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Unformatted text preview: 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 efficient? Explain why. Answer: We know that the TTC algorithm with an efficient 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 ,h 3 ,h 4 ). Step 1: We have initial assignment ( h 1 ,h 2 ,h 3 ,h 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 find a trading cycle h 2 by 2 h 2 (self trading cycle) So the TTC algorithm assigns h 1 ,h assigned 2 ,h 3 ,h 4 . Step 2: We have unassigned rooms { h 1 ,h 3 ,h 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 Then, we can find a trading cycle h 1 by 1 h 4 by 4 h 3 by 3 h 1 1 So, the TTC algorithm assigns h assigned 4 ,h assigned 2 ,h assigned 1 ,h assigned 3 . Clearly, the above allocation is different from initial allocation ( h 1 ,h 2 ,h 3 ,h 3 ). Thus, initial allocation is not efficient. (b) Find all efficient allocations. Explain how you found all the efficient allocations. Answer: As we discussed Problem Set I Problem 2 (a), there are two methods to find all efficient 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 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 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 ,h 1 ,h 3 ,h 4 ) I 2 1 3 4 ( h 4 ,h 2 ,h 1 ,h 3 ) V 1 2 4 3 ( h 2 ,h 1 ,h 4 ,h 3 ) II 2 1 4 3 ( h 4 ,h 2 ,h 1 ,h 3 ) V 1 3 2 4 ( h 2 ,h 3 ,h 1 ,h 4 ) III 2 3 1 4 ( h 4 ,h 2 ,h 1 ,h 3 ) V 1 3 4 2 ( h 2 ,h 4 ,h 1 ,h 3 ) IV 2 3 4 1 ( h 4 ,h 2 ,h 1 ,h 3 ) V 1 4 2 3 ( h 2 ,h 1 ,h 4 ,h 3 ) II 2 4 1 3 ( h 4 ,h 2 ,h 1 ,h 3 ) V 1 4 3 2 ( h 2 ,h 4 ,h 1 ,h 3 ) IV 2 4 3 1 ( h 4 ,h 2 ,h 1 ,h 3 ) V Order of Dictatorships Allocation Type Order of Dictatorships Allocation Type 3 1 2 4 ( h 2 ,h 3 ,h 1 ,h 4 ) III 4 1 2 3 ( h 2 ,h 1 ,h 4 ,h 3 ) II 3 1 4 2 ( h 2 ,h 4 ,h 1 ,h 3 ) IV 4 1 3 2 ( h 2 ,h 4 ,h 1 ,h 3 ) IV 3 2 1 4 ( h 4 ,h 2 ,h...
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 Spring '08
 Serra

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