Puzzle: How often do "Condorcet cycles" arise in real elections with rank-order ballots? Give some prominent real historical examples of such elections.
Answer 1: For public elections, this is unknown. It is very difficult to find an example both because they are rare (few percent?) and because, even when they do occur, how are you going to know it? You could know it if there were polls assessing all pairwise preferences, but such polls are only conducted rarely.
You might be able to make a stab at this by examining Irish rank-order-ballot data (which the Irish have recently started releasing on the internet in some cases 1).
You would have to examine at least 1000 elections to get good-enough statistical error bars. (And, even then your results would be attackable as not "real" since really these votes were almost all or all cast in nonCondorcet, multiwinner elections.) We have summarized a good deal of what is known here.
Answer 2: But for legislative votes, we do know that Condorcet cycles are very common, and a major problem! Why? Because different factions often intentionally try to create them in order to defeat some proposal.
To see what I mean, consider some proposal A. The legislature can either vote "Yea" or vote "Nay" – and in the latter case we stay with the status quo Q. Now suppose a majority feel A>Q so that A would (under normal circumstances) pass. But of course, the opponents of A (whom we suppose can control the order in which votes are taken, i.e. have "agenda setting power") don't want that to happen. So they contrive some amended proposal P, known as the "poison pill," which the majority feel is superior to A, i.e. P>A. The majority dutifully get rid of A and replace it with the amended proposal P. But then P fails to pass because – and this was the point – there is a Condorcet cycle P>A>Q>P and so the majority feels Q>P and defeats the proposal.
OK? This is so common that the word "poison pill" had to be invented to describe such amendments! Some US Congressional recent examples are this, this, this, and this. One of the latest stunts of this nature was the use of the gun lobby as a poison pill to torpedo giving Washington DC taxpayers voting rights; this was actually a variant which successfully poisoned the bill in spite of the fact that the leadership with agenda-setting power actually was in favor of the bill. A parade of poison pills were proposed by Republican Party leaders (including Bill Frist) for attachment to bills to raise the minimum wage for the first time in over 10 years (during which time, Congress had raised its own salaries by over $32,000). These succeeded several times in killing the minimum wage increase.
W.H.Riker [Arrow's theorem and some samples of the paradox of voting, Arnold Foundation Monograph, Dallas, Southern Methodist University 1965] gave several less-recent Congressional examples. For example, the 17th Amendment to the US Constitution (direct election of senators) would have passed Congress 10 years before it actually did if not for a poison pill amended-Amendment which was devised to make it become less acceptable to Southerners. The only reason the 17th eventually passed was the composition of Congress eventually changed,
Of course, if Congress employed range voting among A,P, and Q, then this problem would vanish.
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