Section 7.5: Majority Criterion

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Learning Objectives

• Decide if an election violates the Majority criterion

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The Borda count method prioritizes ordinal position across the entire ranking rather than cardinal first-place dominance. Because of this, the Borda count method has its own serious flaws that can produce arguably unfair outcomes.

• A candidate with a clear majority of first-place votes can still lose.
• The "Compromise Candidate" Problem. Borda count systematically favors candidates who are nobody's favorite but everyone's acceptable second/third choice.
• Vulnerability to Strategic Manipulation. Strategic voting (also called tactical voting) occurs when voters rank candidates insincerely to achieve a better outcome for their preferred candidate. Voters can dramatically improve their favorite's chances by strategically ranking rivals lower than their true preferences.

These flaws mean that Borda count is inappropriate for many applications, particularly:

• High-stakes public elections where democratic legitimacy is essential
• Competitive environments where strategic voting incentives are strong
• Contexts where majority rule is considered non-negotiable

However, Borda count may still be suitable for:

• Small deliberative bodies seeking consensus
• Low-stakes preference surveys
• Situations where strategic voting is discouraged by transparency or social norms
• Contexts where compromise and broad acceptability are explicitly valued over majority preference

Borda count's simplicity and appealing use of complete preference information come at the cost of serious structural flaws. These flaws don't make it universally "bad," but they do make it contextually inappropriate for many applications and require careful consideration of whether its particular trade-offs align with the values and needs of the specific decision-making situation.

This brings us to our second fairness criterion.

Majority Criterion (Fairness Criterion #2)

If a candidate wins by a majority, then that candidate should win the election by another method. If no majority candidate exists, then the majority criterion does not apply.

The violation occurs because Borda count and the majority Criterion answer different questions:

• Borda count asks: "Which candidate has the highest average desirability across all voters?"
• Majority criterion asks: "Which candidate is the first choice of the most voters?"

These questions have different answers when:

• The most popular first choice is strongly unpopular among others
• A less popular first choice is moderately popular among everyone

Neither is objectively "correct". They represent different values about collective decision-making. The "violation" simply reveals that these values conflict. 

 

Example #7.5.1 🤔

Consider the following preference table with Candidates A, B, & C:

Number of Voters	7	4	2
First Choice	C	B	B
Second Choice	B	A	C
Third Choice	A	C	A

Note: When using the Borda count method, Candidate B would be the winner with 32 points. (Candidate C = 29, Candidate A = 17). When using the plurality method, Candidate C is the winner.

1. Does the election violate the majority criterion when using the Borda count method?
2. Does the election violate the majority criterion when using the plurality method?

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✅ Solution:

• Candidate A received 0 out of 13 first place votes or 0%
• Candidate B received 6 out of 13 first place votes or 46.2%
• Candidate C received 7 out of 13 first place votes or 53.8%

So, a majority exists here. Candidate C has a majority of first place votes.

1. Candidate B wins using the Borda count method, yet Candidate C has a majority of first-place votes. Thus, the method violates the majority criterion. (YES)
2. Candidate C wins by both the plurality method and by majority. Thus, the method is not a violation of the majority criterion. (NO)

Note: Plurality ALWAYS satisfies the majority criterion—if someone has a majority, they automatically have the most votes.

Example #7.5.2 🤔

Consider the following preference table with Candidates A, B, & C:

Number of Voters	5	4	3
First Choice	A	B	C
Second Choice	B	A	A
Third Choice	C	C	B

Note: When using the Borda Count method, Candidate A was the winner. When using the plurality method, Candidate A was also the winner.

1. Does the election violate the majority criterion when using the Borda count method?
2. Does the election violate the majority criterion when using the plurality method?

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✅ Solution:

• Candidate A received 5 out of 12 first place votes or 41.7%
• Candidate B received 4 out of 12 first place votes or 33.3%
• Candidate C received 3 out of 12 first place votes or 25%

So, no majority exists here. 

1. Since, no majority exists here, you cannot apply the majority criterion. Thus, there is no violation of the majority criterion. (NO)
2. Since, no majority exists here, you cannot apply the majority criterion. Thus, there is no violation of the majority criterion. (NO)

Example #7.5.3 🤔

A survey of 23 customers was conducted asking them to rank which protein they prefer at a local Chipotle restaurant. The results are listed below:

Number of Voters	8	7	5	3
First Choice	Steak	Carnitas	Chicken	Steak
Second Choice	Chicken	Barbacoa	Carnitas	Barbacoa
Third Choice	Barbacoa	Chicken	Steak	Chicken
Fourth Choice	Carnitas	Steak	Barbacoa	Carnitas

Note: When using the Borda count method, Chicken would be the winner with 64 points. (Steak = 61, Carnitas = 54, Barbacoa = 53). When using the plurality method, Steak is the winner.

1. Does the election violate the majority criterion when using the Borda count method?
2. Does the election violate the majority criterion when using the plurality method?

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✅ Solution:

• Steak received 11 out of 23 first place votes or 47.8%
• Carnitas received 7 out of 23 first place votes or 30.4%
• Chicken received 5 out of 23 first place votes or 21.7%
• Barbacoa received 0 out of 23 first place votes or 0%

So, no majority exists here. 

• Since, no majority exists here, you cannot apply the majority criterion. Thus, there is no violation of the majority criterion. (NO)
• Since, no majority exists here, you cannot apply the majority criterion. Thus, there is no violation of the majority criterion. (NO)

Example #7.5.4 🤔

A student organization with 100 members is voting on which city to host their national conference. The results are listed below:

Number of Voters	52	18	15	10	5
First Choice	Austin	Boston	Chicago	Denver	Seattle
Second Choice	Boston	Chicago	Denver	Seattle	Denver
Third Choice	Chicago	Denver	Seattle	Boston	Boston
Fourth Choice	Denver	Seattle	Boston	Chicago	Chicago
Fifth Choice	Seattle	Austin	Austin	Austin	Austin

Does the election violate the Majority criterion when using the Borda count method?

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✅ Solution:

• Austin received 52 out of 100 first place votes or 52%
• Boston received 18 out of 100 first place votes or 18%
• Chicago received 15 out of 100 first place votes or 15%
• Denver received 10 out of 100 first place votes or 10%
• Seattle received 5 out of 100 first place votes or 5%

So, a majority exists here. Austin has a majority of first place votes. Thus, Austin wins using the Majority criterion.

Now, use the Borda count method:

• Step 1: Assign the points based on position. Since there are 5 candidates, a candidate receives.
  • 5 points for each 1st-place ranking
  • 4 points for each 2nd-place ranking
  • 3 points for each 3rd-place ranking
  • 2 points for each 4th-place ranking
  • 1 point for each 5th-place ranking
• Step 2: Add up all the points each candidate receives.
  • Austin's Rankings:
    • 1st place rankings: Column 2 → 52 votes × 5 points = 260 points
    • 2nd place rankings: Column 1 → 0 votes × 4 points = 0 points
    • 3rd place rankings: None → 0 votes × 3 points = 0 points
    • 4th place rankings: None → 0 votes × 2 points = 0 points 
    • 5th place rankings: Columns 2, 3, 4, & 5 → (18 + 15 + 10 + 5) = 48 votes × 1 point = 48 points
      • Austin's total: 260 + 0 + 0 + 0 + 48 = 308 points
  • Boston's Rankings:
    • 1st place rankings: Column 2 → 18 votes × 5 points = 90 points
    • 2nd place rankings: Column 1 → 52 votes × 4 points = 208 points
    • 3rd place rankings: Columns 4 and 5 → (10 + 5) = 15 votes × 3 points = 45 points
    • 4th place rankings: Column 3 → 15 votes × 2 points = 30 points 
    • 5th place rankings: None → 0 votes × 1 point = 0 points
      • Boston's total: 90 + 208 + 45 + 30 + 0 = 373 points
  • Chicago's Rankings:
    • 1st place rankings: Column 3 → 15 votes × 5 points = 75 points
    • 2nd place rankings: Column 2 → 18 votes × 4 points = 72 points
    • 3rd place rankings: Column 1 → 52 votes × 3 points = 156 points
    • 4th place rankings: Columns 4 and 5 → (10 + 5) = 15 votes × 2 points = 30 points 
    • 5th place rankings: None → 0 votes × 1 point = 0 points
      • Chicago's total: 75 + 72 + 156 + 30 + 0 = 343 points
  • Denver's Rankings:
    • 1st place rankings: Column 4 → 10 votes × 5 points = 50 points
    • 2nd place rankings: Columns 3 & 5 → (15 + 5) = 20 votes × 4 points = 80 points
    • 3rd place rankings: Column 2 → 18 votes × 3 points = 54 points
    • 4th place rankings: Column 1 → 52 votes × 2 points = 104 points 
    • 5th place rankings: None → 0 votes × 1 point = 0 points
      • Denver's total: 50 + 80 + 54 + 104 + 0 = 288 points
  • Seattle's Rankings:
    • 1st place rankings: Column 5 → 5 votes × 5 points = 25 points
    • 2nd place rankings: Column 4 → 10 votes × 4 points = 40 points
    • 3rd place rankings: Column 3 → 15 votes × 3 points = 45 points
    • 4th place rankings: Column 2 → 18 votes × 2 points = 36 points 
    • 5th place rankings: Column 1 → 52 votes × 1 point = 52 points
      • Seattle's total: 25 + 40 + 45 + 36 + 52 = 198 points
• Step 3: The candidate with the highest total wins.
  • Austin's total score = 308 points
  • Boston's total score = 373 points
  • Chicago's total score = 343 points
  • Denver's total score = 288 points
  • Seattle's total score = 198 points

Thus, Boston is the winner by the Borda Count method.

Boston wins using the Borda count method, yet Austin has a majority of first-place votes. Thus, the method violates the majority criterion. (YES)

 

Now, let's see if the Borda count method (Section 7.4) can violate the Condorcet criterion (Section 7.3).

Example #7.5.5 🤔

Consider the following preference table with Candidates A, B, & C:

Number of Voters	4	2	1
First Choice	B	A	A
Second Choice	A	C	B
Third Choice	C	B	C

1. Use the Borda count method to determine a winner.
2. Does the election violate the Condorcet criterion?

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✅ Solution:

1. Borda count method:

• Step 1: Assign the points based on position. Since there are 3 candidates, a candidate receives:
  • 3 points for each 1st-place ranking
  • 2 points for each 2nd-place ranking
  • 1 point for each 3rd-place ranking

• Step 2: Add up all the points each candidate receives.
  • Candidate A's Rankings:
    • 1st place rankings: Columns 2 & 3 → (2 + 1) = 3 votes × 3 points = 9 points
    • 2nd place rankings: Column 1 → 4 votes × 2 points = 8 points
    • 3rd place rankings: None → 0 votes × 1 point = 0 points
      • Candidate A's total: 9 + 8 + 0 = 17 points
  • Candidate B's Rankings:
    • 1st place rankings: Column 1 → 4 votes × 3 points = 12 points
    • 2nd place rankings: Column 3 → 1 vote × 2 points = 2 points
    • 3rd place rankings: Column 2 → 2 votes × 1 point = 2 points
      • Candidate B's total: 12 + 2 + 2 = 16 points
  • Candidate C's Rankings:
    • 1st place rankings: None → 0 votes × 3 point = 0 points
    • 2nd place rankings: Column 2 → 2 votes × 2 points = 4 points
    • 3rd place rankings: Columns 1 & 3 → (4 + 1) = 5 votes × 1 point = 5 points
      • Candidate C's total: 0 + 4 + 5 = 9 points

• Step 3: The candidate with the highest total wins.
  • Candidate A's total score = 17 points
  • Candidate B's total score = 16 points
  • Candidate C's total score = 9 points

Thus, Candidate A is the winner by the Borda count method.

2. Condorcet criterion: We need to check the one-on-one scores for each possible pair of candidates. The match ups are A vs. B, A vs. C, and B vs. C.

Use a table for each match-up to record your results.

\(\begin{array}{r|l} \text{A}&\text{B} \\ \hline &4 \\ 2& \\ 1& \\ \hline \large \textbf{3} & \large \textbf{4} \\ \end{array} \)           \(\begin{array}{r|l} \text{A}&\text{C} \\ \hline 4& \\ 2& \\ 1& \\ \hline \large \textbf{7} & \large \textbf{0} \\ \end{array} \)           \(\begin{array}{r|l} \text{B}&\text{C} \\ \hline 4& \\ &2 \\ 1& \\ \hline \large \textbf{5} & \large \textbf{2} \\ \end{array} \)

B wins A           A wins C          B wins C

 

Here are the head-to-head records:

    A: 1 win (beat C only)
    B: 2 wins (beat A and C)
    C: 0 wins

Thus, Candidate B beats everyone head-to-head. Candidate B is the Condorcet winner.

Candidate A wins using the Borda count method, yet Candidate B is the Condorcet winner. Thus, the method violates the majority criterion. (YES)

Note: Looking at the first head-to-head matchup when B wins A, technically, there is no need to check the rest of the head-to-head matchups, since Candidate A cannot win from this point on. The violation has been established.

 

So, let's summarize what we have learned from the examples above.

Does the Borda count method violate the Condorcet criterion?

(As we saw in Example #7.5.5). YES. The Borda count violates the Condorcet criterion, because a candidate can defeat every other candidate in head-to-head comparisons yet lose the election due to lower total point scores.

Does the Borda count method violate the Majority criterion?

(As we saw in Examples #7.5.1a & 7.5.4). YES. The Borda count violates the Majority criterion, because a candidate with a majority of first‑place votes can lose to another candidate who earns more total ranking points.

Does the plurality method violate the Majority criterion?

The plurality method NEVER violates the Majority criterion, because any candidate who receives a majority of the first‑place votes must necessarily have the most votes and therefore win the election. (We did see a case in Example #7.5.1b, however there is not a case where a violation would exist).

Thus, here is our updated fairness criteria table from Section 7.3:

Fairness Criterion \(\Large\longrightarrow\) Major Voting Method \(\Large\downarrow\)	Condorcet Criterion	Majority Criterion	Monotonicity Criterion	Independence of Irrelevant Alternatives Criterion
Plurality	Violation Possible	\(\checkmark\)	?	?
Borda Count	Violation Possible	Violation Possible	?	?
Plurality-with-Elimination	?	?	?	?
Pairwise Comparison	?	?	?	?

So, we are still looking for a major voting method that satisfies all four fairness criterion's and unfortunately, we have determined that the plurality method and the Borda count method have at least one possible violation for each.

Section 7.5: Majority Criterion [In-Class Exercises]

1. Consider the following preference table with Candidates X, Y, & Z.

Number of Voters	5	4	2	1
First Choice	X	Y	X	Z
Second Choice	Y	X	Z	Y
Third Choice	Z	Z	Y	X

1. Use the Borda count method to determine the winner.
2. Does the election violate the majority criterion?
3. Does the election violate the Condorcet criterion?

2. A diner of 15 customers is voting for their favorite waiter/waitress. The candidates and ranked choices are listed below.

Number of Voters	8	4	3
First Choice	Flo	Mel	Vera
Second Choice	Alice	Vera	Mel
Third Choice	Vera	Alice	Alice
Fourth Choice	Mel	Flo	Flo

1. Use the Borda count method to determine the winner.
2. Does the election violate the majority criterion?
3. Does the election violate the Condorcet criterion?

Answers

1. 1. Candidate X = 30 points, Candidate Y = 26 points, Candidate Z = 16 points. Candidate X wins.
   2. Candidate X wins by both the plurality method and by majority. Thus, the method is not a violation of the majority criterion. (NO)
   3. Candidate X wins by both the Borda county method and by majority. Thus, the method is not a violation of the majority criterion. (NO)
2. 1. Alice = 38 points, Flo = 39 points, Mel = 33 points, Vera = 40 points. Vera wins.
   2. Vera wins using the Borda count method, yet Flo has a majority of first-place votes. Thus, the method violates the majority criterion. (YES)
   3. Vera wins using the Borda count method, yet Flo is the Condorcet winner. Thus, the method violates the majority criterion. (YES)

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