In this system, advocated by fairvote.org for use in Oakland and San Francisco, voters rank their top three choices in order as their vote. Then the election proceeds like Instant Runoff Voting from then on.
Unfortunately, this "top three" restriction makes the voting system considerably worse even than regular "rank all candidates" IRV. It suffers all of IRV's flaws plus more flaws. (Its only advantage is slightly greater simplicity.) We do not recommend it.
We'll now demonstrate a few flaws top-3-IRV suffers from, that ordinary IRV does not suffer from (or not as much). Also, you can follow this link to see that such flaws occur VERY COMMONLY in REAL LIFE, and this link to see a 2010 legal brief (pdf) arguing Top-3-only IRV is unconstitutional. (This argument is not due to us, and not necessarily endorsed by us.)
Example: suppose there are 5 Whigs and 3 Tories running. 53% of the voters are Whigs and rank their favorite three among the five Whigs 1-2-3. (Which three, depends on the voter and, let us suppose, seems random.) The remaining 47% of the voters are Tories and rank the three Tories 1-2-3. Result: A Tory wins even though a clear majority of both the voters and candidates are Whigs!! And not only that, but the Whig voter-majority has zero input into the question of which Tory it will be! (Ordinary IRV exhibits neither problem.)
Also, contrary to IRV-for-Oakland propaganda that the top-3-IRV system elects a "majority winner" – well we just showed that was false, but now we'll show it is false in more ways – (which they billed as an improvement over plurality voting where it is easily possible for no candidate to get a majority, for example a 30, 33, 37 three-way split) in fact it still is entirely possible for no candidate to ever get a majority over any other. E.g. in this 1000-voter 7-candidate election (candidate names A,B,C,D,E,F,G)
|139||A > B > D|
|141||C > D > F|
|142||D > E > G|
|143||B > C > E|
|144||E > F > A|
|145||F > G > B|
|146||G > A > C|
using top-3-only IRV, the elections proceed as follows. We eliminate first A, then C, then E, then G, then B, then F, at which point we suppose D must be the winner since D is the only candidate left. However, only 422 of the 1000 voters voted for D, well below a majority (which would be 501). Furthermore, let us consider all 6 pairwise D-versus-somebody matchups:
|D vs who?||What voters said about it|
|D versus A||429 say A>D, 283 say D>A, 288 say nothing|
|D versus B||427 say B>D, 283 say D>B, 290 say nothing|
|D versus C||430 say C>D, 281 say D>C, 289 say nothing|
|D versus E||287 say E>D, 422 say D>E, 291 say nothing|
|D versus F||289 say F>D, 422 say D>F, 289 say nothing|
|D versus G||291 say G>D, 422 say D>G, 287 say nothing|
In the first 3 of the 6 cases, the other candidate beat D by a plurality, while in the last 3, D beat the other by a plurality (albeit a smaller one). But no candidate beat any other candidate by a majority.
Here's the same example all over again but with the numbers altered so we can make a different point (there are still 7 candidates and 1000 voters):
|107||A > B > D|
|109||C > D > F|
|110||D > E > G|
|111||B > C > E|
|162||E > F > A|
|163||F > G > B|
|238||G > A > C|
using top-3-only IRV, the elections proceed as follows. We eliminate first A, then C, then E, then B, then G, then F, at which point we suppose D must be the winner since D is the only candidate left. However, only 326 of the 1000 voters voted for D in either 1st, 2nd, or 3rd place, well below a majority (which would be 501). Furthermore, let us consider all 6 pairwise D-versus-somebody matchups:
|D vs who?||What voters said about it|
|D versus A||507 say A>D, 219 say D>A, 274 say nothing|
|D versus B||381 say B>D, 219 say D>B, 400 say nothing|
|D versus C||458 say C>D, 217 say D>C, 325 say nothing|
|D versus E||273 say E>D, 326 say D>E, 401 say nothing|
|D versus F||325 say F>D, 326 say D>F, 349 say nothing|
|D versus G||401 say G>D, 326 say D>G, 273 say nothing|
Now D still wins under top-3-choices-IRV despite the fact that D would lose pairwise to A,B,C, and G (more pairwise losses than every candidate except E, which also has 4 pairwise losses but not as severe losses as D) and despite the fact that now there are 6 genuine majority-defeats (A over B, C, and D; and G over A,B, and C), none of which favor D.
G would appear to be the best choice based on these votes (beats every candidate except F pairwise; more pairwise victories than any other candidate; highest number of top-ranking supporters by far), but it is not IRV's choice.
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