Reversal symmetry

Reversal symmetry

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 the Schulze method and the 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.

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 BWith 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 CWith 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.


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