Rights Transitions in Dynamic Assignment: Safety, Investment, and Future Exchange

Dynamic assignment rules must choose whether a current allocation becomes the endowment of a later market. We study this rights transition in finite Shapley-Scarf economies with entry. A rights state is support-safe when it produces the same terminal allocation under top trading cycles as original rights for every supported extension and entrant report. Safety is characterized by a finite basis of elementary entrant probes. For transitions that vest current trading cycles, safety factorizes cycle by cycle and yields a unique maximal safe settlement. We then separate exchange title from licences that authorize relationship-specific investment. On a canonical stochastic domain, future trades induce a directed title-insurance hypergraph: each head is an exposed investment and its tail contains the titles that block reassignment. Singleton tails make the planner’s reduced objective supermodular for every licence cost even when allocation gains have either sign; with free licences and quadratic investment, one minimum cut finds an optimum. Tails of size two already generate substitution and NP-hardness. Public information creates value by identifying the title that protects a licensed investment.