1
A profile is a list of rankings
Each voter ranks all candidates from best to worst. The list of rankings, with how many voters hold each, is the profile, and it is the only input every voting rule here uses.
Lesson · Justice and Collective Choice · Collective choice
Majority Rule and the Condorcet Paradox
Count, for each pair of candidates, how many voters prefer one to the other, and a Condorcet winner is a candidate who beats every rival by majority. With three voters and three candidates the majorities can form a cycle, though each voter's own ranking is in order. The wrong model is that majority rule always produces a ranking.
A beats B if more voters rank A above B
a Condorcet winner beats every rival
A beats B, B beats C, C beats A a cycle
transitive voters, intransitive majority
Pairwise majorities
Core ideas
What this lesson establishes
Collective choice begins with the simplest rule there is, and with the case in which it fails to say anything.
2
A Condorcet winner beats everyone
Compare two candidates by counting the voters who rank one above the other. A candidate who wins every such contest is the Condorcet winner. There is at most one, and there may be none.
3
A cycle is not a tie
In a cycle every contest has a decisive winner, and the winners go round: the first beats the second, the second the third, the third the first. No candidate is undefeated, and nothing in the votes breaks the circle.
The material
Counting pairs
A table of pairwise counts, a profile in which it has a winner, and one in which it does not.
When two candidates are on the ballot, majority rule has a plain meaning: the one more voters prefer wins. With three or more candidates there are several contests, and the question is whether majority rule can be extended to rank them all. The first thing to do is to count, for each pair, the voters on either side.
Definition · Condorcet winner
A Condorcet winner is a candidate who is preferred to each other candidate by a majority of voters. It is named for Condorcet, who observed in the eighteenth century that there may be none.
Take a profile with seven voters: three rank the candidates A B C, two rank them B C A, and two rank them C B A. Candidate A meets B: three voters put A first and four put B above A, so B wins 4 to 3. Candidate B meets C: the three A B C voters and the two B C A voters rank B higher, five against two, and B wins. Since B beats A and B beats C, B is the Condorcet winner. Notice that it is not the candidate with the most first places, which is A.
Now change the voters. Let there be three of them, one with each ranking A B C, B C A and C A B. Each ranking is a perfectly ordinary order of the three candidates.
voter 1 A B C
voter 2 B C A
voter 3 C A B
Count the contests. A against B: voters one and three rank A higher, so A wins 2 to 1. B against C: voters one and two rank B higher, so B wins 2 to 1. C against A: voters two and three rank C higher, so C wins 2 to 1. The majority preference is that A beats B, B beats C and C beats A.
That is the Condorcet paradox. No voter prefers A to B, B to C and C to A, so no voter is irrational in the sense of the money pump from “Preference, Transitivity and the Money Pump”. The group, taken as a chooser, has exactly the cyclic preference that lesson showed can be exploited: offer it B in exchange for A and a majority accepts, then C for B, then A for C.
How close a profile is to a cycle
Add two voters who both rank A B C to the cycle's three, so that five voters rank A B C three times, B C A once and C A B once, and A wins every contest: it beats B four to one and C three to two. Now move one of the three A B C voters to C A B. The contest between A and C flips to three against two for C, and the circle closes. A cycle can be one voter away.
The lab prints the table of counts and, when there is no Condorcet winner, the cycle it found. Two limits apply. It reports the cycle for the profile typed; how likely such profiles are in real elections is a different question. And it is not an argument against majority rule in the two-candidate case, where no cycle can occur.
Interactive
Collective choice
six voting rules, the majority relation, judgment aggregation and the jury theorem
Method
Finding a Condorcet winner or a cycle
Five moves for any profile with three candidates, and the last one checks the work.
1
Write the profile as a list of rankings with counts
Each ranking runs from best to worst and appears with the number of voters who hold it.
2
Pick a pair and count the voters on each side
For A against B, count the voters whose ranking puts A above B. The rest put B above A, since each ranking is complete.
3
Do this for every pair
Three candidates give three contests, and each contest has a winner unless the electorate is even and splits evenly.
4
Look for a candidate who won every contest it was in
If there is one, it is the Condorcet winner. If not, follow the winners round: if the chain returns to the start, there is a cycle.
5
Check that each voter's own ranking is consistent
Each ranking is a straight order. If the group's preference is a circle, the circle is a property of the group and not of any voter.
Counting contests
Score 0 / 0Choose the answer you can justify.
Three voters, three contests
Voter one ranks A B C, voter two B C A, voter three C A B.
A against B: voters 1 and 3 prefer A -> A wins 2-1
B against C: voters 1 and 2 prefer B -> B wins 2-1
C against A: voters 2 and 3 prefer C -> C wins 2-1
cycle: A beats B, B beats C, C beats A
Every contest is decisive, and each voter's list is in order. The circle belongs to the majority, which is why no change of mind by one voter and no tie-break can be said to have repaired a fault in them.
Common mistakes
1
Believing majority rule always produces a rankingWith three voters ranking A B C, B C A and C A B, A beats B, B beats C and C beats A, each by 2 to 1. Majority preference between pairs need not be transitive, so there may be no best candidate and no order. Where there is no cycle the rule works, and a rule that always produced a ranking would need something more than majorities.
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Taking a cycle for a tieEvery contest in the cycle has a clear winner. The three candidates are not equal; each is beaten by one rival and beats another. A tie would mean the counts were equal, and none of them is.
3
Blaming the votersEach of the three voters ranks the candidates in a consistent order. The inconsistency is created by combining three consistent orders, and no voter has to be fickle for it to appear.
Check your understanding
✓
Finish when you can tabulate pairwise majorities and say whether a profile has a Condorcet winner or a cycle.
Given a profile of rankings with counts, compute the count for each pair of candidates, name the Condorcet winner if there is one, and otherwise write the cycle; then build a three-voter profile that cycles.
Note: The lab's rule is shipped as the Condorcet rule; the same profile can be read under the other rules by changing it, and the next lesson does exactly that. Try changing the C A B voter in the cycle profile to C B A and watch the cycle disappear: B now beats A and beats C, each 2 to 1, and is the Condorcet winner. One ranking moved, and the circle is gone.