Conflict-free and Pareto-optimal Allocations 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 partnership and decide how to share the surplus created. Thus, in this market, an outcome involves a matching and a vector of payoffs. Contrary to the two-sided assignment game, stable outcomes often fail to exist in the one-sided assignment game. We introduce the idea of conflict-free outcomes: they are individually rational outcomes where no matched agent can form a blocking pair with any other agent, neither matched nor unmatched. We propose the set of Pareto-optimal (PO) conflict-free outcomes, which is the set of the maximal elements of the set of conflict-free outcomes, as a natural solution concept for this game. We prove several properties of conflict-free outcomes and PO conflict-free outcomes. In particular, we show that each element in the set of PO conflict-free payoffs provides the maximum surplus out of the set of conflict-free payoffs, the set is always non-empty and it coincides with the core when the core is non-empty. We further support the set of PO conflict-free outcomes as a natural solution concept by suggesting an idealized partnership formation process that leads to these outcomes. In this process, partnerships are formed sequentially under the premise of optimal behavior and two agents only reach an agreement if both believe that more favorable terms will not be obtained in any future negotiations.