Tintin reaches Rastapopoulos’s island, but they are informed he has moved out, and the situation is pretty explosive there. Alonso, Bianca, and Coco are running for the post of the island head. On the ballot, each voter lists the three candidates in order of their preference. Counting only the first preferences results, dramatically, in a three-way tie. To break the deadlock, the second preferences are counted, but again there is a three-way tie. Alonso notes that, since the number of voters is odd, they can make two-way decisions without ties. He proposes that the voters first choose between Bianca and Coco, then the winner faces Alonso for the position. Bianca thinks it’s a good resolution, since they only want to identify the winner, not the runner-up. Coco disagrees and complains that this is giving Alonso an advantage. Who is right? Assuming the voters never change their preferences, what is Alonso’s chance of winning under his proposed voting system?

Options

A. Alonso
B. Coco
C. Both are wrong
D. None of these

Explanation

From the conditions of the problem, the number of voters is 3n for some odd positive  integer n. Each of the candidates A, B and C received n first preferences and n second  preferences. Let the number of voters who voted A gt B gt C be x and the number of voters  who voted `A gt C gt B` be y. Then x + y = n and the size of the remaining voting groups can be  easily computed: `A gt B gt C : x, A gt C gt B : y, B gt C gt A: x, B gt A gt C : y, C gt A gt B : x, C gt B gt A:  y`. When only comparing B against C, we have `B gt C : x + x + y = n + x, C gt B : y + x + y = n + y`.  Similarly (noting the cyclic symmetry), we have `A gt B : n + x, B gt A: n + y, A gt C : n + y, C gt A:  n + x`. Since n is odd, `x 6= y. If x gt y`, then B wins against C but goes on to lose against A. If `y gt  x`, then C beats B then also goes on to lose against A. Therefore Christie is right, Adrian will  always win under the proposed system.