I'll define the following criteria here:
1. Condorcet's Criterion
2. Smith Criterion
3. Majority Criterion
4. Condorcet Loser Criterion
5. Majority Loser Criterion
6. Mutual Majority Criterion
7. Monotonicity Criterion
8. Participation Criterion
I want to say, first of all, that I don't consider these criteria as important as the criteria that I've already described in the previous articles. These criteria in this article are too undemanding, too unproductive in benefit. It's no coincidence, therefore, that they're met by a number of methods. I include this article only for completeness. It's topic is the least important of this series of articles, so feel free to skip it for now, and maybe come back to it later, perhaps, for reference.
Not all of these are widely used. The 1st 3 and Monotonicity are the ones most often encountered.
These criteria are sometimes spoken of in terms of voters' psychology, and sometimes in terms of actual votes. When defined by psychology, no method meets these criteria. When defined by actual votes, even Plurality meets all of them. Probably the best way to define them would be in terms of actual votes, but with the (understood) stipulation that all of the above criteria except for Monotonicity are intended to apply only to rank-balloting methods.
Here are their definitions:
1. Condorcet's Criterion:
If there's a candidate who, when compared separately to each one of the other candidates, is voted above him/her by more voters than vice-versa, then that candidate should win. (That's the candidate that I've been calling a "BeatsAll winner").
(Of course if one ranks A, and doesn't rank B, that counts as ranking A over B).
2. Smith Criterion:
If there's a set of candidates such that every candidate in the set beats every candidate outside the set, then the winner should come from that set.
(A beats B if more voters prefer A to B than vice-versa).
3. Majority Criterion:
This is the very familiar criterion that says that if a majority of all the voters vote X alone in 1st place (or vote for him in 1-vote Plurality), then X should win. It's met by every proposed method except for the notorious Borda point system.
4. Condorcet Loser Criterion:
If there's a candidate who, when compared separately to each one of the others, is ranked below that other candidate by more voters than vice-versa, then he/she shouldn't win.
5. Majority Loser Criterion:
If a candidate is ranked last by a majority of all the voters, then he shouldn't win.
6. Mutual Majority Criterion:
This one sounds similar to Smith, but is different. If a specific group of voters consisting of a majority of all the voters vote every candidate in set S over every candidate outside that set, then the winner should come from S.
7. Monotonicity Criterion:
Changing your vote so as to vote someone higher should never make him lose when he'd have otherwise won. Changing your vote so as to vote someone lower should never make him win when he'd have otherwise lost.
Adding to the count one or more ballots that vote X over Y should never change the winner from X to Y.