Condorcet.org
Arrow starts by suggesting some reasonable sounding criteria for a good election method. The criteria I'm going to give are a slight simplification, but they get his point across. First of all, he felt that in a two person election, the candidate with the most votes should win.
A 55
B 45
this wouldn't be the case if the election was largely random, if the real decision was made by a dictator or monarch, or if we always gave the victory to the candidate with fewer votes. Not surprisingly, Arrow found those possibilities unsatisfying.
Next he suggested that if you have an election where A wins, and you introduce a new candidate C. Then either A should still win, or C should now win. After all, the theory is, either the public prefers A to C, or C to A. If they prefer C to A, and A to everyone else, then C should win. On the other hand, if they prefer A to C, and A to everyone else, then A should win. It wouldn't make sense from this perspective if the addition of C caused B to win.
The problem is shown in the following example:
40 A B C
35 B C A
25 C A B
The method has to choose either A, B, or C. I'm going to show that no matter which it chooses, it will violate one of Arrow's criteria. I can then conclude that no method, not just no method I can think of, can pass Arrow's criteria. This isn't an original result, and is in fact what Arrow is famous for proving.
If the method chooses A, then you have to consider, "what if B wasn't running?"
You would have had an election like this
40 A C
60 C A
So, C would have won. But that is exactly the kind of thing that isn't allowed to happen. The introduction of B should have either caused B to win, or kept the result the same.
The obvious conclusion is that a method can't choose A in the above situation, and meet Arrows criteria. Unfortunately, the same argument can be used with any of the candidates winning.
If B wins, you have the example with the removal of C
65 A B
35 B A
If C wins, you have the example with the removal of A
75 B C
25 C B
So, no method can meet Arrow's rather sensible sounding criteria.
Let's reconsider a passage from above:
After all, the theory is, either the public prefers A to C, or C to A. If they prefer C to A, and A to everyone else, then C should win. On the other hand, if they prefer A to C, and A to everyone else, then A should win. It wouldn't make sense from this perspective if the addition of C caused B to win.
Substitute the name "Arrow" for "the public" and it seems irrefutable. Arrow, we assume, is a sensible individual. If he prefers A to B and C to A, we know that he must prefer C to B. And yet, "the public" seems not to have behaved so rationally in the example above.
One conclusion is that "the public" is irrational. But each of their individual preferences could be defended as rational, or not self-contradictory, from the individual's point of view. Let me suggest that the problem is that the phrase "the public" is allowing us to play a kind of linguistic trick that is not surprising would end in absurdity.
But let me give another example where this is more obvious. Imagine a fictional nation of Arrovia. In this nation of 100 citizens, some citizens have a car, others have a bicycle, some have both.
48 Bike only
10 Bike and Car
42 Car
The following statements are true.
The majority of Arrovians have a Bike
The majority of Arrovians have a Car
Can I then conclude that the majority of Arrovians have a Bike and a Car?
Obviously not, since this isn't true. But it would work if I replaced "the majority of Arrovians" with "Arrow"
Arrow has a Bike
Arrow has a Car
Therefore Arrow has a Bike and a Car
The problem is that we are tempted to create an imaginary individual called "the majority of Arrovians" and ascribe various attributes to her. The same thing was done with the phrase "the public" above. It is no surprise that "the public" might have contradictory views. "The public" isn't an individual, it's just a concept we made up.
Arrow felt, that the addition of an alternative (or candidate) that doesn't win shouldn't affect the outcome. He called such an alternative an irrelevant alternative, and the property he was looking for "independence from irrelevant alternatives". Since this isn't satisfiable it's worth considering if these alternatives are truly irrelevant.
40 A C
60 C A
C is the Ranked Pairs winner here. But with the introduction of B, we get
40 A B C
35 B C A
25 C A B
And A is the Ranked Pairs winner. So, from the Ranked Pairs perspective, the introduction of B and the new information provided is decisive. What is that new information? Well, we know no that there is an alternative that beats C. We know that this alternative is beaten by A. In the A vs C example, all we knew was that most people preferred C to A.
Now, we have contradictory information from the fact that A is preferred to B and B is preferred to C. So, it isn't a surprise that this might tip the weight of evidence in favour of A.
A 40%
B 35%
C 25%
40% of people say that A is the best candidate. The other 60% say that A is not the best candidate. Similarly 65% say that B is not the best candidate, and 75% say that C is not the best. 100% say that the candidates are not equal. The majorities that say that A is not the best, B is not the best, and C is not the best contradict the reality that one of them has to be the best. Therefore, these majorities are in conflict. Once again, this doesn't mean that voters are irrational, just that some of them are wrong.
In this way, a comparison can be made between the standard criterion that if a candidate gets a majority of the first place votes that candidate should win, and the Condorcet Criterion, which says that if a single candidate is preferred to each other candidate considered individually, this candidate should win.
If the majorities revealed by 1st place votes are not in conflict, use the majority choice.
If the majorities revealed by considering voter preferences between every pair of two candidates are not in conflict, use the majority choice.
All methods sometimes behave in ways that are counter-intuitive. Advocates of a method are fond of pointing out all the strange ways that other methods behave. However, if we are seriously concerned with finding the method that finds the best candidates (in general), we have to use more than just intuition. Often people present examples where some change in the ballots has created different results. The question should not be, "is this intuitive," but, "might the change in evidence provided to the voting method sensibly cause the change in outcomes?"
I support Ranked Pairs because I believe it is always possible to answer yes to this question.