Before beginning, to start getting in the right frame of mind, check out this completely idiotic Instant Runoff Voting (IRV) election.
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! That argument is based on the theory that, if a voter does not know whether the IRV-spoiler scenario will arise (but figures it will with probability ≈19.6%) but does know that historically with IRV, one of the two major-party candidates wins with probability>99.7%, said voter would probably consider it rational to vote one major top, other major bottom, even if that voter honestly prefers a third-party candidate. If voters do this, then the 19.6% IRV-spoiler risk is avoided, but at the cost of sacrificing the <0.3% chance to elect the good third-party candidate. This sacrifice seems worth the benefit, on average. But it is a theorem that if >75% of voters adopt this top+bottom "naive exaggeration" voting strategy, then it becomes mathematically impossible for a third-party candidate to win... thus amplifying the pathology in a vicious cycle...
However, this "100%" theory has been (justifiably) attacked by Jameson Quinn as resting on a probabilistic delusion. E.g. if majors are winning with probability>99.7% (which they have, historically, in the Australian House in recent decades) that means the "random election model" and the 19.6% spoiler probability estimate were wrong; in a more-correct voting model, the spoiler-risk is much lower, thereby incentivizing honest voting. However, the more honest voters there are, the more the "random election model" becomes correct again... so one might expect an "equilibrium" might develop. It would be interesting to examine that mathematically, but as a matter of reality rather than mathematical models, the fact is, the last three Australian House elections (2001, 2004, 2007) combined elected zero out of 450 third-party-member seatholders despite an average of about 7 candidates per seat.
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, Burlington 2009, France 2007, and Chile 1970, see these: core, simpler, 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), hence it perhaps was wrong of me to label it as "less common."
Here is a simple and mildly 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 – yielding a 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 increasing-distance (from someplace) order.
Question: In this cheesy 1D model (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 more.
IRV non-monotonicity
[raising X in your vote decreases X's winning
chances – or lowering X in your vote increases his winnign chances]:
this happens with probability (2.3+12.2=14.5)% 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).
That same page shows that in
N-candidate elections with N large, IRV becomes overwhelmingly likely to suffer non-monotonicity.
Concrete example.
Also occurred in
Burlington 2009.
This also can occur in extreme forms where e.g. raising X all the way 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 IRV's "best winner" is the same as IRV's "worst loser," see this; it happens about 2.6% of the time in random 3-candidate elections.
In the "cheesy 1D model" above, an "extremist" A or C always wins the vote-reversed election. The forward-vote election is won by the opposite extremist when B is eliminated, any by B when either A or C is eliminated, hence the forward and reverse winners are equal zero percent of the time asymptotically and reversal failure is almost a nonproblem in that model.
Related drop-out⇒reversal paradox.
IRV exponential amplification chaos.
IRV "Irrelevant losers" who aren't.
IRV exhibited a lot of the above pathologies simultaneously in the Burlington VT 2009 mayor election.
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 much 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.
IRV paradox probabilities (up to date, world's most accurate & extensive list)