1
Unanimity and non-dictatorship
Unanimity says that if every voter ranks A above B, so does the group. Non-dictatorship says there is no voter whose ranking becomes the group's ranking whatever the others think. Both are modest conditions.
Lesson · Justice and Collective Choice · Collective choice
Independence and Arrow's Theorem
Arrow asked what a rule that turns voters' rankings into a group ranking should satisfy, and proposed three conditions: unanimity, independence of irrelevant alternatives and non-dictatorship. On a profile the independence condition can be tested by removing a losing candidate and recounting. The theorem says that with three or more candidates no such rule meets all three on every profile. The wrong model is that Arrow proved democracy impossible.
unanimity: all prefer A, so the group does
IIA: A against B depends on A against B
non-dictatorship: no voter always wins
strike a loser and the winner can change
3+ candidates: no ranking rule has all
Arrow's conditions
Core ideas
What this lesson establishes
Arrow's theorem is about a kind of rule and three conditions on it. The conditions are easier to see on a profile than to state in the abstract, and one of them can be tested directly.
2
Independence of irrelevant alternatives
The group's ranking of A against B should depend only on how each voter ranks A against B. A third candidate who is ranked above or below them by various voters is irrelevant to that comparison, and should not change it.
3
The test is a removal
Strike a losing candidate from every ballot, keeping the order of the others, and recount under the same rule. No voter has changed their view of the candidates who remain, so if the winner changes, independence has failed on that profile.
The material
Striking a loser
Two profiles on which the winner changes when someone who lost is removed, and what the theorem says about rules that cannot avoid it.
The Condorcet paradox is a failure of a particular rule, majority rule, to produce a ranking. Arrow's question was larger: is there any rule that turns every profile of rankings into a group ranking and behaves sensibly? He made “sensibly” precise with three conditions, and the second is the one that fails in practice.
- Unanimity. If every voter ranks A above B, the group ranks A above B.
- Independence of irrelevant alternatives (IIA). The group's ranking of A and B depends only on how each voter ranks A and B.
- Non-dictatorship. No one voter is such that the group ranking always equals theirs.
Return to the nine voters of the last lesson: four rank the candidates A B C, three rank them B C A, and two rank them C B A. Under plurality A wins with 4 first places, against 3 for B and 2 for C. Now strike C from every ballot. The four A B C voters become A B, the three B C A voters become B A, and the two C B A voters become B A. Plurality now counts 4 for A and 5 for B, and B wins.
No voter has changed a view about A and B. Before the removal, 4 voters ranked A above B and 5 ranked B above A; after it, the same 4 and the same 5. Yet the group's ranking of the pair has reversed, because the voters who ranked C first have had their first places handed to B. The presence of C decided the contest between A and B, and C was never going to win.
The lab finds this when the removal menu is set to nobody: it tries each losing candidate in turn and names the first whose removal changes the winner. Here that is B, since without B the contest is between A with 4 and C with 5, and C wins. Set the menu to C and the winner becomes B, as above.
Definition · Independence of irrelevant alternatives
A rule satisfies independence of irrelevant alternatives when the group's ranking of two candidates depends only on the voters' rankings of those two. The lab's test is a close and visible cousin: it asks whether the winner changes when a losing candidate is struck from every ballot.
A rule that counts more fails too
It is tempting to blame plurality for reading only first places. Take nine voters: two rank them A C B, two B A C, two B C A and three C B A. Plurality elects B, with 4 first places. Borda also elects B, with 11 points against 10 for C and 6 for A. But C beats B by 5 to 4, C beats A by 5 to 4, and C is the Condorcet winner.
Strike A, who loses under every rule. With two candidates left, both plurality and Borda reduce to majority rule between B and C, and C wins 5 to 4. Both rules fail the removal test on this profile, and the lab says so under either rule.
The theorem
Theorem · Arrow's theorem
Suppose there are at least three candidates, and a rule turns every profile of complete, transitive rankings into a complete, transitive group ranking. If the rule satisfies unanimity and independence of irrelevant alternatives, it is a dictatorship: some one voter's ranking is the group ranking on every profile.
The proof is not reproduced here. The conditions, on the other hand, can all be read on a profile, and the theorem says that no choice of rule stops the removal test from failing somewhere. It is a statement about a specific kind of rule, and three things it does not say are worth stating.
- It does not say elections cannot be held. It says no single rule of this kind has all three properties on all profiles.
- It does not apply to two candidates. Majority rule between two satisfies all three conditions.
- It does not say every rule fails the same way. Borda gives up independence, majority rule gives up always producing a ranking, and restricting the profiles, for example to voters who place the candidates on one left-to-right line, lets majority rule work.
The lab's rules all respect unanimity, and the only dictatorship it can show is the degenerate one of a single voter. Its removal test is about the winner, while Arrow's condition is about the ranking of a pair, so a pass on one profile is a pass on that profile and nothing more.
Interactive
Collective choice
six voting rules, the majority relation, judgment aggregation and the jury theorem
Method
Testing independence on a profile
Five moves. The first four are the removal test and the last relates it to the theorem.
1
Find the winner with everyone standing
Use the rule you are testing and write the winner down. A candidate who wins cannot be the one removed.
2
Pick a loser and strike it from every ballot
Delete it from each ranking and close the gap. Everyone else keeps their relative order, so every voter's view of every remaining pair is unchanged.
3
Recount under the same rule
The counts, and for Borda the points, are recomputed from the shortened ballots.
4
Compare the winners
If the new winner differs, independence fails on this profile, and no voter changed their mind about the pair that now ranks differently. If every loser's removal leaves the winner alone, the test passes on this profile only.
5
Ask which condition the rule gave up
A rule that fails the test is one of the rules the theorem says must give something up. It does not contradict the theorem, and it does not show that a better rule is missing from the profile.
Conditions and removals
Score 0 / 0Choose the answer you can justify.
Striking C from nine ballots
Four voters rank A B C, three B C A, two C B A, under plurality.
with C: A 4, B 3, C 2 -> A wins
strike C: A B 4, B A 3 and 2 -> A 4, B 5
without C: B wins
A over B: 4 voters, before and after
B over A: 5 voters, before and after
The group's ranking of A and B reversed, and the voters' rankings of A and B did not move. The failure is a property of the rule applied to the whole ballot, which is what independence forbids.
Common mistakes
1
Believing that Arrow proved democracy impossibleThe theorem is about rules that turn every profile of rankings into a full group ranking and satisfy three stated conditions. With two candidates, majority rule satisfies all three. With more, each real rule gives up something visible: plurality and Borda give up independence, as the removal of one loser above shows, and majority rule gives up always producing a ranking, as the cycle did. What the theorem forbids is getting everything at once.
2
Reading an independence failure as a voter changing their mindWhen C is struck from the nine-voter profile, the same 4 voters still rank A above B and the same 5 still rank B above A. The group's verdict reversed because the rule gave the voters who ranked C first to B, and no one's view of A or B moved. The failure belongs to the rule.
3
Hearing “dictator” as a tyrantNon-dictatorship rules out a voter whose ranking is the group's on every profile. The dictator need not be powerful or malicious: a rule that copies the first voter's ballot and ignores the rest is a dictatorship by the definition. With one voter alone, as in the last preset, every rule here is one.
Check your understanding
✓
Finish when you can test independence by removing a loser and state what Arrow's theorem says and does not say.
Given a profile and a rule, strike a losing candidate, recount, and report whether the winner changed; then state the three conditions and the theorem, and name two things the theorem does not claim.
Note: Set the rule to Borda on the first preset and strike each candidate in turn: the Borda winner is B, and the removal test passes on that profile. Independence fails on a profile, not for a rule in general, and the lab can show you only the profiles you type.