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The favorite betrayal or sincere favorite criterion is a voting system criterion which requires that no incentive to vote for someone else over their favorite". is always safe for a voter to give their true favorite maximum support, without having to worry that doing so will cause a worse outcome.

Systems which allow the voter to give each candidate an independent evaluation, known as cardinal voting systems, typically satisfy this criterion. ^{[1] [2]} These systems include approval voting, median voting, and score voting. ^{[3] [4] [5]}

Most ordinal voting systems do not satisfy this criterion. For instance, Borda count, Copeland's method, instant runoff voting, minimax condorcet, ranked pairs, and the Schulze method all fail this criterion. It is also failed by plurality voting and two-round runoff voting. ^{[3] [4] [6]}

Definition

The favorite betrayal criterion is defined as follows:

A voting system satisfies the favorite betrayal criterion if there cannot exist a situation where a voter is forced to insincerely list another candidate ahead of their sincere favorite in order obtain a more preferred outcome in the election overall (i.e. the election of a candidate that they prefer to the current winner).

The criterion permits the strategy of insincerely ranking another candidate equal to one's favorite. A related but stronger criterion, the strong favorite betrayal criterion, disallows this. ^[7]

Importance

With the existence of Arrow's theorem and Gibbard's theorem, no perfect voting system may exist, and so advocates are forced to choose among a set of flawed options. ^[8] Proponents of cardinal systems argue that the favorite betrayal criterion is of high importance, because it enables voters to honestly give their true favorite maximum support without need for worry. Arguments against the criteria contend that other, conflicting criteria are more important.

Equal Vote, a major proponent of STAR voting, argues that a voting system must reward voter honesty, both on the individual level and in the aggregate. Thus they consider the favorite betrayal criterion to be of high importance. ^[9] The Center for Election Science has argued that systems that fail the favorite betrayal criterion is a major flaw in both first-past-the-post and instant runoff voting, incentivizing voters to cast dishonest ballots and resulting in lower voter satisfaction. ^{[10] [11]}

Democracy Chronicles, an online publication whose mission is to strengthen democracy worldwide, argues that the favorite betrayal criterion is paramount, since failure can open the door for misinformation campaigns to push voters to cast insincere ballots. They argue that the current media landscape and public perception of election systems requires that giving one's preferred candidate maximum support must always be the best strategy, so that no one can be convinced to do otherwise for any reason. ^[12]

The Sightline Institute, an organization dedicated to making Cascadia "a global model of sustainability," argues that the criteria's importance depends on a voter's relative perceived value between it and later-no-harm, a mutually-exclusive criterion which states that it must always be safe to give second-favorite candidates support after a top favorite. ^[13] According to Sightline, if voters feel strongly that their favorite should win but still want to express opinions about other candidates, they should be more drawn towards systems which satisfy later-no-harm. If they feel that any of their preferred candidates winning would provide sufficient satisfaction, then later-no-harm is not as necessary and the favorite betrayal criterion is helpful to ensure that showing support for their true favorite cannot possibly backfire. Voters of the first variety may choose to bullet vote under systems which pass the favorite betrayal criterion, not wanting to accidentally cause a less preferred candidate to beat their favorite. However, under a system which fails the favorite betrayal criterion, any voter may choose to strategically betray their favorite candidate in order to secure a more favorable result. ^[5]

Demonstration of satisfactory methods

Score voting

The following will provide an example and explain how score voting satisfies the favorite betrayal criterion. Score voting functions as follows:

1. Voters give each candidate a score to indicate support. Scores are numerical values from 0 to a maximum value, usually 5, 7, 10 or 100.
2. Candidates left unassigned are automatically given a score of 0.
3. The candidate with the highest cumulative score wins.

Assume there are three candidates: Agnew, BillyJoe, and Cletus. The voters may give scores up to 5. The vote totals appear as follows:

Assume there is one final voter who has yet to cast their vote, and they know the results as they currently stand. This voter could have any of the following six preferences among the candidates:

1. Agnew > BillyJoe > Cletus
2. Agnew > Cletus > BillyJoe
3. BillyJoe > Cletus > Agnew
4. BillyJoe > Agnew > Cletus
5. Cletus > Agnew > BillyJoe
6. Cletus > BillyJoe > Agnew

If Agnew or BillyJoe is the voter's true favorite, as in scenarios 1-4, the best strategy is to give their true favorite 5 points. This would cause their favorite to win and so no better outcome is possible.

If the voter's true favorite is Cletus, it is safe to give Cletus 5 points. Doing so would not change the scores or relative standing of Agnew and BillyJoe. If the voter wishes to control the outcome of the election, they must also award points to both Agnew and BillyJoe. In this scenario, the voter must give Agnew 5 points if they wish for them to win, but doing so merely places Agnew at the same level of support as Cletus. Since the voter has no incentive to lower their support for Cletus, score voting passes the favorite betrayal criterion.

Demonstration of unsatisfactory methods

Borda count

This example shows that Borda count violates the favorite betrayal criterion. Borda count functions as follows:

1. Voters rank candidates from most to least preferred.
2. Candidates are given one point for every other candidate they outrank on each ballot.
3. Most points wins.

Assume there are three candidates Alfred, Betty and Carl with 8 voters and the following preferences:

Sincere voting scenario

Assuming all voters vote in a sincere way, the positions of the candidates and computation of the Borda points can be tabulated as follows:

Result: Carl wins with 9 Borda points.

Favorite betrayal scenario

Assuming the voters that favored Alfred (marked bold) realize the situation and insincerely vote for Betty ahead of Alfred:

The positions of the candidates and computation of the Borda points would be:

Result: Betty wins with 10 Borda points.

By insincerely listing Betty ahead of their true favorite, Alfred, the two voters obtained a more preferred outcome, causing Betty to win instead of Carl. There is no way to give Betty more support without putting her ahead of Alfred and there is no way for the same voters to lower their support for Carl. Thus, there is no way for these voters to enable Betty to beat Carl without putting Betty ahead of Alfred. Because the construction of this scenario is possible, Borda count fails the favorite betrayal criterion.

Two-round system

This example shows that the two-round runoff voting system violates the favorite betrayal criterion. This method functions as follows:

1. Voters pick one candidate amongst a field of many.
2. The top two candidates advance to the next round.
3. Voters pick between the two remaining candidates.
4. The candidate with the most votes wins.

Assume there are three candidates: Anna, Beverly, and Clay. There are 17 voters with the following preferences:

Sincere voting

Assuming all voters vote in a sincere way, the results from the first round and the runoff would be:

Thus, Clay is eliminated and there is a runoff between Anna and Beverly. Since all voters of Clay prefer Beverly over Anna, Beverly benefits from Clay's elimination.

Result: By picking up the voters who would have rather voted for Clay, Beverly wins with 9 votes against Anna, who only has 8.

Favorite betrayal

Assuming Anna's voters realize the situation and two of them insincerely vote for Clay instead of their favorite, Anna. The results would be:

Anna and Clay proceed to the runoff, while Beverly is eliminated. Anna benefits from this situation, since the voters which favor Beverly, prefer Anna over Clay.

Result: By acquiring the votes of the voters favoring Beverly, Anna wins easily against Clay with a score of 13 to 4.

By voting for their least preferred candidate, Clay, instead of their favorite, Anna, the insincere voters caused Beverly to lose in the first round. This caused Anna to become the winner in the final round, a more acceptable outcome for these voters. There is no other way for these voters to cause Anna to win, and so the two-round runoff system fails the favorite betrayal criterion.

Instant-runoff voting

This example shows that instant-runoff voting violates the favorite betrayal criterion. Note that the example for the two-round runoff voting system also works as an example for instant-runoff voting. Instant-runoff functions as follows:

1. Voters rank candidates from most to least preferred.
2. Candidates are awarded one vote for every ballot in which they are the preferred candidate.
3. If a candidate is preferred on more than 50% of the ballots which list active candidates, they win. Otherwise, the next step is performed.
4. The candidate with the fewest votes is eliminated and their votes are transferred to the next preferred candidate on each ballot. The process returns to the previous step.

Assume there are four candidates: Amy, Bert, Cindy, and Dan. This election has 41 voters with the following preferences:

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.

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:

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 10 votes. Her votes are transferred to Amy as well. Finally, Amy has 21 votes and wins against Dan, who has 20 votes.

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.

See also

• Comparison of electoral systems

• Electoral systems

• Vote splitting

• Independence of irrelevant alternatives

• Strategic voting

External links

• Collective Decisions and Voting: The Potential for Public Choice
• Chaotic Elections!: A Mathematician Looks at Voting
• Decisions and Elections: Explaining the Unexpected
• Election Methods
• Survey of methods satisfying FBC
• FBC in relation to duopoly
• FBC used in mathematical proofs
• Commentary on FBC in relation to other voting methods

References

Category: voting systems Category: elections Category: mathematics Category: game theory Category: political science