Constrained-Optimal Tradewise-Stable Outcomes in the One-Sided Assignment Game: A Solution Concept Weaker than the Core

Abstract

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 tradewise-stable (t-stable) 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 constrained-optimal t-stable outcomes, which is the set of the maximal elements of the set of t-stable outcomes, as a natural solution concept for this game. We prove several properties of t-stable outcomes and constrained-optimal t-stable outcomes. In particular, we show that each element in the set of constrained- optimal t-stable payoffs provides the maximum surplus out of the set of t-stable payoffs, the set is always non-empty and it coincides with the core when the core is non-empty. The general principle of collective rationality on which our theory is based presupposes that a player only engages in cooperation that is optimal for him/her. That is, whatever the dynamics that underlie the pairwise interactions, any negotiation process in this environment should always arrive to an outcome where every trade is optimal (and stable) for the players involved.