Dark horse plus 3 rivals

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The acronym DH3 refers to the "Dark Horse plus 3" scenario, where a candidate universally considered worst is elected due to a strategic arms race.[1] This can happen when a voting system encourages burying your "true" (strongest) opponent underneath weaker opponents. If this is the case, then in an election between three strong candidates and a weak "dark horse", the "dark horse" can win precisely because they were the weakest. Each strong candidate votes the dark horse in second place, in order to better bury the stronger opponents; and then the dark horse, with second-place support from all voters, is the strategic Condorcet winner.

This is one of the most pathological possible voting scenarios, as it could lead even the most universally-despised candidate to win.

There are documented cases of this happening in elections using the Borda count, a system particularly subject to DH3.[2]

A method that passes dominant mutual third burial resistance provides no incentive to bury an opponent under a dark horse: it either does nothing or backfires. In this sense, it is immune to the scenario because the escalation never occurs.

In a 2001 presentation paper, Burt Monroe defined a Nonelection of Irrelevant Alternatives criterion that requires that a method never elects universally despised candidates in a strategic equilibrium.[3] All methods passing this criterion are automatically immune to the DH3 problem.

James Green-Armytage first discussed the scenario, though not by that name, in the context of Monroe's paper, in 2003. [4]


Here is one possible criticism of DH3: One reason given for the likelihood of DH3 is:

Imagine that my sincere election utilities are

UA=10,    UB=9,    UC=8,    UD=0.

Suppose I believe my burying-vote can either

Have no winner-altering effect. (The most likely possibility, by far.)
If I choose to "bury the rivals" that unfortunately might cause D to win, whereas someone else (whose expected utility is [10+9+8]/3 = 27/3 = 9 assuming equal chances for each of {A,B,C}) would have otherwise won. My utility loss in this case is –9.
If I choose to "bury the rivals" that might work and cause A to win, whereas someone else (whose expected utility is [9+8+0]/3 = 17/3 = 5.7 if all three among {B,C,D} are equally likely; but no matter what the likelihoods the expected utility is at most 9) would have otherwise won. My utility gain in this case is somewhere between +1 and +10.

The expected alteration in value for me got by choosing to bury is   ≥1×P(Z) - 9×P(Y).   If this is positive, then it is strategically wise for me to bury. If Z is viewed as lots more likely than Y then burial is a good idea. (Burying is always a good idea if Z at least 9 times more likely. Burying is never a good idea if P(Y)≥1.25×P(Z). If 0.8×P(Y)≤P(Z)≤9×P(Y) then burying might be a good idea.) [5]

However, similar to Nicholas Nassim Taleb's argument that that the worst thing can happen to you is not you dying, but you and everyone in the world dying[6], the scenario where D wins is likely to be considered so bad by most voters that they would avoid it at all costs; it isn't simply a case of losing 9 points of utility on a personal level, but risking causing major harm to everyone in society.

The other thing is that, with some Condorcet methods, it is difficult to determine when burying will work; for example, with Condorcet//Approval, Approval itself often has a strategic equilibrium on the Condorcet winner.

Methods that pass dominant mutual third burial resistance provide no incentive to bury under a dark horse. If enough voters nevertheless do so, the dark horse might still be elected, but they have no reason to do so.

Condorcet methods whose completion mechanism is insensitive to burial also provide little incentive to bury under a dark horse. DH3 relies on a number of factions each being able to get their candidate elected by burying under the dark horse, but if all of them do so, the dark horse is elected. But if the completion mechanism is insensitive to burial, only one of the factions can get its candidate elected by creating a cycle. Hence the other factions have no reason to bury, and the dark horse doesn't win. This property enables Smith,IRV and Smith//IRV to resist DH3 even though they don't strictly pass dominant mutual third burial resistance.

Strictly speaking, any method that requires a winner to have at least one first preference will be immune to DH3, because burial can't give a candidate more first preferences and a universally loathed candidate would have none. This is what makes the voice of reason immune. However, such immunity is not robust: it only takes a single voter to vote the dark horse at top to break this protection unless the method also has a more substantial defense against DH3.

References and footnotes

  1. Smith, Warren D. (2006). "The DH3 "Dark Horse plus 3 rivals" pathology". Center for Range Voting. Retrieved 2021-07-07.
  2. Smith, Warren D. (2006). "Why Range Voting is better than Borda Voting". Center for Range Voting. Retrieved 2021-07-07.
  3. Monroe, Burt L. (September 2001). "Raising Turkeys: An extension and devastating application of Myerson-Weber voting equilibrium" (PDF). Presentation to the American Political Science Association in San Francisco.
  4. Green-Armytage, J. (2003-08-17). "serious strategy problem in Condorcet, but not in IRV?". Election-methods mailing list archives.
  5. Smith, Warren D. (2008). "DH3 utility calculation". Center for Range Voting. Retrieved 2021-11-30.
  6. Taleb, Nassim N. (2017-08-25). "The Logic of Risk Taking". Incerto (medium.com). Retrieved 2021-11-30. In fact I’ve sampled ninety people in seminars and asked them: "what’s the worst thing that happen to you?" Eighty-eight people answered 'my death'. This can only be the worst case situation for a psychopath. For then, I asked those who deemed that the worst case is their own death: "Is your death plus that of your children, nephews, cousins, cat, dogs, parakeet and hamster (if you have any of the above) worse than just your death?" Invariably, yes. "Is your death plus your children, nephews, cousins (…) plus all of humanity worse than just your death?" Yes, of course. Then how can your death be the worst possible outcome?