# Sincere favorite criterion

The sincere favorite or no favorite-betrayal criterion is a property of some voting systems that says voters should have no incentive to vote for someone else over their favorite. It protects voters from having to engage in lesser-evil voting or a strategy called "decapitation" (removing the "head" off a ballot).
Most rated voting systems, including score voting, satisfy the criterion.
Duverger's law says that systems vulnerable to this strategy will typically (though not always) develop two-party systems, as voters will abandon minor-party candidates to support stronger major-party candidates.

## US Presidential elections

The "sincere favorite criterion" suggests that a voter should always rank their sincere favorite candidate as their top choice, without strategizing based on the likely outcomes. However, in certain voting systems, this strategy can lead to suboptimal results, which makes the criterion less applicable.
The U.S. presidential election is a prime example where voters might avoid using a sincere favorite criterion. This is due to the Electoral College system, which is structured as a "first-past-the-post" (FPTP) election within each state. Here, if a voter's sincere favorite has no realistic chance of winning, it may be rational for them to vote for a more viable candidate to prevent a less preferred option from winning. This is known as "tactical voting."
Several sources discuss how the FPTP system (like the one used in U.S. presidential elections) can disincentivize the use of sincere voting strategies:
1. Gary Cox's "Making Votes Count: Strategic Coordination in the World’s Electoral Systems" explores strategic voting in FPTP systems and how they encourage tactical voting.
2. Steven Brams and Peter Fishburn's "Approval Voting" touches on how non-ranking systems like approval voting can mitigate the issues with sincere voting in FPTP contexts.
3. William H. Riker's "Liberalism against Populism" provides an analysis of the implications of various voting systems on strategic behaviors, including in U.S. elections.
These references can provide a deeper theoretical grounding on why the sincere favorite criterion is frequently not practical in FPTP and U.S. presidential elections.

## Definition

A voting rule satisfies the sincere favorite criterion if there is never a need to "betray" a perfect candidate—i.e. if a voter will never achieve a worse result by honestly ranking their favorite candidate first.

## Arguments for

The Center for Election Science argues systems that violate the favorite betrayal criterion strongly incentivize voters to cast dishonest ballots, which can make voters feel unsatisfied or frustrated with the results despite having the opportunity to participate in the election.
Other commentators have argued that failing the favorite-betrayal criterion can increase the effectiveness of misinformation campaigns, by allowing major-party candidates to sow doubt as to whether voting honestly for one's favorite is actually the best strategy.

## Compliant methods
### Rated voting

Because rated voting methods are not affected by Arrow's theorem, they can be both spoilerproof (satisfy IIA) and ensure positive vote weights at the same time. Taken together, these properties imply that increasing the rating of a favorite candidate can never change the result, except by causing the favorite candidate to win; therefore, giving a favorite candidate the maximum level of support is always the optimal strategy.
Examples of systems that are both spoilerproof and monotonic include score voting, approval voting, and highest medians.

### Anti-plurality voting

Interpreted as a ranked voting method where every candidate but the last ranked gets one point, anti-plurality voting passes the sincere favorite criterion. Because there is no incentive to rank one's favorite last, and the method otherwise does not care where the favorite is ranked, the method passes.
Anti-plurality voting thus shows that the sincere favorite criterion is distinct from independence of irrelevant alternatives, and that ranked voting methods do not necessarily fail the criterion.

## Non-compliant methods
### Instant-runoff voting

This example shows that instant-runoff voting violates the favorite betrayal criterion. Assume there are four candidates: Amy, Bert, Cindy, and Dan. This election has 41 voters with the following preferences:
| # of voters | Preferences |
| 10 | Amy > Bert > Cindy > Dan |
| 6 | Bert > Amy > Cindy > Dan |
| 5 | Cindy > Bert > Amy > Dan |
| 20 | Dan > Amy > Cindy > Bert |

#### Sincere voting

Assuming all voters vote in a sincere way, Cindy is awarded only 5 first place votes and is eliminated first. Her votes are transferred to Bert. In the second round, Amy is eliminated with only 10 votes. Her votes are transferred to Bert as well. Finally, Bert has 21 votes and wins against Dan, who has 20 votes.
| Votes in round/ Candidate | 1st | 2nd | 3rd |
| Amy | 10 | 10 | – |
| Bert | 6 | 11 | 21 |
| Cindy | 5 | – | – |
| Dan | 20 | 20 | 20 |
Result: Bert wins against Dan, after Cindy and Amy were eliminated.

#### Favorite betrayal

Now assume two of the voters who favor Amy (marked bold) realize the situation and insincerely vote for Cindy instead of Amy:
| # of voters | Ballots |
| 2 | Cindy > Amy > Bert > Dan |
| 8 | Amy > Bert > Cindy > Dan |
| 6 | Bert > Amy > Cindy > Dan |
| 5 | Cindy > Bert > Amy > Dan |
| 20 | Dan > Amy > Cindy > Bert |
In this scenario, Cindy has 7 first place votes and so Bert is eliminated first with only 6 first place votes. His votes are transferred to Amy. In the second round, Cindy is eliminated with only 7 votes. Her votes are transferred to Amy as well. Finally, Amy has 21 votes and wins against Dan, who has 20 votes.
| Votes in round/ Candidate | 1st | 2nd | 3rd |
| Amy | 8 | 14 | 21 |
| Bert | 6 | – | – |
| Cindy | 7 | 7 | – |
| Dan | 20 | 20 | 20 |
Result: Amy wins against Dan, after Bert and Cindy has been eliminated.
By listing Cindy ahead of their true favorite, Amy, the two insincere voters obtained a more preferred outcome (causing their favorite candidate to win). There was no way to achieve this without raising another candidate ahead of their sincere favorite. Thus, instant-runoff voting fails the favorite betrayal criterion.