Condorcet.orgIn approval, each voter simply votes each candidate as "approved" or "not approved". The candidate with the greatest number of "approval" votes wins.
It is quite possible, therefore for a candidate with the majority of first preferences to lose, since first preferences are not expressed. Consider the following preferences
25 A B | C
35 A C | B
10 B C | A
30 C | A B
I have placed a line between the approved and non-approved candidates of each voter. The total approval votes are
A 60
B 35
C 65
So, even though A is the first choice of most people (and the Condorcet Winner), C wins.
Approval advocates divide on the question of whether this is a good result. Those who agree with the result argue that more people approve of C then of A. C satisfies more people, so C should win. Those who dislike the result argue that voters will use strategy to avoid a result like this.
Here is another example
31 A | B C
29 B | A C
20 C | A B
20 C | B A
In this example, everyone only voted for their first choice, so C won. This is despite the fact that a majority rank C last.
One possibility is that A and Bare similar in policies, and that C is quite different. A and B's unwillingness to co-operate cost them the election.
For example, if the A voters decided to compromise, this result could occur.
31 A B | C
29 B | A C
20 C | A B
20 C | B A
Now B wins. The problem is that either the A-1st or B-1st voters have the ability to give the victory to the other, but both have an incentive to wait for the other to capitulate. So, the more stubborn are likely to win.
But consider that both compromise in the hope of defeating C.
29 A B | C
2 A | B C
26 B A | C
3 B | A C
20 C | A B
20 C | B A
In this example, most of the A-1st voters and B-1st voters compromise. A few do not. The result is that one of A or B wins, and which one is decided by the few people who are unwilling to compromise. Since the purpose of elections is to have decisions made by large groups, this seems like a problem. Approval forces voters to decide which opinions they want to express, and which they don't. This makes it much more likely that some decisions will be made by a proportionally small number of people.
See my Average Ratings page for a criticism of using social utility. As well, Approval is a pretty rough estimate. Utility assigns a value of each candidate to each voter. But approval forces all the votes to be simple Up/Down decisions.
One of the most often used arguments refers to the following result. Assume you have two groups of voters who are voting on the same candidates, using Approval. Assume they both select the same candidate as winner. If you instead find the winner of the combined group, the winner will be the same as the winner from each of the smaller groups. This is refereed to as the property of consistency. That is, consistency between the result from the whole, and any subdivisions.
This is not true of Ranked Pairs. It is possible to have two groups, who separately would choose the same leader, but put together choose someone different. This is, admittedly a surprising, result. There is a strong sense of intuition that a reasonable method would have this kind of consistency.
However, let me first point out that our intuition often betrays us. For example, most people intuitively believe that majorities cannot conflict (see Arrow's Theorem). The motivation was always to use the information provided by voter's ballots to choose the likely best candidate. The ballots of the two smaller groups, when considered separately, don't provide the same information as the ballots of both together. Is it possible that the extra information might provide a reason to declare a different (and more likely accurate) best guess for the combined groups?
Consider the following example: 40 ACB, 35 BAC, 25 CBA
| A | B | C | |
| A | X | 40 | 75 (50) |
| B | 60 (20) | X | 35 |
| C | 25 | 65 (30) | X |
Lock A>C (50)
Lock C>B (30)
Skip B>A (20)
Result A>C>B, A wins
| A | B | C | |
| A | X | 51 (2) | 100 (100) |
| B | 49 | X | 100 (100) |
| C | 0 | 0 | X |
Lock A>C (100)
Lock B>C (100)
Lock A>B (2)
Result A>B>C, A wins, but put the two together:
| A | B | C | |
| A | X | 91 | 175 (150) |
| B | 109 (18) | X | 135 (70) |
| C | 25 | 65 | X |
<p>Lock A>C 150
Lock B>A 18
Lock B>C 70
Result B>A>C, B wins.
So, in group 1, A won. A majority preferred B to A, but because B did so poorly against C, Ranked Pairs decided that A was the best choice. In group 2, C wasn't even a contender, A won by narrowly beating B. Putting the two together, however, we find that over all, B beats both.
To help this make sense, consider the following situation.
Imagine that you are asking a magical oracle about which is the best candidate. The oracle only accepts questions comparing two candidates. The oracle is sometimes wrong, but always gives an accurate prediction of its likelihood of being correct. This is obviously meant to be analogous to the situation of relying on majority decisions.
The oracle claims
A>B has a 80% chance of being correct
B>C has a 70% chance of being correct
C>A has a 60% chance of being correct
Clearly the oracle is wrong about one of these answers. Since the oracle is least certain about C>A, it makes sense to guess that this is where the mistake is made. We can then declare A the likely best candidate.
Now, someone else goes to the oracle to ask about the same candidates. The oracle gives the following information
A>C 51%
A>B 100%
C>B 100%
On the basis of this information alone, it is of course natural to pick A as best candidate.
Let me point out, that the two sessions with the oracle are not inconsistent. They may be unlikely, but they both could happen. If we used the combined information, would we still pick A as the best guess? We now have the combined statements.
A>B 80% chance
B>C 70% chance
C>A 60% chance
A>B 100% chance
C>B 100% chance
Since we know the 100% statements are true, the B>C can be dropped as false, and the b A>B 80% can be dropped as redundant.
A>B 100% chance
C>B 100% chance
A>C 51% chance
C>A 60% chance
So, clearly, since C>A is more likely than A>C, if follows that C is most likely the best candidate. That may be surprising, but it's true. If we think that peoples ranks also give potentially incorrect information about which of two candidates is better, then we have to acknowledge that the same thing can happen, and that it is in fact quite sensible.