1. Introduction
Ranked-choice voting occurs when each member of a voting population ranks k candidates from 1 to k in strict order, 1 best to k worst, where
. Candidates can be politicians, proposals, or consumer products like automobiles or laundry detergents. The idea here is that a ranked permutation of candidates provides much more information about voter preference than a singleton choice, even though we are often interested in selecting a single winner. The objective of ranked-choice voting is to elicit an aggregate social choice ranking of candidates beyond a simple plurality (most first-place votes) and to designate the aggregate top-ranked winner utilizing all available information about voter preference.
Various voting methods, or social choice functions, have been proposed, going back to Eighteenth Century France with competing methods proposed by Borda and Condorcet. Both of these methods depend on an aggregated, anonymous voter profile which summarizes counts of voters associated with various candidate rank permutations. Borda Count, and other positional scoring functions, depend simply on rank counts for each candidate. The Borda Count score winner, in particular, has a number of commendable attributes, although it is considered suboptimal when compared to the Condorcet winner who beats all pairwise opponents, ignoring others. The Condorcet winner, although generally considered optimal, doesn’t always exist, although other robust pairwise methods exist, the most comprehensive and complex being the Kemeny method. In our age of high-powered computation, algorithmic complexity is not really an issue, but understandability of a voting method remains an issue. Hence, Borda Count and other simple scoring methods have continuing allure. Moreover, it is useful to explore where Condorcet and Borda yield the same winner.
Part I of Robinson and Ullman (2017) provides an excellent overview of ranked-choice voting, including all pairwise and positional voting rules and various social choice criteria to evaluate them; they also have a vast array of pedological problems, one of which we highlight later. Saari (2000) and Saari (2023) warn about pitfalls associated with all such voting methods and he champions Borda Count as retaining the most information about voter preference as well as being least impacted by paradoxical behaviors, including strategic manipulative voting, which is further examined in Mata-Perez et al. (2014) .
Heilman (2022) deduces, under certain technical conditions, that Borda Count is the stablest voting method in the face of randomly corrupted votes. Conitzer and Sandholm (2005) show that simple scoring rules, like Borda Count, correspond to maximum likelihood estimators for certain voter noise models, unlike some competing pairwise methods. Young and Levenglick (1978) and Young (1995) support Kemeny as the natural embodiment of Condorcet’s approach. Crisman (2014) uses representation theory of the symmetry group of the permutahedron to define an algebraic spectrum of voting rules connecting Borda to Kemeny. Other general overviews of ranked-choice voting are provided by Nurmi (2010) and Zhang (2020) .
Our focus is twofold. First, we reinterpret the classic Borda Count voting method and derive a family of alternative Borda methods that correspond to various weighted rank means. Our purpose here is not to detract from Borda Count, but rather to reinforce it. Second, we address the issue of voter population composition and size to ensure statistical accuracy of results. Most textbook problems have very small voter populations by design, and they certainly highlight divergences among voting methods. However, these divergences tend to disappear for larger populations that are at once ergodic and sizable.
In the next section, we reinterpret Borda Count as the weighted arithmetic mean across ranks
with minimum score the winner. This opens the door to other weighted means like geometric, harmonic, root-mean-square, and indeed to an infinity of weighted power means. In section 3, we address the issue of voter population composition and size. We demonstrate that size alone does not ensure consistency across voting methods. Ergodicity is also required. Given ergodicity, we indicate how to ensure sufficient voter population size, assuming that is possible. In section 4, we provide an ergodic simulation which demonstrates convergence of Condorcet and Borda variations for sufficiently large population size. In our simulation, only the Plurality method ultimately diverges from the others. Section 5 contains our summary and conclusions and the Appendix contains our R-scripts/functions for Condorcet, Plurality, and alternative Borda methods along with scripts for Problem 2.1 in Robinson & Ullman (2017) and our simulation. Throughout we ignore winner ties as being unlikely in the real world.
2. Borda Variations
We have
distinct candidates and a population of voters, each of whom ranks all k candidates in strict order, from 1 best to k worst. Hence, each voter’s ranking is a permutation of
and assuming equal, anonymous voters, we may aggregate results into a voter profile with voter counts in the first column and specific candidate rank permutations in columns
. This is a transposed version of the usual textbook profile with candidates labeled by numbers rather than letters.
Table 1 profiles the results of an election with 5 ranks and 11 total votes with only 5 of 120 = 5! possible rank permutations represented. Nevertheless, it is quite instructive. The first row indicates 4 votes associated with a permutation that puts candidate 1 in rank 1, candidate 3 in rank 2, candidate 4 in rank 3, and so on. Candidate 1 is the Plurality winner with 4 first-place (Rank 1) votes. Candidate 3 is the Condorcet winner who beats all other candidates in pairwise, head-to-head competition, i.e., 6:5 against each contender 1, 2, 4, 5. Positional methods like Borda Count (and Plurality) involve a
count matrix
where
is the count of votes for candidate i in rank position j.
Table 1. Candidate rank permutations and votes from Problem 2.1 in Robinson & Ullman (2017) .
Borda Count involves weighted candidate scores
and the candidate with maximum score is designated the winner. From Table 1, we compute scores 19, 27, 24, 25, 15 so candidate 2 is the designated Borda Count winner. Calculations for this problem are simple and straightforward but our R-scripts/functions automate the process for this and any other voter profile.
Let n denote the total number of voters and define matrix
. This matrix is doubly stochastic (or doubly Markov, cf Cinlar (1975) ) in that its rows and columns all sum to 1. If we replace
with
in the Borda scoring formula, the winner is the same. If we further replace
by j in the scoring formula, the same winner emerges if we switch from maximum to minimum score.
This revised (and simplified) scoring formula
is recognizable as the weighted arithmetic mean across ranks
for weights
. While this seems logical, we have no assurance that the voting population has in mind a linear scale for ranks. Consequently, it seems advisable to compute alternative weighted means to see if they designate the same winner; if so, the Borda case is bolstered.
There is an infinity of weighted power means to consider; we highlight a few in the following Table 2. See Hong et al. (2007) for application of weighted power mean operators in fuzzy information retrieval.
Table 2. Weighted mean variations on the Borda Count scoring formula. Refer to planetmath.org/weightedpowermean and en.wikipedia.org/wiki/Generalized_mean for general relationships among these weighted means.
Referring to the general power mean formula, Arithmetic corresponds to a = 1, Geometric corresponds to a = 0 (in the limit), Harmonic corresponds to a = −1, and RMS corresponds to a = 2. Referring back to the profile in Table 1, and utilizing the PROFILE2.1 function in the Appendix, Geometric and Harmonic yield the same winner as Arithmetic, candidate 2. RMS, on the other hand, yields candidate 4 as winner, yet another divergence. We next turn to the impact of voter population composition and size.
3. Voter Population
Referring back to the voting profile in Table 1, we note that adding N zeros to every element in the votes column changes absolutely nothing. Voter population size doesn’t matter if there are large swathes of voters who rank candidates in locked step, by virtue of either mania or intimidation. We need more to ensure that size matters and that it may cure lumps and divergences, as in inferential statistics.
Definition. A voter population is ergodic if (1) individual voter rank-permutations are stochastically independent and identically distributed and (2) every voter rank-permutation has positive probability.
This definition is adapted from the usual notion of ergodicity (cf Breiman (1969) ) that a statistical average will converge to the true population mean as size increases. Condition 1 ensures large-sample convergence. Condition 2 ensures that convergence encompasses the entire universe of possibilities. We shall see an example later in Section 4.
Once again, let n denote total voter population size and denote P elements as
. Under ergodicity, we will have independent voter multinomial trials by candidate and we will have
as
by the Law of Large Numbers. We might then ask how large the population should be to ensure a certain level of statistical precision in our estimates. The rows of P are interrelated, so we will focus on any particular row. Goodman (1965) derived simultaneous large-sample confidence intervals of the form
with confidence level
and where Q is the upper
percentile of the chi-square distribution with one degree of freedom. If, for instance,
(to achieve 95% confidence) and
, then
.
Once an election is run, we know the
so we can solve the quadratic equation
for upper and lower confidence limits
or we may simply estimate
by
and obtain confidence limits
. Before an election is run, we have no estimates, but we
can use conservative limits with specified precision to compute required voter population size. Since
is maximized at
, we can examine a conservative solution to
or
where precision
is stipulated and we solve for population size n . For instance, if
,
, and
, then
is the required voter population size. Even for a purely ergodic voter population, we may need significant voter population size to ensure a reasonable degree of statistical precision.
We are not asserting that all voter populations are ergodic. In reality, voter populations are probably some mix of hidebound and random rankings. Moreover, we are not asserting that rank-choice voting should be avoided for small voter populations, only that we acknowledge pitfalls and strive for adequate population size (and randomization) when possible. In the next section, we examine these issues further in the context of an ergodic simulation model.
4. Ergodic Simulation Model
Our simulation randomizes independent and identically distributed voters sequentially with a specified starting random number seed. We first randomize a voting bloc (tendency), followed by 4 ranks according to first-place candidate rank probabilities (without replacement) to generate each voter’s candidate rank permutation. We then aggregate the voting profile and compute winners for Condorcet, Plurality, and Borda variations.
If we denote the first column of Table 3 as v and the remaining 4×4 square matrix as R, then the
vector represents the unconditional first-place candidate rank probabilities for comparison to the first column of the P-matrix simulation results. We designed inputs so that candidates 1 and 2 had close unconditional first-place probabilities while the last two candidates were more “spoilers.” Despite all this machinery, the simulation is ergodic with independent multinomial trials summarized in the voting profile, the C-matrix, and the P-matrix.
Table 4 results were generated by the SIMULATION function in the Appendix. For only 10 voters, there is no Condorcet winner and all Borda variations are consistent with the Plurality winner, candidate 1. For 100 voters, the only difference is that Condorcet comes in, again with winning candidate 1. With 1,000 voters and up, however, candidate 2 wins across the board except for Plurality. This shows that true voter preferences were obscured by noise for smaller population sizes.
All 24 = 4! possible rank permutations are represented Table 5, some more than others of course. This lends further credibility to the notion that we are capturing true social preference across voters. Table 6 is the P-matrix that drives the Plurality and Borda winners for a particular random number seed.
We duplicated the simulation for other random number seeds and got different results for smaller voter populations. For random number seed 17, candidate 2 was the winner for all voting methods up through 10,000 votes. Only a tad above 11,000 votes did Plurality revert to candidate 1. For random number seed 23, we got mixed results below 1,000 voters. Even at 1,000, the uniform winner was candidate 1. We had to get up to 1,900 voters and above to duplicate the 1,000 voter pattern from Table 4. For random number seed 29, we had to get up to 23,000 for Borda-Harmonic to switch from candidate 1 to 2. For an ergodic population, random noise declines naturally as size increases, but in close elections like this one, definitive results depend crucially on a large voter population.
Table 3. Simulation inputs specified by the author to reflect a close election.
Table 4. Simulated winning candidates for random number seed 7. Note the transition at 1,000 voters.
Table 5. Simulated voting profile for 1,000 voters and random number seed 7.
Table 6. Simulated P-matrix for 1,000 voters and random number seed 7. Note that the first column is reasonably close to the stipulated candidate first-place unconditional rank probabilities.
5. Summary & Conclusion
We have introduced a family of weighted-mean Borda Count variations for comparison to Condorcet and Plurality voting methods. We also introduced the notion of an ergodic voter population and discussed the importance of voter population size in that context, together with a statistical confidence interval approach to estimating requisite size. Via simulation, we demonstrated that ergodic population size is crucial in terms of voting method consistency and elicitation of overall voter/social preference, particularly when an election is likely to be plurality-wise close.
As in normal statistical inference, we should strive, when possible, for voter populations that are sufficiently large and randomized to ensure that election results reflect true voter/social preference. We acknowledge that we can’t always achieve these ideals, but we should always be cognizant of inferential pitfalls when they are absent. As we have seen, principal voting methods, other than Plurality, tend to converge for large, ergodic voting populations. There is undoubtedly opportunity for future research on this interplay between voting method and voter population composition and size.
Appendix. R Scripts/Functions
Our analytical and simulation functions are delineated within R-scripts. Once those scripts are loaded into an R workspace, either manually or via the source function, they can be run repeatedly for different inputs and parameters. Moreover, the entire workspace can be saved for repeated use.
CONDORCET <- function(M){
# Script CONDORCET.R
# Produces a Condorcet winner (if available, else 0) from voting profile matrix M
v <- M[,1] # Vote vector
sv<- sum(v) # Total votes
R <- M[,-1] # Candidate rank permutations
n <- length(v)
k <- dim(R)[2] # Number of candidates and positions
# Tally of pairwise rank orderings
C <- matrix(0,nrow=k,ncol=k)
for (i in 1:n){C <- C + v[i]*outer(order(R[i,]),order(R[i,]),"<")}
# Determine which (if any) candidate wins all pairwise contests
t <- NULL
for (i in 1:k){t <- c(t,min(C[i,i!=(1:k)]))}
# Winner must achieve majority against every rival
ifelse(max(t) <= sv/2,0,which.max(t))}
COUNT <- function(M){
# Script COUNT.R
# Produces candidate by position square count matrix from voting profile matrix M
v <- M[,1] # Vote vector
R <- M[,-1] # Candidate rank permutations
n <- length(v)
k <- dim(R)[2] # Number of candidates and positions
C <- matrix(0,nrow=k,ncol=k)
for (i in 1:n){C <- C + v[i]*outer(1:k,R[i,],"==")}
C}
PLURALITY <- function(M){
# Script PLURALITY.R
# Produces plurality scores and a winning candidate for voter profile matrix M
# This is the familiar most-first-place votes method
n <- sum(M[,1]) # Total votes
k <- dim(M)[2]-1 # Number of candidates and ranks
P <- COUNT(M)/n # Candidate-by-rank doubly stochastic matrix corresponding to M
scores<- P%*%c(1,rep(0,(k-1))) # Only rank 1 votes are counted
# Ignoring ties
which.max(scores)}
BORDA.ARITHMETIC <- function(M){
# Script BORDA_ARITHMETIC.R
# Produces Borda Arithmetic scores and a winning candidate for voter profile matrix M
# This is equivalent to the familiar Borda Count voting method
n <- sum(M[,1]) # Total votes
k <- dim(M)[2]-1 # Number of candidates and ranks
P <- COUNT(M)/n # Candidate-by-rank doubly stochastic matrix corresponding to M
scores<- P%*%(1:k) # Weighted arithmetic mean rank by candidate
# Ignoring ties
which.min(scores)}
BORDA.GEOMETRIC <- function(M){
# Script BORDA_GEOMETRIC.R
# Produces Borda Geometric scores and a winning candidate for voter profile matrix M
# This is the natural geometric mean companion to Borda Arithmetic
n <- sum(M[,1]) # Total votes
k <- dim(M)[2]-1 # Number of candidates and ranks
P <- COUNT(M)/n # Candidate-by-rank doubly stochastic matrix corresponding to M
scores<- exp(P%*%log(1:k)) # Weighted geometric mean rank by candidate
# Ignoring ties
which.min(scores)}
BORDA.HARMONIC <- function(M){
# Script BORDA_HARMONIC.R
# Produces Borda Harmonic scores and a winning candidate for voter profile matrix M
# This is the natural harmonic mean companion to Borda Arithmetic
# This is equivalent to the familiar Dowdall/Nauru voting method
n <- sum(M[,1]) # Total votes
k <- dim(M)[2]-1 # Number of candidates and ranks
P <- COUNT(M)/n # Candidate-by-rank doubly stochastic matrix corresponding to M
scores<- 1/(P%*%(1/(1:k))) # Weighted harmonic mean rank by candidate
# Ignoring ties
which.min(scores)}
BORDA.RMS <- function(M){
# Script BORDA_RMS.R
# Produces Borda Root-Mean-Square scores and a winning candidate for voter profile matrix M
# This is the natural RMS companion to Borda Arithmetic
n <- sum(M[,1]) # Total votes
k <- dim(M)[2]-1 # Number of candidates and ranks
P <- COUNT(M)/n # Candidate-by-rank doubly stochastic matrix corresponding to M
scores<- sqrt(P%*%(1:k)^2) # Weighted root mean square rank by candidate
# Ignoring ties
which.min(scores)}
PROFILE2.1 <- function(){
# Script PROFILE21.R
# Produces and stores transposed version of voter profile in Robinson & Ullman, Problem 2.1
# Designates winners for different voting methods
v <- c(4,3,2,1,1)
R <- c(1,3,4,2,5,2,5,4,3,1,3,2,4,1,5,4,2,5,3,1,5,4,2,1,3)
R <- matrix(R,nrow=5,ncol=5)
M <- t(rbind(v,R))
print("Voting Profile")
print(M)
#write.table(M,file="c:/VOTING/PROFILE21.csv",col.names=FALSE,row.names=FALSE,sep=",")
source("c:/VOTING/CONDORCET.R")
source("c:/VOTING/COUNT.R")
source("c:/VOTING/PLURALITY.R")
source("c:/VOTING/BORDA_RMS.R")
source("c:/VOTING/BORDA_ARITHMETIC.R")
source("c:/VOTING/BORDA_GEOMETRIC.R")
source("c:/VOTING/BORDA_HARMONIC.R")
print("C-Matrix")
print(COUNT(M))
P <- COUNT(M)/11
print("P-Matrix")
print(P)
print(paste0("CONDORCET WINNER = ",CONDORCET(M)))
print(paste0("PLURALITY WINNER = ",PLURALITY(M)))
print(paste0("BORDA-ARITHMETIC WINNER = ",BORDA.ARITHMETIC(M)))
print(paste0("BORDA-GEOMETRIC WINNER = ",BORDA.GEOMETRIC(M)))
print(paste0("BORDA-HARMONIC WINNER = ",BORDA.HARMONIC(M)))
print(paste0("BORDA-RMS WINNER = ",BORDA.RMS(M)))}
SIMULATION <- function(NUM,SEED){
# Script SIMULATION.R
# Produces and stores simulated voter profile
# Designates winners for different voting methods
# NUM = total number of independent voters
# SEED = random number seed for reproducible results
# Simulation for independent voters with probabilistic voting bloc tendencies
# All candidate rank permutations are possible
library(dplyr)
# Randomized voting blocs and independent voter rank permutations within blocs
BLOC_PROBS <- c(.4,.3,.2,.1) # Voting bloc probabilities
# Candidate first-place rank probabilities by voting bloc
# Used also for subsequent rankings
RANK_PROBS <- c(.6,.1,.2,.1) # Candidate rank probs for Bloc 1
RANK_PROBS <- rbind(RANK_PROBS,c(.1,.6,.2,.1)) # for Bloc 2
RANK_PROBS <- rbind(RANK_PROBS,c(.1,.2,.1,.6)) # for Bloc 3
RANK_PROBS <- rbind(RANK_PROBS,c(.1,.3,.5,.1)) # for Bloc 4
set.seed(SEED) # Set random number seed
# Simulated independent voter rank permutations
M <- NULL
for (i in 1:NUM){
a <- sample.int(4,size=1,replace=FALSE,prob=BLOC_PROBS)
b <- sample.int(4,size=4,replace=FALSE,prob=RANK_PROBS[a,])
M <- rbind(M,b)}
# Aggregation into candidate rank permutation counts
df<- data.frame(M)
A <- data.matrix(df %>% group_by_all() %>% count)
M <- cbind(A[,5],A[,-5])
print("Simulation Voting Profile")
print(M)
# write.table(M,file="c:/VOTING/SIM_PROFILE.csv",col.names=FALSE,row.names=FALSE,sep=",")
source("c:/VOTING/CONDORCET.R")
source("c:/VOTING/COUNT.R")
source("c:/VOTING/PLURALITY.R")
source("c:/VOTING/BORDA_RMS.R")
source("c:/VOTING/BORDA_ARITHMETIC.R")
source("c:/VOTING/BORDA_GEOMETRIC.R")
source("c:/VOTING/BORDA_HARMONIC.R")
print("Unconditional First-Place Probabilties")
print(BLOC_PROBS%*%RANK_PROBS)
print("C-Matrix")
print(COUNT(M))
P <- COUNT(M)/NUM
print("P-Matrix")
print(P)
# write.table(P,file="c:/VOTING/SIM_P_MATRIX.csv",col.names=FALSE,row.names=FALSE,sep=",")
print(paste0("CONDORCET WINNER = ",CONDORCET(M)))
print(paste0("PLURALITY WINNER = ",PLURALITY(M)))
print(paste0("BORDA-ARITHMETIC WINNER = ",BORDA.ARITHMETIC(M)))
print(paste0("BORDA-GEOMETRIC WINNER = ",BORDA.GEOMETRIC(M)))
print(paste0("BORDA-HARMONIC WINNER = ",BORDA.HARMONIC(M)))
print(paste0("BORDA-RMS WINNER = ",BORDA.RMS(M)))}