Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count

Borda count

The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...

, the Kemeny-Young method

Kemeny-Young method

The Kemeny–Young method is a voting system that uses preferential ballots and pairwise comparison counts to identify the most popular choices in an election...

, and the Schulze method

Schulze method

The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners...

. Methods that fail include Bucklin voting

Bucklin voting

Bucklin voting is a class of voting systems that can be used for single-member and multi-member districts. It is named after its original promoter, James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system...

, instant-runoff voting

Instant-runoff voting

Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

 and Condorcet method

Condorcet method

A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...

s that fail the Condorcet loser criterion

Condorcet loser criterion

In single-winner voting system theory, the Condorcet loser criterion is a measure for differentiating voting systems. It implies the majority loser criterion....

 such as Minimax

Minimax Condorcet

In voting systems, the Minimax method is one of several Condorcet methods used for tabulating votes and determining a winner when using preferential voting in a single-winner election...

.

For cardinal voting systems which can be meaningfully reversed, approval voting

Approval voting

Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

 and range voting

Range voting

Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

satisfy the criterion.

Example

Consider a preferential system where 11 voters express their preferences as:

• 5 voters prefer A then B then C
• 4 voters prefer B then C then A
• 2 voters prefer C then A then B

With the Borda count A would get 23 points (5×3+4×1+2×2), B would get 24 points, and C would get 19 points, so B would be elected. In instant-runoff, C would be eliminated in the first round and A would be elected in the second round by 7 votes to 4.

Now reversing the preferences:

• 5 voters prefer C then B then A
• 4 voters prefer A then C then B
• 2 voters prefer B then A then C

With the Borda count A would get 21 points (5×1+4×3+2×2), B would get 20 points, and C would get 25 points, so this time C would be elected. In instant-runoff, B would be eliminated in the first round and A would as before be elected in the second round, this time by 6 votes to 5.