Explanation of the (not recommended) "BTR-IRV" voting system
Each "vote" is a rank-ordering of all the N candidates, for example
"Nader>Gore>Bush>Buchanan" would be a possible vote (with N=4).
After collecting the votes, the N-candidate election proceeds in a sequence
of N-1 "rounds."
In each round one candidate is eliminated
and he is erased from all votes. For example, if "Bush" were eliminated,
then the above vote would become
"Nader>Gore>Buchanan."
The one to eliminate is
found as follows.
Find the two candidates A and B whom the fewest voters top-rank.
Now, ignoring all candidates except A and B in all the votes,
i.e. based solely on the A>B and B>A relations in those votes,
perform a 2-candidate majority election among A and B only.
The loser of that "election" is the one we eliminate.
After N-1 rounds only one candidate remains standing. He is the
one that BTR-IRV elects.
Warning.
Like a lot of voting system proposals, this sounds on its face like a pretty good
method. (Heck, every voting system ever seriously proposed sounds at first like
it is a pretty good method.) Two properties that sound good are that BTR-IRV always
elects a Condorcet winner if one exists (and indeed always elects a "Smith set" member;
IRV cannot claim either);
and BTR-IRV, like IRV, is immune to the DH3 pathology.
Nevertheless, a lot of good-sounding methods have bad properties.
And indeed, BTR-IRV has some bad properties:
Like IRV, it suffers from "add-top failure."
(On the other hand: like IRV, if you add new votes that rank the current winner W top and
all others co-equal bottom, then W still wins; and if you alter votes that used to
rank W co-equal bottom, so that they now rank W one rung above bottom but below
all others, then W
still wins.)
It abandons a known alleged-advantage of IRV called "later no harm." That is, in IRV,
by casting a vote "later" in your top-to-bottom candidate ordering,
you cannot hurt the election chances of somebody you ranked "earlier."
This advantage is supposed to be important because it encourages voters
to rank, i.e. give their opinion of, every candidate, as opposed to
just "truncating" their ballot and say voting for only
one ("plurality style"). This was a clear "selling point" for IRV over plurality voting.
Supposedly
the main reason the "Bucklin system" (which was used in the early 1900s to elect Governors
in several US states) was jettisoned, was that it failed later-no-harm in a major way so that most
voters just voted plurality-style, defeating the whole purpose of the Bucklin system.
[Actually, I am skeptical of these alleged and supposed claims; and in IRV it can be
strategic to "truncate" your ballot in spite of IRV's later-no-harm property.]
BTR-IRV abandons that selling point.