I found the following fairness criterion in the exercises of Excursions in Modern Mathematics: If a majority of the voters have candidate X ranked last, then candidate X should not be a winner of the election.
I have two questions about this criterion:
1) Does it have an official name at all? Like the way that the Majority Criterion has a name?
2) The textbook claims that plurality with elimination (PwE) violates this criterion. I don't see why. Here is my counterargument: Suppose that candidate X has a majority of last place votes. Then, in the preference table for the election, there are several columns in which X is ranked last. In order for X to eventually win according to PwE, he is going to have to rise to the top of at least one of those columns. But in order to rise to the top of any column when you start at the bottom, all of the candidates above you need to have been eliminated. But this would mean that for X to win, all of the other candidates need to be eliminated. So a candidate with a majority of last place votes could never eventually get a majority of first place votes after rounds of elimination. Can someone tell me what is wrong with my argument?
Thanks!
Your argument is correct. To make it a little more precise, suppose that candidate $X$ is ranked last by a majority $m$ of the $n$ voters. In order for $X$ to win under plurality with elimination, $X$ must be one of the last two candidates. If $Y$ is the other surviving candidate, the preferences schedules at this point look like this:
$$\begin{array}{ccc} \underline{m}&\underline{n-m}\\ Y&X\\ X&Y \end{array}$$
Since $m>n-m$, $Y$ wins.
I’ve not seen a name for that fairness criterion. (I don’t think that it was even in some of the earlier editions of Tennenbaum.)