01:640:103 Chap.2 Fairness Criteria and Arrow's Impossibility Theorem
Fairness Criteria and Arrow's Impossibility Theorem
Criteria judge methods; they do not elect anyone
A fairness criterion turns an intuition about who ought to win into an exact rule. The Majority Criterion says a candidate with more than half the first choices should win. The Condorcet Criterion says a candidate who beats every rival head to head should win.
The Monotonicity Criterion concerns a revote in which the only changes move the original winner higher, and says that winner should still win.
The Independence of Irrelevant Alternatives Criterion concerns a revote after a losing candidate withdraws, and says the original winner should still win.
Never versus sometimes
A method sometimes violates a criterion if a single election, or a single before-and-after pair, can be built in which it does. A method never violates a criterion only if an argument covers every possible election.
Plurality and Instant Runoff never violate the Majority Criterion because a majority candidate wins in the first count; Pairwise Comparisons never violates the Condorcet Criterion because a Condorcet candidate wins every pairing.
The Borda Count can defeat a majority candidate, Instant Runoff can punish a winner who gains support, and removing a losing candidate can change a Borda or Pairwise winner.
Arrow's theorem and its limits
Kenneth Arrow proved in 1949 that no voting method can satisfy all four of these criteria in every election, and he was awarded the 1972 Nobel Prize in Economics.
The theorem does not say that every election is unfair or that democracy fails. It says only that no method can be guaranteed never to break one of the four intuitions, so the choice of method always involves a trade-off.
Building a counterexample yourself
To show that a method sometimes violates a criterion, you need one concrete election.
For the Majority Criterion, give one candidate a bare majority of first places and put that candidate last on every other ballot, then check the Borda totals. For monotonicity under Instant Runoff, find an election where the eventual winner needs a particular rival to be eliminated first, then move the winner up on a few ballots so that a different rival drops out instead.
For independence of irrelevant alternatives, remove a losing candidate and recount. Each counterexample is a proof of sometimes; none of them says anything about how often the failure happens in practice.
What this chapter covers
- 01
Fairness criteria versus voting methods
- 02
The Majority and Condorcet criteria
- 03
Monotonicity and revotes
- 04
Independence of irrelevant alternatives
- 05
Arrow's impossibility theorem
Worked example · free
A majority candidate who loses
- 1A has 12 of 21 first choices, more than half, so A is the majority candidate and the criterion applies.
- 3Borda points are A 45, B 47, C 34, which total 126 = 6 × 21.
- 1B wins under Borda although A holds a majority, so this election is a violation.
Key terms
- Fairness criterion
- A fairness criterion spells out exactly who ought to win an election, and is used to judge voting methods.
- Condorcet candidate
- A Condorcet candidate beats every other candidate in head-to-head competition; an election may have none.
- Monotonicity Criterion
- The Monotonicity Criterion requires a winner to stay the winner when a revote only moves that winner higher.
- Arrow's theorem
- Arrow's theorem shows that no voting method can satisfy all four fairness criteria in every election.
- Strategic voting
- Strategic voting is ranking candidates insincerely to change the outcome, such as burying a strong rival.
- Majority Criterion
- The Majority Criterion says a candidate holding more than half of the first choices should win.
Fairness Criteria and Arrow's Impossibility Theorem FAQ
Does a Condorcet candidate always have a majority of first choices?
No. A candidate can win every head-to-head contest while being few voters' first choice, even nobody's, if most ballots rank that candidate second. The reverse direction does hold: a majority candidate always beats each rival head to head.
What makes a revote count as a monotonicity case?
Every change between the two ballot sets must move the original winner higher, and nothing else may move up. If voters also promote a losing candidate, a new winner proves nothing about monotonicity, although it may be an example of strategic voting.
Does Arrow's theorem mean elections are unfair?
It means only that each voting method, applied to enough elections, will eventually break at least one of the four criteria. Most individual elections break none, and methods are chosen for the situations where their risks matter least.
Can a method violate the Majority Criterion in an election with no majority candidate?
No. A violation requires a majority candidate who then loses. If no candidate holds more than half of the first choices, the criterion simply does not apply, whatever the method decides.
Exam move
For each method keep a two-column note: criteria it never violates with the one-line reason, and criteria it sometimes violates with a small counterexample you built yourself.
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