Studying RCV methods with real election data

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This page gives summary statistics for the results from our experiments looking at real-world data from America(n=567), Australia (n=327), and Scotland (n=1128).

Uniqueness

We compare the various voting methods to determine how often the elect the same winner. This allows us to create a “similarity matrix” which we can then use to create heatmaps and dendrograms to visualize the “closeness” of the voting methodologies. Results are shown for each of American, Australia, and Scotland, as well as across all three countries.

Fringe winners

One hopes that an election method will avoid electing “fringe” candidates. Unfortunately there isn't consensus on what makes a candidate fringe. In our case, we labeled a candidate “fringe” based on their (average) Borda score in comparison to the maximum (average) Borda score. We counted fringe winners for each of 9 ways of setting a “fringe threshold” and displayed those results in a graph.

First Place Rank

American voters are used to the winner being the candidate with the number of first place votes, so it might be concerning for voters if the winning candidate is “too far” from being a winning candidate under plurality. These graphs show the number of elections where the winning candidate was not the plurality winner or plurality runner-up.

Top 4 stability

Top-k voting methods --- where one voting method is used to winnow down the candidate pool, and a second is used to select the winner --- have some appeal in voting reform circles. We determined the number of elections where the winner would change if you reduced the candidate pool to the top four plurality winners. Note that Scotland has a large number of such elections since their elections are actually multi-winner, and so a single party will often be represented by a number of candidates on the ballot (and, therefore, voters aligned with that party might have ballots that rank those candidates differently).

Majority winner/Condorcet Loser

Majority winner: when a majority winner exists, the method does not elect the majority winner. Condorcet loser: the method elects a condorcet loser (candidate that is defeated in a head to head compairson with every other candidate). These graphs show the number of elections where a particular method was susceptible to these failures.

Cloning

A “clone” candidate is one which aligns with a leading candidate in many ways; in plurality elections, a competing party might try to incentivize such a clone candidate to run in the hopes of ciphoning off support from the original candidate. We simulated the insertion of one clone candidate in elections, giving the clone candidate a probability (either 0%, 10%, 20%, 30%, 40%, or 50%) of appearing ahead of the original candidate on the ballot. We then checked to see whether the insertion of this clone candidate would spoil the election (i.e., change the result of the election). These graphs show the proportion of elections which were spoiled in this way.

Spoiler

A candidate is called a “spoiler” if they are a losing candidate, but their removal from the pool of candidates results in a different winner in the election. In these graphs we determined the number of elections in which at least one spoiler candidate existed.

America

Dendrogram America
Heatmap America

Australia

Dendrogram Australia
Heatmap Australia

Scotland

Dendrogram Scotland
Heatmap Scotland
*Only 13 Scottish elections have a majority winner

Fringeness is decided by percentage of the first place Borda average score. I.e. a winner is considered fringe if it's Borda score is less than x percent of the first place Borda score.

Condensation Methods:

Goal: Condense each Scottish election so that every party only has 1 representative

borda score – for each party, only keep the candidate with the highest borda score
mention score – for each party, only keep the candidate with the highest mention score
first place score – for each party, only keep the candidate with the highest first place score
borda tiebreaker – for each ballot, use borda score as a measure to determine which candidate to keep from each party
mention tiebreaker – for each ballot, use mention score as a measure to determine which candidate to keep from each party
first place tiebreaker – for each ballot, use first place score as a measure to determine which candidate to keep from each party
keep first mentioned – for each ballot, only keep the first candidate mentioned (from rank 1 to rank n) from each party
keep last mentioned – for each ballot, only keep the last candidate mentioned (from rank 1 to rank n) from each party

All Condensation Methods

Compare Single Condensation Method

Condensation Methods:

Goal: Condense each Scottish election so that every party only has 1 representative

borda score – for each party, only keep the candidate with the highest borda score
mention score – for each party, only keep the candidate with the highest mention score
first place score – for each party, only keep the candidate with the highest first place score
borda tiebreaker – for each ballot, use borda score as a measure to determine which candidate to keep from each party
mention tiebreaker – for each ballot, use mention score as a measure to determine which candidate to keep from each party
first place tiebreaker – for each ballot, use first place score as a measure to determine which candidate to keep from each party
keep first mentioned – for each ballot, only keep the first candidate mentioned (from rank 1 to rank n) from each party
keep last mentioned – for each ballot, only keep the last candidate mentioned (from rank 1 to rank n) from each party

All Condensation Methods

Compare Single Condensation Method

Let W be the winning candidate and L be a losing candidate.

  • UpMono - W is moved up in some ballots and subsequently loses the election
  • DownMono - L is moved down in some ballots and subsequently wins the election
  • NoShow - Some ballots that rank L over W are removed and L becomes the new winner
  • LaterNoHarm - Every ballot that prefers W to L moves L up so that it is immediately behind W, and L becomes the winner
  • Strat_compromise - Every ballot that prefers L to W moves L to the top and L wins the election
  • Strat_truncate_L - Every ballot truncates immediately after ranking L and L wins the election
  • Strat_truncate_W - Every ballot that ranks L above W truncates immediately above W and L wins the election
  • Strat_bury_shallow - Every ballot that ranks L above W removes W from the ballot and L wins the election
  • Strat_bury_deep - Every ballot that ranks L above W removes W, fills in any unranked candidates in reverse order of first-place votes, and ranks W in dead last, and then L wins the election

America (n=347)

Australia (n=325)

Scotland (n=1128)

Scotland Condensed (n=1128)

Let W be the winning candidate and L be a losing candidate.

  • Strat_compromise - Every ballot that prefers L to W moves L to the top and L wins the election
  • Strat_truncate_L - Every ballot truncates immediately after ranking L and L wins the election
  • Strat_truncate_W - Every ballot that ranks L above W truncates immediately above W and L wins the election
  • Strat_bury_shallow - Every ballot that ranks L above W removes W from the ballot and L wins the election
  • Strat_bury_deep - Every ballot that ranks L above W removes W, fills in any unranked candidates in reverse order of first-place votes, and ranks W in dead last, and then L wins the election
Note:
  • Expected_values.csv - The expected value for voters of a random losing candidate L if L convinces all their supporters to vote strategically
  • Prob_change_winner.csv - The probability that if a random losing candidate L convinces all their supporters to vote strategically, the winner of the election will change (not necessarily to L)

America

Expected_values.csv

Prob_change_winner.csv

Australia

Expected_values.csv

Prob_change_winner.csv

Scotland

Expected_values.csv

Prob_change_winner.csv

Scotland Condensed

Expected_values.csv

Prob_change_winner.csv

Polling Strategies:

  • Voters_compromise - Of the top 4 candidates in the poll (contenders), voters move their two favorite contenders to the top of their ballots, fill in the rest of their ballots, and rank their two least favorite contenders at the very bottom of their ballots
  • Voters_bullet - If a voter's top candidate is in the top 2 in the polls, they bullet vote
  • Voters_protect - Voters protect their top 2 candidates by filling in a complete ballot and moving any candidates who outperform their top 2 in the polls to the very bottom of the ballot.
  • Voters_score - Voters give all candidates a score based on the sum of the Borda score from their sincere ballot plus the Borda score from the poll, and then ranks candidates according to this score
Note:
  • Expected_values.csv - The expected value for voters of a random losing candidate L if L convinces all their supporters to vote strategically
  • Prob_change_winner.csv - The probability that if a random losing candidate L convinces all their supporters to vote strategically, the winner of the election will change (not necessarily to L)
  • Winner_poll_positions.csv - The expected rank of the eventual winner when all voters adopt a poll-based voting strategy, where candidates are ranked by their first-place votes. The candidate with the most first-place votes is 0, the second most is 1, and so on.

America

Expected_values.csv

Prob_change_winner.csv

Winner_poll_positions.csv

Australia

Expected_values.csv

Prob_change_winner.csv

Winner_poll_positions.csv

Scotland

Expected_values.csv

Prob_change_winner.csv

Winner_poll_positions.csv

Scotland Condensed

Expected_values.csv

Prob_change_winner.csv

Winner_poll_positions.csv

Let W be the winning candidate and L be a losing candidate.

  • Strat_compromise - Every ballot that prefers L to W moves L to the top and L wins the election
  • Strat_truncate_L - Every ballot truncates immediately after ranking L and L wins the election
  • Strat_truncate_W - Every ballot that ranks L above W truncates immediately above W and L wins the election
  • Strat_bury_shallow - Every ballot that ranks L above W removes W from the ballot and L wins the election
  • Strat_bury_deep - Every ballot that ranks L above W removes W, fills in any unranked candidates in reverse order of first-place votes, and ranks W in dead last, and then L wins the election
Note: For a given election method and voting strategy, given that a candidate W loses to a candidate L by convincing their voters to vote strategically, we show the probability that W can regain their seat by also encouraging their voters to vote with the same strategy.

America

Australia

Scotland

Scotland Condensed