I would have thought I’d had enough of these sorts of mental torture after reading Inevitable Illusions by Massimo Piatelli-Palmarini. However, I had to stop and work through this a few times when I discovered it on the website of the Montreal Economic Institute:

Question: Suppose that three alternatives can be put before the Québec voters: alternative A is the constitutional status quo; alternative B is a renewed federalism with more powers to the Québec government; alternative C is the separation of Québec from Canada. Is it possible that, even if each voter is individually rational, the majority would cycle between A and C – i.e., would choose both a united Canada (alternative A) and an independent Québec (alternative C)?

Hint: Here, rationality means transitive preferences: if, or example, a voter prefers A to B and B to C, he will prefer A to C.

Answer: Yes. The electorate will not necessarily be irrational, but it may very well be. To see this, assume that the electorate is made of three equal sub-groups of voters: the Xs are centrists who prefer B to A to C; the Ys are separatists who prefer C to B to A; and the Zs, afraid of “political uncertainty” generated by the unstable middle, prefer A to C to B. It is easy to see that, if the electorate is presented with a choice between A and B, the majority (two thirds in the present example) will vote for B. Similarly, if the choice put to the voters is between B and C, the majority will chose C. We would tend to conclude that the electorate, which prefers C to B and B to A, would rationally prefer C to A. Now, if we ask the electorate to vote on C and A, it can be seen that the majority will chose A – as both the Xs and the ’s prefer A to C. Thus the voters prefer both separation (C) to the status quo (A), and the status quo (A) to separation (C).