Dark horse plus 3 rivals

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. 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 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 two-candidate plus dark horse strategic equilibrium. If a method passes the criterion, and its resistance generalizes to any number of candidates, it will automatically pass DH3 as well.

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

Explanation

Warren Smith explains the DH3 scenario in this way:

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.)

Criticisms

Strategy leading to the worst-case outcome is not a pure equilibrium

If voters reason like Warren Smith's example, then P(Y) increase simply as a consequence of the voters' knowledge that the other voters will be reasoning in the same way. Formally speaking, DH3 leads to a game of Chicken, with driving straight being analogous to burial and swerving being analogous to voting honestly. Chicken does not have a Nash equilibrium where everybody drives straight, precisely because the players will take into account the others' expectations. Thus it is unlikely that everybody would bury in a DH3 scenario, assuming game-theoretical rationality.

This criticism can be seen as similar to Nicholas Nassim Taleb's argument that the worst thing that can happen to you is not you dying, but you and everyone in the world dying. As voters reason that they would bury, they consider the outcome of D winning to be sufficiently bad that they only bury with some probability less than one.

Honesty may be evolutionarily stable

Marcus Ogren argues that strategies that are likely to be used in large elections need to exhibit the property that a small number of voters must immediately benefit. That is, the starting strategy of honesty must be susceptible to an invasion by a minority who play the defection strategy if the strategy is to become commonplace.

If strategy doesn't yield an immediate benefit to small groups of voters, then honesty is an w:evolutionarily stable strategy. In the game of Chicken, the mixed equilibrium where everybody buries with some probability is the only ESS. Marcus argues that unlike the game of Chicken, the voters have a preference for honesty.

Thus, he argues, in Condorcet methods, an initially small group of voters do not benefit from burial since they're not large enough to create a Condorcet cycle. However, they do pay a cost by no longer voting honestly. This cost serves as a deterrent and prevents burying groups from growing large enough to threaten the outcome.

Mitigating DH3

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 would still be elected, but they have no incentive to do so (as doing so would be individually irrational).

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.