In this 24-voter IRV election, A wins after the IRV process eliminates C. But now suppose every voter reverses her preference order (i.e. they are now attempting to choose the worst candidate rather than the best). In that case A still wins after B is eliminated. I.e. IRV contradicts itself; IRV's unambiguously "best" candidate A is here the same as its "worst"! (Such embarrassing winner=loser reversal failures are, however, never exhibited in range, approval, Borda, Tideman Ranked Pairs, or Schulze beatpath voting, all of which are logically self-consistent in this respect.)
This failure is really just a mere symptom of the flawed logic underlying IRV.
This example also illustrates a bizarre kind of strategic voting. Suppose 3 of the B>C>A voters reverse their votes to A>C>B. In that case B is eliminated whereupon C wins 13-to-11 over A. Incredibly, the reversal's raising of A from bottom-to-top in their vote caused A to lose – and voting maximally dishonestly as though they were suicidally trying to elect the worst candidate, was actually optimal strategy that caused the election result to improve from their point of view. (And that pathology happened in Peru 2006.)
Finally, this election also illustrates a "no show paradox": If those three B>C>A voters had simply refused to vote, then C would have won, an improvement in their view. Different way of saying the same thing: these three voters' decision to cast an honest A-last vote caused A to win. (Amazing, but true.)
In the 14-voter 4-candidate IRV election on the left, the winner is A while the last-place finisher is D (knocked out in round 1).
However, if all ballots are reversed so that every voter expresses exactly the opposite opinion on everything, then we get the election at right. You might think that now D would win and A finish last. Wrong – that would be sane. What the IRV system delivers is instead the psychotic result that A still wins, while D still finishes last.
These examples also nicely lay to rest the silly myth that IRV "always elects true-majority winners." If you believe that myth, then you must believe that the true-majority choice of the best candidate, is the same as the true-majority choice of the worst candidate. We recommend not believing that myth.
Family of reversal-failure example elections by Saari & Barney
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