# Dark horse plus 3 rivals

The acronym DH3 refers to the "Dark Horse plus 3" scenario. 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.

## Notes

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

X.
Have no winner-altering effect. (The most likely possibility, by far.)
Y.
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.
Z.
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.) [1]

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[2], 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.

## References and footnotes

1. "DH3 utility calculation".
2. "The Logic of Risk Taking". 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?