 Receiver-pessimal stable matching

For fixed strict preferences, the same output of the Gale-Shapley algorithm gives each receiver its worst partner obtainable in any stable matching, treating unmatched as last. "Worst" is restricted to stable possibilities: a receiver might still receive its favourite acceptable partner if every stable matching assigns it that partner. Reversing the proposing side gives receivers their best stable partners instead.

For a short proof, let $M$ be a proposer-optimal stable matching and $S$ any other stable matching. If a receiver $r$ preferred $M(r)=\ell$ to $S(r)$, then proposer optimality would make $\ell$ prefer $r=M(\ell )$ to $S(\ell )$, since the partners differ and preferences are strict. The pair $(\ell ,r)$ would block $S$. Therefore no receiver prefers its partner in $M$ to its partner in $S$. In the worked example, right-side ranks in $M_L$ are $2,2,3$ for $X,Y,Z$, whereas $M_R$ gives them $1,1,2$. Neither outcome is universally best for both sides.