Black's method: Difference between revisions

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'''Black's method''' chooses the [[Condorcet winner]] if it exists, but uses the [[Borda count]] instead if there is an ambiguity ~~(the~~ method is named for [[w:Duncan Black|Duncan Black]]).<ref>{{Cite book| isbn = 978-94-009-4225-7| last = Black| first = Duncan| title = The Theory of Committees and Elections| accessdate = 2020-03-31| date = 1987| url = http://link.springer.com/openurl?genre=book&isbn=978-94-010-8375-1|publisher=|year=|location=|pages=66}}</ref>

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'''Black's method''' chooses the [[Condorcet winner]] if it exists, but uses the [[Borda count]] instead if there is an ambiguity. This method is named for [[w:Duncan Black|Duncan Black]].<ref>{{Cite book| isbn = 978-94-009-4225-7| last = Black| first = Duncan| title = The Theory of Committees and Elections| accessdate = 2020-03-31| date = 1987| url = http://link.springer.com/openurl?genre=book&isbn=978-94-010-8375-1|publisher=|year=|location=|pages=66}}</ref> Note that the winner of the Borda count is not necessarily in the Smith set.<ref>{{Cite web| title = Is there formal proof that Duncan Black's Electoral System is a Condorcet System?| work = ResearchGate| accessdate = 2020-03-31| url = https://www.researchgate.net/post/Is_there_formal_proof_that_Duncan_Blacks_Electoral_System_is_a_Condorcet_System|date=March 12, 2014|last=di Fenizio|first=Pietro Speroni|archive-url=|archive-date=|url-status=live}}</ref>

Example:

== Examples ==

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25 A>B>C

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40 B>C>A

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35 C>A>B

=== Example one ===

Borda scores are A 185, B 205, C 210. A pairwise beats B beats C beats A, so there is no Condorcet winner (because everyone has at least one defeat). So the Borda winner, C, wins.

In this example, the candidates are ranked as follows:

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25: A>B>C

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40: B>C>A

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35: C>A>B

==== Stage 1 of example one ====

Note that the winner of the Borda count is not necessarily in the Smith set.<ref>{{Cite web| title = Is there formal proof that Duncan Black's Electoral System is a Condorcet System?| work = ResearchGate| accessdate = 2020-03-31| url = https://www.researchgate.net/post/Is_there_formal_proof_that_Duncan_Blacks_Electoral_System_is_a_Condorcet_System|date=March 12, 2014|last=di Fenizio|first=Pietro Speroni|archive-url=|archive-date=|url-status=live}}</ref> For example, if there are 8 candidates and 3 voters, with ballots:

Evaluating these ballots pairwise, we see the following

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A > B > X > D1 > D2 > D3 > D4 > C

B > C > X > D1 > D2 > D3 > D4 > A

A (1 win, 1 loss): defeats B (60 to 40) but loses to C (75 to 25)

C > A > X > D1 > D2 > D3 > D4 > B

B (1 win, 1 loss): defeats C (65 to 35) but loses to A (40 to 60)

C (1 win, 1 loss): defeats A (75 to 25) but loses to B (65 to 35)

Then A, B, and C form the Smith Set / Condorcet cycle (with 16 points each), while X is the Borda winner (with 18 points).

This demonstrates a Condorcet cycle (where A defeats B, B defeats C, and C defeats A), so we advance to the Borda stage.

==== Stage 2 of example 1 ====

A [[Smith-efficient]] variant of Black's method could be used instead, to ensure someone in the [[Smith set]] will win.

The points allocated in the Borda stage can be expressed using [[ABIF]] in the following text:

25: A/2 > B/1 > C/0

40: B/2 > C/1 > A/0

35: C/2 > A/1 > B/0

Counting up the Borda scores using the points above, we see:

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== References ==

A: ((25 * 2) + (35 * 1)) = 85

B: ((40 * 2) + (25 * 1)) = 105

C: ((35 * 2) + (40 * 1)) = 110

C earns 110 points in the second round, so C is the winner.

=== Example two ===

What follows is an example of Borda winner that is not in the Smith set, with 8 candidates ("A", "B", "C", "De", "Df", "Dg", "Dh", and "X") and 3 voters. The rankings are expressed below:

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1: A > B > X > De > Df > Dg > Dh > C

1: B > C > X > De > Df > Dg > Dh > A

1: C > A > X > De > Df > Dg > Dh > B

==== Stage 1 of example two ====

A, B, and C form a Condorcet cycle. which can be seen by removing the candidates other than A, B, and C:

1: A > B > C

1: B > C > A

1: C > A > B

So the next stage is a Borda count.

==== Stage 2 of example two ====

The points allocated in the Borda round are expressed using [[ABIF]] in the following text:

1: A/7 > B/6 > X/5 > De/4 > Df/3 > Dg/2 > Dh/1 > C/0

1: B/7 > C/6 > X/5 > De/4 > Df/3 > Dg/2 > Dh/1 > A/0

1: C/7 > A/6 > X/5 > De/4 > Df/3 > Dg/2 > Dh/1 > B/0

This results in the following point allocations

* A receives 7+0+6 points (13 points)

* B receives 6+7+0 points (13 points)

* C receives 0+7+6 points (13 points)

* X receives 5+5+5 points (15 points)

This X wins the election in the Borda-based tiebreaker, even though X wasn't part of the Condorcet cycle in the first round.

== Similar methods==

A [[Smith-efficient]] Condorcet method could be used in the first stage (such as [[Copeland's method]]), where only the candidates that are tied using the first method advance to the second method. This would ensure that someone in the [[Smith set]] advances from round one, only resorting to a Borda-based tiebreaker when a [[Condorcet winner criterion|Condorcet winner]] doesn't defeat all other candidates pairwise.

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==References==

[[Category:Condorcet-Borda hybrid methods]]