Chicken dilemma

The "chicken dilemma" refers to a situation where two similar candidates share a majority, but are opposed by one candidate which has a plurality against either of the two. This can happen when there is a majority split into two subfactions (below called A and B), competing against a united minority (below called C) that is bigger than either of the subfactions.

This scenario has been called the "chicken dilemma" because in many election systems, the two majority subfactions are in a situation that resembles the classic "chicken" or "snowdrift" game (especially if voters are not sure which of the two subfactions is larger). A method that encourages cooperation by threatening to punish defectors is said to pass the chicken dilemma criterion. See Analysis for more information.

Definition

Below are two definitions of the Chicken Dilemma criterion; "CD" and "CD2".

CD

The A voters are the voters who vote A over everyone else. The B voters are the voters who vote B over everyone else. The C voters are the voters who vote C over everyone else.

Premise

1. There are 3 candidates: A, B, and C.
2. The A voters and the B voters, combined, add up to more than half of the voters in the election.
3. The A voters are more numerous than the B voters. The C voters are more numerous than the A voters, and more numerous than the B voters.
4. The A voters vote B over C. The B voters refuse to vote A over anyone.
5. None of the C voters vote A or B over the other.

B doesn't win.

CD2

CD is sufficient, as-is, but here is a non-numerical definition:

The A voters are the voters who vote A over everyone else. The B voters are the voters who vote B over everyone else. The C voters are the voters who vote C over everyone else.

Premise:

1. There are 3 candidate: A, B, and C.

2. If the A voters and B voters all voted both A and B over C, then C couldn't win.

3. The ballot set is such that if C withdrew from the election and the count, A would win.

4. The A voters vote B over C.

5. The B voters don't vote A over anyone.

Requirement:

B doesn't win.

Analysis

If we assume each faction has a single, coordinated strategy defined as "cooperate" (vote both candidates A and B above bottom) or "defect" (bullet vote, with only the favorite above bottom); and that each faction values its preferred choice at 10, its less-preferred choice at 8, and candidate C at 0, many voting systems lead to the following payoff matrix:

 cooperate defect cooperate 9, 9 8, 10 defect 10, 9 0, 0 Fig. 2: Chicken with numerical payoffs

There are various ways to deal with this situation. For instance:

1. Some voting systems, such as approval voting, ignore the problem. Perhaps the assumption here is that it will be impossible to organize a defection without prompting a retaliation, and thus that both sides will prefer to cooperate. ("Mutual assured destruction"?)
2. Some voting systems, such as Majority Choice Approval, try to exploit the fact that each faction is not a single coordinated entity, but a group of individual voters. The idea is that if a small number of voters defect, they should be ignored; hopefully, in that situation, majority cooperation will be a stable strategy.
3. Other voting systems, such as ICT, try to exploit the fact that in a real-world election, A and B are never perfectly balanced; one subfaction is always larger. In this case, a voting system can encourage the smaller group to cooperate by threatening to elect C (punishing both groups) if the smaller group defects. The chicken dilemma criterion is usually passed by this kind of voting system.
4. Reciprocal Score Voting addresses the problem by explicitly rewarding reciprocal cooperation and punishing the defecting sides. Whichever subfaction defects receives no support from the other faction, even if it's larger. The only way to maximize the odds of winning is by cooperating as much as possible with similar factions.

The chicken dilemma happens when there is a Condorcet winner and a majority Condorcet loser, but not a majority Condorcet winner. In many voting systems, supporters of one of the two similar candidates have a dilemma, like a game of "chicken": they can either "cooperate" and support both similar candidates, helping to ensure the opposing plurality candidate loses but risking a win by the less-preferred of the similar ones; or they can "betray" and support only their favorite candidate, trying to take advantage of cooperation by the other side.

An example of a chicken dilemma scenario, in the format of "#voters:true preferences" (see Strong/weak preference option for notation):

• 33: A>B>>C
• 22: B>A>>C
• 45: C>>A=B

In the chicken dilemma scenario described in the premise of the Chicken Dilemma Criterion (CD), defined above, if B won, then the B voters would have successfully taken advantage of the A voters' cooperativeness. The A voters wanted to vote both A and B over the candidate disliked by both the A voters and B voters. Thereby they helped {A,B} against the worse candidate. But, with methods that fail CD, the message is "You help, you lose".

Methods passing criterion

Some methods that pass the Chicken Dilemma Criterion:

ICT, Symmetrical ICT, MMPO, MDDTR, IRV and most Condorcet-IRV hybrid methods (Benham's method, Woodall's method).

Reciprocal Score Voting soft-fails if C is about as large as A+B and there is insufficient mutual support, but passes if A+B is a sufficiently large majority with sufficient mutual support, which is always encouraged by the system.

Notes

One direct implication of the chicken dilemma criterion is a spoiler effect. This is because if A drops out of the race, then B becomes a majority's 1st choice and wins in any majority criterion-passing method. Further, there is also usually incentive for Favorite Betrayal, since A-top voters generally benefit from putting B 1st. That would be the case in IRV for:

26 A>B

25 B

49 C

If two A-top voters instead vote B-top, B wins instead of C.

Most advocates of the CD criterion would likely counter that it's a more common issue to encounter CD-type scenarios because of strategic voting rather than honest voting, therefore these are worthy prices to pay for the strategic resistance.

Another major issue is that the chicken dilemma is indistinguishable based on the ballots from a situation where a majority subfaction begins to support a 3rd party. The above example with different candidate names:

26 Green Party>Democrats

25 Democrats

49 Republicans

The chicken dilemma criterion assumes that the Democrat-top voters are trying to withhold their support from the Green Party, and thus it punishes their "lack of cooperation" by electing the Republicans. Yet really, the Democrats don't support the Green Party in this example, and the Green Party-top voters merely wished to elect either the Green Party or the Democrats, and would not have wanted their support for their 1st choice to hurt their 2nd choice. So it can be argued that in fact, a voting method should fail the CD criterion and always elect the "non-cooperating" faction's preferred candidate to avoid this spoiler effect.

Most defeat-dropping Condorcet methods, rated methods, and to some extent, Condorcet-cardinal hybrid methods fail the criterion.

Reciprocal Score Voting would not give the Democrats a benefit by not supporting the Greens, as they would get no support themselves, so this is not a valid strategy then. Such situations would necessarily have to be honest. But presumably, if a significant portion of Green Party voters support Democrats then a significant portion of Democrats would also support Greens, as the overlap between voter bases should be mostly symmetrical. Therefore, it can be argued such strong asymmetries are unrealistic to begin with, and the system should encourage this mutual support to appear whenever it exists.

See also Asset voting for some discussion on this; the majority can be thought of as a "majority semi-solid coalition".