Please, complete this by 11 pm possibly
Math 103:18 Post-class Assignment 6 Second Chancesdue 11pm Apr 4, 2021 There were three learning goals for the course that were associated with Post-class Assignment 6, and for which second chance problems are available, namely 16Power-LG2: In the setting of numerical weighted voting systems, you can list sequential coalitions, and determine when a player is pivotal. You can compute and interpret the Shapley-Shubik power distribution. 18Power-LG4: In the setting of non-numerical weighted voting systems, you can list sequential coalitions, and determine when a player is pivotal. You can compute and interpret the Shapley-Shubik power distribution. 20Power-LG6: You can list sequential coalitions and answer questions about the number of these. 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. 16Power-LG2: Shapley-Shubik power in the setting of numerical weighted voting systems. Consider the weighted voting system [14: 8, 6, 6, 4], which has players A, B, C, and D respectively. (a) List all sequential coalitions for this weighted voting system, and underline the pivotal player in each one. (b) Find the Shapley-Shubik power distribution for this weighted voting system. (c) How many times more actual power (as measured by Shapley-Shubik) does A have than D? 18Power-LG4: Shapley-Shubik power in the setting of non-numerical weighted voting systems Consider a simplified version of the UN Security Council in which there are seven players (council members) A, B, C, D, E, F, and G. In order for a resolution to pass, a majority of council members must vote in favor of it. Moreover, A and B have veto power while the others do not. (a) How many sequential coalitions are there for this weighted voting system? (b) In each of the following sequential coalitions, underline the pivotal player: < a,="" c,="" e,="" g,="" b,="" d,="" f=""> < g,="" f,="" e,="" d,="" c,="" b,="" a="">< d,="" e,="" f,="" a,="" b,="" c,="" g="">< f,="" a,="" c,="" e,="" g,="" b,="" d=""> (c) Give an example of a sequential coalition in which a player who doesn’t have veto power is pivotal, and underline the pivotal player. (d) What can you say about all the sequential coalitions in which a player who doesn’t have veto power is pivotal? 20Power-LG6: Time required to list sequential coalitions. This question is about how long it takes to do part of the computation of the Shapley-Shubik power distribution, when there are 18 players. The part of the computation we’re looking at is just listing all the sequential coalitions. (a) Suppose that your computer can list one sequential coalition per second. How many years would it take the computer to list all the sequential coalitions of 18 players? (b) Suppose that you also have access to a much faster computer, that can list one hundred million (108) sequential coalitions per second. How many years will it take to list all the sequential coalitions? Round your answer to the nearest year. (c) In light of the computation in part (b), if someone tells you that they need you to compute the Shapley-Shubik power distribution for a weighted voting system with 18 players, and they need the results next week, what would you say? Math 103:18 Post-class Assignment 7due 11pm Apr 1, 2021 1. Queen’s University has just bought 500 new desktop computers, and must apportion them among its four campus computer labs: · College Boulevard · Duggloose · Lovingstone · Schrubb [Any resemblance to an actual university and its campuses is purely coincidental.] The number of computers apportioned to each campus is based on the number of students living on that campus, given in the following table: campus # of students College Blvd 8123 Duggloose 4832 Lovingstone 3821 Schrubb 7224 (a) Which are the seats, which are the states, and what plays the role of population in this apportionment problem? You are not asked here for numbers; instead answer in terms of students, computers, and campuses. seats:number of Prostates:campus population: number of students (b) Find the standard divisor for this apportionment problem. What units is the standard divisor measured in? (c) Explain in one or two sentences what the standard divisor represents in this problem (not how to calculate it); use the language of computers, students, and campuses to explain. (d) Find the standard quota for each campus, to 3 decimal places. campus # of students standard quota College Blvd 8123 Duggloose 4832 Lovingstone 3821 Schrubb 7224 (e) What is the sum of the standard quotas? 2. This problem refers to the Queen’s University scenario described in problem 1 above. (a) Find the Hamilton apportionment of computers to campuses based on number of students on the campus (as presented in problem 1 above). campus # of students Standard quota Lower standard quota Decimal part Hamilton apportionment (actual number of computers given to each campus) College Blvd 8123 Duggloose 4832 Lovingstone 3821 Schrubb 7224 Total: (b) Based on the number of computers each campus receives in the Hamilton apportionment, compute for each campus the average number of computers per student. This need not be a whole number, since it is an average, and will be different for each campus. (c) Which campus has the most students per computer? Is this the best campus on which to be a student, or the worst? 3. This problem refers to the Queen’s University scenario described in problem 1 above. (a) Find the Jefferson apportionment of computers to campuses based on number of students on the campus. [Hint: One of the following modified divisors will work: 47.6, 47.8, 48, or 48.2. Before trying any of these, first think whether the modified divisor should be greater than the standard divisor or less than the standard divisor.] (b) Based on the number of computers each campus receives in the Jefferson apportionment, compute for each campus the average number of students per computer. This need not be a whole number, since it is an average, and will be different for each campus. (c) Which campus has the most students per computer? 4. This problem refers to the Queen’s University scenario described in problem 1 above. (a) When you compare the Hamilton and Jefferson apportionments, which one is better for the campus with the most students (the state with highest population), or is there no difference? (b) When you compare the Hamilton and Jefferson apportionments, which one is better for the campus with the fewest students (the state with lowest population), or is there no difference? 5. This problem pertains to the Alabama paradox. (a) Suppose that the Queen’s University described in problem 1 above acquires one more computer, for a total of 501, but the number of students on each campus remains the same. Find the Hamilton apportionment. (b) Does increasing the number of computers from 500 to 501 create an instance of the Alabama paradox? Explain how you can tell. 6. Based on a hypothetical apportionment given below, where 1000 seats are being apportioned: State Population Standard Quota Modified quota (when modified divisor equals 3.443) Apportionment under Jefferson’s method A 296 85.80 85.97 85 B 400 115.94 116.18 116 C 850 246.38 246.88 246 D 1904 551.88 553.01 553 Do any violations of the Quota Rule occur in this apportionment? Explain clearly how you can tell.