Mergers. Sometimes in a weighted voting system two or more players decide to merge-that is to say, to combine their votes and always vote the same way. (Note that a merger is different from a coalition-coalitions are temporary, whereas mergers are permanent.) For example, if in the weighted voting system
a. Consider the weighted voting system
b. Consider the weighted voting system
c. Rework the problem in (b) for the weighted voting system
d. What are your conclusions from (a), (b), and (c)?
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Excursions in Modern Mathematics (9th Edition)
- Why is it impossible, with an odd number of voters, to have two distinct candidates win the same election using Condorcet's method.arrow_forwardA. Determine the number of winning coalitions in the weighted voting system [100: 50, 45, 45, 3]. 3 1 2 5 4 B. Which player(s) are critical in the coalition {P1, P2, P3}? P2 and P3 only P2 only P1 and P2 only There are no critical players P3 only P1 only P1, P2, and P3arrow_forwardA student organization at Penn State is voting to pass its budget. Susan has 6 votes as the President, Joe has 4 votes as the Vice-President, Carol has 2 votes as an assistant and Margo has 1 vote as a committee member. At least 8 of the 13 votes are needed to pass a resolution. How is the weighted voting system represented mathematically? Group of answer choices [8: 6, 2, 1, 1] [13: 8, 4, 3, 2, 1 ] [13: 6, 4, 2, 1] [8: 6, 4, 2, 1]arrow_forward
- A father and his three children decide to hold a vote to select an after dinner activity. They will either see a play or a movie. If the majority of the family agrees on a preferred activity, then they will do that activity. If two family members vote to see a play and the other two family members vote to watch a movie, then (since the father will pay for the activity), the father's vote will be used to break the tie. (a) How many winning coalitions are there in this situation? (b) Find one of the children's Banzhaf power index.arrow_forwardThe weighted voting systems for the voters A, B, C, ... are given in the form {a: w, w2, wy, wa, .., w,}. The weight of voter A is w1, the weight of voter B is w2, the weight of voter C is w3, and so on. A weighted voting system is given by (18: 9, 7, 4, 2, 1). (a) What is the quota? 18 (b) How many voters are in this system? x voters 41 (c) What is the weight of voter C? (d) What is the weight of the coalition {B, C}? 11 (e) Is {B, C, D, E} a winning coalition? Yes No (f) Which voters are critical voters in the coalition {A, B, D}? A A and B A and D B and D A, B, and D (g) How many coalitions can be formed? 6 x coalitions (h) How many coalitions consist of exactly three voters? 7 x coalitions O O Oarrow_forwardA group of fun-loving people have decided to play a practical joke on one of their friends, but they can't decide which friend, Alice (A), Betty (B), or Connie (C). Their preferences are: (ABC) (CBA) (BCA) 9 ,10, 4 Who wins the election using the Hare methodarrow_forward
- Use the Banzhaf power index to determine the power index for each player. Consider the voting system {18: 9, 6, 3, 3, 2}. Use the winning coalitions below to find each player's power index. . {P1, Р2, Р3} • {P1, P2, P4} . {P1, P2, Р3, Р4} . {P1, Р2, Р3, Р5} . {P1, P2, Р4, Р5} . {Р1, Р2, Р3, Р4, P5} P1 = 33.3%, P2 = 33.3%, P3 = 11.1%, P4 = 11.1%, P5=0% a P1 = 37.5%, P2 = 31.5%, P3 = 11.1%, P4 = 11.1%, P5 = 8.8% P1 = 33.3%, P2 = 33.3%, P3 = 11.1%, P4 = 11.1%, P5 = 3% d. P1 = 33.5%, P2 = 37.5%, P3 = 11.1%, P4 = 11.1%, P5 = 6.8% O O O Oarrow_forwardScientific Research Corporation has offices in Boston and Chicago. The number of employees at each office is shown in the following table. There are 22 vice presidents to be apportioned between the offices. Chicago 1220 Office Boston Employees 154 (a) Use the Hamilton method to find each office's apportionment of vice presidents. Boston Chicago (b) The corporation opens an additional office in San Francisco with 145 employees and decides to have a total of 24 vice presidents. If the vice presidents are reapportioned using the Hamilton method, will the new states paradox occur? Explain. O Yes. Boston lost a vice president and Chicago gained one. O Yes. Chicago lost a vice president and Boston gained one. O Yes. Both Boston and Chicago lost a vice president. O No. Both Boston and Chicago gained a vice president. O No. The number of vice presidents in Boston and Chicago remained the same.arrow_forwardConsider the weighted voting system (17: 15, 8, 4, 3}. (a) Fill in the table below. If a coalition does not win, use O for the number of critical players. Coalition (P1, P2, P3, P4} Weight Number of Critical Players {P1, P2, P3} {P1, P2, P4} {P1, P3, P4} {P2, P3, P4} (P1, P2) (P1, P3) (P1, P4} {P2, P3) {P2, P4} (P3, P4} (b) Find the Banzhaf score (critical count) for each player: P1: P2: P3: P4: (c) Find the Banzhaf index for each player (please enter your answer as a fraction): P1: P2: P3: P4:arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage