Plurality-with-runoff (b) violates monotonicity criterion.
Nbr of voters 1st 2nd 3rd
7 A B C
8 B C A
10 C A B
4 A C B
Runoff is between A and C. Striking B from table gives result for first place votes:
A --- 11,
C --- 18.
C wins.
1
Now suppose in a re-election, the voters in the last column change the order of A and C. New preference schedule.
Nbr of voters 1st 2nd 3rd
7 A B C
8 B C A
14 C A B
Now runoff is between B and C. Results:
B --- 15,
C --- 14.
B wins.
This violates the monotonicity C criterion: had the election won. In the re-election all changes were in favor ONLY of C. But C did NOT win the re-election.
2
(c) Plurality-with-runoff violates the Condorcet criterion. Nbr of voters 1st 2nd 3rd 10 A B C 8 C B A 6 B C A
The runoff is between A and C, and C wins 14 to 10. Is there a Condorcet candidate? Compare: B to A: 14 to 10. B wins that comparison. B to C: 16 to 8. B wins that comparison. Hence, B is a Condorcet candidate. Thus, this election violates the Condorcet criterion.
3