Chapter 1: The Mathematics of Voting



Chapter 1: The Mathematics of Voting (Students Notes)

ESSENTIAL QUESTIONS

Section 1.1: What are preference ballots and schedules?

Section 1.2: How do we use the Plurality Method in an election?

Section 1.3: How do we use the Borda Count Method in an election? What ballot information is important?

Section 1.4: When and how do you implement the Plurality with Elimination Method?

Section 1.5: How does the Method of Pairwise Comparisons determine victor?

Section 1.6: What are Extended and Recursive Ranking Methods?

WORD WALL:

PREFERENCE SCHEDULE

PREFERENCE BALLOT

PLURALITY

MAJORITY CRITERION

CONDORCET CRITERION

BORDA COUNT

PLURALITY WITH ELIMINATION

MONOTONICITY CRITERION

PAIRWISE COMPARISONS

INDEPENDENCE OF IRRELEVANT ALTERNATIVES CRITERION

RECURSIVE RANKING

EXTENDED RANKING

ARROW’S IMPOSSIBILITY THEOREM

FAIR VOTING

Section 1.1: Preference Ballots and Schedules

• Election Components

o Voters:

o Candidates:

o Ballot:

▪ Preference Ballot:

▪ Linear Ballot:

|1st Choice |A |B |A |C |B |

|1st | | | | | |

|2nd | | | | | |

|3rd | | | | | |

|4th | | | | | |

• Preference Schedule:

• Preference Ballot Important Characteristics

o Transitive: If A is preferred to B and B is preferred to C, then A is preferred to C.

o Elimination of Candidates: preference is not affected by removal of a candidates

|Voters | | | |

|1st | | | |

|2nd | | | |

|3rd | | | |

EXAMPLE: A political convention has 1500 voting delegates choosing among three possible party platforms Liberal (L), Moderate (M), and Conservative (C). Seventeen percent of the delegates prefer L to M and M to C. Thirty-two percent of the delegates like C the most and L the least. The rest of the delegates like M the most and C the least. Write out the preference schedule.

|# voters |14 |10 |8 |4 |1 |

|1st |A |C |D |B |C |

|2nd |B |B |C |D |D |

|3rd |C |D |B |C |B |

|4th |D |A |A |A |A |

Exploratory Question: If there are 3 candidates in an election, is there a maximum number of columns a preference schedule can have? If yes, what is that maximum number?

Section 1.2 The Plurality Method

• Plurality Method:

• Majority Rule: in an election b/w 2 candidates, the majority (greater than ½) of the votes wins

Would you rather be a candidate in an election that requires plurality or majority method? Why?

• PROBLEM: Consider an election of 721 voters.

o What is the smallest number of votes needed to be a majority candidate? _______

o If there are 5 candidates, what is the smallest number of votes that a plurality candidate could have?

o If there are 10 candidates, what is the smallest number of votes that a plurality candidate could have?

• MAJORITY V. PLURALITY:

• FAIRNESS CRITERION (2 of 4 Criterion to Judge an Election)

o #1: MAJORITY CRITERION:

If candidate X has _________________________________________ of the first-place voters, then candidate X should be the __________________ of the election.

▪ The Plurality Method ____________________ the Majority Criterion.

|# |49 |48 |3 |

|1st |R |H |C |

|2nd |H |S |H |

|3rd |C |O |S |

|4th |O |C |O |

|5th |S |R |R |

Example: A marching band has been invited to 5 different bowl games: the Rose Bowl, Cotton Bowl, Hula Bowl, Sugar Bowl, and Orange Bowl. The 100 members of the marching band vote to decide which they will attend. Using the Plurality method, at which bowl game will the marching band perform?

Does this seem like a good representation of where the marching band really wanted to go? Why/why not?

o #2: CONDORCET CRITERION:

If a candidate X is ______________________ by the voters over each of the other candidates in a _________________ comparison, then candidate X should be the _____________ of the election.

▪ The Plurality Method ___________________________ the Condorcet Criterion.

• Insincere Voting or Strategic Voting:

• Drawbacks or Flaws of the Plurality Method:

HOMEWORK: p 30 and p 32 # 1, 3, 5, 11, 13, 14

Section 1.3 Borda Count Method

• Borda Count Method:

o Borda Winner:

Based on your knowledge of the definition, do you think ties can happen in a Borda Count Method? What are real life examples of this method? (Hint: Sports World)

• What’s good about this method?

• Calculation Example:

|# voters |14 |10 |8 |4 |

|1st |A |C |D |B |

|2nd |B |B |C |D |

|3rd |C |D |B |C |

|4th |D |A |A |A |

A = _________

B = _________

C = ________

D = _________

FORMULA BASED TECHNIQUE OF CALCULATION:

For each candidate, sum pairs of points of a rank times total number of votes for that rank

| |5 |8 |6 |3 |9 |

|1st |Red |Blue |Red |Blue |Green |

|2nd |Green |Green |Blue |Pink |Blue |

|3rd |Blue |Pink |Green |Red |Red |

|4th |Pink |Red |Pink |Green |Pink |

• Try Borda Count on your own, using either technique.

| |6 |2 |3 |

|1st |A |B |C |

|2nd |B |C |D |

|3rd |C |D |B |

|4th |D |A |A |

A = _______

B = ________

C = ________

D = ________

o Does this violate any of the fairness criterions?

EXPLORATORY QUESTIONS:

1) In an election of 3 candidates and 100 voters with Borda Count method.

a. What is the max number of points a candidate can receive?

b. What is the minimum number of points a candidate can receive?

2) In an election with 4 candidates (A, B, C, D),

a. How many points are given out by one ballot?

b. If there are 110 votes, what is the total number of points given out to the candidates?

c. If A has 320 points, B has 290 points, and C has 180 points, then how many points does D have?

3) Is there an easy way you could know if you made a mistake while doing this method?

Section 1.4 Plurality with Elimination Method (Instant Runoff Voting)

• Plurality with Elimination Method: ** MAJORITY is what we WANT***

o START OF METHOD:

Calculate the value to be called a majority.

o ELIMINATION STEP:

o CHECK WINNER:

o REPEAT:

PROBLEM: Use Plurality with Elimination to determine a winner

| |48% |24% |16% |12% |

|1st |A |C |B |B |

|2nd |D |B |D |A |

|3rd |B |D |A |D |

|4th |C |A |C |C |

SHORTHAND: DO NOT RE-DRAW NEW SCHEDULES

Keep track of first place votes of each candidate outside of table and re-distribute who inherits the first place votes from eliminated candidates only

| |20 |14 |24 |18 |8 |10 |6 |

|1st |Adidas |Asics |Nike |Reebok |Nike |Adidas |Mizuno |

|2nd |Nike |Reebok |Mizuno |Mizuno |Asics |Asics |Asics |

|3rd |Reebok |Mizuno |Reebok |Nike |Adidas |Reebok |Nike |

|4th |Asics |Adidas |Asics |Adidas |Mizuno |Mizuno |Reebok |

|5th |Mizuno |Nike |Adidas |Asics |Reebok |Nike |Adidas |

Majority Votes =

ADIDAS =

NIKE =

REEBOK =

ASICS =

MIZUNO =

Example 2: Use plurality with elimination to determine the winner of the straw poll and also official election

| |7 |8 |10 |4 |

|1 |A |B |C |A |

|2 |B |C |A |C |

|3 |C |A |B |B |

| |7 |8 |14 |

|1 |A |B |C |

|2 |B |C |A |

|3 |C |A |B |

Straw Poll (Attempts to predict election) Official Election

• MONOTONICITY CRITERION Criterion (3rd Fairness Criterion)

If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor _________________________________________ then candidate X should remain the winner of the election.

HOMEWORK: pp 32 - 35 # 19 - 23 (odd), 26 , 29, 31, 34

Section 1.5 Method of Pairwise Comparisons (Copeland’s Method)

• Pairwise Comparison (PWC):

• Method of Pairwise Comparisons:

o Win = _________ points

o Tie = _________ points

o Lose = __________ points

EXAMPLE #1:

| |14 |10 |8 |4 |1 |

|1 |A |C |D |B |C |

|2 |B |B |C |D |D |

|3 |C |D |B |C |B |

|4 |D |A |A |A |A |

What are all the pairings?

A = _________ B = _________ C = _________ D = _________

EXAMPLE #2:

| |10 |5 |12 |7 |

|1 |A |C |B |A |

|2 |B |B |A |C |

|3 |C |A |C |B |

EXPLORATORY: How many Pairwise Comparisons must you compute in general?

o The Number of Pairwise Comparisons for N Candidates =

o Find the following.

▪ 1 + 2 + 3 + … + 56 + 57 =

▪ 27 candidates are running for office in a large election. How many PWC’s exist?

EXAMPLE #3: A coaching staff is considering drafting football players A, B, C, D, or E with the following coaches preferences described below. Who should the team draft?

| |2 |6 |4 |1 |1 |4 |4 |

|1st |A |B |B |C |C |D |E |

|2nd |D |A |A |B |D |A |C |

|3rd |C |C |D |A |A |E |D |

|4th |B |D |E |D |B |C |B |

|5th |E |E |C |E |E |B |A |

A = __________ B = __________

C = __________ D = __________

E = __________

• EXAMPLE #3 continuation: What would happen if C dropped out of the draft to pursue a Rhodes Scholarship? (Myron Rolle, FSU)

| |2 |6 |4 |1 |1 |

|1st |A |C |D |B |C |

|2nd |B |B |C |D |D |

|3rd |C |D |B |C |B |

|4th |D |A |A |A |A |

• Extended Plurality:

• Extended Borda Count:

• Extended Plurality w/ Elimination:

• Extended Pairwise Comparisons:

EXTENDED RANKING PRACTICE PROBLEMS:

| |10 |6 |5 |4 |2 |

|1st |Apples |Bananas |Bananas |Cherries |Oranges |

|2nd |Cherries |Oranges |Cherries |Apples |Cherries |

|3rd |Bananas |Cherries |Apples |Oranges |Bananas |

|4th |Oranges |Apples |Oranges |Bananas |Apples |

1) Perform ALL EXTENDED Ranking Methods for the preference schedule of favorite fruit.

2) Perform Extended Pairwise Comparison on the preference schedule for favorite sports league.

| |8 |7 |6 |2 |1 |

|1st |NHL |NFL |NFL |MLB |MLS |

|2nd |NBA |NBA |NBA |NHL |NHL |

|3rd |MLB |NHL |MLS |NBA |NFL |

|4th |NFL |MLB |MLB |NFL |NBA |

|5th |MLS |MLS |NHL |MLS |MLB |

3) Perform Extended Plurality-with-Elimination and Extended Plurality Rankings on the preference schedule for favorite music genre.

| |12 |8 |9 |6 |3 |

|1st |Hip Hop |Country |Rock |Pop |Country |

|2nd |Country |Rock |Country |Rock |Pop |

|3rd |Rock |Hip Hop |Hip Hop |Hip Hop |Rock |

|4th |Pop |Pop |Pop |Country |Hip Hop |

Section 1.6 RECURSIVE RANKINGS

• Recursive: the same operation or step is performed over and over again

• Recursive Ranking Methods:

o RANKING =

EXAMPLES

1) Recursive Plurality

| |21 |15 |12 |7 |

|1ST |A |C |D |B |

|2ND |B |B |C |D |

|3RD |C |D |B |C |

|4TH |D |A |A |A |

2) Recursive Borda Count

| |10 |9 |7 |

|1ST |G |I |J |

|2ND |H |H |G |

|3RD |J |J |H |

|4TH |I |G |I |

3) Recursive Plurality with Elimination

| |7 |7 |8 |5 |

|1ST |P |R |S |Q |

|2ND |Q |Q |R |P |

|3RD |R |P |Q |R |

|4TH |S |S |P |S |

4) Recursive Pairwise Comparison

| |8 |6 |5 |5 |2 |

|1ST |C |A |E |D |D |

|2ND |B |E |C |C |A |

|3RD |A |D |D |A |E |

|4TH |D |B |B |E |B |

|5TH |E |C |A |B |C |

| |14 |10 |8 |4 |1 |

|1ST |A |C |D |B |C |

|2ND |B |B |C |D |D |

|3RD |C |D |B |C |B |

|4TH |D |A |A |A |A |

Comparing Recursive Plurality and Recursive Plurality with Elimination:

BE CAREFUL!!!

o Recursive Plurality:

o Recursive Plurality with Elimination:

FAIR VOTING: All 4 fairness criteria are satisfied

ARROW’S IMPOSSIBILITY THEOREM: It is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria.

HOMEWORK: Finish 1.6 Worksheet and BRING YOUR BALLOT COLLECTION

Bonus: p. 36 #41, 48

-----------------------

| |14 |10 |8 |4 |1 |

|1st |A |C |D |B |C |

|2nd |B |B |C |D |D |

|3rd |C |D |B |C |B |

|4th |D |A |A |A |A |

Is there a majority?

If not, Who is eliminated?

Write the new preference schedule.

| |14 |10 |8 |4 |1 |

|1st | | | | | |

|2nd | | | | | |

|3rd | | | | | |

Is there a majority?

If not, who is eliminated?

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