Homework 2, A little math.

Short Description:

This assignment is worth 40 points.

Goals

• Solved some problems relating to the Gale Shapley Algorithm

Formal Description

1. Given the following data, using the techniques from class
   • M = {A,B,C,D}, W = {1,2,3,4}
   • A: 3, 4, 1, 2
   • B: 4, 1, 2, 3
   • C: 1, 2, 3, 4
   • D: 4, 3, 2, 1
   • 1: A, B, C, D
   • 2: A, B, C, D
   • 3: B, C, A, D
   • 4: C, A, B, D
   1. Trace the algorithm for this data when the men propose.
   2. Trace the algorithm for this data when the women propose.
2. Decide whether you think the following statements are true or false. If true, give a short proof, otherwise give a counter example.
   1. In every instance of the stable matching problem, there is a stable matching containing a pair $(m,w)$ such that $m$ is ranked first on the preference list of $w$ and $w$ is ranked first on the preference list of $m$.
   2. Consider an instance of the stable matching problem in which there exists a man $m$ and a woman $w$ such that $m$ is ranked first on the preference list of $w$ and $w$ is ranked first on the preference list of $m$. Then in every stable matching $S$ for the instance, the pair $(m,w)$ belongs to $S$.

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