Question 2 Take any marriage problem in which agents on one side, say, women, have the same preferences over the men. Without loss of generality, assume that the preference of

any woman w is Pwтi, т,, тѣ, .., Mz.%3D For simplicity we assumethat men and women are mutually acceptable. Wedo not needto make any assumption about the number of men (z) and number of women (y). The resultwill hold regardless of whether we have z < y, z = y or z > y. The objective of this exercise is to show that in this case there is a unique stable matching. 1. Let w be m's most preferred woman. Show that if a matching u is such that µ(wi) /=mi then u cannot be stable. 2. Consider now m.. Let wz be m's most preferred woman in W\{wi}. Show that at any stable matching m is necessarily matched to wr. 3. Consider now m:. Using the previous question, to which woman will m: be matched at any stable matching? 4. Make a short (3-4 lines max) argument to deduce what happens with m., ms, ...