Condorcet.orgFor the sake of simplicity, the definition of Ranked Pairs in the introduction didn't give any mention of how to handle ties.
To handle ties, you'll want a complete ranking of the candidates to act as a tie-breaker. In the simplest case, you could just randomly order the candidates. For example, by a draw, or using a random number generator. Or, you could designate one person to provide the ranking, preferably as soon as the nominations are in.
Once you have the tiebreaking ranking of the candidates, it allows you to designate a precedence between victories at the same margin. Then, you can use the normal procedure or definition.
Let's say you have two different victories, like A vs. B and C vs. D this involves the candidates A, B, C, and D. One of these candidates is ranked highest on the tiebreaker. Whichever victory involves that candidate is considered higher after the tiebreaker. The same candidate can appear in both victories, as long as it isn't highest. For example, given the tiebreaker A>B>C>D>E, C>B would beat D>E, since B is ranked highest on the tiebreaker.
But it might be, that you have the same candidate repeated, and this candidate is the highest. Not surprisingly, you look at the candidates the victories don't share, and see which is higher of those. That victory is considered higher after the tiebreaker. For example, with the tiebreaker A>B>C>D>E, C>D would beat E>C, since D is higher than E on the tiebreaker.
Next, you consider pairwise ties. Score those as a victory for the higher ranked over the lower ranked candidate. So, with tiebreaker A>B>C>D>E, B=E would cause B to be ranked over E.
This procedure has some advantages over other possible procedures.
1. It is final. The procedure will always give a result, which would not be the case if it could itself result in a tie.
2. It maintains important properties of Ranked Pairs.
3. It is fairly easy to carry out, by a human or computer.
I have programs for carrying out Ranked Pairs, and many other voting procedures. They can be found at vote.sourceforge.net . You will even find a web page that tabulates Ranked Pairs elections over the internet.
Often it is desired that the Status Quo (SQ) be given some sort of preference. Often legislatures have rules that a quorum is required to pass legislation. In some cases a 2/3 vote may be required to over-rule some other body. On the Internet, the uk.* Usenet hierarchy requires that the vote to create a newsgroup exceeds the vote not to by a margin of 12.
It is not obvious how these requirements might be integrated into Ranked Pairs.
One approach, the approach used by uk.* is to eliminate the candidates that fail the test against the SQ. Note that they do not use Ranked Pairs, but instead call a re-vote if there is no Condorcet winner once this is done. However, consider a cycle condition. In this example, each candidate loses to one other candidate by more than 20.
40 A B C
35 B C A
25 C A B
No matter which candidate is the SQ, it will lose. If the vote were repeated, the winner would cycle between all the candidates. I'm going to suggest as an important criterion that this should not happen, in other words
Stability Criterion - If candidate X wins for some candidate as the Status Quo, or when there is no Status Quo, it must win when it is the Status Quo.
Another sensible criterion is:
Pareto Criterion - If every voter prefers candidate X to candidate Y, then candidate Y must not win, even if candidate Y is the Status Quo.
I also suggest that since the preference is meant to be the margin required, if this margin was decreased to 0, the method should be equivalent to normal Ranked Pairs.
Here then, is my suggested method. Assume that there is a required margin called m.
Before you start Ranked Pairs(RP), consider all the pairwise contests involving the SQ. If any of them have a margin less than m, with either winning, set them as a victory for the SQ over that candidate, with margin m-1/2. Do RP as normal.
This has the counter-intuitive quality that the SQ can lose to a candidate who does not pairwise beat it by the required margin. This can be explained by noting that some candidate does beat it by the required margin, and that you could arrange a series of contests, each of which giving the required margin, and where the end result is the winner given by the method I suggest. In other words, there is a path of the required margin from the winner to the SQ.
Of course, we might not want to define a required margin. We might want to require a particular fraction of the vote, like 2/3. A vote of 2/3 may have a lower margin than one without in the pairwise matrix. To apply the method I just mentioned, you need to convert the fraction into a margin that is in some sense equivalent. This is the margin equal to a 2/3 pairwise vote with all voters participating. That is, a margin of 1/6 of the voters is equivalent to a contest with full participation and a 2/3 majority. In fact, since it is undesirable to have an important vote decided by a small number, it might be better to set the required margin based on the total number in the legislature, rather than participants in the vote.
Another potential requirement is for a quorum. For example many legislatures require that some number be in attendance for the votes to be valid. This requirement has no effect on the Ranked Pairs procedure. However, it is a somewhat odd requirement. It means that sometimes it is possible to defeat a proposal by not attending, rather than voting against it. We might, therefore, want people voting against the proposal not to count towards meeting the quorum. We might want to convert this kind of requirement into a margin for use with Ranked Pairs as well.
To do so, consider the total number voting. If this is lower that the quorum, obviously the Status Quo wins. If not, we want to know the minimum margin which will be scored against the SQ in a contest with full participation, and where the winning side is more that the quorum. Calling the number of voters (v), and the quorum (q), this gives us m=2q-v (after a little algebra). If m drops below 1, it has no effect.
So, if we decide that the Chief Executive should only be removed by a quorum corresponding to an absolute majority of the legislature of 101, and 70 people vote, then, the margin required is m=2*51-70=31.
To understand why this makes sense, consider what happens if a new ballot is entered that ranks the SQ last. This increases v by 1, so m decreases by 1. However, the margin of all of the SQ's defeats are also increased, so the effect is neutral with regard to quorum, which is what was desired. As well, although there does not need to be an absolute majority against the CEO, there needs to be a majority either against or neutral. Both these are legitimately part of a quorum.
As well, often legislative bodies require that the winner get more than 1/2, with the SQ winning otherwise. This corresponds to a minimum margin of 1.