How Markets End: Insurance Exhaustion and the Limits of Priority

Centralized assignment is often followed by an aftermarket that reallocates vacated and unfilled positions. We model the reliability of this second stage as a design choice. Reliable rescue insures main-round failure and creates its own demand: along an equilibrium exhaustion locus, each additional surviving seat induces another failed-reach claimant after attrition. At a closed-form reliability threshold, the safety tier switches from oversubscription to undersubscription. The best priority rule preserves main-round discipline up to that threshold, and no non-wasteful rule can preserve it beyond: priority reorders the queue but cannot shorten it. Under score priority, welfare has an exact phase-by-phase derivative, and the global optimum is one of three candidates selected by two closed-form comparisons. Residual-seat quality can further discipline reaching, while commitment is required to implement any welfare-improving rescue cap. The model predicts cutoff reversal in supplementary admissions and a calculable demographic transition date at which aftermarket volume changes slope.