Math in Politics: Get Stronger

Voting Theory

  1. To decide on a new website design, the designer asks people to rank three designs that have been created (labeled A, B, and C). The individual ballots are shown below. Create a preference table.

    ABC, ABC, ACB, BAC, BCA, BCA, ACB, CAB, CAB, BCA, ACB, ABC

  2. To decide on a movie to watch, a group of friends all vote for one of the choices (labeled A, B, and C). The individual ballots are shown below. Create a preference table.

    CAB, CBA, BAC, BCA, CBA, ABC, ABC, CBA, BCA, CAB, CAB, BAC

  3. The planning committee for a renewable energy trade show is trying to decide what city to hold their next show in. The votes are shown below.
[latex]\begin{array}{|c|c|c|c|c|} \hline \textbf { Number of voters } & \textbf { 9 } & \textbf { 19 } & \textbf { 11 } & \textbf { 8 } \\ \hline \textbf { 1st choice } & \text { Buffalo } & \text { Atlanta } & \text { Chicago } & \text { Buffalo } \\ \hline \textbf { 2nd choice } & \text { Atlanta } & \text { Buffalo } & \text { Buffalo } & \text { Chicago } \\ \hline \textbf { 3rd choice } & \text { Chicago } & \text { Chicago } & \text { Atlanta } & \text { Atlanta } \\ \hline \end{array}[/latex]
  1. How many voters voted in this election?
  2. How many votes are needed for a majority? A plurality?
  3. Find the winner under the plurality method.
  4. Find the winner under the Borda Count Method.
  5. Find the winner under the Instant Runoff Voting method.
  6. Find the winner under Copeland’s method.
  1. A non-profit agency is electing a new chair of the board. The votes are shown below.
[latex]\begin{array}{|c|c|c|c|c|} \hline \textbf { Number of voters } & \mathbf{1 1} & \mathbf{5} & \mathbf{1 0} & \mathbf{3} \\ \hline \textbf { 1st choice } & \text { Atkins } & \text { Cortez } & \text { Burke } & \text { Atkins } \\ \hline \textbf { 2nd choice } & \text { Cortez } & \text { Burke } & \text { Cortez } & \text { Burke } \\ \hline \textbf { 3rd choice } & \text { Burke } & \text { Atkins } & \text { Atkins } & \text { Cortez } \\ \hline \end{array}[/latex]
  1. How many voters voted in this election?
  2. How many votes are needed for a majority? A plurality?
  3. Find the winner under the plurality method.
  4. Find the winner under the Borda Count Method.
  5. Find the winner under the Instant Runoff Voting method.
  6. Find the winner under Copeland’s method.
  1. The student government is holding elections for president. There are four candidates (labeled A, B, C, and D for convenience). The preference schedule for the election is:
[latex]\begin{array}{|c|c|c|c|c|c|c|} \hline \textbf {Number of voters} & \mathbf{1 2 0} & \mathbf{5 0} & \mathbf{4 0} & \mathbf{9 0} & \mathbf{6 0} & \mathbf{1 0 0} \\ \hline \textbf {1st choice } & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{D} \\ \hline \textbf{2nd choice } & \mathrm{D} & \mathrm{C} & \mathrm{A} & \mathrm{C} & \mathrm{D} & \mathrm{B} \\ \hline \textbf{3rd choice } & \mathrm{B} & \mathrm{A} & \mathrm{B} & \mathrm{B} & \mathrm{C} & \mathrm{A} \\ \hline \textbf{4th choice } & \mathrm{A} & \mathrm{D} & \mathrm{C} & \mathrm{D} & \mathrm{B} & \mathrm{C} \\ \hline \end{array}[/latex]
  1. How many voters voted in this election?
  2. How many votes are needed for a majority? A plurality?
  3. Find the winner under the plurality method.
  4. Find the winner under the Borda Count Method.
  5. Find the winner under the Instant Runoff Voting method.
  6. Find the winner under Copeland’s method.
  1. The homeowners association is deciding a new set of neighborhood standards for architecture, yard maintenance, etc. Four options have been proposed. The votes are:
[latex]\begin{array}{|c|c|c|c|c|c|c|} \hline \textbf { Number of voters } & \mathbf{8} & \mathbf{9} & \mathbf{1 1} & \mathbf{7} & \mathbf{7} & \mathbf{5} \\ \hline \textbf { 1st choice } & \text { B } & \text { A } & \text { D } & \text { A } & \text { B } & \text { C } \\ \hline \textbf { 2nd choice } & \text { C } & \text { D } & \text { B } & \text { B } & \text { A } & \text { D } \\ \hline \textbf { 3rd choice } & \text { A } & \text { C } & \text { C } & \text { D } & \text { C } & \text { A } \\ \hline \textbf { 4th choice } & \text { D } & \text { B } & \text { A } & \text { C } & \text { D } & \text { B } \\ \hline \end{array}[/latex]
  1. How many voters voted in this election?
  2. How many votes are needed for a majority? A plurality?
  3. Find the winner under the plurality method.
  4. Find the winner under the Borda Count Method.
  5. Find the winner under the Instant Runoff Voting method.
  6. Find the winner under Copeland’s method.
  1. Consider an election with [latex]129[/latex] votes.
    1. If there are [latex]4[/latex] candidates, what is the smallest number of votes that a plurality candidate could have?
    2. If there are [latex]8[/latex] candidates, what is the smallest number of votes that a plurality candidate could have?
  1. Consider an election with [latex]953[/latex] votes.
    1. If there are [latex]7[/latex] candidates, what is the smallest number of votes that a plurality candidate could have?
    2. If there are [latex]8[/latex] candidates, what is the smallest number of votes that a plurality candidate could have?
  1. Does this voting system having a Condorcet Candidate? If so, find it.
[latex]\begin{array}{|c|c|c|c|} \hline \textbf { Number of voters } & \mathbf{1 4} & \mathbf{1 5} & \mathbf{2} \\ \hline \textbf { 1st choice } & \mathrm{A} & \mathrm{C} & \mathrm{B} \\ \hline \textbf { 2nd choice } & \mathrm{B} & \mathrm{B} & \mathrm{C} \\ \hline \textbf { 3rd choice } & \mathrm{C} & \mathrm{A} & \mathrm{A} \\ \hline \end{array}[/latex]
  1. Does this voting system having a Condorcet Candidate? If so, find it.
[latex]\begin{array}{|c|c|c|c|} \hline \textbf { Number of voters } & \mathbf{8} & \mathbf{7} & \mathbf{6} \\ \hline \textbf { 1st choice } & \mathrm{A} & \mathrm{C} & \mathrm{B} \\ \hline \textbf { 2nd choice } & \mathrm{B} & \mathrm{B} & \mathrm{C} \\ \hline \textbf { 3rd choice } & \mathrm{C} & \mathrm{A} & \mathrm{A} \\ \hline \end{array}[/latex]
  1. The marketing committee at a company decides to vote on a new company logo. They decide to use approval voting. Their results are tallied below. Each column shows the number of voters with the particular approval vote. Which logo wins under approval voting?
[latex]\begin{array}{|c|c|c|c|} \hline \textbf { Number of voters } & \mathbf{8} & \mathbf{7} & \mathbf{6} \\ \hline \textbf { 1st choice } & \mathrm{A} & \mathrm{C} & \mathrm{B} \\ \hline \textbf { 2nd choice } & \mathrm{B} & \mathrm{B} & \mathrm{C} \\ \hline \textbf { 3rd choice } & \mathrm{C} & \mathrm{A} & \mathrm{A} \\ \hline \end{array}[/latex]
  1. The downtown business association is electing a new chairperson, and decides to use approval voting. The tally is below, where each column shows the number of voters with the particular approval vote. Which candidate wins under approval voting?
    [latex]\begin{array}{|c|c|c|c|c|c|c|c|} \hline \textbf { Number of voters } & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} \\ \hline \mathbf{A} & \mathrm{X} & \mathrm{X} & & & \mathrm{X} & & \mathrm{X} \\ \hline \mathbf{B} & \mathrm{X} & & \mathrm{X} & \mathrm{X} & & & \mathrm{X} \\ \hline \mathbf{C} & & \mathrm{X} & \mathrm{X} & \mathrm{X} & & \mathrm{X} & \\ \hline \mathbf{D} & \mathrm{X} & & \mathrm{X} & & \mathrm{X} & \mathrm{X} & \\ \hline \end{array}[/latex]
  2. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and C decided to drop out of the election, which caused B to become the winner, which is the primary fairness criterion violated in this election?
  3. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and many people who had voted for C switched their preferences to favor A, which caused B to become the winner, which is the primary fairness criterion violated in this election?
  4. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If in a head-to-head comparison a majority of people prefer B to A or C, which is the primary fairness criterion violated in this election?
  5. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If B had received a majority of first place votes, which is the primary fairness criterion violated in this election?

Apportionment

In exercises [latex]1-8[/latex], determine the apportionment using

  • Hamilton’s Method
  • Jefferson’s Method
  • Webster’s Method
  • Huntington-Hill Method
  • Lowndes’ method
  1. A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. If the college can only afford to hire [latex]15[/latex] tutors, determine how many tutors should be assigned to each subject.
    • Math: [latex]330[/latex]
    • English: [latex]265[/latex]
    • Chemistry: [latex]130[/latex]
    • Biology: [latex]70[/latex]
  2. Reapportion the previous problem if the college can hire [latex]20[/latex] tutors.
  3. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift. Apportion [latex]20[/latex] salespeople given the information below.
    [latex]\begin{array}{|l|l|l|l|l|}
    \hline \text { Shift } & \text { Morning } & \text { Midday } & \text { Afternoon } & \text { Evening } \\
    \hline \begin{array}{l}
    \text { Average number of } \\
    \text { customers }
    \end{array} & 95 & 305 & 435 & 515 \\
    \hline
    \end{array}[/latex]
  4. Reapportion the previous problem if the store has [latex]25[/latex] salespeople.
  5. Three people invest in a treasure dive, each investing the amount listed below. The dive results in [latex]36[/latex] gold coins. Apportion those coins to the investors.
    • Alice: [latex]$7,600[/latex]
    • Ben: [latex]$5,900[/latex]
    • Carlos: [latex]$1,400[/latex]
  6. Reapportion the previous problem if [latex]37[/latex] gold coins are recovered.
  7. A small country consists of five states, whose populations are listed below. If the legislature has [latex]119[/latex] seats, apportion the seats.
    • A: [latex]810,000[/latex]
    • B: [latex]473,000[/latex]
    • C: [latex]292,000[/latex]
    • D: [latex]594,000[/latex]
    • E: [latex]211,000[/latex]
  8. A small country consists of six states, whose populations are listed below. If the legislature has [latex]200[/latex] seats, apportion the seats.
    • A: [latex]3,411[/latex]
    • B: [latex]2,421[/latex]
    • C: [latex]11,586[/latex]
    • D: [latex]4,494[/latex]
    • E: [latex]3,126[/latex]
    • F: [latex]4,962[/latex]
  9. A small country consists of three states, whose populations are listed below.
    • A: [latex]6,000[/latex]
    • B: [latex]6,000[/latex]
    • C: [latex]2,000[/latex]
    1. If the legislature has [latex]10[/latex] seats, use Hamilton’s method to apportion the seats.
    2. If the legislature grows to [latex]11[/latex] seats, use Hamilton’s method to apportion the seats.
    3. Which apportionment paradox does this illustrate?
  10. A state with five counties has [latex]50[/latex] seats in their legislature. Using Hamilton’s method, apportion the seats based on the [latex]2000[/latex] census, then again using the [latex]2010[/latex] census. Which apportionment paradox does this illustrate?
    [latex]\begin{array}{|l|l|l|}
    \hline \textbf { County } & \mathbf{2 0 0 0} \textbf { Population } & \mathbf{2 0 1 0} \textbf { Population } \\
    \hline \text { Jefferson } & 60,000 & 60,000 \\
    \hline \text { Clay } & 31,200 & 31,200 \\
    \hline \text { Madison } & 69,200 & 72,400 \\
    \hline \text { Jackson } & 81,600 & 81,600 \\
    \hline \text { Franklin } & 118,000 & 118,400 \\
    \hline
    \end{array}[/latex]
  11. A school district has two high schools: Lowell, serving [latex]1715[/latex] students, and Fairview, serving [latex]7364[/latex]. The district could only afford to hire [latex]13[/latex] guidance counselors.
    1. Determine how many counselors should be assigned to each school using Hamilton’s method.
    2. The following year, the district expands to include a third school, serving [latex]2989[/latex] students. Based on the divisor from above, how many additional counselors should be hired for the new school?
    3. After hiring that many new counselors, the district recalculates the reapportion using Hamilton’s method. Determine the outcome.
    4. Does this situation illustrate any apportionment issues?
  12. A small country consists of four states, whose populations are listed below. If the legislature has [latex]116[/latex] seats, apportion the seats using Hamilton’s method. Does this illustrate any apportionment issues?
    • A: [latex]33,700[/latex]
    • B: [latex]559,500[/latex]
    • C: [latex]141,300[/latex]
    • D: [latex]89,100[/latex]

Weighted Voting

  1. Consider the weighted voting system [latex][47: 10,9,9,5,4,4,3,2,2][/latex] 
    1. How many players are there?
    2. What is the total number (weight) of votes?
    3. What is the quota in this system?
  2. Consider the weighted voting system [latex][q: 7,5,3,1,1][/latex] 
    1. What is the smallest value that the quota [latex]q[/latex]  can take?
    2. What is the largest value that the quota [latex]q[/latex]  can take?
    3. What is the value of the quota if at least two-thirds of the votes are required to pass a motion?
  3. Consider the weighted voting system [latex][13: 13, 6, 4, 2][/latex]
    1. Identify the dictators, if any.
    2. Identify players with veto power, if any
    3. Identify dummies, if any.
  4. Consider the weighted voting system [latex][19: 13, 6, 4, 2][/latex]
    1. Identify the dictators, if any.
    2. Identify players with veto power, if any
    3. Identify dummies, if any.
  5. Consider the weighted voting system [latex][15: 11, 7, 5, 2][/latex]
    1. What is the weight of the coalition [latex]\left\{P_{1}, P_{2}, P_{4}\right\}[/latex]
    2. In the coalition [latex]\left\{P_{1}, P_{2}, P_{4}\right\}[/latex] which players are critical?
  6. Find the Banzhaf power distribution of the weighted voting system [latex][27: 16, 12, 11, 3][/latex]
  7. Consider the weighted voting system [latex][q: 15, 8, 3, 1][/latex]. Find the Banzhaf power distribution of this weighted voting system,
    1. When the quota is [latex]15[/latex]
    2. When the quota is [latex]16[/latex]
    3. When the quota is [latex]18[/latex]
  8. Consider the weighted voting system [latex][17:13,9,5,2][/latex]. In the sequential coalition [latex]\left\{P_{3}, P_{2}, P_{1}, P_{4}\right\}[/latex] which player is pivotal?
  9. Find the Shapley-Shubik power distribution for the system [latex][24: 17, 13, 11][/latex].
  10. Consider the weighted voting system [latex][q: 7, 3, 1][/latex]
    1. Which values of [latex]q[/latex] result in a dictator (list all possible values)
    2. What is the smallest value for [latex]q[/latex] that results in exactly one player with veto power but no dictators?
    3. What is the smallest value for [latex]q[/latex] that results in exactly two players with veto power?
  11. Using the Shapley-Shubik method, is it possible for a dummy to be pivotal? 
  12. Consider a weighted voting system with three players. If Player 1 is the only player with veto power, there are no dictators, and there are no dummies:
    1. Find the Banzhaf power distribution.
    2. Find the Shapley-Shubik power distribution
  13. An executive board consists of a president (P) and three vice-presidents [latex]\left(\mathrm{V}_{1}, \mathrm{V}_{2}, \mathrm{V}_{3}\right)[/latex]. For a motion to pass it must have three yes votes, one of which must be the president’s. Find a weighted voting system to represent this situation.
  14. In a corporation, the shareholders receive 1 vote for each share of stock they hold, which is usually based on the amount of money the invested in the company. Suppose a small corporation has two people who invested [latex]$30,000[/latex] each, two people who invested[latex]$20,000[/latex] each, and one person who invested [latex]$10,000[/latex]. If they receive one share of stock for each [latex]$1000[/latex] invested, and any decisions require a majority vote, set up a weighted voting system to represent this corporation’s shareholder votes.
  15. The United Nations Security Council consists of [latex]15[/latex] members, [latex]10[/latex] of which are elected, and [latex]5[/latex] of which are permanent members. For a resolution to pass, [latex]9[/latex] members must support it, which must include all [latex]5[/latex] of the permanent members. Set up a weighted voting system to represent the UN Security Council and calculate the Banzhaf power distribution.