"BTR-IRV" is a voting method intended to try to combine the virtues of Instant Runoff Voting (IRV) with Condorcet: BTR-IRV will always elect a Condorcet-winner if one exists. The BTR-IRV method is this. Each "vote" is a rank-ordering of the N candidates. The two candidates with the fewest top-rank votes ("the two worst") are compared head-to-head based on all the votes with the remaining N-2 candidates erased from the picture. Whoever loses this comparison ("the worst") is eliminated from the election and from all votes. We then continue doing such eliminations until only one candidate remains, and declare him winner.
There are many many voting methods out there, and (like this one) all sound like reasonably good ideas, at least at first. We are just considering BTR-IRV here as a typical example. Because BTR-IRV is a Condorcet method, a general-purpose theorem tells us it must exhibit "add-top failure." Our goal on this page is to make that completely concrete by actually exhibiting a BTR-IRV election in which that happens. Here it is:
| #voters | Their Vote |
|---|---|
| 3 | A > D > B > C |
| 3 | A > D > C > B |
| 4 | B > C > A > D |
| 5 | D > B > C > A |
| 4 | C > A > B > D |
The proof of the "general-purpose theorem" is not at all mysterious. It simply exhibits about 7 different fully-concrete election scenarios just like the table above (in fact this is one of them) and argues via a case analysis that no matter what election method you use, if it is a "Condorcet method," then it must exhibit add-top failure (that is: adding some new votes all ranking the current winner top, causes him to lose) in at least one of those 7 scenarios.
Here is another example of add-top failure, this time for "Rouse's voting method," another (fairly complicated to describe) elimination method that elects Condorcet winners whenever they exist.
| #voters | Their Vote |
|---|---|
| 5 | C > A > B |
| 4 | A > B > C |
| 2 | B > C > A |
In this 11-voter example, A wins (C then B are eliminated). But if we add 2 new voters each of whom votes A>C>B, i.e. all ranking the current winner A top... then C wins! (B then A are eliminated.) One of the new votes ranking the current-winner top actually causes him to lose!
Warren D. Smith: The voting impossibilities of Arrow, Gibbard & Satterthwaite, and Young, (survey) paper #79 here.