Consistency in one-sided assignment problems

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One-sided assignment problems combine important features of two well-known matching models. First, as in roommate problems, any two agents can be matched and second, as in two-sided assignment problems, the division of payoffs to agents is flexible as part of the solution. We take a similar approach to one-sided assignment problems as Sasaki (Int J Game Theory 24:373–397, 1995) for two-sided assignment problems, and we analyze various desirable properties of solutions including consistency and weak pairwise-monotonicity. We show that for the class of solvable one-sided assignment problems (i.e., the subset of one-sided assignment problems with a non-empty core), if a subsolution of the core satisfies [Pareto indifference and consistency] or [invariance with respect to unmatching dummy pairs, continuity, and consistency], then it coincides with the core (Theorems 1 and 2). However, we also prove that on the class of all one-sided assignment problems (solvable or not), no solution satisfies consistency and coincides with the core whenever the core is non-empty (Theorem 4). Finally, we comment on the difficulty in obtaining further positive results for the class of solvable one-sided assignment problems in line with Sasaki’s (1995) characterizations of the core for two-sided assignment problems.