The DH3 "Dark Horse plus 3 rivals" pathology

(Executive summary)

Many voting system "pathologies" or "paradoxes" or "bad behaviors" have been noticed over the years. But some are more worrisome and damaging than others:

The DH3 pathology is a very very bad pathology by those measures, and it affects a lot of voting systems. (In view of its frequency and seriousness, some would not use the word "affects." They would use the word "destroys.") In particular most (every?) "Condorcet method with full A>B>C>······>Z rank-orderings as votes" suffers from DH3. (That includes BTR-IRV, Tideman's ranked pairs, and Schulze's beatpaths systems.) Also, so does the Borda system. Permitting equalities in vote-rankings, using either winning votes or margins, does not avoid the DH3 pathology, although I initially thought they might work for that purpose. (Answers to questions about that.) But range voting, plurality, and IRV are immune to DH3. In the example below, strategic Range Voters would simply rank one of A,B,C max and everybody else min. C would win.

What is DH3?

It is simply this. Suppose there are 3 main rival candidates A, B, & C, who all have some good virtues. This happens a lot. (In fact, whenever it doesn't happen, the situation is uninteresting – only 2 real contenders – and we almost might as well just be using the plurality voting system.) Let us suppose support is roughly equally divided among those three, say 31%, 32%, and 37%, although the precise numbers do not matter much. Suppose also there are one or more additional "dark horse" candidates whom nobody takes seriously as contenders because they stink. For simplicity assume there is only one dark horse D, but what we are going to say also works (indeed works even more powerfully) with more than one.

Now, what happens? The A-supporters say to themselves: "We are in trouble. Polls suggest A is going to lose if we just vote A>B>C>D as is our honest opinion. But if we exaggeratedly vote A>D>B>C downgrading A's main rivals as far as we can, then maybe A will have a chance." The B-supporters say "those rotten A-supporters for sure are going to exaggerate and effectively get twice the A-versus-B discriminating power as if they were honest. We cannot sit still and just take that. We have to fight back by also exaggerating: B>D>C>A." And similarly the C-supporters say "we will not just sit back and be robbed of our deserved victory by those dishonest exaggerating scum. We will also exaggerate: C>D>A>B." (And by the way, they are completely right. C would definitely lose to A or B if they just sat there.)

Pictorial example under Borda; careful examination in several kinds of Condorcet systems.

Incidentally, some purists may quibble: why did the A-fan voters decide to exaggerate? Well the C-fans felt forced to do so because, given that the A- and B-fans already chose to exaggerate, the C-voters knew that C could not win without exaggeration. But all three kinds of voters do not know what the others are going to do and how many of them are going to do it (nor even how many of them there are), and hence have to guess, and their guess is "most of those rotters are probably going to exaggerate"! So based on this guess, they feel they too must exaggerate to get any chance of victory. (And that feeling is always accurate in the sense that, if some appropriate fraction of the opposing voters exaggerated, then [1] our candidate would be sure to lose to a rival, but [2] by such exaggeration we could regain the victory.)

The result of the new exaggerated votes is: D, the worst candidate in the eyes of all, wins the election. Guaranteed. As we said, this happens with the Borda system and also with every Condorcet method.

In contrast, with range voting the A voters will exaggerate thus: A=99, B=C=D=0, and if everybody acts that way, then C will (deservedly) be elected. (There is no advantage a range voter gains by dishonestly scoring D above B and C – this can't help A win versus B and/or C and simply incurs risk that D will win. But there is sometimes advantage for Condorcet or Borda voters.) C also wins with IRV and plurality. This is an example of the fact that range voting is designed to exhibit only a mild degradation in reaction to dishonestly-exaggerated "strategic voting." With Condorcet and Borda, the allergic reaction is not mild: it is "anaphylactic shock."

How common is this? Very common – every situation with 3 near-equal good rivals and one or more other (less-good) candidates is fodder for DH3. I.e. whenever it is not merely essentially a 2-man race. Here is the (amusing?) story of how DH3 happened in the only serious-stakes ($100,000+) ranked-ballot vote I was ever involved in. DH3-like phenomena were also immediately observed in the only government in the world (Kiribati) that tried Borda voting.

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