Another “small election exhibiting some obscure feature” post. I hope you’re not bored of these yet (but if you are I recommend you just don’t read them).

Consider the following election:

• 1, 2, 3, 4
• 2, 1, 4, 3
• 3, 4, 1, 2

The Kemeny-Young winner for this election is 1, as that is the unique Condorcet winner.

But if the second voter wants to vote tactically and changes their vote from 2, 1, 4, 3 to 2, 3, 4, 1 so the votes are now as follows:

• 1, 2, 3, 4
• 2, 3, 4, 1
• 3, 4, 1, 2

There is no longer a Condorcet winner. The majority preferences go 1 > 2 > 3 > 4 > 1 so there is a cycle.

However there is still an unambiguous Kemeny-Young winner (you’ll have to trust me on this one or work it out yourself. It’s a little hard to show the working). The new Kemeny-Young winner is 2, which our tactical voter strictly prefers to 1, so they should make that change.

The Gibbard-Sattherthwaite theorem guarantees that an election like this must exist, but I was pleasantly surprised at how simple the example was.