In this setup there are 4 candidates running. For a candidate to be eliminated, the candidate needs to receive less than 1/3 of the votes when paired up with another candidate. This process continues until only one candidate remains. The problem asks to prove that regardless of how the preferences are structured, it is not possible that every candidate could become a winner just based on the order of voting.
For a little context, one can consider where just a simple minority vote leads to elimination. Here such a case could occur. It is possible if A defeats B and B defeats C for C to then defeat A. In the case with 1/3 votes it is not so clear.
I can show that if A defeats B and B defeats C that C cannot defeat A. This is because more than 2/3 of the votes show that A > B. Also, more than 2/3 of the votes show B > C. Therefore, at worst there is 1/3 voters who voted both A > B and B > C. By transitivity, these voters choose A > C. Therefore, it is impossible to get a more than 2/3 vote for C to kill A. For the three candidate case it is easy then.
However, with four candidates it is possible to have A kills B, B kills C, C kills D, and then D kills A without violating the logic above. By choosing the order carefully, this would allow any candidate to win. So my guess is that D kills A is not possible but I can't prove it. Any help??