25  Scoring Rules

A scoring rule is a voting method where each voter submits a ranking of the candidates. Based on this ranking, each voter assigns a score to each candidate. The overall score of a candidate is the sum of the scores assigned by all voters. The candidates with the highest overall score are declared the winners.

Here are two well-known examples of scoring rules:

• Plurality: Each voter assigns a score of 1 to the candidate ranked in first place and 0 to all other candidates (this score is called the Plurality score). The candidates with the largest overall Plurality score are declared the winners.

• Borda: If there are \(n\) candidates, each voter assigns a score of \(n-1\) to the candidate ranked in first place, \(n-2\) to the candidate ranked in second place, and so on, assigning 0 to the candidate ranked last (this score is called the Borda score). The candidates with the largest overall Borda score are declared the winners.

We will illustrate these two methods on the following profile:

\[\begin{array}{cccc} 7 & 5 & 4 & 3 \\\hline a & b & d & c \\ b & c & b & d \\ c & d & c & a \\ d & a & a & b \\ \end{array}\]

There are four candidates in this election. The first step to apply any scoring rule is to determine how many voters rank each of the candidates in first, second, third and fourth place.

	\(a\)	\(b\)	\(c\)	\(d\)
Number ranking the candidate in first place	7	5	3	4
Number ranking the candidate in second place	0	11	5	3
Number ranking the candidate in third place	3	0	11	5
Number ranking the candidate in fourth place	9	3	0	7

Using the above table, the Plurality Score of the candidates are defined as follows:

\[\begin{array}{rclcl} PluralityScore(a) & = & 1 \times 7 + 0 \times 0 + 0 \times 3 + 0 \times 9 & = & 7\\ PluralityScore(b) & = & 1 \times 5 + 0 \times 11 + 0 \times 0 + 0 \times 3 & = & 5\\ PluralityScore(c) & = & 1 \times 3 + 0 \times 5 + 0 \times 11 + 0 \times 0 & = & 3\\ PluralityScore(d) & = & 1 \times 4 + 0 \times 3 + 0 \times 5 + 0 \times 7 & = & 4\\ \end{array}\]

Thus, candidate \(a\) is the unique Plurality winner since it has the greatest Plurality score.

The Borda Score of the candidates are defined as follows:

\[\begin{array}{rclcl} BordaScore(a) & = & 3 \times 7 + 2 \times 0 + 1 \times 3 + 0 \times 9 & = & 24\\ BordaScore(b) & = & 3 \times 5 + 2 \times 11 + 1 \times 0 + 0 \times 3 & = & 37\\ BordaScore(c) & = & 3 \times 3 + 2 \times 5 + 1 \times 11 + 0 \times 0 & = & 30\\ BordaScore(d) & = & 3 \times 4 + 2 \times 3 + 1 \times 5 + 0 \times 7 & = & 23\\ \end{array}\]

Thus, candidate \(b\) is the unique Borda winner since it has the greatest Borda score.

25.1 Exercises

1. Find the Plurality and Borda scores for each candidate in the following profile. Who are the winners according to each voting method?

   \[\begin{array}{cccc} 9 & 6 & 2 & 8\\\hline b & c & c & a\\ a & a & b & c\\ c & b & a & b \end{array}\]

   Show Answer

   We first determine how many voters rank each of the candidates in the different possible positions:

   	\(a\)	\(b\)	\(c\)
   Number ranking the candidate in 1st place	8	9	8
   Number ranking the candidate in 2nd place	15	2	8
   Number ranking the candidate in 3rd place	2	14	9

   The Plurality and Borda scores are then calculated as follows:

   \[\begin{array}{rclllcc} BordaScore(a) & = & 2 \times 8 & + & 1 \times 15 & + & 0 \times 2 & = & 31 \\ BordaScore(b) & = & 2 \times 9 & + & 1 \times 2 & + & 0 \times 14 & = & 20 \\ BordaScore(c) & = & 2 \times 8 & + & 1 \times 8 & + & 0 \times 9 & = & 24 \\ \end{array}\]

   The Borda winner is \(a\).

   \[\begin{array}{rclllcc} PluralityScore(a) & = & 1 \times 8 & + & 0 \times 15 & + & 0 \times 2 & = & 31 \\ PluralityScore(b) & = & 1 \times 9 & + & 0 \times 2 & + & 0 \times 14 & = & 20 \\ PluralityScore(c) & = & 1 \times 8 & + & 0 \times 8 & + & 0 \times 9 & = & 24 \\ \end{array}\]

   The Plurality winner is \(b\).

2. Find the Plurality and Borda scores for each candidate in the following profile. Who are the winners according to each voting method?

   \[\begin{array}{cccc} 2 & 1 & 1 & 3\\\hline b & d & a & c\\ d & b & d & a\\ a & a & b & d\\ c & c & c & b \end{array}\]

   Show Answer

   We first determine how many voters rank each of the candidates in the different possible positions:

   	\(a\)	\(b\)	\(c\)	\(d\)
   Number ranking the candidate in 1st place	1	2	3	1
   Number ranking the candidate in 2nd place	3	1	0	3
   Number ranking the candidate in 3rd place	3	1	0	3
   Number ranking the candidate in 4th place	0	3	4	0

   The Plurality and Borda scores are then calculated as follows:

   \[\begin{array}{rcllllcc} BordaScore(a) & = & 3 \times 1 & + & 2 \times 3 & + & 1 \times 3 & + & 0 \times 0 & = & 12 \\ BordaScore(b) & = & 3 \times 2 & + & 2 \times 1 & + & 1 \times 1 & + & 0 \times 3 & = & 9 \\ BordaScore(c) & = & 3 \times 3 & + & 2 \times 0 & + & 1 \times 0 & + & 0 \times 4 & = & 9 \\ BordaScore(d) & = & 3 \times 1 & + & 2 \times 3 & + & 1 \times 3 & + & 0 \times 0 & = & 12 \\ \end{array}\]

   The Borda winners are \(a\) and \(d\).

   \[\begin{array}{rcllllcc} PluralityScore(a) & = & 1 \times 1 & + & 0 \times 3 & + & 0 \times 3 & + & 0 \times 0 & = & 12 \\ PluralityScore(b) & = & 1 \times 2 & + & 0 \times 1 & + & 0 \times 1 & + & 0 \times 3 & = & 9 \\ PluralityScore(c) & = & 1 \times 3 & + & 0 \times 0 & + & 0 \times 0 & + & 0 \times 4 & = & 9 \\ PluralityScore(d) & = & 1 \times 1 & + & 0 \times 3 & + & 0 \times 3 & + & 0 \times 0 & = & 12 \\ \end{array}\]

   The Plurality winner is \(c\).

3. Find the Borda winners in the following profile:

   \[\begin{array}{cccc} 4 & 1 & 1 & 1\\\hline c & d & a & a\\ b & b & b & d\\ d & a & d & b\\ a & c & c & c \end{array}\]

   Show Answer

   We first determine how many voters rank each of the candidates in the different possible positions:

   	\(a\)	\(b\)	\(c\)	\(d\)
   Number ranking the candidate in 1st place	2	0	4	1
   Number ranking the candidate in 2nd place	0	6	0	1
   Number ranking the candidate in 3rd place	1	1	0	5
   Number ranking the candidate in 4th place	4	0	3	0

   Then, we calculate the Borda scores:

   \[\begin{array}{rcllllcc} BordaScore(a) & = & 3 \times 2 & + & 2 \times 0 & + & 1 \times 1 & + & 0 \times 4 & = & 7 \\ BordaScore(b) & = & 3 \times 0 & + & 2 \times 6 & + & 1 \times 1 & + & 0 \times 0 & = & 13 \\ BordaScore(c) & = & 3 \times 4 & + & 2 \times 0 & + & 1 \times 0 & + & 0 \times 3 & = & 12 \\ BordaScore(d) & = & 3 \times 1 & + & 2 \times 1 & + & 1 \times 5 & + & 0 \times 0 & = & 10 \\ \end{array}\]

   The Borda winner is \(b\).

4. Is there always a unique winner according to the Borda voting method?

   Show Answer

   No, there are many examples of elections where there is a tie according to the Borda method. Below is an example:

   \[\begin{array}{cc} 2 & 1\\\hline a & c\\ c & b\\ b & a \end{array}\] \[\begin{array}{rclllcc} BordaScore(a) & = & 2 \times 2 & + & 1 \times 0 & + & 0 \times 1 & = & 4 \\ BordaScore(b) & = & 2 \times 0 & + & 1 \times 1 & + & 0 \times 2 & = & 1 \\ BordaScore(c) & = & 2 \times 1 & + & 1 \times 2 & + & 0 \times 0 & = & 4 \\ \end{array}\]

   The Borda winners are \(a\) and \(c\).

5. Is there always a unique winner according to the Plurality voting method?

   Show Answer

   No, there are many examples of elections where there is a tie according to the Plurality method. Below is an example:

   \[\begin{array}{ccc} 1 & 1 & 1\\\hline a & c & b\\ c & a & a\\ b & b & c \end{array}\] \[\begin{array}{rclllcc} PluralityScore(a) & = & 1 \times 1 & + & 0 \times 2 & + & 0 \times 0 & = & 4 \\ PluralityScore(b) & = & 1 \times 1 & + & 0 \times 0 & + & 0 \times 2 & = & 2 \\ PluralityScore(c) & = & 1 \times 1 & + & 0 \times 1 & + & 0 \times 1 & = & 3 \\ \end{array}\]

   The Plurality winners are \(a\), \(b\), and \(c\).

6. True or False: In any profile, if a candidate \(a\) is the absolute majority winner, then \(a\) is the unique Plurality winner.

7. True or False: In any profile, if a candidate \(a\) is the absolute majority winner, then \(a\) is the unique Borda winner.