Abstract
Roth (Roth AE (1985) Conflict and coincidence of interest in job matching: Some new results and open questions. Math. Oper. Res. 10(3):379–389) claimed that (i) if each firm is allowed to select its most preferred subset of employees from those that assigned to it at two different stable matchings, then the choices result in a stable matching; and (ii) the set of stable matchings is a lattice under the partial order of the firms' common interests. Here, we provide counterexamples that show that these claims are incorrect, and we explain the flaws in Roth's reasoning.
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