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Mathematics LibreTexts

7.3: Exercises

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    22347
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    1. An organization recently made a decision about which company to use to redesign its website and host its members’ information. The Board of Directors will vote using preference ballots ranking their first choice to last choice of the following companies: Allied Web Design (A), Ingenuity Incorporated (I), and Yeehaw Web Trends (Y). The individual ballots are shown below. Create a preference schedule summarizing these results.

    AIY, YIA, YAI, AIY, YIA, IAY, IYA, IAY, YAI, YIA, AYI, YIA, YAI

    1. A group needs to decide where their next conference will be held. The choices are Kansas City (K), Lafayette (L), and Minneapolis (M). The individual ballots are shown below. Create a preference schedule summarizing these results.

    KLM, LMK, MLK, LMK, MKL, KLM, KML, LMK, MKL, MKL, MLK, MLK

    1. A book club holds a vote to figure out what book they should read next. They are picking from three different books. The books are labeled A, B, and C, and the preference schedule for the vote is below.

    Number of voters

    12

    9

    8

    5

    10

    1st choice

    A

    B

    B

    C

    C

    2nd choice

    C

    A

    C

    A

    B

    3rd choice

    B

    C

    A

    B

    A

    1. How many voters voted in the election?
    2. How many votes are needed for a majority?
    3. Find the winner using the Plurality Method.
    4. Find the winner using the Borda Count Method.
    5. Find the winner using the Plurality with Elimination Method.
    6. Find the winner using the Pairwise Comparisons Method.
    1. An election is held for a new vice president at a college. There are three candidates (A, B, C), and the faculty rank which candidate they like the most. The preference ballot is below.

    Number of voters

    8

    10

    12

    9

    4

    1

    1st choice

    A

    A

    B

    B

    C

    C

    2nd choice

    B

    C

    A

    C

    A

    B

    3rd choice

    C

    B

    C

    A

    B

    A

    1. How many voters voted in the election?
    2. How many votes are needed for a majority?
    3. Find the winner using the Plurality Method.
    4. Find the winner using the Borda Count Method.
    5. Find the winner using the Plurality with Elimination Method.
    6. Find the winner using the Pairwise Comparisons Method.
    1. A city election for a city council seat was held between 4 candidates, Martorana (M), Jervey (J), Riddell (R), and Hanrahan (H). The preference schedule for this election is below.

    Number of voters

    60

    73

    84

    25

    110

    1st choice

    M

    M

    H

    J

    J

    2nd choice

    R

    H

    R

    R

    M

    3rd choice

    H

    R

    M

    M

    R

    4th choice

    J

    J

    J

    H

    H

    1. How many voters voted in the election?
    2. How many votes are needed for a majority?
    3. Find the winner using the Plurality Method.
    4. Find the winner using the Borda Count Method.
    5. Find the winner using the Plurality with Elimination Method.
    6. Find the winner using the Pairwise Comparisons Method.
    1. A local advocacy group asks members of the community to vote on which project they want the group to put its efforts behind. The projects are green spaces (G), city energy code (E), water conservation (W), and promoting local business (P). The preference schedule for this vote is below.

    Number of voters

    12

    57

    23

    34

    13

    18

    22

    39

    1st choice

    W

    W

    G

    G

    E

    E

    P

    P

    2nd choice

    G

    P

    W

    E

    G

    P

    W

    E

    3rd choice

    E

    E

    E

    W

    P

    W

    G

    W

    4th choice

    P

    G

    P

    P

    W

    G

    E

    G

    1. How many voters voted in the election?
    2. How many votes are needed for a majority?
    3. Find the winner using the Plurality Method.
    4. Find the winner using the Borda Count Method.
    5. Find the winner using the Plurality with Elimination Method.
    6. Find the winner using the Pairwise Comparisons Method.
    1. An election is held and candidate A wins. A mistake was found that showed that candidate C was not qualified to run in the election. The candidate was removed, and the election office determined the winner with candidate C removed. Now candidate D wins. What fairness criterion was violated?
    1. An election is held and candidate C wins. Before the election is certified the ballots are misplaced. Another election is held, and the only change in the ballots was that more people put C as their first choice. When the winner is determined, candidate A now wins. What fairness criterion was violated?
    1. An election is held and candidate B has the majority of first-place votes. However, candidate B does not win the election. What fairness criterion was violated?
    1. An election is held and candidate D is favored in a head-to-head comparison to every other candidate. However, D does not win the election. What is D called, and what fairness criterion was violated?
    1. Consider the weighted voting system yvYjDyN432k6OVvjHPSyiChtmnCaa8mo4bYglQhbrCFLGinBeXcNS6ZZXbCExiVT2L9WqbHAiLOgstNTjzZiblfHt-2X-yOddp-lzBI3-Gt06RS6fIsHzN8rP7Z8DVdjlHP6Zqk.
    1. How many players are there?
    2. List the weight of each player.
    3. What is the quota?
    4. Is this a valid system? Why or why not?
    1. Consider the weighted voting system -d2MB1HpS6TYEegfeyjmoIRUEyy61aJCBBE4_0M5O7U9HcyMy7aSHY8K25HmjCXTOqyJ6ufJ101-4QIXjt8dDV1padXUlWNZTCXS5DKpwc8Ynlke7d1M5m1eJmb58-9zfW6Dee4.
    1. How many players are there?
    2. List the weight of each player.
    3. What is the quota?
    4. Is this a valid system? Why or why not?
    1. Consider the weighted voting system hJbAxkqQe8R7K-wuAPSkoUOg8VUOazMOEIgDst7oSIybJcqq6rEeMR_Fzl6UxvBQgSYGYu63PptikxfaquRc8OoV7uYX25vd-Pq7SPkvh7g74dCd1vfH7h8B_u50XXA6ap4VUzw.
    1. How many players are there?
    2. List the weight of each player.
    3. What is the quota?
    4. Is this a valid system? Why or why not?
    1. Consider the weighted voting system nsGVzYCpHg0SRcuDblF-0HTLMLr504DH3ilbs4iLLxZY-cNWhGhMyiuwedJ_ze2fQQ_PJIb9B5AnST9O_dEyJ5bZ5FLm3Ws-dcOi5lnt9TZ6O3iMSC7d36rXbGuidEnflDOHjE0.
    1. How many players are there?
    2. List the weight of each player.
    3. What is the quota?
    4. Is this a valid system? Why or why not?
    1. Consider the weighted voting system 3-4FNlNrBLxvNd7DE1eoqjw3aux1sCl-hAJzcrdZYYEKsOw7DQrP9YRIM_YRFMmsr4elfW7Wp3ZPB6rdLKDoNcawAw8sHBV6CMAYvz86jto1tx4F_Lq6BdVvenOfbBmy2RRUc60.
      1. What is the minimum value of the quota q?
      2. What is the maximum value of the quota q?
      3. What is the quota q if a motion can only pass with 2/3’s of the vote?
      4. What is the quota q if a motion can only pass with more than 2/3’s of the vote?
    1. Consider the weighted voting system OBraNbbGPYIsXl_ecy8swBioRINVyTt1JAvo5_x1QBLvwu2qhuVipnKz_moLv2oBAoYDvRp9UT97qneo9f7cKybZMpZUFOpJjeA29FByB7n5y5z1rx8_8R-lvp1S_L-9REhW2G8.
      1. What is the minimum value of the quota q?
      2. What is the maximum value of the quota q?
      3. What is the quota q if a motion can only pass with 2/3’s of the vote?
      4. What is the quota q if a motion can only pass with more than 2/3’s of the vote?
    1. Consider the weighted voting system oxfdLw5_DXFDXuX-kvq_gUkFO9sfBKBY0Pfzb8-5qrTPzCEK-coPoPmbHUQFs-mupQrYTlyfyJbFp9E21fewk-corxweBxFxovmHiwcCQBfi0TXKOZ7b69Ta71FJn7ke959BhgQ. Are any players dictators? Explain.
    1. Consider the weighted voting system gRcsEk98b5lUu_Kn1wVfKiJKg6TFrZ_gU0xI3tWVr5HGkBW6XL3jdBDGzqPLLOJt7XQekb5osYBn-FctAwCKeQpwz2A9JAONaqbVPIyOeyN8YVXZ7w7ymPxkVdN0QyrDR-hFOag. Do any players have veto power? Explain.
    1. Consider the weighted voting system Klfb7Q1yRnskbdgAaLZkXeuV9-zIRi5fs8bnrd5GZnCy9kOH27G4xzHqVghi0UaZ1pWdiZXXuSyDtjR1L-kKd3JdShD32tW18DPyr5lm5560_M65ZBBeUclnYs2EQjOsELJFBOs.
    1. Find the Banzhaf power index for each player.
    2. Find the Shapely-Shubik power index for each player.
    3. Are any players a dummy?
    1. Consider the weighted voting system LKHHVp8hs5qj4dFX-qfyHZ4cIzH-6hH9jnybRrEuN_cjvCyJZ_Yf80p5-fzweuxAZqYJlfekPJZIrL0-EL9rR87ecXiO73xYw3jhljqiZ-VNRMgUJzo4bw65w21q9-pb6Kq5ShU.
    1. Find the Banzhaf power index for each player.
    2. Find the Shapely-Shubik power index for each player.
    3. Are any players a dummy?
    1. Consider the weighted voting system NDsiPQ-5ccVgbZIy3tkJAD5-oYB1mjN8miXpsf3RqO3-Fv5c4a2WZnwwocpOPyyhzZtdcdXB_VyC4__ecANrz3mt8B_4_WpOlIcLwudJh6zHcVdv27d25ndhMs2jsUqEgWvyEIY.
    1. Find the Banzhaf power index for each player.
    2. Find the Shapely-Shubik power index for each player.
    3. Are any players a dummy?
    1. Consider the weighted voting system pMSQfLoLWXEH_NPl1hCIyGbihTuy-uC0LTJRkN0SnT8kaqWHJn7lafLaEcWowdB5pqFeAcdcozcUHbvEJc8GZjVTb97a6sAMHOAKtIm8MOpBe4ey6gwGEqSI2TbbBjXIDgskbB0.
    1. Find the Banzhaf power index for each player.
    2. Find the Shapely-Shubik power index for each player.
    3. Are any players a dummy?
    1. Consider the weighted voting system kiCRcoOP5RPPUli2f6rJXYZA50Y8vdZF-0sW2MlWT2CyOiYPGlkpbTmxnAlL9w81H29ia_jb4rZaGCfBdFtyhidwj0XF-kMoUwMtnRpmW5OapuHr2xoJWHkEr351IYhzkG1bPiM. Find the Banzhaf power index for each player?
    1. Consider the weighted voting system cQ-kvT44VkgWvOzEMnkcZbTLbh9fAvhFd3Ix8VluO3jxBBbKEoxTvbVPDyfg6eN2322bIEAQiobKjlyb6w0lBkEeZzru5OVw0tJRW6KwX2O3rlKk2-RYdU-LubG-EsNwd11oAMY. Find the Banzhaf power index for each player?
    1. The United Nations (UN) Security Council consists of five permanent members (United States, Russian Federation, the United Kingdom, France, and China) and 10 non-permanent members elected for two-year terms by the General Assembly. The five permanent members have veto power, and a resolution cannot pass without nine members voting for it. Set up the weighted voting system for the UN.
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