I am teaching my students about the fairness criteria for voting system, working up towards arrow's impossibility theorem.
One of the voting methods is called the pairwise comparison method: voters rank each of the candidates from most to least favourite. To tally the votes, talliers compare each pair of candidates. If candidate X is more often preferred than candidate Y, then X receives a point. (If they tie, they each get half a point.) At the end of the comparisons, the candidate with the most points is selected.
We discuss criteria for a voting system to be fair. One criterion in particular is the "irrelevant alternative criteria" which states:
If an election is held and a winner is declared, this winning candidate should remain the winner in any recalculation of votes as a result of one or more of the losing candidates dropping out.
Can anyone think of an example of when the pairwise comparison method violates the irrelevant alternative criterion?
Suppose we have three candidates $A,B,C$ and four voters with preferences as below:
$$\begin{array}{c | c | c | c} 1 & 2 & 3 & 4 \\ \hline A & A & C & B\\ C & C & B & A\\ B & B & A & C \end{array}$$
Then $A$ ties with $B$ and beats $C$ so has $3/2$ points, while $B$ ties with $A$ and loses to $C$ so has $1/2$ points, and $C$ loses to $A$ and beats $B$ so has $1$ point, thus $A$ wins. But if $C$ drops out, $A$ and $B$ tie.
This is the worst that can happen with three candidates, since if $B$ beats $A$ head-to-head then $B$ has at least $1$ point and $A$ has at most $1$ point. However, with four candidates $A,B,C,D$ and six voters we can have:
$$\begin{array}{c | c | c | c | c} 1 & 2 & 3 & 4 & 5 & 6\\ \hline A & A & C & C & B & B\\ C & D & D & D & A & A\\ D & B & B & B & D & D\\ B & C & A & A & C & C \end{array}$$ Then $A$ has $2$ points, $B$ has $3/2$ points, $C$ has $1$ point and $D$ has $3/2$ points, so $A$ wins. But if $D$ drops out we have:
$$\begin{array}{c | c | c | c | c} 1 & 2 & 3 & 4 & 5 & 6\\ \hline A & A & C & C & B & B\\ C & B & B & B & A & A\\ B & C & A & A & C & C\\ \end{array}$$
Then $A$ has $1$ point, $B$ has $3/2$ points and $C$ has $1/2$ point so $B$ wins.