv, in Subsection Script for function apv, runs APV and returns numbers identifying the elected candidates. Two potentially-large inputs, V and w, of apv are prepared as Class ptr objects (see Subsection Script for object-class ptr) to reduce data storage requirements and run time. Function pmf, in Subsection Script for function pmf, is used internally by apv to convert nonnegative functions on sets of candidates to probability mass functions. The script in Subsection Script to run example 1 yields the results of Example 1. The script in Subsection Script to run example 2 yields the results of Example 2. Each script may be copied from this document, pasted into a new blank script in the Octave Editor, and saved as a file with the desired file name and “.m” extension. The script files may be in the same directory. The Octave interpreter is freely available for download and use from https://www.octave.org/download.html or https://ftp.gnu.org/gnu/octave/ [13].

Apportioned-voting examples

This subsection gives election examples illustrating how APV behaves in practice.

The first example, Example 1: proportional representation, illustrates that APV provides proportional representation of voting blocs in election results. The results of two-party partisan bloc voting, as a function of the total number of votes cast and the number of votes cast for each party, are compared.

The second example, Example 2: efficiency, illustrates the efficiency of APV when applied to elections with large numbers of voters (20 million), candidates (20), and positions to be filled (10).

Subsection Octave scripts of Section Materials and methods gives listings of the Octave scripts used to obtain the example results. The Octave interpreter is freely available for download and use from https://www.octave.org/download.html and https://ftp.gnu.org/gnu/octave/ [13].

Example 1: Proportional representation.

A partisan bloc vote is of the form, V(x,:), where, x is a row (vote) index of V and, for every column (candidate) index, y, of V, V(x, y) is positive if the yth column of V represents a candidate of a certain political party, and zero otherwise. Using the APV procedure, the following example compares the results of two-party partisan bloc voting, as a function of the total number of votes cast and the number of votes cast for each party.

Suppose the following:

• The number of positions to be filled by the election is n = 2.
• The vector of numeric identifiers of the candidates available to fill the positions is c = 1: 5 = [1 2 3 4 5].
• Candidates one and two represent the R party, and candidates three, four, and five represent the B party.
• The number of votes cast in the election is m.
• RBvotes is the 10-by-5 random-sample matrix of independent and uniformly-distributed pseudo-random numbers on the unit interval generated by first calling rand("state", 1) to initialize the Octave pseudo-random number generator “rand”; then setting RBvotes equal to rand(10,5).
• The m -by-5 matrix, V, of votes cast consists of the first nRvotes rows of matrix Rvotes and the first nBvotes = m − nRvotes rows of matrix Bvotes, where
  • Rvotes consists of the first two columns of RBvotes followed by three all-zero columns, and
  • Bvotes consists of two all-zero columns followed by the last three columns of RBvotes.
• The vote-weight vector, w, is the row vector with exactly m elements each of which equals one.
• The number of R-party candidates elected is nRElected.

Then the following runs of the APV procedure, apv(n, c, V, w), were made and results obtained:

Table 1 gives the results of applying the APV procedure to the election for nRvotes = 0: 8 when m = 8.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Results for Example 1’s eight-vote election.

https://doi.org/10.1371/journal.pone.0317580.t001

Table 2 gives the results of applying the APV procedure to the election for nRvotes = 0: 9 when m = 9.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Results for Example 1’s nine-vote election.

https://doi.org/10.1371/journal.pone.0317580.t002

Table 3 gives the results of applying the APV procedure to the election for nRvotes = 0: 10 when m = 10.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Results for Example 1’s ten-vote election.

https://doi.org/10.1371/journal.pone.0317580.t003

All election results were nonrandom because Instruction 1.6.3 of the APV procedure was not executed.

The candidates elected are the first two to be identified as elected, i.e., given nondeficient weighted-vote totals.

When the proportion, nRvotes/m, of the votes cast for the R party is below 1/3, all candidates elected belong to the B party because nRvotes < m/3 = m/(n + 1) = t₀ = the minimum possible weighted-vote total of an elected candidate, so, R-party votes are not sufficient to elect any candidate while B-party votes can elect two (2t₀ < nBvotes).

When nRvotes/m is strictly between 1/3 and 2/3, one candidate from each party is elected because t₀ < nRvotes < 2t₀ and t₀ < nBvotes < 2t₀, so, each party can elect one, but not two, candidates.

When nRvotes/m exceeds 2/3, all candidates elected belong to the R party because B-party votes are not sufficient to elect any candidate (nBvotes < t₀) while R-party votes can elect two (2t₀ < nRvotes).

When nRvotes/m equals 1/3, at least one candidate elected will belong to the B party because R-party votes can elect no more than one candidate (nRvotes = t₀) while B-party votes can elect up to two candidates (nBvotes = 2t₀). Similarly, when nRvotes/m equals 2/3, at least one candidate elected will belong to the R party because B-party votes can elect no more than one candidate (nBvotes = t₀) while R-party votes can elect up to two candidates (nRvotes = 2t₀). In these two cases, the conditions on the votes-cast matrix, V, that determine whether or not the elected candidates belong to the same party and whether or not this outcome is random have not been identified.

Discussion

This section discusses various features, capabilities, and benefits of APV.

APV assumes that, on each submitted ballot (vote cast), the voter has assigned each candidate a nonnegative number representing the voter’s level of support for or rating of the candidate relative to the other candidates. Under this assumption, if a candidate is removed from the ballot, there is no need to change already submitted ballots: the ballots continue to satisfy the assumption after the candidate is removed (e.g., see Instruction 1.1 of the APV procedure).

APV can accept data from any completed ballot that can be interpreted as representing a nonnegative level-of-support function on the set of candidates. For example, APV could accept data from single-vote plurality, approval-voting, List-PR, approximate rank-voting or STV, CV, or grade-voting election ballots [14, 15].

A ballot designed specifically for APV might have the following form:

• The voter could use a stand-alone computer app or script to complete a ballot and print an official paper copy of the completed ballot. The app or script would permit the voter to do the following:
  • Specify a nonnegative number to represent a level of support.
  • Click on or tap a candidate’s name to automatically assign the specified level of support to the candidate and, if the candidate represents a political party, update the level of support assigned to the party (note that a party’s level of support is the sum of the levels of support assigned to the party’s candidates).
  • Click on or tap a party’s name to automatically assign the specified level of support to the party and update the levels of support assigned to the party’s candidates by normalizing them so that they sum to one; then, multiplying them by the specified level of support.
  • Undo the last action if necessary.
  • Repeat these actions until satisfied with the result.
  The initial and default level of support assigned to every candidate and party is zero. A bar chart could be used as a visual aid. A candidate’s or party’s name and level of support would be displayed along each bar. The length of the bar would be proportional to the level of support displayed along the bar. When satisfied with the result, the voter would print an official paper copy of the completed ballot.
• To cast a vote, the voter would submit the completed paper ballot at a polling place. A scan of the paper ballot would supply the data needed by APV. APV uses only candidate data.

Note that each party has a list of candidates running in the election. The parties’ candidate lists must not overlap. Every candidate must represent at most one party. A voter can assign relative levels of support to candidates either directly or by supporting their political parties if any. A party’s support is distributed among its candidates so that the party’s level of support equals the sum of the levels of support assigned to its candidates. APV determines how many and which candidates are elected from each party’s list of candidates.

The weights that apply to APV’s votes are not voter inputs to APV, but are determined by the authorities governing the election.

An important advantage of APV over other voting systems, except grading, is that it offers voters a greater range of options for expressing candidate support [14, 15]. APV and grading can offer the same range of options for expressing candidate support, but APV and grading process their inputs differently [15]. See [15], the description of a possible APV ballot above, and Subsection APV description.

Other problems with voting systems that APV can correct include the following:

Grading does not weight votes equally when computing candidate vote totals because it does not normalize votes before adding them [15]. APV normalizes votes automatically before computing weighted-vote totals.

CV requires that each vote report relative support for candidates with the sum of its numbers equal to its allowed weight [4–6]. This puts an added burden on the voter. APV only requires voters to grade or rate the candidates on whatever nonnegative scale the voter desires. It does not require voters to distribute a fixed amount of support among the candidates.

Unlike APV, CV does not identify defeated candidates for elimination and reallocate their support to the remaining candidates and, if the election is to fill more than one position, CV does not reallocate excess support from elected candidates to candidates that need it to fill remaining unfilled positions [4–6].

Under STV, voters complete their ballots by ranking the candidates on the ballot by preference [7–10]. A candidate ranking cannot represent a nontrivial level-of-support function on the set of candidates. Professor Nicolaus Tideman, who has reviewed this paper, observed that “apportioned voting becomes equivalent to STV in the limit as voters allocate their votes in larger and larger sequential proportions: One … for the first candidate, one thousandth … for the second candidate, one millionth … for the third candidate, and so on.” Conceptually, APV can approximate STV as closely as desired. Therefore, APV is capable of offering voters a much greater range of options for expressing candidate support than STV.

Voter inputs to STV can be derived from voter inputs to APV, but not vice versa. STV’s voter inputs provide STV with only the directions of each-voter’s preferences, not their relative strengths. APV’s voter inputs provide APV with both the directions and relative strengths of each-voter’s preferences. APV bases its election results on more information about voter preferences than STV.

STV requires that each voter rank the candidates. To rank the candidates, the voter must sort them by preference. Sorting k candidates is of order k log k or k² hard, depending on the sorting method. APV requires that each voter grade or rate the candidates. Rating k candidates is of order k hard because rating each candidate requires only that the voter compare the candidate to one previously-rated candidate whose rating is positive, if one exists. APV does not require that a voter sort the candidates.

Under APV, the voters determine how many and which of each party’s candidates are elected. Under List PR, the voters determine how many of each party’s candidates are elected, but each party may determine which of its candidates are elected.

APV takes into account every-voter’s secondary preferences when computing candidate vote totals because each vote may distribute the voter’s support among the candidates and, when computing candidate vote totals, APV uses a quota or threshold system and a process of candidate elimination that reallocate levels of voter support from candidates that do not need them to candidates that do until all open positions are filled. List PR doesn’t take into account any voter’s secondary preferences when computing party or candidate vote totals because each vote must give all the voter’s support to one party or candidate.

APV eliminates the need for primary and runoff elections. All candidates can participate directly in the general election and APV will fill all open positions whenever there are enough candidates to fill them.

Voting-district boundaries are often politically manipulated to affect election outcomes. APV provides the option of consolidating voting districts. Periodic at-large elections that provide proportional representation can help eliminate the need for voting districts. For example, suppose that a certain legislative body has 100 members and that an at-large APV election is held annually to fill 20 seats. Then each seat would be up for election every five years without the need to partition the seats and voters into voting districts. At-large APV elections can also be expected to improve the legislative process and its output because they hold each lawmaker accountable to every voter affected by the lawmaker’s performance as a legislator. Regional interests could be represented by political parties with regionally-restricted memberships. Like every other party, they would provide candidates to compete in general elections. Regional governments and associations might perform this function. If districts are required, they can be managed in the same way they are now. The voting systems of current general elections could be replaced with APV, permitting the elimination of primary and runoff elections.

The use of subjective and politically-biased criteria to qualify candidates for the ballot limits the political diversity of the candidates on the ballot. The political diversity of the candidates on the ballot can be expected to increase with the number of candidates on the ballot and the objectivity of the criteria used to qualify them for the ballot. Examples of objective criteria that might be used to qualify candidates for the ballot include the following:

• Level of support for the candidate in recent public opinion polls.
• Number of recent individual ballot-access-petition signers in support of the candidate.
• Number of recent individual donors to the candidate.
• Size of the candidate’s constituency if the candidate holds an elective office.
• Level of public support for the candidate’s political party if the candidate represents a political party.

A legislature could use APV in the legislative process to select a proposal from among multiple legislative proposals; then use an up-or-down vote to determine whether or not to approve (pass) the selected proposal. Effectively, the goal is to determine which proposal has the most support and, then, determine whether this support is sufficient to pass it. For example, this process might be used to select and pass a budget proposal. The proposals need not be related to one another.