Done by same expert
New assignment 1. Consider the weighted voting system [15: 8, 5, 5, 2], which has players A, B, C, and D respectively. List all winning coalitions for this weighted voting system, and underline the critical players in each one. 2. In the previous question, who has veto power (if anyone), and how can you tell? · Who is a dictator (if anyone), and how can you tell? · Who is a dummy (if anyone), and how can you tell? 3. A weighted voting system with 4 players has the following complete list of winning coalitions: {W, X, Y, Z} {W, Y, Z} {W, X, Y} {W, X, Z} Fill in the table with each player's percentage of the actual power according to Banzhaf. Banzhaf. Banzhaf power index W X Y Z 4. Write out all the sequential coalitions for the weighted voting system [17: 9, 8, 7], with players A, B, and C, and underline all pivotal players. 5. Part 1: Suppose you need to find the Shapley-Shubik power distribution for 15 players. If a slow computer can list 1 sequential coalition per second, how many years would it take to list all the sequential coalitions for the 15 players? Round to the nearest year, and show your work. Part 2: If a faster computer can list 10,000 sequential coalitions per second, how many years would it take to list all the sequential coalitions for the 15 players? Round to the nearest year. 6. A hospital uses Hamilton's method to apportion 550 nurses to four shifts based on the number of patients on each shift, as follows: shift patients nurses Morning 1762 179 Afternoon 2047 209 Evening 1240 126 Late night 351 36 Total: 5400 550 If one more nurse is added, and Hamilton's method is used again, complete the following table to find the new apportionment. Round the standard quotas to 3 decimal places. Shift # of patients Standard quota (to 3 decimal places) Hamilton apportionment Morning 1762 Afternoon 2047 Evening 1240 Late night 351 Total: 5400 7. Based on your answer to the previous question, about the apportionment of one extra nurse to the hospital, does the Alabama paradox occur? Explain clearly how you can tell. 8. Three siblings, Henrik, Kendrick, and Schmendrick, inherit a house, but each one has a different idea of its value, as follows: Henrik: $450,000, Kendrick: $400,000 Schmendrick: $360,000 Is the following a fair division? Explain clearly how you can tell. Kendrick gets the house, pays Henrik $150,000, and pays Schmendrick $130,000. 9. A hospital wants to use Jefferson's method to apportion 350 nurses to four shifts based on the number of patients on each shift. If they use a modified divisor of 15.4, find the modified quotas, rounded to 3 decimal places: shift patients modified quotas Morning 1742 Afternoon 2067 Evening 1220 Late night 371 Total: 10. In the previous question, do you obtain a Jefferson apportionment? Explain clearly how you can tell, i.e. show that you understand how to take the modified quotas and see whether you get a Jefferson apportionment. 11. This problem considers how 34 Congressional seats would be apportioned to the 4 states NY, PA, NJ, and CT, based on the populations in the table below, and whether the standard divisor and standard quotas will result in a Webster apportionment. state population standard quota NY 19,300,000 14.779 PA 12,700,000 9.725 NJ 8,800,000 6.739 CT 3,600,000 2.757 If these quotas work, what is the Webster apportionment? And if they don't work, should the modified divisor be greater than or less than the standard divisor? 12. When Jefferson's method is used to apportion 39 buses to 4 bus routes based on the number of riders per day, the following are the results: riders standard quota modified quota Jefferson apportionment A 3,100 27.603 29.245 29 buses B 550 4.897 5.189 5 buses C 420 3.740 3.962 3 buses D 310 2.760 2.925 2 buses total 4,380 39 39 buses Does a Quota Rule violation occur here? Explain clearly how you can tell. 13. This problem considers how 7 Congressional seats would be apportioned to the 3 states PA, NJ, and CT, based on the populations in the table below, and whether the standard divisor and standard quotas will result in a Huntington-Hill apportionment. Fill in each of the Huntington-Hill rounding cutoffs (to 3 decimal places), and what the corresponding number of seats will be for that state. state population standard quotas Huntington-Hill rounding cutoff (to 3 decimal places) Huntington Hill apportionment? PA 12,800,000 3.542 NJ 8,900,000 2.462 CT 3,600,000 0.996 Do these rounded quotas result in a Huntington-Hill apportionment of the 7 seats? (Enter yes or no) Math 103:18 Post-class Assignment 8 Second chances There were three learning goals for the course that were associated with Assignment 8, and for which rewrite problems are available, namely 24Apportionment-LG3: Given an apportionment problem, you can perform all the steps in computing the Webster apportionment. 27Apportionment -LG6: Given an apportionment problem, you can perform all the steps in computing the Huntington-Hill apportionment. 28Fair Division-LG1: You can explain the meaning of our technical definition of fairness (proportionality), and can determine in particular cases whether a division is fair in this sense. For each of the learning goals listed above, there is a new problem below which is associated with that learning goal. You can pick one learning goal on which you received a score less than Mastered, and submit a solution to the new problem below that is associated with that learning goal. If you submit more than one solution, the first one will be graded, unless you make clear which one you wish to be graded. In order to show mastery level understanding, you must show your work, i.e. you must show enough steps of your thought process to make it clear that you have understood the process of obtaining your answer. 24Apportionment-LG3: Webster’s method. This problem considers how 75 Congressional seats would be apportioned to the 4 states NY, PA, NJ, and CT, based on the populations in the table in part (a). (a) Do the standard divisor and standard quotas result in a Webster apportionment? Show your work, and explain briefly how you can tell whether you obtain a Webster apportionment. State population Standard quota Webster apportionment?? NY 19,400,000 PA 12,819,000 NJ 8,900,000 CT 3,600,000 Total: 44,719,000 (b) Does a modified quota of 600,000 result in a Webster apportionment? Explain briefly how you can tell (and be sure to compute the modified quotas). State population Modified quotas using 600,000 as divisor Webster apportionment?? NY 19,400,000 PA 12,819,000 NJ 8,900,000 CT 3,600,000 Total: 44,719,000 27Apportionment-LG6: The Huntington-Hill method. This problem considers how 75 Congressional seats would be apportioned to the 4 states NY, PA, NJ, and CT, based on the populations in the table in part (a). (c) Do the standard divisor and standard quotas result in a Huntington-Hill apportionment? Show your work, and explain briefly how you can tell whether you obtain a Huntington-Hill apportionment. State population Standard quota Huntington-Hill rounding cutoff Huntington-Hill apportionment?? NY 19,400,000 PA 12,819,000 NJ 8,900,000 CT 3,600,000 Total: 44,719,000 (d) Does a modified quota of 596,500 result in a Huntington-Hill apportionment? Explain briefly how you can tell (and be sure to compute the modified quotas). State population Modified quotas using 596,500 as divisor Huntington-Hill rounding cutoff Huntington-Hill apportionment?? NY 19,400,000 PA 12,819,000 NJ 8,900,000 CT 3,600,000 Total: 44,719,000 28Fair Division-LG1: The notion of fairness Three roommates jointly own a desktop computer, but now they are all moving out. Each roommate values the computer differently, as follows: valuation of the computer Chandler $3000 Joey $1500 Ross $2100 (a) What is each roommate’s minimum fair share amount? (b) Is the following a fair division? Explain clearly how you can tell. Chandler gets the computer, and pays $900 each to Joey and Ross. (c) Is the following a fair division? Explain clearly how you can tell. Ross gets the computer, and pays $1000 each to Chandler and Joey.