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HW8Spring08answer - Homework 8(Total 20 points Due on April...

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Homework 8 (Total 20 points), Due on April 29, Tuesday . Q1 : There are 6 individuals. Each individual i = 1 , 2 , 3 , 4 , 5 , 6 is endowed with an indivisible object ω i . Their preferences over objects are listed as 1 2 3 4 5 6 ω 2 ω 6 ω 4 ω 6 ω 2 ω 5 ω 5 ω 3 ω 3 ω 5 ω 5 ω 3 ω 4 ω 1 ω 2 ω 1 ω 1 ω 6 ω 1 ω 5 ω 5 ω 4 ω 4 ω 1 ω 3 ω 2 ω 6 ω 3 ω 6 ω 4 ω 6 ω 4 ω 1 ω 2 ω 3 ω 2 Perform the (Gale’s) top-trading-cycle algorithm to find the core allocation. Answer : Round 1: 1 ω 2 , 2 ω 6 , 3 ω 4 , 4 ω 6 , 5 ω 2 , 6 ω 5 . There is one cycle 2 6 5 2. 2 gets ω 6 , 6 gets ω 5 , 5 gets ω 2 and leave. Round 2: 1 ω 4 , 3 ω 4 , 4 ω 1 . There is one cycle 1 4 1. 1 gets ω 4 , 4 gets ω 1 and leave. Round 3: 3 ω 3 . 3 gets ω 3 . Summing up, the core allocation is ( ω 4 , ω 6 , ω 3 , ω 1 , ω 2 , ω 5 ). Q2 : There are four workers and four firms. Workers are denoted by w 1 , w 2 , w 3 , w 4 respectively, firms are denoted by f 1 , f 2 , f 3 , f 4 respectively. Each worker can work for at most one firm, and each firm can hire at most one worker. Their preferences over the opposite side are as follows: w 1 w 2 w 3 w 4 f 1 f 2 f 3 f 4 f 2 f 1 f 1 f 2 w 1 w 3 w 1 w 3 f 1 f 3 f 2 f 3 w 2 w 4 w 4 w 1 f 3 f 2 f 4 f 4 w 3 w 1 w 2 w 4 f 4 f 4 f 3 f 1 w 4 w 2 w 3 w 2

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HW8Spring08answer - Homework 8(Total 20 points Due on April...

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