Core Nonemptiness of Stratified Pooling Games: A Structured Markov Decision Process Approach

Abstract

We study several service providers that keep spare parts in stock to protect for downtime of their high-tech machines and that face different downtime costs per stockout. Service providers can cooperate by forming a joint spare parts pool, and we study the allocation of the joint costs to the individual service providers by studying an associated cooperative game. In extant literature, the joint spare parts pool is typically controlled by a suboptimal full-pooling policy. A full-pooling policy may lead to an empty core of the associated cooperative game, and we show this result in our setting as well. We then focus on situations where service providers apply an optimal policy: a stratification that determines, depending on the real-time on-hand inventory, which service providers may take parts from the pool. We formulate the associated stratified pooling game by defining each coalitional value in terms of the minimal long-run average costs of a Markov decision process. We present a proof demonstrating that stratified pooling games always have a nonempty core. This five-step proof is of interest in itself, because it may be more generally applicable for other cooperative games where coalitional values can be defined in terms of Markov decision processes.