Reversal symmetry

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Reversal symmetry is a voting system criterion that is stated as follows: If a candidate A is the unique winner, and the individual preferences of each voter are inverted, then candidate A must not be elected. Methods that satisfy reversal symmetry include Schulze method, Borda count. Methods that fail reversal symmetry include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion like Minimax.

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

[edit] 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.