Some of us have recently been referring to criteria that suggest the ranking given by Ranked Pairs is in some way more internally consistent than other methods. This is going to be an overview of those criteria. The first, I'll call the Ranking Reinforcement criterion. What this says is, if the method gives you a ranking of the candidates where X is ranked above Y, then on any ballot where Y is ranked immediately above X, and both are ranked alone, you can switch the two, without affecting the result. The idea is that the switch only gives more information in favour of the conclusion already reached, so it shouldn't change the conclusion. I can already imagine Demorep tapping away about how changing votes is election fraud, so I'll just say this. The idea of this criterion, is not that ballots will actually be changed. It is simply to suggest that some methods behave in ways that don't make sense, and we can see this in how they respond to additional information. This is also the reasoning behind monotonicity. Also, it could be argued that a criterion based on ranking doesn't matter unless we want a complete ranking. But if a method has an obvious interpretation in terms of giving a complete ranking, and that ranking doesn't make sense in some way, then we might not want to trust it when it comes to picking a winner either, unless there is some good reason to. The second property, I can't think of a name for. This is that if you take an RP ranking, and you eliminate the top ranked candidate, the new RP ranking places the remaining candidates in the same order as they had in the old ranking. The same is true if you eliminate the bottom ranked candidate. This is certainly what intuition says should happen, although intuition is often wrong. For example, you still can't remove any arbitrary candidate, and expect the ranking to hold. You can only remove the topmost or bottommost candidate. Of course, since you can do this repeatedly, you could actually remove any number of topmost and bottommost candidates, and expect the ranking to hold for the remaining candidates. In general, we have tended to favour methods that meet independence of irrelevant alternatives as much as possible (given ranked ballots). This is another kind of independence, beyond the local independence and independence of clones criteria. The result is independent of dropping any number of last ranked candidates. Also, it has a modest practical benefit. It would often be nice when electing two people, to hold one election and pick the top two ranked, rather than hold two elections. Unless voter's opinions change between elections, it doesn't matter which you do in Ranked Pairs, so the decision can be made on the basis of practicality. In other methods, it can affect the outcome, so the choice might made to achieve a particular result. The third is the following, lets say a method gives a complete ranking A>B>C>D, but B pairwise loses to C. We can change the ranking in agreement with the B>C victory without affecting agreement with any other majority. That is, we could have A>C>B>D just as easily, from this perspective. So, the third property of the RP ranking is that each candidate pairwise beats the next lowest ranked candidate. --------------------------------------------------------------------- Blake Cretney http://www.fortunecity.com/meltingpot/harrow/124/path Ranked Pairs gives the ranking of the options that always reflects the majority preference between any two options, except in order to reflect majority preferences with greater margins. (B. T. Zavist & T. Tideman, "Complete independence of clones in the ranked pairs rule", Social choice and welfare, vol 6, 167-173, 1989)
