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.^[1] It protects voters from having to engage in a kind of strategy called lesser evil voting or decapitation (i.e. removing the "head" off a ballot).^[2]

Most rated voting systems, including score voting, satisfy the criterion.^[3]^[4]^[5] By contrast, instant-runoff, traditional runoffs, plurality, and most other variants of ranked-choice voting (including all strictly-Condorcet-compliant methods) fail this criterion.^[4]^[6]^[7]

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.^[8]

Instant-runoff voting fails the favorite-betrayal criterion whenever it fails to elect the Condorcet winner, a situation referred to as center-squeeze.

Definition

The no favorite betrayal criterion is defined as follows:

A voting system satisfies the no 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 to obtain a more preferred outcome in the election overall (i.e. the election of a candidate that they prefer to the current winner).

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 even after having the opportunity to participate in the election.^[9]^[10]^[11]

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.^[12]

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 10 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.

Condorcet methods

External links

References

^ Alex Small, “Geometric construction of voting methods that protect voters’ first choices,” arXiv:1008.4331 (August 22, 2010), http://arxiv.org/abs/1008.4331.
^ Merrill, Samuel; Nagel, Jack (1987-06-01). "The Effect of Approval Balloting on Strategic Voting under Alternative Decision Rules". American Political Science Review. 81 (2): 509–524. doi:10.2307/1961964. ISSN 0003-0554. JSTOR 1961964.
^ Baujard, Antoinette; Gavrel, Frédéric; Igersheim, Herrade; Laslier, Jean-François; Lebon, Isabelle (September 2017). "How voters use grade scales in evaluative voting" (PDF). European Journal of Political Economy. 55: 14–28. doi:10.1016/j.ejpoleco.2017.09.006. ISSN 0176-2680. A key feature of evaluative voting is a form of independence: the voter can evaluate all the candidates in turn ... another feature of evaluative voting ... is that voters can express some degree of preference.
^ ^a ^b Wolk, Sara; Quinn, Jameson; Ogren, Marcus (2023-03-20). "STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform". Constitutional Political Economy (Journal Article). 34 (3): 310–334. doi:10.1007/s10602-022-09389-3. Retrieved 2023-07-16.
^ Eberhard, Kristin (2017-05-09). "Glossary of Methods for Electing Executive Officers". Sightline Institute. Retrieved 2023-12-31.
^ Woodall, Douglas (1997-06-27). "Monotonicity of single-seat preferential election rules". Discrete Applied Mathematics. 77 (1): 81–98. doi:10.1016/S0166-218X(96)00100-X. Retrieved 2024-05-02.
^ Fishburn, Peter; Brams, Steven (1983-09-01). "Paradoxes of Preferential Voting". Mathematics Magazine. 56 (4): 207–214. doi:10.1080/0025570X.1983.11977044. JSTOR 2689808. Retrieved 2024-05-02.
^ Volić, Ismar (2024-04-02). "Duverger's law". Making Democracy Count. Princeton University Press. Ch. 2. doi:10.2307/jj.7492228. ISBN 978-0-691-24882-0.
^ Hamlin, Aaron (2015-05-30). "Top 5 Ways Plurality Voting Fails". Election Science. The Center for Election Science. Retrieved 2023-07-17.
^ Hamlin, Aaron (2019-02-07). "The Limits of Ranked-Choice Voting". Election Science. The Center for Election Science. Retrieved 2023-07-17.
^ "Voting Method Gameability". Equal Vote. The Equal Vote Coalition. Retrieved 2023-07-17.
^ Ossipoff, Michael (2013-05-20). "Schulze: Questioning a Popular Ranked Voting System". Democracy Chronicles. Retrieved 2024-01-01.
^ Hamlin, Aaron; Hua, Whitney (2022-12-19). "The case for approval voting". Constitutional Political Economy. 34 (3): 335–345. doi:10.1007/s10602-022-09381-x.
^ Sullivan, Brendan (2022). An Introduction to the Math of Voting Methods. 619 Wreath. ISBN 9781958469033.

[small-1] Alex Small, “Geometric construction of voting methods that protect voters’ first choices,” arXiv:1008.4331 (August 22, 2010), http://arxiv.org/abs/1008.4331.

[2] Merrill, Samuel; Nagel, Jack (1987-06-01). "The Effect of Approval Balloting on Strategic Voting under Alternative Decision Rules". American Political Science Review. 81 (2): 509–524. doi:10.2307/1961964. ISSN 0003-0554. JSTOR 1961964.

[3] Baujard, Antoinette; Gavrel, Frédéric; Igersheim, Herrade; Laslier, Jean-François; Lebon, Isabelle (September 2017). "How voters use grade scales in evaluative voting" (PDF). European Journal of Political Economy. 55: 14–28. doi:10.1016/j.ejpoleco.2017.09.006. ISSN 0176-2680. A key feature of evaluative voting is a form of independence: the voter can evaluate all the candidates in turn ... another feature of evaluative voting ... is that voters can express some degree of preference.

[STAR_Analysis-4] Wolk, Sara; Quinn, Jameson; Ogren, Marcus (2023-03-20). "STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform". Constitutional Political Economy (Journal Article). 34 (3): 310–334. doi:10.1007/s10602-022-09389-3. Retrieved 2023-07-16.

[Sightline-5] Eberhard, Kristin (2017-05-09). "Glossary of Methods for Electing Executive Officers". Sightline Institute. Retrieved 2023-12-31.

[Woodall_Monotonicity-6] Woodall, Douglas (1997-06-27). "Monotonicity of single-seat preferential election rules". Discrete Applied Mathematics. 77 (1): 81–98. doi:10.1016/S0166-218X(96)00100-X. Retrieved 2024-05-02.

[Fishburn-7] Fishburn, Peter; Brams, Steven (1983-09-01). "Paradoxes of Preferential Voting". Mathematics Magazine. 56 (4): 207–214. doi:10.1080/0025570X.1983.11977044. JSTOR 2689808. Retrieved 2024-05-02.

[:1-8] Volić, Ismar (2024-04-02). "Duverger's law". Making Democracy Count. Princeton University Press. Ch. 2. doi:10.2307/jj.7492228. ISBN 978-0-691-24882-0.

[9] Hamlin, Aaron (2015-05-30). "Top 5 Ways Plurality Voting Fails". Election Science. The Center for Election Science. Retrieved 2023-07-17.

[10] Hamlin, Aaron (2019-02-07). "The Limits of Ranked-Choice Voting". Election Science. The Center for Election Science. Retrieved 2023-07-17.

[11] "Voting Method Gameability". Equal Vote. The Equal Vote Coalition. Retrieved 2023-07-17.

[12] Ossipoff, Michael (2013-05-20). "Schulze: Questioning a Popular Ranked Voting System". Democracy Chronicles. Retrieved 2024-01-01.

[Hamlin-13] Hamlin, Aaron; Hua, Whitney (2022-12-19). "The case for approval voting". Constitutional Political Economy. 34 (3): 335–345. doi:10.1007/s10602-022-09381-x.

[14] Sullivan, Brendan (2022). An Introduction to the Math of Voting Methods. 619 Wreath. ISBN 9781958469033.

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