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chap 2 questions

chap 2 questions - player in any winning coalition d...

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Chap 2 questions 18 c B1 60% B2 20% B3 20% B4 0% E B1 33 1/3 % b2 33 1/3% b3 33 1/3% b4 0% 20. a. {p1 p2} {p1 P2 p3} {p1 P2 P4} {p1 p2 p3 p4} b. B1 50% B2 50% B3 0% B4 0% 24. a. {P1 p2} {p1 p3} {p1 p4} {p1 p2 p3} {p1 p2 p4} {p1 p3 p4} {p2 p3 p4 } {p1 p2 p3 p4} b. B1 66 2/3% B2 11 1/9% B3 11 1/9% Bd 11 1/9% 28. d σ 1 33 1/3% σ 2 33 1/3% σ 3 33 1/3% σ 4 0% 34. σ a 50% σ b 16 2/3% σ c 16 2/3% σ d 16 2/3% 36

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a 479,001,600 b 1.307674 x 10^12 c 2.432902 x 10^18 d 2,432,902,008 sec or 4,022.66 weeks chap 2 supplemental problem 1. a. {p1 P2 } {p1 p3 } {p1 p2 p3} {p1 p2 p4} {p1 p3 p4} {p1 p2 p3 p4} b. Banzhaf power index (as percentage, rounded to 2 decimal places) Share of population (as percentage, rounded to 2 decimal places) Difference P 1 60% 16/29 = 55.17% 4.83% P 2 20% 8/29 = 27.59% 7.59% P 3 20% 4/29 = 13.79% 6.21% P 4 0% 1/29 = 3.45% 3.45% c. no dictator. P1 has veto power. P4 is a dummy. The player who has veto power is identified as someone who is a critical player in every winning coalition. The player who is the dummy is identified as someone who is never a critical
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Unformatted text preview: player in any winning coalition. d. district 2 – 7.59% unaccounted for when using Banzhaf vs actual population %. e. {p1 p2 } {p1 p2 p3} {p1 p2 p4} {p1 p3 p4 } {p1 p2 p3 p4} Banzhaf power index (as percentage, rounded to 2 decimal places) Share of population (as percentage, rounded to 2 decimal places) Difference P 1 50% 16/29 = 55.17% 5.17% P 2 30% 8/29 = 27.59% 2.41% P 3 10% 4/29 = 13.79% 3.79% P 4 10% 1/29 = 3.45% 6.55% f. g. no dictator. p1 has veto power – b/c is critical player in every winning coalition no dummy. h. district 1 has 5.17% unaccounted for using the banzhaf method in comparison to its actual population. i. The higher quota (21) is fairer because it results in the lowest maximum difference between a district’s banzhaf power index percentage and its actual population percentage. Also, there is no dummy in the higher quota election....
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