Favorite betrayal criterion

The favorite betrayal criterion or sincere favorite criterion is a voting system criterion which requires that "voters should have no incentive to vote someone else over their favorite".

It is passed by Approval voting, Range voting, and Majority Judgment. All these are examples of cardinal voting systems.

On the other hand, most ordinal voting systems do not pass this criterion. For instance, Borda Count, Copeland's method, Instant runoff voting (IRV, known in the UK as the Alternative Vote), Kemeny-Young, Minimax Condorcet, Ranked Pairs, and Schulze all fail this criterion. A few ordinal methods, like Antiplurality, pass it. Some Condorcet methods pass it when combined with the tied at the top rule, though this means they may not be Condorcet-efficient when some voters equally rank multiple candidates.

It is also failed by Plurality voting and two-round runoff voting.

Definition
It is defined as follows:


 * A voting system satisfies the Favorite Betrayal Criterion (FBC) if there do not exist situations where a voter is only able to obtain a more preferred outcome (i.e. the election of a candidate that he or she prefers to the current winner) by insincerely listing another candidate ahead of his or her sincere favorite.

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.

Complying methods
Approval voting, range voting, Majority Judgment, MinMax(pairwise opposition), MCA (except MCA-A and some versions of MCA-R), MAMPO, Majority Defeat Disqualification Approval, and Improved Condorcet Approval comply with the favorite betrayal criterion, as do ICT and Symmetrical ICT.

Borda count, plurality voting, Condorcet methods (except for Improved Condorcet methods, such as Kevin Venzke's ICA, and Chris Benham's ICT, and Symmetrical ICT) and instant-runoff voting do not comply.

Borda count
This example shows that Borda count violates the favorite betrayal criterion. Assume three candidates A, B and C with 8 voters and the following preferences:

Sincere voting
Assume all voters would vote in a sincere way. The positions of the candidates and computation of the Borda points can be tabulated as follows:

Result: C wins with 9 Borda points.

Favorite betrayal
Now, assume, the voters with favorite A (marked bold) realize the situation and insincerely vote for candidate B instead of their favorite A:

Now, the positions of the candidates and computation of the Borda points would be:

Result: B wins with 10 Borda points.

Conclusion
By insincerely listing B ahead of their sincere favorite A, the two voters obtained a more preferred outcome. There was no way to achieve this without raising another candidate ahead of their sincere favorite. Thus, Borda count fails the favorite betrayal criterion.

Copeland
This example shows that Copeland's method violates the favorite betrayal criterion. Assume four candidates A, B, C and D with 6 voters and the following preferences:

Sincere voting
Assume all voters would vote in a sincere way. The results would be tabulated as follows:

Result: D can defeat two of the three opponents, whereas no other candidate wins against more than one opponent. Thus, D is elected Copeland winner.

Favorite betrayal
Now, assume, the voters with favorite A (marked bold) realize the situation and insincerely vote for candidate C instead of their favorite A:

The results would be tabulated as follows:

Result: C is the Condorcet winner and thus, C is Copeland winner, too.

Conclusion
By insincerely listing C ahead of their sincere favorite A, the two voters obtained a more preferred outcome, that is C is the winner instead of the least preferred candidate D. There was no way to achieve this without raising another candidate ahead of their sincere favorite. Thus, the Copeland method 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 is an example for instant-runoff voting, too.

Now, assume four candidates A, B, C and D with 41 voters and the following preferences:

Sincere voting
Assume all voters would vote in a sincere way.

C has only 5 first place votes and is eliminated first. Its votes are transferred to B. Now, A is eliminated with its 10 votes. Its votes are transferred to B, too. Finally, B has 21 votes and wins against D with 20 votes.

Result: B wins against D, after C and the Condorcet winner A has been eliminated.

Favorite betrayal
Now, assume, two of the voters with favorite A (marked bold) realize the situation and insincerely vote for candidate C instead of their favorite A:

Now, C has 7 first place votes and thus, B with its only 6 first place votes is eliminated first. Its votes are transferred to A. Now, C is eliminated with its 10 votes. Its votes are transferred to A, too. Finally, A has 21 votes and wins against D with 20 votes.

Result: A wins against D, after B and C has been eliminated.

Conclusion
By insincerely listing C ahead of their sincere favorite A, the two voters obtained a more preferred outcome, that is they achieved, that their favorite wins. 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.

Two-round system
This example shows that the two-round runoff voting system violates the favorite betrayal criterion. Assume three candidates H, S and L with 17 voters and the following preferences:

Sincere voting
Assume all voters would vote in a sincere way, i.e. they vote for their favorites in the first round. The results from the first round and the runoff would be:

Thus, L would be eliminated and there would be a runoff between H and S. Since all voters of L prefer S over H, S would benefit from the elimination of L.

Result: By acquiring the votes of the voters which favor L, S wins with 9 to 8 votes against H.

Favorite betrayal
Now, assume, the voters of H realize the situation and two of them insincerely vote for candidate L instead of their favorite H. The results would be: H and L proceed to the runoff, while S is eliminated. H benefits from that, since the voters which favor S, prefer H over L.

Result: By acquiring the votes of the voters favoring S, H wins clearly against L with 13 against 4 votes.

Conclusion
By voting for their least preferred candidate L instead of their favorite H, the voters changed their favorite from loser to winner and changed the outcome from a least preferred alternative to a more (in this case even the most) preferred alternative. There was no other way for them to accomplish this and still vote for their favorite in the first round. Thus, the two-round system fails the favorite betrayal criterion.

Commentary
Election methods that meet this criterion provide no incentive for voters to betray their favorite candidate by voting another candidate over him or her.

An interpretation of this criterion applied to votes as cast is the Sincere Favorite criterion.

Favorite Betrayal Criterion video
A video titled "How our voting system (and IRV) betrays your favourite candidate" by Dr. Andy Jennings at Center for Election Science explains favorite betrayal in plurality and instant-runoff voting:

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Jennings refers to the dominant sample parties as the "Good Party" and "Bad Party", where the "Good Party" frequently beats the "Bad Party" candidate 55% to 45%. Then a new third party emerges: the "Ideal Party", a small set of voters who prefer the Good Party to the Bad Party. A voter that prefers the "Ideal Party" to the "Good Party" will naturally want to rank:


 * 1) Ideal Party
 * 2) Good Party
 * 3) Bad Party

This works well, so long as the "Ideal Party" doesn't get very popular, and the Ideal Party voters rank the Good Party as their second choice (thus ensuring that the Good Party candidates

However, if the "Ideal Party" gets popular, then the Ideal Party candidate can cause the Good Party candidate to get eliminated. If the all of the voters that prefer the Good Party ranked the Ideal Party candidate as their second choice, then the Ideal Party candidate can still win. But it only takes a small portion of Good Party voters to tip the election to the Bad Party candidate by voting these preferences:


 * 1) Good Party
 * 2) Bad Party
 * 3) Ideal Party

Stronger forms of the criterion
FBC simply requires that for a given election, a voter always has some kind of strategy they can use to vote in such a way that they most support their favorite candidate. However, this means that some voting methods that fail FB can allow a voter to benefit by doing FB, even though they didn't actually have to. For example, several voting methods which pass FBC because they allow a voter to protect themselves by equally ranking multiple candidates 1st (implying that the voter has a simple way to always avoid FB i.e. equal-ranking, as opposed to some FBC-compliant voting methods where the non-FB strategy may be opaque or difficult to figure out and thus less useful for avoiding FB) are like this. Score voting passes a stronger form of FBC, which says that voters can never benefit by doing FB i.e. there is no possible strategy involving FB that can benefit a voter.

Least favorite betrayal criterion
A criterion related to FB is whether or not a voter can be hurt by giving no support to their least favorite candidates. Approval and Score voting pass this criterion, since if never benefits you to help your least favorite beat other candidates on approvals/points.

Criterion failure rates
Though many voting methods fail FB, they tend to decrease the incentive to do FB relative to FPTP. Example for Condorcet: 25 A>B

26 B

49 C In FPTP, C would win with 49 votes to B's 26 and A's 25, so the A-top voters would have to vote B>A instead (i.e. put B as their 1st choice) to ensure B wins with 51 votes, rather than C with 49. In Condorcet methods, this isn't necessary, since the pairwise table is: and B is the Condorcet winner. See also the chicken dilemma.

Reducing FB incentive in a chicken dilemma
26 A>B

25 B

49 C A pairwise beats B beats C beats A, so there is a cycle. C would win in several methods, such as some Condorcet methods, and IRV. To prevent A-top voters from having to vote B>A to make B the CW/majority criterion winner, it can essentially be observed that they have the incentive to do so, and if they do this, nobody else has incentive/ability to vote differently to get a different outcome, so B would automatically win.

However, for Condorcet methods, note that if the 49 C voters instead had voted C>A, then they make it so that now the voting method recognizes the C-top voters have an incentive to do FB to elect A rather than B. So a cycle would be formed all over again in terms of deciding which factions can force their preferences through strategic voting, meaning that this trick doesn't always work (see Algorithmic Asset Voting for discussion on this).