PracticeQuestionsIAnswerW10

PracticeQuestionsIAnswerW10 - Econ 145: Practice Questions...

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Econ 145: Practice Questions I Suggested Answer February 8, 2010 1. Consider a room assignment problem with 4 students and 4 rooms { h 1 ,...,h 4 } . Room h i is initially assigned to student i. Suppose that the preferences of the students are as follows h 2  1 h 4  1 h 1  1 h 3 h 2  2 h 1  2 h 3  2 h 4 h 1  3 h 3  3 h 4  3 h 2 h 3  4 h 4  4 h 1  4 h 2 where  i is student i ’s (strict) preference over the rooms. Answer following questions. (a) Is the initial allocation e cient? Explain why. Answer: We know that the TTC algorithm with an e cient initial allocation leads the same allocation (See Problem Set I Problem 1 (a)). So we here simply apply the TTC algorithm, then compare the resulting allocation with the initial allocation ( h 1 ,h 2 3 4 ). Step 1: We have initial assignment ( h 1 2 3 4 )/ Each student wants to obtain h 1 by 1 h 2 h 2 by 2 h 2 h 3 by 3 h 1 h 4 by 4 h 3 Then, we can f nd a trading cycle h 2 by 2 h 2 (self trading cycle) So the TTC algorithm assigns ³ h 1 assigned 2 3 4 ´ . Step 2: We have unassigned rooms { h 1 3 4 } . Each remaining student (student 1, 3, and 4) wants to obtain h 1 by 1 h 4 h 3 by 3 h 1 h 4 by 4 h 3 1
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Then, we can f nd a trading cycle h 1 by 1 h 4 by 4 h 3 by 3 h 1 · So, the TTC algorithm assigns ³ h assigned 4 ,h assigned 2 assigned 1 assigned 3 ´ . Clearly, the above allocation is di f erent from initial allocation ( h 1 2 3 3 ). Thus, initial allocation is not e cient. (b) Find all e cient allocations. Explain how you found all the e cient allocations. Answer: As we discussed Problem Set I Problem 2 (a), there are two methods to f nd all e cient allocations Method I: TTC Algorithm with All Possible Initial Assignments Method II: Serial Dictatorship with All Possible Orders We here apply Method II. Rewriting preferences h 2 Â 1 h 4 Â 1 h 1 Â 1 h 3 h 2 Â 2 h 1 Â 2 h 3 Â 2 h 4 h 1 Â 3 h 3 Â 3 h 4 Â 3 h 2 h 3 Â 4 h 4 Â 4 h 1 Â 4 h 2 We generate all possible order of serial dictatorships. There are 4 students and 4 × 3 × 2 × 1=24 ways of orders. Order of Dictatorships Allocation Type Order of Dictatorships Allocation Type 1 2 3 4( h 2 1 3 4 )I 2 1 3 h 4 2 1 3 )V 1 2 4 3( h 2 1 4 3 I 2 1 4 h 4 2 1 3 1 3 2 h 2 3 1 4 I I 2 3 1 h 4 2 1 3 1 3 4 2( h 2 4 1 3 V 2 3 4 1( h 4 2 1 3 1 4 2 h 2 1 4 3 I 2 4 1 h 4 2 1 3 1 4 3 h 2 4 1 3 V 2 4 3 h 4 2 1 3 Order of Dictatorships Allocation Type Order of Dictatorships Allocation Type 3 1 2 h 2 3 1 4 I I 4 1 2 h 2 1 4 3 I 3 1 4 h 2 4 1 3 V 4 1 3 h 2 4 1 3 V 3 2 1 h 4 2 1 3 4 2 1 h 4 2 1 3 3 2 4 h 4 2 1 3 4 2 3 h 4 2 1 3 3 4 1 h 2 4 1 3 V 4 3 1 h 2 4 1 3 V 3 4 2 h 4 2 1 3 4 3 2 h 4 2 1 3 Thus, there are f ve e cient allocations ( h 2 1 3 4 ) | {z } type I , ( h 2 1
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PracticeQuestionsIAnswerW10 - Econ 145: Practice Questions...

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