Persons A, B, and C have Euclidean preferences centered around their ideal points, marked A, B, and C. Recall that Euclidean preferences means that they are equally unhappy at any points equidistant from their ideal points, so indifference curves are circular around the ideal points, with circles further from the ideal points representing lower levels of utility than circles closer to the ideal points. The triangle A-B-C is called the Pareto Set; we'd like to keep chosen outcomes within the Pareto Set (though nothing necessarily restricts outcomes to being within the Pareto Set).
Ok. Say that point Z is our status quo point denoting a combination of X and Y chosen by the group of three people. Any point within the lightly-shaded "petals" surrounding point Z will be majority preferred to point Z (we call that the win-set). Why? Because 2 of 3 people are happier within one of the shaded areas than at the status quo point Z. Taking the large shaded area to the left of Z, we see that the shaded area is closer to the ideal points of people A and B than point Z is, so A and B will vote for any point within the shaded area rather than point Z. Point Q lies within the shaded area; if we put it to a vote, Q>Z by 2-1. But, point Q has its own win-set -- the darkly-shaded petals surrounding Q. Any point within those darkly-shaded petals will be majority preferred to point Q. Taking the big area to the right of Q, we see that persons B and C will vote for any point in that area as they prefer any of those points to point Q. So, if we have a vote pitting point R against point Q, R>Q by 2-1. But, again, R has its own win-set; here, it's the area bordered by the dark lines. Now, we see that our starting point, Z, is contained within R's win-set. That means that Z will beat R 2-1, with A and C voting for Z. So, we have big problems in making social choice! Q>R>Z>Q>R>Z>Q>R>Z and so on forever by majority voting.
McKelvey proves that any point chosen by an appropriately-empowered agenda setter can be reached by a series of majority votes. This result is called the "McKelvey result"; we have global cycling over the entire domain.
Another interesting aspect is Buchanan's conjecture that cycling is desirable; if we have cycling, then we can't have the tyranny of the majority, where the majority turns the minority into its slaves. Say that A and B wanted to gang up and choose a result wholly unacceptable to C. Well, C could always find some other point that it would prefer, and that at least one of A or B would also prefer. Cycling, according to Buchanan, may be a desirable feature of democracy rather than a cause for despair. Something to think about!
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