Favorite betrayal criterion: Difference between revisions

The '''favorite betrayal criterion''' is a [[voting system criterion]] which 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.<ref>Alex Small, “Geometric construction of voting methods that protect voters’ first choices,” arXiv:1008.4331 (August 22, 2010), http://arxiv.org/abs/1008.4331.</ref>

It is passed by [[Approval voting]], [[Range voting]], and [[Majority Judgment]]. All these are examples of [[cardinal voting systems]]; on the other hand, most [[Ranked voting systems|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 method|Kemeny-Young]], [[Minimax Condorcet]], [[Ranked Pairs]], and [[Schulze method|Schulze]] all fail this criterion. It is also failed by [[Plurality voting]] and two-round [[runoff voting]].

== Examples ==

=== Borda count ===

{{Main|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:

{| class="wikitable"

|-

! # 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:

{| class="wikitable"

! candidate !! #1. !! #2. !! #last !! computation !! Borda points

|-

| A || 2 || 3 || 3 || 2*2 + 3*1 || bgcolor=#ffbbbb|''7''

|-

| B || 3 || 2 || 3 || 3*2 + 1*1 || bgcolor=#ffbbbb|''8''

|-

| C || 3 || 3 || 2 || 3*2 + 3*1 || bgcolor=#bbffbb|'''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:

{| class="wikitable"

|-

! # 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:

{| class="wikitable"

! candidate !! #1. !! #2. !! #last !! computation !! Borda points

|-

| A || 0 || 5 || 3 || 0*2 + 5*1 || bgcolor=#ffbbbb|''5''

|-

| B || 5 || 0 || 3 || 5*2 + 0*1 || bgcolor=#bbffbb|'''10'''

|-

| C || 3 || 3 || 2 || 3*2 + 3*1 || bgcolor=#ffbbbb|''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|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:

{| class="wikitable"

|-

! # 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:

{| class=wikitable border=1

|+ Pairwise election results

|-

| colspan=2 rowspan=2 |

| colspan=4 bgcolor="#c0c0ff" align=center | X

|-

| bgcolor="#c0c0ff" | A

| bgcolor="#c0c0ff" | B

| bgcolor="#c0c0ff" | C

| bgcolor="#c0c0ff" | D

|-

| bgcolor="#ffc0c0" rowspan=4 | Y

| bgcolor="#ffc0c0" | A

|

| bgcolor="#c0c0c0" | [X] 3 <br>[Y] 3

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

|-

| bgcolor="#ffc0c0" | B

| bgcolor="#c0c0c0" | [X] 3 <br>[Y] 3

|

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

|-

| bgcolor="#ffc0c0" | C

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

|

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

|-

| bgcolor="#ffc0c0" | D

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

|

|-

| colspan=2 bgcolor="#c0c0ff" | 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:

{| class="wikitable"

|-

! # 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:

{| class=wikitable border=1

|+ Pairwise election results

|-

| colspan=2 rowspan=2 |

| colspan=4 bgcolor="#c0c0ff" align=center | X

|-

| bgcolor="#c0c0ff" | A

| bgcolor="#c0c0ff" | B

| bgcolor="#c0c0ff" | C

| bgcolor="#c0c0ff" | D

|-

| bgcolor="#ffc0c0" rowspan=4 | Y

| bgcolor="#ffc0c0" | A

|

| bgcolor="#c0c0c0" | [X] 3 <br>[Y] 3

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

|-

| bgcolor="#ffc0c0" | B

| bgcolor="#c0c0c0" | [X] 3 <br>[Y] 3

|

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

|-

| bgcolor="#ffc0c0" | C

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

|

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

|-

| bgcolor="#ffc0c0" | D

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

| bgcolor="#ffe0e0" | [X] 2 <br>[Y] 4

| bgcolor="#e0e0ff" | [X] 4 <br>[Y] 2

|

|-

| colspan=2 bgcolor="#c0c0ff" | 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|Instant-runoff voting}}

This example shows that instant-runoff voting violates the favorite betrayal criterion. Note, that the [[Favorite_betrayal_criterion#Two-round_system|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:

{| class="wikitable"

|-

! # 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.

{| class="wikitable"

|-

!Votes in round/<br />Candidate !! 1st !! 2nd !! 3rd

|-

| A || bgcolor=#ddffbb|10 || bgcolor=#ffbbbb|''10'' ||

|-

| B || bgcolor=#ddffbb|6 || bgcolor=#ddffbb|11 || bgcolor=#bbffbb|'''21'''

|-

| C || bgcolor=#ffbbbb|''5'' ||

|-

| D || bgcolor=#ddffbb|20 || bgcolor=#ddffbb|20 || bgcolor=#ffbbbb|''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:

{| class="wikitable"

|-

! # 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.

{| class="wikitable"

|-

!Votes in round/<br />Candidate !! 1st !! 2nd !! 3rd

|-

| A || bgcolor=#ddffbb|8 || bgcolor=#ddffbb|14 || bgcolor=#bbffbb|'''21'''

|-

| B || bgcolor=#ffbbbb|''6'' ||

|-

| C || bgcolor=#ddffbb|7 || bgcolor=#ffbbbb|''7'' ||

|-

| D || bgcolor=#ddffbb|20 || bgcolor=#ddffbb|20 || bgcolor=#ffbbbb|''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|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:

{| class="wikitable"

|-

! # 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:

{| class="wikitable"

|-

! !! colspan=2|# of voters

|-

! Candidate !! 1st round !! Runoff

|-

| H || 8 || bgcolor=#ff9999|8

|-

| S || 5 || bgcolor=#99ff99|9

|-

| L || bgcolor=#ff9999|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:

{| class="wikitable"

|-

! !! colspan=2|# of voters

|-

! Candidate !! 1st round !! Runoff

|-

| H || 6 || bgcolor=#99ff99|13

|-

| S || bgcolor=#ff9999|5

|-

| L || 6 || bgcolor=#ff9999|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.

==Further reading==

* [http://books.google.com/books?id=RN5q_LuByUoC ''Collective Decisions and Voting: The Potential for Public Choice'']

* [http://books.google.com/books?id=i5IPnH1C0hoC ''Chaotic Elections!: A Mathematician Looks at Voting'']

* [http://books.google.com/books?id=V6P9IV0EkxQC ''Decisions and Elections: Explaining the Unexpected'']

* [http://www.barnsdle.demon.co.uk/vote/stfree.html Strategy Criteria by Mike Ossipoff]

* [http://www.icann.org/presentations/guillard-ccnso-carthage-28oct03.pdf Election Methods]

* [http://www.rangevoting.org/FBCsurvey.html Survey of methods satisfying FBC]

* [http://www.rangevoting.org/EscapingDuopoly.html FBC in relation to duopoly]

* [http://www.rangevoting.org/SimmonsSmithPf.html FBC used in mathematical proofs]

* [http://www.rangevoting.org/BackAtKlarreich.html Commentary on FBC in relation to other voting methods]

* [http://nodesiege.tripod.com/elections/#critfbc Kevin Venzke's statement on FBC]

== References ==

{{reflist}}