Math in Politics: Get Stronger Answer Key

Voting Theory

    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]
    2.  
      1. [latex]9+19+11+8 = 47[/latex]
      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)
      3. Atlanta, with [latex]19[/latex] first-choice votes
      4. Atlanta [latex]94[/latex], Buffalo [latex]111[/latex], Chicago [latex]77[/latex]. Winner: Buffalo
      5. Chicago eliminated, [latex]11[/latex] votes go to Buffalo. Winner: Buffalo
      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.
    4.  
      1. [latex]120+50+40+90+60+100 = 460[/latex]
      2. [latex]231[/latex] for majority; [latex]116[/latex] for plurality
      3. A with [latex]150[/latex] first choice votes
      4. A [latex]1140[/latex], B [latex]1060[/latex], C [latex]1160[/latex], D [latex]1240[/latex]. Winner: D
      5. B eliminated, votes to C. D eliminated, votes to A. Winner: A
      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.

      Winner would probably be C since C was preferred over D

    6.  
      1. [latex]33[/latex]
      2. [latex]17[/latex]
    8.  
    9. Yes, B
    10.  
    11. B, with [latex]17[/latex] approvals
    12.  
    13. Independence of Irrelevant Alternatives Criterion
    14.  
    15. Condorcet Criterion

Apportionment

      1. Math: [latex]6[/latex], English: [latex]5[/latex], Chemistry: [latex]3[/latex], Biology: [latex]1[/latex]
      2. Math: [latex]7[/latex], English: [latex]5[/latex], Chemistry: [latex]2[/latex], Biology: [latex]1[/latex]
      3. Math: [latex]6[/latex], English: [latex]5[/latex], Chemistry: [latex]3[/latex], Biology: [latex]1[/latex]
      4. Math: [latex]6[/latex], English: [latex]5[/latex], Chemistry: [latex]3[/latex], Biology: [latex]1[/latex]
      5. Math: [latex]6[/latex], English: [latex]5[/latex], Chemistry: [latex]2[/latex], Biology: [latex]2[/latex]
    2.  
      1. Morning: 1, Midday: 5, Afternoon: 6, Evening: 8
      2. Morning: [latex]1[/latex], Midday: [latex]4[/latex], Afternoon: [latex]7[/latex], Evening: [latex]8[/latex]
      3. Morning: [latex]1[/latex], Midday: [latex]5[/latex], Afternoon: [latex]6[/latex], Evening: [latex]8[/latex]
      4. Morning: [latex]1[/latex], Midday: [latex]5[/latex], Afternoon: [latex]6[/latex], Evening: [latex]8[/latex]
      5. Morning: [latex]2[/latex], Midday: [latex]5[/latex], Afternoon: [latex]6[/latex], Evening: [latex]7[/latex]
    4.  
      1. Alice: [latex]18[/latex], Ben: [latex]14[/latex], Carlos: [latex]4[/latex]
      2. Alice: [latex]19[/latex], Ben: [latex]14[/latex], Carlos: [latex]3[/latex]
      3. Alice: [latex]19[/latex], Ben: [latex]14[/latex], Carlos: [latex]3[/latex]
      4. Alice: [latex]19[/latex], Ben: [latex]14[/latex], Carlos: [latex]3[/latex]
      5. Alice: [latex]18[/latex], Ben: [latex]14[/latex], Carlos: [latex]4[/latex]
    6.  
      1. A: [latex]40[/latex], B: [latex]24[/latex], C: [latex]15[/latex], D: [latex]30[/latex], E: [latex]10[/latex]
      2. A: [latex]41[/latex], B: [latex]24[/latex], C: [latex]14[/latex], D: [latex]30[/latex], E: [latex]10[/latex]
      3. A: [latex]40[/latex], B: [latex]24[/latex], C: [latex]15[/latex], D: [latex]30[/latex], E: [latex]10[/latex]
      4. A: [latex]40[/latex], B: [latex]24[/latex], C: [latex]15[/latex], D: [latex]30[/latex], E: [latex]10[/latex]
      5. A: [latex]40[/latex], B: [latex]24[/latex], C: [latex]15[/latex], D: [latex]29[/latex], E: [latex]11[/latex]
    8.  
    9.  
    10.  
    11.  
    12.  

    Weighted Voting

      1. [latex]9[/latex] players
      2. [latex]10+9+9+5+4+4+3+2+2 = 48[/latex]
      3. [latex]47[/latex]
      1. [latex]9[/latex], a majority of votes
      2. [latex]17[/latex], the total number of votes
      3. [latex]12[/latex], which is [latex]2/3[/latex] of [latex]17[/latex], rounded up 
      1. P1 is a dictator (can reach quota by themselves)
      2. P1, since dictators also have veto power
      3. P2, P3, P4
      1. none
      2. P1
      3. none 
      1. [latex]11+7+2 = 20[/latex]
      2. P1 and P2 are critical
    6. 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]

      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
      Power: [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]
       

      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]
      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]
      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]
    8. [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.
    9. 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]

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

      1. [latex]6,7[/latex]
      2. [latex]8[/latex], given P1 veto power
      3. [latex]9[/latex], given P1 and P2 veto power
    11. 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.
    12. We know P2+P3 can’t reach quota, or else P1 wouldn’t have veto power. P1 can’t reach quota alone. P1+P2 and P1+P3 must reach quota or else P2/P3 would be dummy.
      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
      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] 
    13. [latex][4: 2,1,1,1][/latex] is one of many possibilities
    14. [latex][56: 30,30,20,20,10][/latex]
    15. [latex][54: 10,10,10,10,10,1,1,1,1,1,1,1,1,1,1][/latex] is one of many possibilities