An election method satisfies the Majority Criterion ("MC") if a candidate favored by a majority of voters is guaranteed to win. (Note: all Condorcet methods satisfy the MC.)

## 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.

1. Consider a voting method which satisfies the MC.

2. 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.