Reversal of Social Preference in Plurality Voting

I continued working through the exercises in Chapter 1 of Donald Saari's Basic Geometry of Voting, and found another interesting problem in Exercise 1.1.3.3 (just one exercise after my last posts' excursion, considering when a plurality winner can be a Condorcet loser).

This time, we're given some information about voters' preferences in an election. All we're told is how many voters ranked each alternative first. The example in the text has four alternatives; A, B, C, and D with 9, 8, 7, and 6 votes, respectively. The election uses the plurality method, so that the social preference is A beats B beats C beats D.

We're then told that if a candidate drops out of the election, all of those people who voted for them instead vote for their second choice preference. It turns out that such an occurrence can actually reverse the resultant social preference. For example, removing candidate D can lead to the plurality result:

C beats B beats A!

I found this quite strange, and decided to dig deeper. I determine an algorithm which "optimizes" this strange result, and find conditions to see what elections permit such a result. You can read about it all right here:

Reversal of Social Preference in Plurality Voting

Oh, and I managed to finish the rest of the exercises in the section without feeling the need to write another blog post. Good thing, otherwise I might never get past chapter 2. The introductory material does, however, give one a lot of interesting things to think about (obviously)!