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

Abstract

The random-order interpretation of the Shapley value specifies both a terminal allocation and a payment path: when a player enters, she receives her marginal contribution at that moment. We ask whether players would voluntarily follow this path. A player may prefer to wait if her marginal contribution is expected to rise as the coalition grows. This creates a tension in convex games. The Shapley allocation belongs to the core, but, except in additive games, the associated marginal-contribution payment path does not support voluntary entry. We separate these two objects by introducing payment flows that divide the surplus created at each coalition transition. Every efficient allocation can be implemented by a local, budget-balanced voluntary flow when signed payments are available. We select a canonical implementation by minimizing the distance from the Shapley flow, and show that the problem simplifies sharply in symmetric cardinality games. We then consider equal residual sharing, under which any transition surplus not paid to the entrant is divided equally among incumbents. The closest voluntary flow in this class selects an allocation endogenously. With two player types, the selected allocation is a game-dependent affine combination of the Shapley and equal-division allocations. Examples show both the role of payment restrictions and a possible conflict between voluntary entry and coalitional stability.