Minimal enforceability and indirect domination relations in the Shapley–Scarf economy

We consider the von Neumann-Morgenstern (vNM) stable sets based on a farsighted version of the weak domination relation for the barter model with indivisible goods. When defining a farsighted version of weak domination, it is necessary to define more precisely what coalitions can or cannot do internally, an issue which was not analyzed for other farsighted domination relations or for myopic domination relations. In this paper, we impose a minimality condition, where in each step of the deviation, only one trading cycle which is minimal with respect to set inclusion can deviate. With this restriction, we show first that the strict core, if nonempty, is the unique vNM stable set with respect to this farsighted weak domination. Moreover, under a mild assumption where each agent is not indifferent between the endowment and any other good, we show that a set consisting of a single allocation is a vNM stable set if and only if this allocation is a Pareto efficient allocation in the core.