| #voters | their vote |
|---|---|
| 30 | A>B>C>D |
| 26 | B>A>C>D |
| 24 | C>A>D>B |
| 10 | D>B>A>C |
| 10 | D>C>A>B |
In this 100-voter 4-candidate election by Toby Pereira (candidates named A,B,C,D) the winner under IRV (instant runoff voting) is B:
Round #1: A(30) - B(26) - C(24) - D(20) eliminated
Round #2: A(30) eliminated - B(36) - C(34)
Round #3: B(66) - C(34) eliminated
Winner: B.
However A seems a superior choice:
• A is the winner under plain plurality voting, with 30 votes, beating B(26), C(24), and D(20).
• More strongly, A gets the most top-rankings with 30, and the most {top & second} with 80 (and hence A also is the "Bucklin winner"), and the most {top & second & third} with 100. I.e, no matter at what ranking you draw a "line," A enjoys the most rankings above that line (and the fewest below it).
• Also A beats every rival pairwise by landslide supermajorities, e.g.
A beats B 64-36 (meaning 66 voters rank A above B versus only 34 who rank B above A); A beats C 66-34; A beats D 80-20.
Incidentally, Howard Dean [Vermont Governor 1991-2003, once the frontrunning USA presidential candidate in 2004, chair of Democratic Party 2005-2009, founder of Democracy For America] at the end of his 7 Oct. 2016 op-ed for the New York Times claimed IRV was the "best [way] to uphold majority rule." Sorry Howard: If A beats every rival by landslide supermajorities, but IRV demands that A lose, then IRV does not "uphold majority rule."
• A also wins under the Borda count system, with score 210 versus B(158), C(148), and D(84).
• Also A would win using score voting with top-ranked candidates getting score W, second-ranked getting score X, third-ranked getting score Y, and last-ranked getting score Z, for any W≥X≥Y≥Z, not all equal.
In this election, if S dropped out of the race, then A would win, for S=B, or S=C (but not S=D). In other words C is a "spoiler," meaning that by the act of running, C prevented A from winning, "spoiling" it for A. (Don't ever let anybody tell you that Instant Runoff prevents the "spoiler effect." For example Dean in that same op-ed falsely said "major parties don't have to fear being 'spoiled'." Sorry: if A were a major party candidate, then he'd be spoiled with IRV, so he would need to "fear" that.)
IRV also does not prevent "wasted votes" (no matter how many times somebody says it does, e.g. again Dean falsely implied this in his op-ed) – as we see because the 24 votes for C were assuredly wasted since they caused their most-hated candidate B to win. Those voters of course would have preferred A to win – and they futilely said so in their votes – but the very act of them voting caused A to lose – if they either had not voted or if they had appropriately lied in their vote, then A would have won.
More realistic(?) election example by Goldman
Comprehensive table of how often various IRV pathologies happen