Constrained-optimal tradewise-stable outcomes in the one-sided assignment game: a solution concept weaker than the core

In the one-sided assignment game, any two agents can form a trade; they can decide to form a partnership and agree on how to share the surplus created. Contrary to the two-sided assignment game, stable outcomes often fail to exist in the one-sided assignment game. Hence the core, which coincides with the set of stable payoffs, may be empty. We introduce the idea of tradewise-stable (t-stable) outcomes: they are individually rational outcomes where all trades are stable; that is, no matched agent can form a blocking pair with any other agent, neither matched nor unmatched. We propose the set of constrained-optimal (optimal) t-stable outcomes, the set of the maximal elements of the set of t-stable outcomes, as a natural solution concept for this game. We prove that this set is non-empty, it coincides with the set of stable outcomes when the core is non-empty, and it satisfies similar properties to the set of stable outcomes even when the core is empty. We propose a partnership formation process that starts with the outcome where every player stands alone, goes through steps where the set of active players expands, always forming t-stable outcomes, and ends in an (in any) optimal t-stable outcome. Finally, we also use the new concept to establish conditions under which the core is non-empty.