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Unformatted text preview: Econ 145: Problem Set II Suggested Answer February 16, 2010 1. Kidney Exchange Suppose that there are 5 patient-donor pairs ((1 ,d 1 ) ,..., (5 ,d 5 )) . Let k i be the kidney of donor i. Each patient has the following preference over { k ,...,k 5 } ( k is the option to be given some priority in the waiting list). k 5 1 k 2 1 k 3 1 k 1 k 1 1 k 4 k 1 2 k 5 2 k 4 2 k 2 k 2 2 k 3 k 4 3 k 1 3 k 3 k 2 3 k 5 3 k 3 k 1 4 k 4 4 k 2 4 k 3 4 k 4 k 5 k 4 5 k 2 5 k 3 5 k 5 5 k 5 k 1 (a) Apply TTCC (Top-Trading-Cycle-and-Chain) algorithm. What is the final allocation? Answer: Apply the TTCC algorithm (see Lecture Note of Kidney Exchange p.14-15) Step 1: We have available kidneys ( k 1 ,k 2 ,k 3 ,k 4 ,k 5 ) and waiting list k . Each patient wants to obtain k 1 by 1 k 5 k 2 by 2 k 1 k 3 by 3 k 4 k 4 by 4 k 1 k 5 by 5 k 4 Then, we can Find a trading cycle k 1 by 1 k 5 by 5 k 4 by 4 k 1 (trading chain) The TTCC algorithm assigns k assigned 5 ,k 2 ,k 3 ,k assigned 1 ,k assigned 4 . 1 Step 2: See footnote 1 We have available kidneys ( k 2 ,k 3 ) and waiting list k . Each remaining patient wants to obtain k 2 by 2 k (trading chain) k 3 by 3 k (trading chain) Assuming that a patient who has lower index has a priority . Then, the TTCC allocates, k assigned 5 ,k assigned ,k 3 ,k assigned 1 ,k assigned 4 ....
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