Minimal Enforceability and Indirect Domination Relations in the Shapley-Scarf Economy

Abstract

We consider the von Neumann-Morgenstern (vNM) stable sets based on a farsighted version of weak domination relation for the barter model with indivisible goods of Shapley and Scarf (1974). Recent research on farsighted stable sets has focused on coalitional sovereignty, a term used by Ray and Vohra (2015) regarding what coalitions can do to other coalitions, and they note the importance of this issue and how they may affect the results in characteristic function form games. However, defining a farsighted version of indirect weak domination introduces an additional issue of what coalitions can or cannot do from the inside, an issue which needed not be analyzed for the stronger versions of indirect domination nor for myopic domination relations in the literature. We first impose a minimality condition on the coalitions that can deviate in just one step, as coalitions in this model represent trading cycles, where such a minimality condition is often imposed. Under the domain of preferences defined by Klaus, Klijn, and Walzl (2010), each Pareto efficient allocation in the core and its Pareto indifferent allocations constitute a vNM stable set defined by a farsighted weak domination that respects the minimal enforceability condition, and those are the only essentially singleton such vNM stable sets.