Condercet was a French social theorist in the opening decades of the 19th century and is credited with first discovering a paradox of majority voting that bears his name. Here is the paradox: Imagine that you have a group of three people (A,B, and C) who are voting on three different alternatives (X, Y, and Z). A prefers X to Y and Y to Z. B prefers Y to Z and Z to X. C prefers Z to X and X to Y. If X is paired in a vote with Y, then X wins (A and C against B). If Y is paired with Z, then Y wins (A and B against C). But – and this is the kicker – if Z is paired with X, then Z wins (B and C against A). In other words, even if the individual preferences of A, B and C are transitive, the collective preferences of A, B, and C are not. Put in starker terms, if you control the order the votes are taken in, then you can get any outcome you want because any choice can be defeated by one of the others. I have often wondered if this paradox might in part account for how Brigham Young became president of the Church.
When Joseph was murdered, it was not clear to the Saints who his successor should be. There were lots of claimants, but in the months immediately after his death the biggies were Sydney Rigdon and the Quorum of the Twelve. In August of 1844 a conference was held. Both Brigham and Sydney spoke, and then Sydney put to the conference the question of whether they would prefer to be led by himself of the Quorum of the Twelve. The conference voted in favor of the Twelve and Sydney left Nauvoo to found his own church, which rapidly fell to pieces. More than three years later, a conference in Winter Quarter’s voted to make Brigham Young President of the Church.
Notice the way the votes were ultimately paired. First it was the Twelve versus Rigdon and the Twelve won. Then it was the Twelve versus Brigham and Brigham won. Condercet teaches us, however, that the mere fact that Brigham beat the alternative that beat Sydney does not necessarily mean that Brigham could have beat Sydney. In other words, the outcome might have been different had the votes been taken in a different order.
Of course, this need not necessarily have been the case. Brigham might have beaten Sydney if the vote had been put that way in 1844. I don’t think that we really know enough about the preferences of the voters to ever be certain one way or the other. Very few people voted in favor of Sydney at the 1844 conference and he probably would have lost regardless of how the votes were structured. Who knows! Still, it is an intriguing little possibility.
Democratic theorists have often been troubled by Condercet’s Paradox, and by its more rigorous theorization by Kenneth Arrow. It seems to suggest that at least under some circumstances, majority will is a fiction. There is simply the order in which the votes are taken. (Or in the absence of such agenda setting, endless cycling of alternatives.) However, if majority will is a fiction we are simply being ruled by the agenda setters.
Interestingly, there is one way of insuring that you never run into the problem of Condercet’s Paradox. In place of a majority wins rule, you substitute a super-majority or unanimity requirement. Interestingly, if what we are told is correct, the Quorum of the Twelve has adopted such a unanimity requirement. No action is taken unless everyone agrees. Obviously, this would create its own interesting dynamics, but it does insure that institutional decisions are not simply the random result of the order in which the votes are taken.