Prescriptive exchange rule given rational preferences and an initial allocation

Determine a prescriptive exchange or decision rule that specifies, for any initial allocation of a finite set of resources among finitely many agents with rational preferences that are asymmetric, transitive, and complete and with well-defined ownership (no overlapping claims and no unowned resources), what actions agents ought to take from that initial allocation.

Background

The paper formalizes the Coase theorem using a set-theoretic framework with finite agents and resources, well-defined ownership, and rational preferences (asymmetry, transitivity, completeness). It shows that certain common interpretations of the theorem fail under a bartering-style exchange constraint (double coincidence of wants), and introduces the notion of ideal exchanges to guarantee convergence to Pareto optimal outcomes.

In the concluding remarks, the author emphasizes that institutional conditions like well-defined ownership and frictionless transactions do not by themselves specify the prescriptive step of how agents should act from a given starting allocation. This leaves unresolved the normative question of what agents ought to do given their preferences and the initial distribution, motivating the need to articulate a formal decision or exchange rule that fills this gap.