Answer
$m_{1}-w_{1}, m_{2}-w_{2}, m_{3}-w_{3} :$ Stable
$m_{1}-w_{1}, m_{2}-w_{3}, m_{3}-w_{2} :$ Not stable
$m_{1}-w_{2}, m_{2}-w_{1}, m_{3}-w_{3} :$ Not stable
$m_{1}-w_{2}, m_{2}-w_{3}, m_{3}-w_{1} :$ Not stable
$m_{1}-w_{3}, m_{2}-w_{1}, m_{3}-w_{2} :$ Stable
$m_{1}-w_{3}, m_{2}-w_{2}, m_{3}-w_{1} :$ Not Stable
Work Step by Step
The assignment is stable if and only if there is a no-man $m$ and no women $w$ such that $m$ prefer $w$ over his current partner or such that $w$ prefers $m$ over her current partner.
$m_{1}: w_{3} > w_{1} > w_{2} $
$m_{2}: w_{1} > w_{2} > w_{3} $
$m_{3}: w_{2} > w_{3} > w_{1} $
$w_{1}: m_{1} > m_{2} > m_{3} $
$w_{2}: m_{2} > m_{1} > m_{3} $
$w_{3}: m_{3} > m_{2} > m_{1} $
We will check out on $6$ of possible matching to see if there is a matching that is stable.
$First$ $Matching$
we note that all women are assigned their preferred partner and thus the matching is stable
$Second$ $Matching$
we note that the matching is not stable because $m_{2}$ prefer to over his current partner and $w_{2}$ prefer $m_{2}$ over her current partner
$Third$ $Matching$
we note that the matching is not stable because $m_{1}$ prefers $w_{1}$ over his current partner and $w_{1}$ prefers $m_{1}$ over her current partner
$Forth$ $Matching$
we note that the matching is not stable because $m_{1}$ prefer $w_{1}$ over his current partner and $w_{1}$ prefers $m_{1}$ over her current partner
$Fifth$ $Matching$
we note that all men are assigned their preferred partner and the matching is stable
$Sixth$ $Matching$
we note that matching is not stable because $m_{3}$ prefers $w_{3}$ over his current partner and $w_{3}$ prefers $m_{3}$ over her current partner
