*(Guest post by Patrick Lapinski)*

Lecture 9

In this lecture, we continued the proof we were doing in the last class. Specifically, we proved Lemma 2: the matching produced by the GS algorithm has no instability. We had already proved Lemma 1: GS algorithm outputs a perfect matching.

We first made two observations:

Observation 1 – Once m is engaged, he keeps getting engaged to “better” women

Observation 2 – If w proposed to m’, and then to m (or never to m), then she prefers m’ to m

We then went on to prove Lemma 2 by contradiction.

We made an assumption:

there is an instability (m,w’)

m prefers w’ to w

w’ prefers m to m’

This is the matching we were working with:

m——-w

m’——w’

There were then two possible cases depending on whether or not w’ had proposed to m.

Case 1: w’ never proposed to m

-w’ prefers m’ to m (by Obs 2)

-w’ prefers m to m’ (by assumption)

These two points cause a contradiction.

Case 2: w’ had proposed to m

Case 2.1: m had accepted w’ proposal

m is now engaged to w (based on the assumed matching)

thus m prefers w to w’ (by Obs 1)

However, we had assumed that m preferred w’ to w, and now have another contradiction

Case 2.2: m had rejected w’ proposal

m was engaged to w” and prefers w” to w’ (note: w” may be the same as w)(by algorithm)

m is finally engaged to w (prefers w to w”)(by Obs 1)

m prefers w to w’

Same as with case 2.1, this causes a contradiction.

So since all possible cases of the assumed opposite of Lemma 2 cause contradictions, Lemma 2 must be true.

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