We consider one-to-one, one-sided matching (roommate) problems in which agents can either be matched as pairs or remain single. We introduce a so-called bi-choice graph for each pair of stable matchings and characterize its structure. Exploiting this structure we obtain as a corollary the "lone wolf" theorem and a decomposability result. The latter result together with transitivity of blocking leads to an elementary proof of the so-called stable median matching theorem, showing how the often incompatible concepts of stability (represented by the political economist Adam Smith) and fairness (represented by the political philosopher John Rawls) can be reconciled for roommate problems. Finally, we extend our results to two-sided matching problems.
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