Condorcet.orgRanked Pairs is a method for conducting elections. Each voter ranks the options, and then the method constructs a complete ranking of the options based on the decisions of the voters. Leaving aside the issue of ties, RP (Ranked Pairs) can be defined fairly simply:
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.
So, for example, if more people rank Mr. X over Mr. Y than vice versa, RP tries to rank Mr. X over Mr. Y in the final result. When doesn't it? Let's say that there are 20 more people who vote Mr. X over Mr. Y than vice versa. Mr. X has a majority of 20 over Mr. Y. Now, let's suggest another candidate, Ms. Z. Mr. Y has a majority of 30 against Ms. Z, but Ms. Z has a majority of 35 against Mr. X. RP would like to reflect these majority opinions in its final ranking, by ranking Mr. Y over Ms. Z, and Ms. Z over Mr. X. But that is obviously incompatible with ranking Mr. X over Mr. Y. Because the two majorities involving Ms. Z have higher margins, RP gives these precedence.
This kind of contradiction of majorities may seem counter-intuitive, but it can happen. I'll talk more about this later, after I give some justification for RP's approach.
Example: Commendation
Consider the following example. A member of a club stands up and suggests that a resolution should be passed commending Bill Smith for his work as Club President. Someone else stands up and suggests that instead a resolution should be passed condemning Bill Smith for his incompetence as Club President. A third person stands up to suggest that no resolution should be passed on the matter as Mr. Smith is deserving of neither commendation nor condemnation.
Robert's Rules of Order specifies what should be done under this circumstance, and would likely be followed, but I'm going to suggest a number of different ways this could be resolved as a way of introducing problems with plurality and benefits of Ranked Pairs.
One way would be to handle the issue as if it were a plurality (or "First Past the Post") election. Each club member would vote for one of the three possible outcomes.
A. Commend
B. No Action
C. Condemn
Now if the results were something like
A 55%
B 20%
C 25%
A would win under plurality, and in fact under most electoral methods. I suspect most people would judge this as fair, as A has a majority of the votes. I will examine the foundation for this in more detail later.
It is possible, however, that the results may end up more like this
A 45%
B 25%
C 30%
A wins, but we might question whether this should be so. Consider that anyone who wants to condemn ( C) would prefer no action ( B) to commendation ( A). Similarly anyone who wants to commend ( A) would prefer no action ( B) to condemnation ( C). This means that in a direct comparison, a majority prefer B to A. As well, a majority prefer B to C. This fact that B loses under plurality (and IRV) seems to contradict our intuitive notion of majority rule.
But we should have more than just an intuitive notion before calling this result unfair. The question is, do we have any reason to believe that A would be a worse result than B or C. This is a difficult question; we don't know anything about Bill Smith, and how he did as club president. We have to rely on what the voters say as an indication. In the first example, the fact that most voters could be convinced in favour of commendation gives good reason to pick this outcome, if we are forced to pick an outcome. It's far from proof, but it seems the safest guess.
Example 2 is a little more murky. But consider the following thought experiment. Obviously Smith annoyed a lot of people. What if he hadn't annoyed so many. What if the C voters instead of being so hostile to Smith were only neutral, then the result would be:
A 45%
B 55%
So, curiously, we can imagine that if Bill was less disliked he might have lost his commendation under plurality or IRV.
It's worth noting how Robert's Rules of Order would have handled this
situation. The resolution to commend would be first, next would come
an amendment to change this to condemn. The voting would first be
on the amendment, it is unclear in this example whether it would pass or
not, but it doesn't matter since either resolution would be voted down.
So, B would win in both example 1 and example 2, which doesn't have
the same inconsistency.
Consider another example. The club members are asked how much they are willing to budget for the upcoming club party. Various club members suggest different amounts. A few suggest $0 because they don't want the club party. Others suggest extravagant amounts. Most people are somewhere in between.
I'm also going to assume for the sake of simplicity that preferences are what mathematicians call "linear". This means that you can arrange the options on a line, with each persons favourite choice being a point on the line. Then, if it is linear, an alternative is always preferred if it is closer to the voter's optimal cost and on the same side.
This is easier to understand by thinking about an example. If someone thinks that $5.15 should be spent, we might expect that they would prefer spending $10.12 over $20.17 and $3.43 over $0, because these options are closer; this is what linear preferences mean. It is less clear that the person would prefer 4$ over 7$, just because 4 is closer to 5 than 7 is to 5; linear preferences does not assume this.
In fact, people's preferences might not be linear. Someone might say, "if we're going to do it, let's do it right. Either $100 or nothing." For the sake of simplicity, I am going to assume that no one thinks like this. But non-linear situations are going to be very important later on.
Anyway, the question is, what is the best result. The plurality choice, the price preferred most often, seems almost random. If 3 people pick 20.25, this has a very good chance of winning, but if one of them had said 20.24 instead, the support would be down to two.
In fact, if the election was held by plurality, voters would use a great deal of strategy to get around this problem. Blocks of voters would compromise ahead of time to support particular costs. The best organized people would be much more likely to win.
But it's worth trying to decide what the best result is without assuming
that voters are using this kind of strategy. I suggest that the best
result is the median cost. This means that a majority of people support
the median cost or lower, and a majority support the median or higher.
So, if there were a two-alternative vote between the median cost and any
other suggested cost, the median would win. Using the median is consistent
with the principle of majority rule. It also means that the winning
cost would not go down because people wanted to spend more money, or up
because they wanted to spend less.
Real elections are much more complicated than the median situation. We can't simply arrange candidates on a line and assume people's preferences based on it. Of course, political scientists often think of candidates as falling of a left to right political spectrum. However, this is, at best, a useful approximation. If we want to know, we have to ask people what their preferences are. And yet, the examples above are still important because the correct answer is more obvious, and because real elections will often be similar.
For example, imagine the club above is electing a "director of club parties", who will decide how much money will be spent on the upcoming party. The various nominees propose different sums to be spent, and the election is held largely on this issue. We would hope that the result of the election would mimic the direct result.
Consider the following example. Three candidates are running. Candidate A, B and C. A wants to spend the least, C the most, and B is somewhere in the middle. Of course, this is similar to the kind of left to right spectrum I mentioned before.
45 A B C
6 B C A
14 B A C
35 C B A
The A-1st voters naturally prefer B to C. The C voters prefer B to A. The B voters are split between A and C for second place.
Clearly, under plurality, A would win. However, if we want to be consistent with the median principle, we have to elect B. That is, B supports the amount of spending that would win if the voters were directly deciding how much money would be spent, and using the median principle.
However, it doesn't make sense to find the median of candidates, so we have to find a new criterion that works for candidates but is consistent with the principle of finding the median choice. Notice that a majority of voters, 55 to 45, prefer B to A. That is, if I look at my table of how people voted, 55 of them rank B over A, and only 45 rank A over B. Likewise, a majority prefer B to C (65-35). This means that B is what is called a Condorcet winner. A candidate that is ranked by a majority over any other candidate. The median winner in a linear example is always also the Condorcet winner. I suggest, therefore, that the Condorcet winner is a way to extrapolate the median situation into more general election scenarios.
But the reason for supporting the Condorcet winner is more basic even than this. The whole point of having a vote is that we think that over all, if given two choices A and B, that people are more likely to vote for the better one. Of course, often people will make bad decisions. Often even the majority will make bad decisions, but if we thought that there was no connection between the way people vote, and the way they should vote, elections would be pointless.
So, if a majority say that B is better than A what that really means is that more people have said that B is better than A, than have said A is better than B. If we think people have a tendency toward correct decisions, then we can say that B is more likely a better outcome than A. If B has a majority when compared one-on-one with each other candidate, it makes sense to say B is the most likely best candidate.
Consider the following example:
40 A B C
5 A C B
6 B C A
14 B A C
31 C B A
4 C A B
This is the same as above, except that now the A-1st and C-1st
voters aren't all voting B in second place, as would be required
by linearity. They must be voting on some other basis than the amount
the candidate wants to spend. But B still has a majority over
either A or C, so it is still reasonable to expect that B
is the best candidate. Of course, this would change if more people
ranked B in last place.
Consider the following example. Nine people are trying to choose who will be their club president. The people running are Anne, Bill, and Carol.
2 A B C
3 B C A
4 C A B
Curiously enough, majorities prefers A to B (6-3), B to C (5-4), and C to A (7-2). But it is not possible that A is better than B is better than C is better than A. Of course, they are different majorities, comprised of different voters. No single person has contradictory preferences, but different majorities clearly are in conflict.
Many people find this a very strange and unsettling result. But in fact, it has a very simple explanation. Sometimes majorities are wrong. Of course, you already knew this. When a result like this happens, it just proves it.
At first glance, one might try to explain this phenomenon by saying that the voters have behaved irrationally, or that their opinions are nonsensical. Actually, their choices may make perfect sense from their individual points of view. It is clear, that these choices can't be arranged on a left to right, high to low, line. But that doesn't mean they are nonsense. Most likely the different voters have chosen different issues as the most important. It is perfectly natural to be able to arrange candidates in many different ways based on different issues.
Most people find this result surprising when they first hear it. This is because it is common to think of "the majority" as the same group of people. How can "the majority" be in conflict with itself? Well, it isn't the same group of people, although there has to be some over-lap between different majorities.
Another problem is that often people assume that the purpose of an election is to discover, and act on, the majority will. Since the majority will can be in conflict, this is not always possible. This is why I was careful, when explaining the desirability of the Condorcet Criterion, to base it on more than an appeal to majority rule. I based it on the more basic principles that:
1. We should be trying to determine the best choice.
2. Voters have a tendency toward picking the correct winner.
It so happens that the result is as close to majority rule as possible, but that isn't the basis. I will discuss conflicting majorities further in Arrow's Theorem.
The method I will be using is called Ranked Pairs, and it works on the following basis:
You start by finding the majority decision decided by the greatest margin. For example, 50 more people might say A is better than B, than say the opposite. We "lock" this choice in. That is, we conclude that in our final ranking, A will be above B.
Then we find the next highest margin, and lock that in. As we lock in subsequent decisions, we keep a watch out that do not lock in any contradictions. For example, if we decide that A will be ranked over B, and B over C, it would be absurd to try to rank C over A, despite a majority opinion to this effect. So, majorities that contradict already locked in majorities are simply skipped.
Here is an example, just for practice:
| A | B | C | D | E | |
| A | X | 30 | 21 | 17 | |
| B | X | 60 | 15 | 18 | |
| C | 10 | X | 16 | 19 | |
| D | X | 20 | |||
| E | X |
Lock B>C 60
Lock A>B 30
Lock A>D 21
Lock D>E 20
Lock C>E 19
Lock B>E 18
Lock C>D 16
Lock B>D 15
Now, we get to C>A 10. Since we have already lock A>B and B>C,
this would be a contradiction. It is therefore skipped.
So, on the basis of all these locked decisions, we get
A>B>C>D>E
as our final ranking. If we only need a single winner, then A wins.
It seems reasonable that the more people support a proposition the more likely it is to be true, and the more who oppose the more likely it is to be false. This isn't an absurd "ad populem" argument where an appeal to the majority takes the place of argument. The nature of a vote is that the only information that the method has to make the decision is the preferences of the voter.
So, the margin of victory gives an indication of a propositions likelihood of being correct. It then follows that if we are confronted with several propositions, and we must discard one, we should discard the least probable, that is the one with the weakest support.
It's worth noting that this procedural definition of Ranked Pairs is equivalent to the definition I give at the top of this page, although this fact may not be obvious.