In what I call the random elections model,
(which is a rather crude probability model,
but in my experience the results you get from it are
surprisingly valid in real life)...
the two most common IRV pathologies seem to be "favorite betrayal"
and "participation" failures.
In Favorite Betrayal [probably the most important-in-practice kind of spoiler], it is unstrategic to vote for your true favorite top. In this model that happens 19.6% of the time (for at least one kind of voters) in 3-candidate elections. There is a nice exact formula for 19.6% namely arccot(√2)/π. (Also for strategic voters acting on imperfect information, it happens, arguably, 100% of the time.)
Concrete examples: simple, realistic, unusual structure, Jeez Louise.
Incidentally these refute the false but unfortunately common claim that IRV "eliminates" the "spoiler" and "wasted vote" defects of plurality voting.
In Participation Failure, the election is such that if you were to add t more co-voting honest voters, (for some t and some vote-type) then the election result would get worse from their point of view, i.e. they were "better off staying home." This happens 16.2% of the time in 3-candidate elections, see my preliminary write-up here; and also, according to analysis by Depankar Ray, exactly 50% of the time in 3-candidate elections in which the IRV and plurality winners differ.
("16.2%" ought to be expressible in close form also, but have not tried; not sure how simple you could get it. I also can semi-prove participation failure happens with probability→1 in the limit of a large number of random voters and large number of candidates, for two Condorcet methods. What happens to these two pathology probabilities for IRV in the limit of a large number C of candidates? I do not know. It certainly increases with C. There might be an easy argument it goes to 1?)
There are additional nasties concerning "top-3 only" bastardized IRV which are very common in real life.
For examples of IRV refusing to elect Condorcet ("beats-all") winners, as happened in Peru 2006, France 2007, and Chile 1970, see these: core, nasty, another. We can make this very extreme. But this pathology is fairly rare in the random elections model; it happens about 3.7% of the time (see puzzle 17).
Incidentally, both this and IRV reversal failure totally refute the false but unfortunately common claim that IRV "always elects true-majority winners."
This pathology is however considerably more common in "1-dimensional politics" (and "2-dimensional politics," see pictures where it happens about 25% of the time).
Here is a simple and somewhat interesting model of 1-dimensional politics. Assume 3-candidates (A, B, and C) and all 4 "singlepeaked 1-dimensional" votes are equally likely: A>B>C, C>B>A, B>A>C, B>C>A – except the latter two have half the likelihood just to keep things interesting – 3-way near-tie in top rank votes. Note, this model is precisely regarding all "interesting" 1D-politics elections as equally likely, where "interesting" means all 3 candidates have a chance – not just a 2-man race – and "1D" means the candidates A,B,C are three points on a line in that order, and the only votes that can happen are orderings of the three points in decreasing-distance (from someplace) order.
Question: Then (in the limit of a large number of independent random voters) what are the probabilities IRV yields a favorite betrayal scenario, or a "center squeeze" scenario where the middle candidate B is not elected by IRV despite being a Condorcet winner?
Answers:
A look at Australia's federal IRV races in 2007 suggests at least 9 of the 150 were pathological, probably in this sort of center-squeeze fashion (we cannot tell with certainty which pathology happened because Australia refuses to provide full-enough election information to allow us to reconstruct the ballots, but as the analysis shows we can still deduce the presence of either paradox A or paradox B in many elections...) and probably a lot more than 9.
IRV non-monotonicity
[raising X in your vote decreases X's winning
chances]:
this happens with probability (2.5+12.2=14.7)% in 3-candidate
large random elections (there are two disjoint kinds of nonmonotonicity,
"winner now loses" and "loser now wins"
hence the two numbers; also Schläfli function method explained in
puzzle #4).
Concrete example.
This also can occur in extreme forms where e.g. raising X from bottom to top causes X to lose. (Another example; see point 5 there). Here's a related weird phenomenon.
For IRV reversal failure where the IRV winner is the same as the IRV loser, see this and this happens about 2.6% of the time in random 3-candidate elections.
Related drop-out⇒reversal paradox.
IRV exponential amplification chaos.
IRV "Irrelevant losers" who aren't.
You can't count IRV elections in precincts because there is no such thing as a "precinct subtotal" (at least nothing that can be recorded in any form more compact than just listing all the votes in that precinct, if there are a reasonable number such as 13 of candidates). Here is a web page to make that clear: IRV non-additivity.