Mutual majority criterion

The mutual majority criterion is a rule used to determine if a decision made by a group is fair. Imagine you and your friends are voting on which game to play at your birthday party. There are three games to choose from: Tag, Hide-and-Seek, and Duck-Duck-Goose.

To use the mutual majority criterion, we need to make sure that the winning game is not just the most popular among the group, but also has the support of a large majority of people. This means that any game that wins the majority vote should also be able to beat any combination of the other games.

Here's how it works: Let's say that six of your friends vote for Tag, three vote for Hide-and-Seek, and two vote for Duck-Duck-Goose. Tag is the clear winner with 60% of the votes, but we still need to make sure that Tag has the support of a majority of the voters.

To do this, we look at all the possible combinations of two games. There are three possible combinations: Tag and Hide-and-Seek, Tag and Duck-Duck-Goose, and Hide-and-Seek and Duck-Duck-Goose. We then check which game has the most support in each of these combinations.

In the first combination, Tag beats Hide-and-Seek with a vote of 6-3. In the second combination, Tag also beats Duck-Duck-Goose with a vote of 6-2. In the third combination, Hide-and-Seek beats Duck-Duck-Goose with a vote of 3-2.

Since Tag has the most support in every possible combination of games, it satisfies the mutual majority criterion and can be considered the fairest choice for the group.

In summary, the mutual majority criterion is a way of making sure that a decision made by a group is fair by requiring that the majority choice is also able to beat any combination of other choices.