## A 9-voter 3-candidate election example suggesting something is wrong with both instant runoff and Condorcet

By Toby Pereira, May 2016

#voters Their Vote
3 A>B>C
2 B>A>C
2 B>C>A
2 C>A>B

A is the Condorcet winner, beating B pairwise by 5:4 and also beating C pairwise by 5:4. Also, A would win using "instant runoff voting" (C eliminated, then A defeats B 5:4). Now: remove two voters each of types "A>B>C," "B>C>A," and "C>A>B" (six removed in all) who all together should constitute a three-way tie. Then you are left with:

#voters Their Vote
1 A>B>C
2 B>A>C

whereupon B becomes the Condorcet winner! Namely, B beats A pairwise by 2:1, and B also beats C pairwise by 3:0. And B also is the winner using "instant runoff voting."

This seems to demonstrate a self-contradiction within the Condorcet philosophy. (Also within the instant runoff philosophy.)

However, it does not demonstrate a contradiction for Borda voting; B is the Borda winner in both elections. To see a Borda self-contradiction, consider this.

Fishburn's anti-Condorcet example