Arrow's Theorem

This scene stress-tests rules that merge many rankings into one. Switch profiles and rules and watch what breaks.

Majority rule can spin in a circle. A beats B, and B beats C. Then C turns around and beats A. The Borda count wobbles for another reason. Delete a hopeless alternative and the ranking flips. No rule keeps all four fairness conditions.

For example, three 🧑 friends run for class president. The class splits into three groups with clashing favorites. Head to head, A beats B and B beats C. Yet C beats A. Majority rule alone cannot pick a president.

\text{no rule satisfies U + P + IIA + non-dictatorship}

Part of these learning paths

General Equilibrium and Welfare

Examples & Exercises

Counterexample

For the profile (1: $A > B > C$ , 2: $B > C > A$ , 3: $C > A > B$ ), show by direct computation that pairwise majority cycles.

A > B,\quad B > C,\quad C > A

Verdict

The claim is false — pairwise majority cycles, $A > B > C > A$ , so it yields no transitive social ranking.

Condorcet paradox — no transitive social ranking.

Show proof

Counterexample and verification

$A$ vs $B$ : voters 1 and 3 support $A > B$ , so $A > B$ (2–1).
$B$ vs $C$ : voters 1 and 2 have $B > C$ , so $B > C$ (2–1). $C$ vs $A$ : voters 2 and 3 have $C > A$ , so $C > A$ (2–1). Hence $A > B > C > A$ cycles — pairwise majority yields no transitive ranking.

Counterexample

For the profile (1, 2: $B > A > C$ , 3: $A > C > B$ ), compute Borda scores and show how the $A$ – $B$ order changes when $C$ is removed (IIA violation).

\text{full: } A = B = 4,\quad \text{without } C\text{: } B = 2 > 1 = A

Verdict

The claim is false — the Borda rule violates IIA: removing the irrelevant alternative $C$ flips the $A$ and $B$ order.

The

A

–

B

order flips when

C

is removed.

Show proof

Counterexample and verification

Full Borda scores (2/1/0): $A = 1+1+2 = 4$ , $B = 2+2+0 = 4$ , $C = 0+0+1 = 1$ — $A$ and $B$ tie.
Without $C$ (1/0): $B = 1+1+0 = 2$ , $A = 0+0+1 = 1$ — $B$ wins. Removing the irrelevant alternative $C$ changed the $A$ – $B$ ranking, violating IIA.

17. Social Choice and Welfare