| #voters | their vote |
|---|---|
| 7 | B>G>N |
| 6 | G>B>N |
| 5 | N>G>B |
| 3 | N>G>B |
In this 21-voter IRV election, B wins (by 15-to-8 after G is eliminated). But if the 3 voters in the last line had not voted, then G would have won (which those voters would have preferred). (Because N is eliminated then G beats B by 11-to-7.).
This scenario also shows how the 3 voters in the last line would be motivated to "betray" their true-favorite N (Nader) by dishonestly voting G>N>B or G>B>N to rank their "lesser evil" G (Gore) dishonestly top; then G would win, whereas their honest vote N>G>B "spoils it" by causing both G and N to lose. This refutes the myth that IRV "cures" that "lesser-evil/spoiler" problem with plurality voting. (Computer simulations show that Favorite Betrayal scenarios arise in 3-candidate random IRV elections about 19.6% of the time.)
Indeed this illustrates how IRV can suffer an even worse kind of spoiler scenario than the usual one. In the usual spoiler scenario, if you vote Nader, then your second-favorite ("Gore") and Nader both lose, but you could've made Gore win by voting Gore. But at least, voting Nader did not make it worse than not voting at all. With IRV, your honest vote can (and here does) actually make it worse than not voting.
How common is that? I.e. how common is it in 3-candidate IRV elections that some set of co-voting voters, by voting honestly, obtain a worse election result (in their view) than if they had not voted at all? This question was analysed by Depankar Ray: On the practical possibility of a "no show paradox" under the single transferable vote, Math'l Social Sciences 11,2 (1986) 183-189. Ray found that in situations where the IRV and plain-plurality winner differ, then (under very mild probabilistic assumptions) the probability of this happening was exactly 50%. That seems a serious indictment of IRV voting as an "improvement" over plurality voting.
Another slightly more complicated example intended to be more realistic (this one was intended to be simple).