PracticeQuestionsIW10

# W 2 â m 1 w 3 â m 1 w 4 â m 1 w 1 m 3 â w 1 m 4

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w 2 Â m 1 w 3 Â m 1 w 4 Â m 1 w 1 , m 3 Â w 1 m 4 Â w 1 m 1 Â w 1 m 2 w 2 Â m 2 w 3 Â m 2 w 4 Â m 2 w 1 , m 4 Â w 2 m 1 Â w 2 m 2 Â w 2 m 3 w 2 Â m 3 w 3 Â m 3 w 4 Â m 3 w 1 , m 1 Â w 3 m 2 Â w 3 m 3 Â w 3 m 4 w 4 Â m 4 w 1 Â m 4 w 2 Â m 4 w 3 , m 2 Â w 4 m 3 Â w 4 m 4 Â w 4 m 1 (a) What is the W-optimal stable matching? De fi ne W-optimal sta- ble matching by words or using mathematical notations. (b) Find the W-optimal stable matching. Explain how you derived it. (c) Find the M-optimal stable matching. Explain how you derived it. 2

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(d) Suppose that Women-proposing DAA is used. Show that every man can misrepresent his preference (including the ranking of staying alone) to generate the M-optimal stable matching, assum- ing that everyone else reveals his or her true preference honestly. (e) Suppose that w 3 ’s preference is given by m 1 w 3 m 2 Â w 3 m 3 Â w 3 m 4 instead, i.e. w 3 is indi ff erent between m 1 and m 2 . We apply DAA by introducing an arti fi cial tie-breaking. Show that, given some tie-breaking rule, DAA leads to an allocation that is not W-optimal . 3
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