Majority Criterion

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.

In these cases, let’s assume you hate Candidate Awful, are okay with Candidate Better, and love Candidate Classy. Let’s give them honest utility values (we’re rating them on a 0-10 scale):

• Awful: 0
• Better: 6
• Classy: 10

Approval Voting Example #1

If approval polls:

• Awful: 50%
• Better: 50%
• Classy: 30%

You want to vote for Better and Classy here. You vote for Better because you want Better to beat Awful. Classy doesn’t have a shot, but you vote for her anyway to show your support and give her ideas more legitimacy.

Approval Voting Example #2

If approval polls:

• Awful: 50%
• Better: 50%
• Classy: 50%

You still vote for Better and Classy. You don’t vote for Classy alone because you have a strong preference for Better against Awful. By only voting for Better or Classy, you risk Awful winning against both of them.

Approval Voting Example #3

If approval polls:

• Awful: 30%
• Better: 50%
• Classy: 50%

You actually only vote for Classy here. When Awful is enough out of the race, you can narrow your sights against Better and show your support for Classy.

When exactly do you only vote for Classy? It depends on how far out of the competition Better is. And it depends on how much you dislike Better along with how likable Better is compared to Awful. If Awful and Better are similarly unlikable (you’re indifferent to which one wins), a voter may be more inclined to vote for Classy alone when she is closer to winning.

Approval Voting Example #4

If approval polls:

Awful: 50%

Better: 30%
Classy: 50%

Again, your only vote is for Classy here. It’s not Better that’s giving competition to Awful anymore; it’s Classy competing against Awful. Whether you include Better in the vote would depend on how much you actually supported Better’s views. Like in the first example where Classy had 30% and was a token vote, support for Better in this case is also a token vote because it likely won’t change the outcome. So, if you wanted to give support for Better because of some view he had that you liked, then you could get away with supporting him and Classy.

Step 1: Who is likely to win? Consider the relative utility of each. Of those candidates, approve all whom you prefer. You may end up voting for more than one candidate within this group depending on whom is challenging your preferred candidate(s).

Step 2: Who is less likely to win? Of those candidates, approve of all you wish to give support.

These examples remove the argument that Approval Voting regresses to Plurality Voting (via bullet voting). There are numerous scenarios (as shown above) when bullet voting simply makes no strategic sense. But notice that when you do only vote for one candidate, it’s done in a way that’s not damaging to the outcome. Also, factoring in who is likely to win is something we do anyway when considering what to do under Plurality Voting. But with Approval Voting, we just have more options on what we can do with that information. Also note that it was always to your advantage to vote your favorite. That will ALWAYS be true with Approval Voting.

Also, when there are more candidates, there are more variations on what to do. Though the concepts are the same. Expectantly, with more candidates, voters will also approve of more candidates on average.  There may also be cross-support from multiple independents/third parties that share certain views.

Finally, even with “tactical” voting, Approval Voting will nearly always choose the candidate that can beat everyone in a head-to-head race. This is called a Condorcet winner. Approval voting does not achieve this flawlessly, but it does an excellent job nonetheless. It is also argued that when Approval Voting doesn’t select the Condorcet winner, it does so for good reason. More on this topic here.