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.

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

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:

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.

This example shows that Majority Judgment violates the Reversal symmetry criterion. Assume two candidates A and B and 2 voters with the following ratings:

Now, the winners are determined for the normal and the reversed ballots.

In the following the Majority Judgment winner for the normal ballots is determined.

The sorted ratings would be as follows:

Result: The median of A is between "Good" and "Poor" and thus is rounded down to "Poor". The median of B is "Fair". Thus, B is elected Majority Judgment winner.

In the following the Majority Judgment winner for the reversed ballots is determined. For reversing, the higher ratings are considered to be mirror-inverted to the lower ratings ("Good" is exchanged with "Poor", "Fair" stays as is).

The sorted ratings would be as follows:

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