Stable Allocations, Unstable Payment Paths: The Shapley Value and Voluntary Coalition Formation

The random-order interpretation of the Shapley value specifies both a terminal allocation and a payment path: each entrant is paid her marginal contribution at the time of entry. We embed this path in a finite dynamic entry problem in which a selected player may accept immediately or defer and allow another outsider to enter first. A flow is fully voluntary when immediate acceptance is optimal at every decision state. Although this criterion is defined through optimal continuation values, backward induction yields an equivalent system of linear inequalities in full-consent continuation values. The distinction between allocation and path is sharp in convex games: the Shapley allocation lies in the core, but the Shapley flow is fully voluntary only in additive games. Nevertheless, every efficient allocation admits a budget-balanced fully voluntary implementation when signed payments are available. We select a canonical implementation by minimizing distance from the Shapley flow, derive a reduced quadratic program for symmetric cardinality games, and study equal residual sharing, which selects the terminal allocation endogenously. With two player types, the selected allocation is an affine combination of the Shapley and equal-division allocations. Examples illustrate the effects of payment restrictions and a possible conflict between full voluntariness and coalitional stability.