Significance

It is frequently claimed that a voting method is errant if it fails to satisfy the Majority Criterion. This is actually a common criticism of Score and Approval Voting. Here is a simple mathematical proof of precisely the opposite.

- Consider a voting method which satisfies the MC.
- Consider an electorate with the following preferences for options X, Y, and Z.

- 35% X>Y>Z
- 33% Y>Z>X
- 32% Z>X>Y

* E.g. the top row, comprising 35% of the voters, prefers X to Y and Z, and Y to Z.

Let us first identify the “right winner”. That is, the candidate who best satisfies the preferences of the electorate as a whole. For our purposes it doesn’t matter which candidate that is, so we can simply look at three possibilities in turn.

X is best

Consider then the following scenario where option Y is removed, with no change in any voter’s preferences, leaving us with this:

- 35% X>Z
- 65% Z>X

Because no voter’s preferences have changed, the electorate necessarily still prefers X. So Z is the Condorcet winner, and the majority winner, but not the best candidate according to the electorate. Therefore any election method satisfying the MC would elect the wrong candidate here. Again, that includes all Condorcet methods.

Y is best

Removing Z, we have:

- 33% Y>X
- 67% X>Y

X is the majority and Condorcet winner, even though Y, not X, is best.

Z is best

Removing X, we have:

- 32% Z>Y
- 68% Y>Z

Y is the majority and Condorcet winner, even though Z, not Y, is best.

Conclusion

Sometimes a candidate who is the Condorcet winner, or even the majority winner, isn’t the favored or “most representative” candidate of the electorate.