Failure of Majority Rule with Runoff:

Consider the following two profiles, which differ only in that twenty-five citizens who ranked C first in profile 1 rank C second in profile 2.  Pairwise preferences for A versus B are the same.
Profile
1
60
Citizens		 25 
Citizens		 50
Citizens		 40
Citizens
First
Choice
A
 C B C
Second
Choice
C
 A A B
Last 
Choice
B
 B C A
Majority rule with runoff:
First round A:60, B:50, C:65
Runoff A: 110, C:65 so A wins.
      
Profile
2		 85 
Citizens		 50 
Citizens		 40
Citizens
First 
Choice		 A B C
Second
Choice		 C A B
Last
Choice		 B C A
Majority rule with runoff:
First round A:85, B:50, C:40 
Runoff A: 85, B:90 so B wins
Simnple Majority rule fails the Unrestricted Domain test in both profiles since no candidate gets the required 88 votes..

Majority with runoff fails Positive Association since 2 citizens raising their ranking of A lowers the social ranking of A

Majority rule with runoff also fails Independence of Irrelevant Alternatives since each citizen's pairwise ranking of alternatives A and B in profile 2 is the same as it is in profile 1, but A and B are NOT in the same order in the two social preference orderings.  The same 85 people prefer A to B and the same 90 people prefer B to A pairwise in both profiles.