The Non-Transitive Voting Paradox

It was around this time of the year that we used to hold the elections for the club president. There are many who scoff at the need for such things and it is true that today most members prefer a much more egalitarian approach to membership and would shy away from any suggestion of a club hierarchy. But it was not always so. There have been 29 successive presidents of the Diogenes Club as indicated on the plaque on the library wall. But that all came to an end the year the elections took a strange turn.

Three people stood for president that year, Anstruther, Baldwin and Caldwell. A poll showed that there was almost equal support for each of the three candidates. One third of the members rated the candidates in order of preference A, B then C. Another third of the members rated the candidates in order of preference B, C then A. The last third of the members rated the candidates in order of preference C, A then B.

ORDER OF PREFERENCE
1/3 A, B, C
1/3 B, C, A
1/3 C, A, B

As a consequence Anstruthers quite rightly claimed that 2/3 of members preferred him to Baldwin. Baldwin claimed that 2/3 of members preferred him to Caldwell. And Caldwell claimed that 2/3 of members preferred him to Anstruthers. And the truth is they were all absolutely correct.

And there in a nutshell was the problem. The more deep thinking amongst us then realised that a clever committee could decide who would get the post by simply deciding the order of the voting. If they decided to vote first between A and B, then A would win, and the final run off vote between A and C would mean that C would win overall. But if we started off with a first vote between B and C, then B would win and the final runoff vote with A would see A winning. It didn't depend on the members wishes but on who was put forward first.

Well, when the members realised this, then they understood that it meant that the whole democratic voting system was faulty. It was not a problem with the three candidates, but a problem with the system itself. How often has a candidate come in first with less than half of the vote, which meant that more people didn't want him than did.

Every voting system was faulty and the members decided to have nothing more to do with it. So we did what any logical club would do. We did away with voting and with presidents altogether. And much better we have all been for it.