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.

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 effect because the ball never gets rolling.

Notes

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

Imagine that my sincere election utilities are

U_A=10,    U_B=9,    U_C=8,    U_D=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.)

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, 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.