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.

Current Definition of FBC:

Michael Ossipoff definition:

Requirements:

The voting system must allow the voter to vote at top as many candidates as s/he wishes.

If the winner is a candidate who is top-voted by you, then moving an additional candidate to top on your ballot shouldn't change the winner to a candidate who is not then top-voted by you.

Supplementary definition:

A candidate is "top-voted" by you, and is "at top" on your ballot, if you don't vote anyone over him/her.

Earlier Definition

The definition written below is the one that FBC's initial proponent had originally written and used. Its problem was that it led to the question of "What if the way of voting that optimizes your outcome without favorite-burial is some complicated, difficult-to-find strategy?". That question led to a better definition, written above on this page. Some time ago, someone else, too, had written that definition, and a link to it is given at the bottom of this page, under a different name (Sincere Favorite Criterion).

Supplementary Definition:

A voter optimizes the outcome (from his/her own perspective) if his vote causes the election of the best possible candidate that can be elected, based on his own preferences, given all the votes cast by other voters.

Earlier FBC definition:

For any voter who has a unique favorite, there should be no possible set of votes cast by the other voters such that the voter can optimize the outcome (from his own perspective) only by voting someone over his favorite.

Complying methods

Approval voting, range voting, Majority Judgment, MinMax(pairwise opposition), MCA (except MCA-A and some versions of MCA-R), MAMPO, 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.

Examples

Borda count

Main article: 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:

# of voters	Preferences
2	A > B > C
3	B > C > A
3	C > A > B

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:

candidate	#1.	#2.	#last	computation	Borda points
A	2	3	3	2*2 + 3*1	7
B	3	2	3	3*2 + 1*1	8
C	3	3	2	3*2 + 3*1	9

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:

# of voters	Preferences
2	B > A > C
3	B > C > A
3	C > A > B

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

candidate	#1.	#2.	#last	computation	Borda points
A	0	5	3	0*2 + 5*1	5
B	5	0	3	5*2 + 0*1	10
C	3	3	2	3*2 + 3*1	9

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

Main article: Copeland's method

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:

# of voters	Preferences
2	A > B > C > D
2	C > D > B > A
1	D > A > B > C
1	D > B > A > C

Sincere voting

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

Pairwise election results

		X
		A	B	C	D
Y	A		[X] 3 [Y] 3	[X] 2 [Y] 4	[X] 4 [Y] 2
	B	[X] 3 [Y] 3		[X] 2 [Y] 4	[X] 4 [Y] 2
	C	[X] 4 [Y] 2	[X] 4 [Y] 2		[X] 2 [Y] 4
	D	[X] 2 [Y] 4	[X] 2 [Y] 4	[X] 4 [Y] 2
Pairwise election results (won-tied-lost):		1-1-1	1-1-1	1-0-2	2-0-1

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:

# of voters	Sincere Preferences	Ballots
2	A > B > C > D	C > A > B > D
2	C > D > B > A	C > D > B > A
1	D > A > B > C	D > A > B > C
1	D > B > A > C	D > B > A > C

The results would be tabulated as follows:

Pairwise election results

		X
		A	B	C	D
Y	A		[X] 3 [Y] 3	[X] 4 [Y] 2	[X] 4 [Y] 2
	B	[X] 3 [Y] 3		[X] 4 [Y] 2	[X] 4 [Y] 2
	C	[X] 2 [Y] 4	[X] 2 [Y] 4		[X] 2 [Y] 4
	D	[X] 2 [Y] 4	[X] 2 [Y] 4	[X] 4 [Y] 2
Pairwise election results (won-tied-lost):		0-1-2	0-1-2	3-0-0	2-0-1

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

Main article: 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:

# of voters	Preferences
10	A > B > C > D
6	B > A > C > D
5	C > B > A > D
20	D > A > C > B

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.

Votes in round/ Candidate	1st	2nd	3rd
A	10	10
B	6	11	21
C	5
D	20	20	20

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:

# of voters	Ballots
2	C > A > B > D
8	A > B > C > D
6	B > A > C > D
5	C > B > A > D
20	D > A > C > B

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.

Votes in round/ Candidate	1st	2nd	3rd
A	8	14	21
B	6
C	7	7
D	20	20	20

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

Main article: 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:

# of voters	Preferences
8	H > S > L
5	S > H > L
4	L > S > H

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:

	# of voters
Candidate	1st round	Runoff
H	8	8
S	5	9
L	4

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:

	# of voters
Candidate	1st round	Runoff
H	6	13
S	5
L	6	4

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:

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.

Notes

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.

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:

Wins are bolded

	B	C	A
B	---	51	26
C	49	---	49
A	25	25	---

and B is the Condorcet winner. See also the chicken dilemma.

Condorcet methods can have their FB incentive eliminated in certain scenarios. For example:

26 A>B

25 B

49 C

A beats B beats C beats A. To prevent A-top voters from voting B>A to make B the CW, it can essentially be observed that they have the incentive to do so, and if they do this, nobody else has incentive to vote differently, so B would automatically win. However, 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. This means that this trick doesn't always work. See Algorithmic Asset Voting.

C>A

Further reading

• Collective Decisions and Voting: The Potential for Public Choice
• Chaotic Elections!: A Mathematician Looks at Voting
• Decisions and Elections: Explaining the Unexpected
• Strategy Criteria by Mike Ossipoff
• 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
• Kevin Venzke's statement on FBC

References

Some parts of this article are derived with permission from text at http://electionmethods.org

This page uses Creative Commons Licensed content from Wikipedia (view authors).