Procedures & Applications:

Academy Awards Nominations and Single Transferable Vote

The Academy of Motion Pictures Arts and Sciences uses a single transferable vote (STV) election procedure for nominating candidates for its most prestigious awards, including Best Picture, Best Actor, and Best Actress.  There are variations of STV procedures, and, as such, STV is considered a class of election procedures.  Because electing a single candidate using STV is equivalent to the Instant Run-Off method, the following description assumes that more than one candidate is elected.

In an STV election, voters submit a ballot in which they rank order any or all of the candidates, depending on the variation used.  The election proceeds through a series of rounds.  In any round, a candidate that has reached or exceeded a required minimum number of votes, known as the quota, is elected.  If not enough candidates have been elected, then votes are transferred from a candidate whose votes exceed the quota or from the candidate(s) with the fewest votes.  The candidate with the fewest votes is then eliminated.  Votes are transferred according to the ranking of candidates on the ballots.  The iterative process continues until the required number of candidates has been elected.  STV is to induce honesty in the ranking of candidates by the voters because votes for candidates that receive either sufficient support to be elected or insufficient support to be eliminated are transferred to the next-ranked candidate on the voter’s ballot.

Variations of STV exist because different quota and different mechanisms for transferring votes exist.  The most common quota is the Droop quota in which a candidate is among the winners if it receives at least \left(\frac{n}{k+1}\right)+1 votes where n is the number of votes cast and k is the number of candidates to be elected.  It may seem reasonable that a candidate should at least 1/k of the votes cast to be elected.  The following example demonstrates the logic of the Droop quota.   For example, if n = 100 and k = 3, then the Droop quota is 26=\left(\frac{100}{3+1}\right)+1. Suppose a candidate receives 26 votes.  No other three candidates can also receive 26 votes because 26 + (3 x  26) = 4 x  26 > 100.  When trying to elect k candidates, if the Droop quota is used, then k + 1 candidates cannot receive \left(\frac{n}{k+1}\right)+1 or more votes because (k+1)\times\left[\left(\frac{n}{k+1}\right)+1\right]=n+k+1>n, the number of votes cast.

The Academy of Motion Pictures Arts and Sciences currently uses a version of STV to elect k = 5 nominees for a category, such as Best Picture.  Members of the Academy receive a list of all possible candidates in a category, as well as a ballot.  Each voter is to rank order his or her 5 most-preferred candidates from the list.  The five candidates then compete against one another for the Oscar in a follow-up election.

Example The following example demonstrates how STV is employed to nominate k = 5 candidates for (the fictitious) Best “Best Picture” of the 20th Century from among the following nine movies that won an Oscar for Best Picture over this time period:

A: Annie Hall (1977)
B: Braveheart (1995)
C: Casablanca (1943)
D: The Deer Hunter (1978)
E: The English Patient (1996)
F: From Here to Eternity (1953)
G: The Godfather (1972)
H: How Green Was My Valley (1941)
I: It Happened One Night (1934)

Suppose n = 30 votes are cast.  Using the Droop quota, a movie makes the final five if it receives at least \left(\frac{30}{5+1}\right)+1=6 votes.  Suppose that the 30 voters cast 10 types of ballots, according to the rank order of the candidates.  Further let the number above the ordering represent the number of Academy members who submitted ballots with that ranking.  For example, six voters rank the candidates G, C, E, F, I in that order.

 

6

3

4

3

1

2

3

2

1

5

1st

G

G

C

A

H

I

B

D

D

F

2nd

C

A

I

B

B

B

A

A

F

D

3rd

E

E

E

E

E

E

E

E

E

E

4th

F

C

A

D

I

H

I

B

C

C

5th

I

H

F

C

D

G

D

G

A

H

Round 1 The only candidate to reach the quota in the first round is G, The Godfather.  The nine votes it receives are three more than the quota, so that the excess three votes are distributed to the second-ranked candidates of the voters who ranked G first.  Of the voters that ranked G first, two-thirds had preferences G, C, E, F, I, while one-third had preferences G, A, E, C, H.  Hence, two-thirds of the three excess votes, or two additional votes, become first-place votes for C, Casablanca.  If C were to be eliminated due to insufficient support, then these 2 votes would be transferred to E.  If C were to be elected with an excess number of votes, then a portion of the excess would be transferred to E.  Similarly, one additional vote is transferred to A, Annie Hall from the voters with preferences G, A, E, C, H.

At this point, only candidate G has been elected.  To view the process, eliminate G from the ballots, as well as six ballots that elected G.  The three excess votes are still present.

 

2

1

4

3

1

2

3

2

1

5

1st

    

C

A

H

I

B

D

D

F

2nd

C

A

I

B

B

B

A

A

F

D

3rd

E

E

E

E

E

E

E

E

E

E

4th

F

C

A

D

I

H

I

B

C

C

5th

I

H

F

C

D

 

D

 

A

H

Rounds 2 and 3 Because C received two transferred votes, Casablanca is elected with the minimum number of votes.  Because there were no excess votes for C, no votes are transferred.  These six voters are removed from the data.  Of the remaining candidates, none has enough first-place votes.  Hence, the candidate with the fewest first-place votes, E, The English Patient, is removed.  This leaves the following tabular data:

 

 

1

 

3

1

2

3

2

1

5

1st

 

 

 

A

H

I

B

D

D

F

2nd

 

A

 

B

B

B

A

A

F

D

3rd

 

 

 

 

 

 

 

 

 

 

4th

 

 

 

D

I

H

I

B

 

 

5th

 

H

 

 

D

 

D

 

A

H

Rounds 4 and 5 Because E had no first-place votes, another candidate with insufficient support must be eliminated.  This is candidate H, How Green Was My Valley.  Its single vote is transferred to B, Braveheart, but is not enough for B to reach the quota.  Hence, I, It Happened One Night, is also eliminated because it only received two first-place votes.  This leaves the following tabular data:

 

 

1

 

3

1

2

3

2

1

5

1st

 

 

 

A

 

 

B

D

D

F

2nd

 

A

 

B

B

B

A

A

F

D

3rd

 

 

 

 

 

 

 

 

 

 

4th

 

 

 

D

 

 

 

B

 

 

5th

 

 

 

 

D

 

D

 

A

 

Rounds 6 and 7 Braveheart now has the bare minimum support to be elected.  Three candidates have been elected, G, C, and B, and three candidates have been eliminated, E, H, and I.  Hence, two of the three candidates, A, D, and F, must be elected.  Because no candidate has enough votes to be elected, the candidate with the fewest votes, D, is eliminated.  Hence, G, C, B, A, and D are all elected.  We conclude the example with transferring D’s votes to A and F which yields:

 

 

1

 

3

 

 

 

2

1

5

1st

 

 

 

A

 

 

 

 

 

F

2nd

 

A

 

 

 

 

 

A

F

 

3rd

 

 

 

 

 

 

 

 

 

 

4th

 

 

 

 

 

 

 

 

 

 

5th

 

 

 

 

 

 

 

 

A

 

Commentary:  Pros and Cons A critique of STV for elections in which more than one candidate is elected is that the outcome may not be politically stable, as the elected candidates may represent different factions of the electorate.  Notice that in the example that all voters ranked E, The English Patient, as third in their rankings.  In fact, candidate E is preferred to every other candidate in a head-to-head, two candidate election.  In general it is not possible to determine a voter’s preference between two-candidates if only a partial ranking of all candidates is given.  But, because E appears among the ranked candidates for all voters in the example, it is possible in this case.  The following data give the results for elections between E and every other candidate.

E

A

 

E

B

 

E

C

 

E

D

 

E

F

 

E

G

 

E

H

 

E

I

19

11

 

24

6

 

20

10

 

22

8

 

24

6

 

21

9

 

29

1

 

25

5

For example, E defeats candidate A in a head-to-head election because 11 of the 30 voters prefer A to E while the remaining 19 voters prefer E to A, by either placing A below E on their ranking of the top five candidates, or by not placing A on the list at all.

The Academy of Motion Pictures Arts and Sciences is most concerned with having the five nominees represent excellence in the category.  Therefore, the critique about stability is not a concern for their process.

Candidate E is known as the Condorcet winner, as it defeats all other candidates in pairwise, head-to-head elections.  The Condorcet winner, if one exists, is viewed as a compromise candidate that garners support from the entire electorate, because it can defeat every other candidate head-to-head.  Hence, a more specific critique is that STV may not elect the Condorcet winner, if one exists.  The Condorcet winner derives its name from the Marquis de Condorcet, Marie Jean Antoine Nicolas Caritat, a French, 18th century mathematician and philosopher, who championed electing a Condorcet winner if one exists.  Unfortunately, the Condorcet winner does not always exist, as demonstrated by a simple three voter example:

1
1
1
 
A
 
B
 
B
 
C
 
C
 
A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A

C

B

 
2

 

1

 
2

 

1

 
2

 

1

B

A

C

 

 

 

 

 

 

 

 

 

 

 

 

C

B

A

 

 

 

 

 

 

 

 

 

 

 

 

No Condorcet winner exists because:
A defeats B in a head-to-head election, but loses to C in such an election,
B defeats C in a head-to-head election, but loses to B in such an election, and
C defeats A in a head-to-head election, but loses to B in such an election.

The Academy Awards counts votes by hand and physically moves ballots into separate piles and sub-piles.  Other variations of STV can become more complicated to determine the outcome of elections and require computers to tally the results.  Although there are always instances in which an election procedure can be manipulated, an advantage of STV procedures is that the computations are too complex to be manipulated by a voter attempting to rank competitors of its most preferred candidate at the bottom of its preference list.