Until quite recently, the phrase "ranked voting" was commonly understood to encompass many voting systems, in fact all voting systems in which one votes by listing the candidates in order of preference. But in the last few years the term has come to more specifically denote IRV (instant runoff voting). But for lack of a good alternative, I will revert to using "ranked voting" in the more general and more traditional way. Voters are mostly concerned with how to vote and they may neglect to give much thought to how votes are to be counted. They just assume their expressed opinions will be taken into good account. Unfortunately, that does not necessarily follow. For example, if the counting is done by only looking at the first (most favored) entry in each voters' list while ignoring all of the other entries then this form of ranked voting would be in no essential way different from plurality voting. Voters would soon realize that and they would only rarely bother to submit a list of more than a single candidate.
Borda voting, along with actually an infinite number of similar ranked voting systems that I called the Borda family were described in an earlier article. There are an infinite number of ranked voting systems. And in a yet earlier article, I introduced yet another ranked voting system that I called IRRV ( Instant-Runoff-Removal Voting ). A recent article added some additional observations and hopefully contributed some further understanding of IRRV. Either of these earlier articles can be read to suggest that IRRV is a better voting system than IRV, but I should point out that is not entirely clear. In the comments section of the more recent article I outlined a way to investigate which system is actually better. But though that study might be interesting, it should be remembered that Arrow's theorem guarantees that there will be significant defects in any ranked voting system ; in that respect, at least, the question of which ranked voting system is best does not matter much. Put another way, Arrow's theorem warns that no ranked voting system is to be trusted; every ranked voting system has the potential of electing the wrong candidate.
But appeals to Arrow's theorem may leave many people unconvinced. An actual example of failure could well be more satisfying. We have seen such an example of how IRV can fail and we might ask whether there is as illustrative an example of IRRV failure. Following a few preliminaries, we will provide such an example, in fact one that happens to be quite similar to the earlier IRV example.
Whenever there are more than just two candidates, plurality voting can fail badly. Plurality voting can and does simply select the wrong winner in some elections and that fact has long been recognized. Historically, this has been used as a persuasive argument for limiting ballot access, perhaps to just the two dominant parties. However, it would surely seem preferable to sacrifice plurality voting than to sacrifice democracy in this very direct and apparent way. IRV is just one attempt to find a good replacement for plurality voting.
IRV has many faults, starting with the very concept that motivated its invention. The thinking behind IRV was that rather than conducting just one plurality election, it might be better to conduct a series of plurality elections, eliminating one candidate at a time. In the end, plurality voting could make a credible decision between just the two finalists. The trouble with this plan is that the best candidate may well be eliminated in one of the earlier rounds of plurality voting. Despite this pretty obvious conceptual flaw, IRV was proposed to mimic such a series of plurality vote-counts while holding only a single election.
But if we wanted to hold a series of elections with the objective of eliminating one candidate in each election we might ask how sensible it is to ask voters "which remaining candidate is your favorite". To eliminate one candidate, would it not seem more sensible to ask the voters, "which candidate is your least favorite" and then eliminate the candidate whom the most voters wish to be removed? That change and otherwise following the same approach used to define IRV, leads us instead to IRRV. A voter is asked to list in order which candidates should be removed. Intuitively at least, IRRV seems to promise a better voting method than IRV and it is possible that IRRV in fact is the better system.
Another puzzling idea behind IRV seems to be that if one plurality voting election is bad then perhaps many of them would improve matters (and in fact it does). But least IRRV attempts to model the use of many elections using a variation on plurality voting (not that the variation is without its own faults). Nonetheless, IRRV is a ranked voting systems and so Arrow's theorem guarantees us that just like IRV, IRRV will also lead to erroneous election outcomes at times.
In the example we gave of an election in which IRV fails to elect the right candidate, a candidate favored by only 10% of the voters would win even though most (90%) voters preferred a different candidate over the actual winner. That election involved eleven candidates but a similar election could have been constructed with a smaller odd number of candidates, even just three. But with fewer candidates, the fact that the outcome is a mistake would be less dramatic and not so apparent. For the same reason, our example to illustrate the failure of IRRV will have nine candidates instead of a fewer number.
As with the IRV example, the consensus candidate for this new IRRV example is the favorite of none of the voters, but that same consensus candidate is second choice of a wide (80%) majority of the voters. Whatever their reasons, the remaining 20% of voters submit ballots listing this consensus candidate as the first to be removed.
The other eight of the nine candidates who share 80% of the (negative) votes in the first count of ballots we will call the issue-candidates. The issue-candidates happen to fall into pairs with one candidate on each side of just one issue. The 80% of voters split evenly on the which of the four issues is most vitally important and also on which side of the issue they support. As a result, in the first round of counting ballots, each of the eight issue-candidates gets 10% of the votes (for removal) while the consensus candidate gets 20%. As in the IRV example, the consensus candidate will be the very first candidate to be removed from contention. The winner of the election will necessarily be one of the eight issue candidates. Most voters, around 70%, would have much preferred the consensus candidate and they will be quite disappointed with this election outcome.
These examples only illustrate what Arrow's theorem tells us to expect, that it would be wise to avoid using any ranked voting system. My own opinion is that the smart choice should be a balanced and evaluative voting system such as BAV (balanced approval voting).