{"id":66,"date":"2025-01-02T23:05:08","date_gmt":"2025-01-02T23:05:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/math-in-politics-get-stronger-answer-key\/"},"modified":"2025-01-02T23:05:08","modified_gmt":"2025-01-02T23:05:08","slug":"math-in-politics-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/math-in-politics-get-stronger-answer-key\/","title":{"raw":"Math in Politics: Get Stronger Answer Key","rendered":"Math in Politics: Get Stronger Answer Key"},"content":{"raw":"\n

Voting Theory<\/h2>\n
    \n\t
  1. \n
      \n\t
    1. [latex]\\begin{array}{|l|l|l|l|l|l|}
      \n\\hline \\text { Number of voters } & 3 & 3 & 1 & 3 & 2 \\\\
      \n\\hline 1^{\\text {st }} \\text { choice } & \\mathrm{A} & \\mathrm{A} & \\mathrm{B} & \\mathrm{B} & \\mathrm{C} \\\\
      \n\\hline 2^{\\text {nd }} \\text { choice } & \\mathrm{B} & \\mathrm{C} & \\mathrm{A} & \\mathrm{C} & \\mathrm{A} \\\\
      \n\\hline 3^{\\text {rd }} \\text { choice } & \\mathrm{C} & \\mathrm{B} & \\mathrm{C} & \\mathrm{A} & \\mathrm{B} \\\\
      \n\\hline
      \n\\end{array}[\/latex]<\/li>\n\t
    2.  <\/li>\n\t
    3. \n
        \n\t
      1. [latex]9+19+11+8 = 47[\/latex]<\/li>\n\t
      2. [latex]24[\/latex] for majority; [latex]16[\/latex] for plurality (though a choice would need a minimum of [latex]17[\/latex] votes to actually win under the Plurality method)<\/li>\n\t
      3. Atlanta, with [latex]19[\/latex] first-choice votes<\/li>\n\t
      4. Atlanta [latex]94[\/latex], Buffalo [latex]111[\/latex], Chicago [latex]77[\/latex]. Winner: Buffalo<\/li>\n\t
      5. Chicago eliminated, [latex]11[\/latex] votes go to Buffalo. Winner: Buffalo<\/li>\n\t
      6. A vs B: B. A vs C: A. B vs C: B. B gets [latex]2[\/latex] pts, A [latex]1[\/latex] pt. Buffalo wins.<\/li>\n<\/ol>\n<\/li>\n\t
      7.  <\/li>\n\t
      8. \n
          \n\t
        1. [latex]120+50+40+90+60+100 = 460[\/latex]<\/li>\n\t
        2. [latex]231[\/latex] for majority; [latex]116[\/latex] for plurality<\/li>\n\t
        3. A with [latex]150[\/latex] first choice votes<\/li>\n\t
        4. A [latex]1140[\/latex], B [latex]1060[\/latex], C [latex]1160[\/latex], D [latex]1240[\/latex]. Winner: D<\/li>\n\t
        5. B eliminated, votes to C. D eliminated, votes to A. Winner: A<\/li>\n\t
        6. A vs B: B. A vs C: A. A vs D: D. B vs C: C. B vs D: D. C vs D: C. A [latex]1[\/latex]pt, B [latex]1[\/latex]pt, C [latex]2[\/latex]pt, D [latex]2[\/latex]pt. Tie between C and D.<\/li>\n<\/ol>\n

          Winner would probably be C since C was preferred over D<\/p>\n<\/li>\n\t

        7.  <\/li>\n\t
        8. \n
            \n\t
          1. [latex]33[\/latex]<\/li>\n\t
          2. [latex]17[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
          3.  <\/li>\n\t
          4. Yes, B<\/li>\n\t
          5.  <\/li>\n\t
          6. B, with [latex]17[\/latex] approvals<\/li>\n\t
          7.  <\/li>\n\t
          8. Independence of Irrelevant Alternatives Criterion<\/li>\n\t
          9.  <\/li>\n\t
          10. Condorcet Criterion<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

            Apportionment<\/h2>\n
              \n\t
            1. \n
                \n\t
              1. \n
                  \n\t
                1. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]3[\/latex], Biology: [latex]1[\/latex]<\/li>\n\t
                2. Math: [latex]7[\/latex], English: [latex]5[\/latex], Chemistry: [latex]2[\/latex], Biology: [latex]1[\/latex]<\/li>\n\t
                3. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]3[\/latex], Biology: [latex]1[\/latex]<\/li>\n\t
                4. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]3[\/latex], Biology: [latex]1[\/latex]<\/li>\n\t
                5. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]2[\/latex], Biology: [latex]2[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
                6.  <\/li>\n\t
                7. \n
                    \n\t
                  1. Morning: 1, Midday: 5, Afternoon: 6, Evening: 8<\/li>\n\t
                  2. Morning: [latex]1[\/latex], Midday: [latex]4[\/latex], Afternoon: [latex]7[\/latex], Evening: [latex]8[\/latex]<\/li>\n\t
                  3. Morning: [latex]1[\/latex], Midday: [latex]5[\/latex], Afternoon: [latex]6[\/latex], Evening: [latex]8[\/latex]<\/li>\n\t
                  4. Morning: [latex]1[\/latex], Midday: [latex]5[\/latex], Afternoon: [latex]6[\/latex], Evening: [latex]8[\/latex]<\/li>\n\t
                  5. Morning: [latex]2[\/latex], Midday: [latex]5[\/latex], Afternoon: [latex]6[\/latex], Evening: [latex]7[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
                  6.  <\/li>\n\t
                  7. \n
                      \n\t
                    1. Alice: [latex]18[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]4[\/latex]<\/li>\n\t
                    2. Alice: [latex]19[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]3[\/latex]<\/li>\n\t
                    3. Alice: [latex]19[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]3[\/latex]<\/li>\n\t
                    4. Alice: [latex]19[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]3[\/latex]<\/li>\n\t
                    5. Alice: [latex]18[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]4[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
                    6.  <\/li>\n\t
                    7. \n
                        \n\t
                      1. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n\t
                      2. A: [latex]41[\/latex], B: [latex]24[\/latex], C: [latex]14[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n\t
                      3. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n\t
                      4. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n\t
                      5. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]29[\/latex], E: [latex]11[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
                      6.  <\/li>\n\t
                      7.  <\/li>\n\t
                      8.  <\/li>\n\t
                      9.  <\/li>\n\t
                      10.  <\/li>\n<\/ol>\n

                        Weighted Voting<\/h2>\n
                          \n\t
                        1. \n
                            \n\t
                          1. [latex]9[\/latex] players<\/li>\n\t
                          2. [latex]10+9+9+5+4+4+3+2+2 = 48[\/latex]<\/li>\n\t
                          3. [latex]47[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
                          4. \n
                              \n\t
                            1. [latex]9[\/latex], a majority of votes<\/li>\n\t
                            2. [latex]17[\/latex], the total number of votes<\/li>\n\t
                            3. [latex]12[\/latex], which is [latex]2\/3[\/latex] of [latex]17[\/latex], rounded up <\/li>\n<\/ol>\n<\/li>\n\t
                            4. \n
                                \n\t
                              1. P1 is a dictator (can reach quota by themselves)<\/li>\n\t
                              2. P1, since dictators also have veto power<\/li>\n\t
                              3. P2, P3, P4<\/li>\n<\/ol>\n<\/li>\n\t
                              4. \n
                                  \n\t
                                1. none<\/li>\n\t
                                2. P1<\/li>\n\t
                                3. none <\/li>\n<\/ol>\n<\/li>\n\t
                                4. \n
                                    \n\t
                                  1. [latex]11+7+2 = 20[\/latex]<\/li>\n\t
                                  2. P1 and P2 are critical<\/li>\n<\/ol>\n<\/li>\n\t
                                  3. Winning coalitions, with critical players underlined:
                                    [latex]\\left\\{\\underline{\\mathrm{P} 1}, \\underline{\\mathrm{P} 2}\\right\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}, \\mathrm{P} 4\\}[\/latex]<\/center>P1: [latex]6[\/latex] times, P2: [latex]2[\/latex] times, P3: [latex]2[\/latex] times, P4: [latex]0[\/latex] times. Total: [latex]10[\/latex] times\nPower: [latex]\\mathrm{P} 1: 6 \/ 10=60 \\%, \\mathrm{P} 2: 2 \/ 10=20 \\%, \\mathrm{P} 3: 2 \/ 10=20 \\%, \\mathrm{P} 4: 0 \/ 10=0 \\%[\/latex]\n <\/li>\n\t
                                  4. \n
                                      \n\t
                                    1. [latex]\\{\\underline{\\mathrm{P} 1}\\}\\{\\mathrm{P} 1, \\mathrm{P} 2\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 3, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}[\/latex] P1: [latex]100\\%[\/latex], P2: [latex]0\\%[\/latex], P3: [latex]0\\%[\/latex], P4: [latex]0\\%[\/latex]<\/li>\n\t
                                    2. [latex]\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 4}\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 3, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}[\/latex] P1: [latex]7\/10 = 70\\%[\/latex], P2: [latex]1\/10 = 10\\%[\/latex], P3: [latex]1\/10 = 10\\%[\/latex], P4: [latex]1\/10 = 10\\%[\/latex]<\/li>\n\t
                                    3. [latex]\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}[\/latex] P1: [latex]6\/10 = 60\\%[\/latex], P2: [latex]2\/10 = 20\\%[\/latex], P3: [latex]2\/10 = 20\\%[\/latex], P4: [latex]0\/10 = 0\\%[\/latex]<\/li>\n<\/ol>\n<\/li>\n\t
                                    4. [latex]\\mathrm{P} 3=5 . \\mathrm{P} 3+\\mathrm{P} 2=14 . \\mathrm{P} 3+\\mathrm{P} 2+\\mathrm{P} 1=27[\/latex], reaching quota. P1 is critical.<\/li>\n\t
                                    5. Sequential coalitions with pivotal player underlined:
                                      [latex]<\\mathrm{P} 1, \\underline{\\mathrm{P} 2}, \\mathrm{P} 3><\\mathrm{P} 1, \\underline{\\mathrm{P} 3}, \\mathrm{P} 2><\\mathrm{P} 2, \\underline{\\mathrm{P} 1}, \\mathrm{P} 3><\\mathrm{P} 2, \\underline{\\mathrm{P} 3}, \\mathrm{P} 1><\\mathrm{P} 3, \\underline{\\mathrm{P} 1}, \\mathrm{P} 2><\\mathrm{P} 3, \\underline{\\mathrm{P} 2}, \\mathrm{P} 1>[\/latex]<\/center>[latex]\\mathrm{P} 1: 2 \/ 6=33.3 \\%, \\mathrm{P} 2: 2 \/ 6=33.3 \\%, \\mathrm{P} 3: 2 \/ 6=33.3 \\%[\/latex]<\/li>\n\t
                                    6. \n
                                        \n\t
                                      1. [latex]6,7[\/latex]<\/li>\n\t
                                      2. [latex]8[\/latex], given P1 veto power<\/li>\n\t
                                      3. [latex]9[\/latex], given P1 and P2 veto power<\/li>\n<\/ol>\n<\/li>\n\t
                                      4. If adding a player to a coalition could cause it to reach quota, that player would also be critical in that coalition, which means they are not a dummy. So a dummy cannot be pivotal.<\/li>\n\t
                                      5. We know P2+P3 can\u2019t reach quota, or else P1 wouldn\u2019t have veto power. P1 can\u2019t reach quota alone. P1+P2 and P1+P3 must reach quota or else P2\/P3 would be dummy.\n\n\n
                                          \n\t
                                        1. [latex]\\left\\{\\underline{\\mathrm{P} 1}, \\underline{\\mathrm{P} 2}\\right\\}\\left\\{\\mathrm{P} 1, \\underline{\\mathrm{P} 3}\\right\\}\\left\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\right\\}[\/latex] P1: 3\/5, P2: 1\/5, P3: 1\/5<\/li>\n\t
                                        2. [latex]<\\mathrm{P} 1, \\underline{\\mathrm{P} 2}, \\mathrm{P} 3><\\mathrm{P} 1, \\underline{\\mathrm{P} 3}, \\mathrm{P} 2><\\mathrm{P} 2, \\underline{\\mathrm{P} 1}, \\mathrm{P} 3><\\mathrm{P} 2, \\mathrm{P} 3, \\underline{\\mathrm{P} 1}><\\mathrm{P} 3, \\underline{\\mathrm{P} 1}, \\mathrm{P} 2><\\mathrm{P} 3, \\mathrm{P} 2, \\underline{\\mathrm{P} 1}>[\/latex] <\/li>\n<\/ol>\n<\/li>\n\t
                                        3. [latex][4: 2,1,1,1][\/latex] is one of many possibilities<\/li>\n\t
                                        4. [latex][56: 30,30,20,20,10][\/latex]<\/li>\n\t
                                        5. [latex][54: 10,10,10,10,10,1,1,1,1,1,1,1,1,1,1][\/latex] is one of many possibilities<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n","rendered":"

                                          Voting Theory<\/h2>\n
                                            \n
                                          1. \n
                                              \n
                                            1. [latex]\\begin{array}{|l|l|l|l|l|l|}
                                              \\hline \\text { Number of voters } & 3 & 3 & 1 & 3 & 2 \\\\
                                              \\hline 1^{\\text {st }} \\text { choice } & \\mathrm{A} & \\mathrm{A} & \\mathrm{B} & \\mathrm{B} & \\mathrm{C} \\\\
                                              \\hline 2^{\\text {nd }} \\text { choice } & \\mathrm{B} & \\mathrm{C} & \\mathrm{A} & \\mathrm{C} & \\mathrm{A} \\\\
                                              \\hline 3^{\\text {rd }} \\text { choice } & \\mathrm{C} & \\mathrm{B} & \\mathrm{C} & \\mathrm{A} & \\mathrm{B} \\\\
                                              \\hline
                                              \\end{array}[\/latex]<\/li>\n
                                            2.  <\/li>\n
                                            3. \n
                                                \n
                                              1. [latex]9+19+11+8 = 47[\/latex]<\/li>\n
                                              2. [latex]24[\/latex] for majority; [latex]16[\/latex] for plurality (though a choice would need a minimum of [latex]17[\/latex] votes to actually win under the Plurality method)<\/li>\n
                                              3. Atlanta, with [latex]19[\/latex] first-choice votes<\/li>\n
                                              4. Atlanta [latex]94[\/latex], Buffalo [latex]111[\/latex], Chicago [latex]77[\/latex]. Winner: Buffalo<\/li>\n
                                              5. Chicago eliminated, [latex]11[\/latex] votes go to Buffalo. Winner: Buffalo<\/li>\n
                                              6. A vs B: B. A vs C: A. B vs C: B. B gets [latex]2[\/latex] pts, A [latex]1[\/latex] pt. Buffalo wins.<\/li>\n<\/ol>\n<\/li>\n
                                              7.  <\/li>\n
                                              8. \n
                                                  \n
                                                1. [latex]120+50+40+90+60+100 = 460[\/latex]<\/li>\n
                                                2. [latex]231[\/latex] for majority; [latex]116[\/latex] for plurality<\/li>\n
                                                3. A with [latex]150[\/latex] first choice votes<\/li>\n
                                                4. A [latex]1140[\/latex], B [latex]1060[\/latex], C [latex]1160[\/latex], D [latex]1240[\/latex]. Winner: D<\/li>\n
                                                5. B eliminated, votes to C. D eliminated, votes to A. Winner: A<\/li>\n
                                                6. A vs B: B. A vs C: A. A vs D: D. B vs C: C. B vs D: D. C vs D: C. A [latex]1[\/latex]pt, B [latex]1[\/latex]pt, C [latex]2[\/latex]pt, D [latex]2[\/latex]pt. Tie between C and D.<\/li>\n<\/ol>\n

                                                  Winner would probably be C since C was preferred over D<\/p>\n<\/li>\n

                                                7.  <\/li>\n
                                                8. \n
                                                    \n
                                                  1. [latex]33[\/latex]<\/li>\n
                                                  2. [latex]17[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                  3.  <\/li>\n
                                                  4. Yes, B<\/li>\n
                                                  5.  <\/li>\n
                                                  6. B, with [latex]17[\/latex] approvals<\/li>\n
                                                  7.  <\/li>\n
                                                  8. Independence of Irrelevant Alternatives Criterion<\/li>\n
                                                  9.  <\/li>\n
                                                  10. Condorcet Criterion<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                    Apportionment<\/h2>\n
                                                      \n
                                                    1. \n
                                                        \n
                                                      1. \n
                                                          \n
                                                        1. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]3[\/latex], Biology: [latex]1[\/latex]<\/li>\n
                                                        2. Math: [latex]7[\/latex], English: [latex]5[\/latex], Chemistry: [latex]2[\/latex], Biology: [latex]1[\/latex]<\/li>\n
                                                        3. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]3[\/latex], Biology: [latex]1[\/latex]<\/li>\n
                                                        4. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]3[\/latex], Biology: [latex]1[\/latex]<\/li>\n
                                                        5. Math: [latex]6[\/latex], English: [latex]5[\/latex], Chemistry: [latex]2[\/latex], Biology: [latex]2[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                        6.  <\/li>\n
                                                        7. \n
                                                            \n
                                                          1. Morning: 1, Midday: 5, Afternoon: 6, Evening: 8<\/li>\n
                                                          2. Morning: [latex]1[\/latex], Midday: [latex]4[\/latex], Afternoon: [latex]7[\/latex], Evening: [latex]8[\/latex]<\/li>\n
                                                          3. Morning: [latex]1[\/latex], Midday: [latex]5[\/latex], Afternoon: [latex]6[\/latex], Evening: [latex]8[\/latex]<\/li>\n
                                                          4. Morning: [latex]1[\/latex], Midday: [latex]5[\/latex], Afternoon: [latex]6[\/latex], Evening: [latex]8[\/latex]<\/li>\n
                                                          5. Morning: [latex]2[\/latex], Midday: [latex]5[\/latex], Afternoon: [latex]6[\/latex], Evening: [latex]7[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                          6.  <\/li>\n
                                                          7. \n
                                                              \n
                                                            1. Alice: [latex]18[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]4[\/latex]<\/li>\n
                                                            2. Alice: [latex]19[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]3[\/latex]<\/li>\n
                                                            3. Alice: [latex]19[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]3[\/latex]<\/li>\n
                                                            4. Alice: [latex]19[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]3[\/latex]<\/li>\n
                                                            5. Alice: [latex]18[\/latex], Ben: [latex]14[\/latex], Carlos: [latex]4[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                            6.  <\/li>\n
                                                            7. \n
                                                                \n
                                                              1. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n
                                                              2. A: [latex]41[\/latex], B: [latex]24[\/latex], C: [latex]14[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n
                                                              3. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n
                                                              4. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]30[\/latex], E: [latex]10[\/latex]<\/li>\n
                                                              5. A: [latex]40[\/latex], B: [latex]24[\/latex], C: [latex]15[\/latex], D: [latex]29[\/latex], E: [latex]11[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                              6.  <\/li>\n
                                                              7.  <\/li>\n
                                                              8.  <\/li>\n
                                                              9.  <\/li>\n
                                                              10.  <\/li>\n<\/ol>\n

                                                                Weighted Voting<\/h2>\n
                                                                  \n
                                                                1. \n
                                                                    \n
                                                                  1. [latex]9[\/latex] players<\/li>\n
                                                                  2. [latex]10+9+9+5+4+4+3+2+2 = 48[\/latex]<\/li>\n
                                                                  3. [latex]47[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                  4. \n
                                                                      \n
                                                                    1. [latex]9[\/latex], a majority of votes<\/li>\n
                                                                    2. [latex]17[\/latex], the total number of votes<\/li>\n
                                                                    3. [latex]12[\/latex], which is [latex]2\/3[\/latex] of [latex]17[\/latex], rounded up <\/li>\n<\/ol>\n<\/li>\n
                                                                    4. \n
                                                                        \n
                                                                      1. P1 is a dictator (can reach quota by themselves)<\/li>\n
                                                                      2. P1, since dictators also have veto power<\/li>\n
                                                                      3. P2, P3, P4<\/li>\n<\/ol>\n<\/li>\n
                                                                      4. \n
                                                                          \n
                                                                        1. none<\/li>\n
                                                                        2. P1<\/li>\n
                                                                        3. none <\/li>\n<\/ol>\n<\/li>\n
                                                                        4. \n
                                                                            \n
                                                                          1. [latex]11+7+2 = 20[\/latex]<\/li>\n
                                                                          2. P1 and P2 are critical<\/li>\n<\/ol>\n<\/li>\n
                                                                          3. Winning coalitions, with critical players underlined:\n
                                                                            [latex]\\left\\{\\underline{\\mathrm{P} 1}, \\underline{\\mathrm{P} 2}\\right\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}, \\mathrm{P} 4\\}[\/latex]<\/div>\n

                                                                            P1: [latex]6[\/latex] times, P2: [latex]2[\/latex] times, P3: [latex]2[\/latex] times, P4: [latex]0[\/latex] times. Total: [latex]10[\/latex] times
                                                                            \nPower: [latex]\\mathrm{P} 1: 6 \/ 10=60 \\%, \\mathrm{P} 2: 2 \/ 10=20 \\%, \\mathrm{P} 3: 2 \/ 10=20 \\%, \\mathrm{P} 4: 0 \/ 10=0 \\%[\/latex]
                                                                            \n <\/li>\n

                                                                          4. \n
                                                                              \n
                                                                            1. [latex]\\{\\underline{\\mathrm{P} 1}\\}\\{\\mathrm{P} 1, \\mathrm{P} 2\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 3, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}[\/latex] P1: [latex]100\\%[\/latex], P2: [latex]0\\%[\/latex], P3: [latex]0\\%[\/latex], P4: [latex]0\\%[\/latex]<\/li>\n
                                                                            2. [latex]\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 4}\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 3, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}[\/latex] P1: [latex]7\/10 = 70\\%[\/latex], P2: [latex]1\/10 = 10\\%[\/latex], P3: [latex]1\/10 = 10\\%[\/latex], P4: [latex]1\/10 = 10\\%[\/latex]<\/li>\n
                                                                            3. [latex]\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 2}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1, \\mathrm{P} 3}, \\mathrm{P} 4\\}\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3, \\mathrm{P} 4\\}[\/latex] P1: [latex]6\/10 = 60\\%[\/latex], P2: [latex]2\/10 = 20\\%[\/latex], P3: [latex]2\/10 = 20\\%[\/latex], P4: [latex]0\/10 = 0\\%[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                            4. [latex]\\mathrm{P} 3=5 . \\mathrm{P} 3+\\mathrm{P} 2=14 . \\mathrm{P} 3+\\mathrm{P} 2+\\mathrm{P} 1=27[\/latex], reaching quota. P1 is critical.<\/li>\n
                                                                            5. Sequential coalitions with pivotal player underlined:\n
                                                                              [latex]<\\mathrm{P} 1, \\underline{\\mathrm{P} 2}, \\mathrm{P} 3><\\mathrm{P} 1, \\underline{\\mathrm{P} 3}, \\mathrm{P} 2><\\mathrm{P} 2, \\underline{\\mathrm{P} 1}, \\mathrm{P} 3><\\mathrm{P} 2, \\underline{\\mathrm{P} 3}, \\mathrm{P} 1><\\mathrm{P} 3, \\underline{\\mathrm{P} 1}, \\mathrm{P} 2><\\mathrm{P} 3, \\underline{\\mathrm{P} 2}, \\mathrm{P} 1>[\/latex]<\/div>\n

                                                                              [latex]\\mathrm{P} 1: 2 \/ 6=33.3 \\%, \\mathrm{P} 2: 2 \/ 6=33.3 \\%, \\mathrm{P} 3: 2 \/ 6=33.3 \\%[\/latex]<\/li>\n

                                                                            6. \n
                                                                                \n
                                                                              1. [latex]6,7[\/latex]<\/li>\n
                                                                              2. [latex]8[\/latex], given P1 veto power<\/li>\n
                                                                              3. [latex]9[\/latex], given P1 and P2 veto power<\/li>\n<\/ol>\n<\/li>\n
                                                                              4. If adding a player to a coalition could cause it to reach quota, that player would also be critical in that coalition, which means they are not a dummy. So a dummy cannot be pivotal.<\/li>\n
                                                                              5. We know P2+P3 can\u2019t reach quota, or else P1 wouldn\u2019t have veto power. P1 can\u2019t reach quota alone. P1+P2 and P1+P3 must reach quota or else P2\/P3 would be dummy.\n
                                                                                  \n
                                                                                1. [latex]\\left\\{\\underline{\\mathrm{P} 1}, \\underline{\\mathrm{P} 2}\\right\\}\\left\\{\\mathrm{P} 1, \\underline{\\mathrm{P} 3}\\right\\}\\left\\{\\underline{\\mathrm{P} 1}, \\mathrm{P} 2, \\mathrm{P} 3\\right\\}[\/latex] P1: 3\/5, P2: 1\/5, P3: 1\/5<\/li>\n
                                                                                2. [latex]<\\mathrm{P} 1, \\underline{\\mathrm{P} 2}, \\mathrm{P} 3><\\mathrm{P} 1, \\underline{\\mathrm{P} 3}, \\mathrm{P} 2><\\mathrm{P} 2, \\underline{\\mathrm{P} 1}, \\mathrm{P} 3><\\mathrm{P} 2, \\mathrm{P} 3, \\underline{\\mathrm{P} 1}><\\mathrm{P} 3, \\underline{\\mathrm{P} 1}, \\mathrm{P} 2><\\mathrm{P} 3, \\mathrm{P} 2, \\underline{\\mathrm{P} 1}>[\/latex] <\/li>\n<\/ol>\n<\/li>\n
                                                                                3. [latex][4: 2,1,1,1][\/latex] is one of many possibilities<\/li>\n
                                                                                4. [latex][56: 30,30,20,20,10][\/latex]<\/li>\n
                                                                                5. [latex][54: 10,10,10,10,10,1,1,1,1,1,1,1,1,1,1][\/latex] is one of many possibilities<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n","protected":false},"author":6,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":3,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/66"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/66\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/66\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=66"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=66"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=66"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=66"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}