math liberal arts question in image topic measuring power

Question not from Textbook SHWl. (3 points per question12 points total) In each of the following weighted voting systems, determine which players, if any, are (i) are dictators; (ii) have veto power; (iii) are dummy players. a. [19: 9, 7, S, 3, 1] b. [15: 16, s, 4, 1] c. [17: 13, 5, 2, 1] d. [25: 12. 8. 4. 21 HWI. Consider the weighted voting system [12: 9, 4, 3, 2]. a. (6 points) Write out all winning coalitions for this weighted voting system, and underline the critical playeds) in each one. b. (4 points) Find the Banzhaf power distribution of the weighted voting system [12: 9, 4, 3, 2]. A table is included below for your convenience. Banzhaf power index A c. (3 points) Identify any dictators, players with veto power, and dummies; justify your answer (briey)! d. (6 points) How \"fair" are the results of the Banzhaf power distribution? Use the share of total weight to determine \"expected" power and then compare it to the actual power according to Banzhaf to determine whether the amount of power each player receives is fair or not. Which player receives more? Less? Exactly what they were expecting? Player Banzhaf power index Share of total weight i..e \"ex . ected\" - uwer _._ _ _ __., "'l"D __ """'D .. -_ _ 'D'' _ HWZ. This problem looks again at the weighted voting system in WI, which was [12: 9, 4, 3, 2], and considers the effect of changing the quota but leaving all the weights unchanged. We will call the 4 players A, B, C, and D, respectively. a. (4 points) Find the Banzhaf power distribution of [11: 9, 4, 3, 2], i.e., the result of taking [12: 9, 4, 3, 2], and reducing the quota by l but changing nothing else. b. (2 points) In part a of this question, does any player change its dummy status from what it was in WI as a result of the change in the quota? Justify your answer (briey)! HWT. In a weighted voting system with four players the winning coalitions are the following: {P13233315}, {P151329 P3}. {P1, P2315}: {P1, 393,134}. {P233315}, {P1, P2}, {Pb P3}. {P1, P4} a. (8 points) Underline the critical player(s) in each winning coalition. The coalitions are provided again below for your convenience. {P1, P2, P1, P4}, {P1, P2. P1}. {P1, P2, P4}, {P1, P3, P4}. {P2, P3, P4}, {P1, P2}, {Pb P1}. {Pb P4} h. (4 points) Find the Banzhaf power distribution of the weighted voting system. c. (2 points) Determine which players, if any, are dictators, and explain briey how you can tell. d. (2 points) Determine which players, if any, have veto power, and explain briey how you can tell. e. (2 points) Determine which players, if any, are dummies, and explain briey how you can tell. weighted voting system [31:12, 8, 6, 5, 5, 5, 2]. a. (7 pointsl point per player) What would be the Banzhaf power distribution for this weighted voting system? b. (3 points) Identify if there are any players that may have veto power, or that may be dictators or dummies. c. (2 points) How close to the \"expected\" distribution of power is the Banzhaf power distribution? Use the table on the next page to help you organize the information if needed. {P15 P2: P3, P49 P59 P6) P7} {P15 P29 P39 P49 P5: P6} {P19 P29 P3: P4: P5: P7} {P19 P25 P39 P49 P6! P7} {P15 P2: P3, P59 P6: P7} {P19 P29 P49 P55 P6: P7} {P19 P3, P49 P59 P6: P7} {P2, P39 P49 P59 P6: P7} {P15 P29 P39 P4: P5} {P1, P2, P33 P49 P7} {P13 P29 P33 P6: P7} {P19 P29 P5: P6, P7} {P15 P2: P3, P4: P6} {P19 P23 P33 P59 P6} {P13 P29 P43 P5: P6} {P19 P3: P4: P5; P6} {P15 P29 P39 P5: P7} {P19 P23 P43 P59 P7} {P13 P29 P43 P6: P7} {P1, P29 P39 P4} {P13 P23 P3: P5} {P13 P29 P3: P6} \f