3.28 Plurality method violates the IIA

In a plurality voting system 7 voters rank alternatives (A, B, …).

• 3 voters rank (A\(\succ\)B)
• 2 voters rank (B\(\succ\)A)
• 2 voters rank (B\(\succ\)A)

Assume that in an election, initially only A and B run: then, B wins with 4 votes to A’s 3. However, the entry of C into the race makes A the new winner.

• 3 voters rank (A\(\succ\)B\(\succ\)C)
• 2 voters rank (B\(\succ\)A\(\succ\)C)
• 2 voters rank (C\(\succ\)B\(\succ\)A)

• The relative positions of A and B are reversed by the introduction of C, an “irrelevant” alternative.