Reversal symmetry

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, the Kemeny-Young method, and the Schulze method. Methods that fail include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.

For cardinal voting systems which can be meaningfully reversed, approval voting and range voting 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.