One situation where there is exactly one matching that is stable is when all the men and women have identical preferences. In particular, if we denote the set of n men as V := \{m_1,\ldots,m_n\} and the n women as W := \{w_1,\ldots,w_n\}, and further:

- every man has the preference w_1 \succ w_2 \succ \cdots w_n and
- every woman has the preference m_1 \succ m_2 \succ \cdots m_n;

then then only stable matching is (w_1,m_1),\cdots,(w_n,m_n).

To see this, suppose a stable matching M matches w_i to m_j where i \neq j, and let i be the smallest index for which this happens (i.e, for all \ell < i, M matches w_\ell with m_\ell). Then: it must be that j > i (since all men m_j with j < i are already matched to w_j). However, this implies that (w_i,m_i) will form a blocking pair (note that m_i is also matched to some w_t with t > i), contradicting our assumption that M is stable.

Food for thought: are there other examples?

Evaluation remark: for full credit, it suffices that the answer describes a valid example, even if there is no justification.