by Clay Shentrup & Warren Smith (See also simpler pocket-sized example and Pereira's example where IRV does worse than plain plurality voting)
Rob Richie, evangelist for the Instant Runoff Voting (IRV) movement, excuses the fact that IRV can fail to elect a Condorcet ("beats all") winner, citing scenarios where the Condorcet winner didn't have many first-place votes. Richie calls the first place votes "core support." That's somewhat arbitrary, because order tells us essentially nothing about intensity. For instance, you could hate or love all 4 candidates, or love 2 and hate 2 – and still have the same ordered preferences. So what's so special about first place support? Nothing really, but Richie believes otherwise.
In any case, here is an anti-Richie example with 28 voters and 4 candidates where
|10||G > C > P > M|
|3||C > G > P > M|
|5||C > P > M > G|
|6||M > P > C > G|
|4||P > M > C > G|
C is the clear "Condorcet winner," meaning he is preferred by a landslide majority over all his individual rivals. C is preferred over G, P, and M all by an 18-10 margin.
But... M wins this election if we use Instant Runoff Voting (IRV),
even though he also has less "core support" (6 voters) than C with 8.
It's amazing how much crazy IRV behavior occurs in this single example, but this is by no means the craziest.
In this example, 18 voters say C>M while 10 say M>C. The reason IRV nevertheless makes M win, is that IRV ignores 10 of the C>M votes, and only looks at 8 of them. Those 8 are not enough to overcome the 10 M>C votes.
Also e.g, 22 voters say P>M while 6 say M>P. The reason IRV nevertheless makes M win, is that IRV ignores 18 of the P>M votes, and only looks at 4 of them. Those 4 are not enough to overcome the 6 M>P votes.
It is about ignoring the preferences of some voters, while counting others.
The example above was intended to be "realistic," perhaps somewhat resembling the situation in the (now evolving) 2008 US presidential race with G="Green", M=McCain, C=Edwards, and P=Paul. But if you are willing to drop realism and construct artificial election scenarios, then this demonstrates how to construct arbitrarily-severe election examples of this kind, where, e.g, the IRV winner (our M) loses pairwise to the Condorcet winner (our C) by an enormous (100:1) margin, and also M has less core support by an enormous (100:1) margin, and also M is ranked bottommost or second-worst by 75% of the voters, and also M is below average in the eyes of 99.9% of the voters, and also M would lose a head-to-head race with every opponent but one by at least a 75-25 landslide, and also M has less core support than that one exception by a 100:1 margin, and also C would beat every opponent pairwise, and also C's pairwise victories would be a 75-25 or larger landslide in every case but one (where C still wins, but not by a landslide)... but still IRV insists on electing M!
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