145hw1W10_answer

145hw1W10_answer - Econ 145: Problem Set I Suggested Answer...

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Suggested Answer February 3, 2010 1. Consider a general room assignment problem with N students and N rooms. Answer the following questions. (a) Suppose that the initial assignment of rooms is already efficient. Show that TTC algorithm leads to the same allocation of rooms, i.e. every student ends up with his or her initially assigned room. Definition: An allocation x X is efficient if there is no allocation y X such that (1) y j % j x j for all j N , and (2) y j ± j x j for some j . Answer: Shorter Proof Assume that the TTC algorithm leads to another allocation x 0 X (then show contradiction later) where x 0 6 = x . Since the TTC algorithm leads to a unique core allocation which is individually rational (see Lecture Note of Room Assignment p.22-26) we have x 0 i ² i x i for all i N (definition of individually rationality). However, since we assumed x 0 6 = x , we must have x 0 i ± i x i for some i N . This contradicts the efficiency of allocation x . Answer: Longer Proof Assume that the TTC algorithm leads to another allocation x 0 X . (then show contradiction later) Also assume that the initial allocation is ( h 1 ,h 2 ,...,h N - 1 ,h N ) We know that in each step of TTC algorithm, it creates (1) trading cycle(s), (2) self-trading cycle(s), or (3) both of (1) and (2) Since x 0 6 = x , in some step(s) of TTC algorithm, there must be at least one trading cycle (otherwise x 0 = x ). In such a trading cycle, a group of members ( k 1 ,k 2 ,...,k J ) creates a trading cycle h k 1 by k 1 h k 2 by k 2 ··· → by k J - 1 h k J by k J h k 1 . (trading cycle) Thus, compared to their initial rooms, each of group members ( k 1 ,k 2 ,...,k J ) get a strictly better room (i.e. x 0 i ± i x i for trading cycle students). Also, students who create self-trading cycles get their initial assignments (i.e. x 0 i i x i for self-trading cycle students). This contradicts the efficiency of allocation x . 1
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This note was uploaded on 04/25/2010 for the course ECON 145 taught by Professor Obara during the Winter '10 term at UCLA.

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145hw1W10_answer - Econ 145: Problem Set I Suggested Answer...

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