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

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.

Bounds on the Cost of Stabilizing a Cooperative Game

A key issue in cooperative game theory is coalitional stability, usually captured by the notion of the core---the set of outcomes that are resistant to group deviations. However, some coalitional games have empty cores, and any outcome in such a game is unstable. We investigate the possibility of stabilizing a coalitional game by using subsidies. We consider scenarios where an external party that is interested in having the players work together offers a supplemental payment to the grand coalition, or, more generally, a particular coalition structure. This payment is conditional on players not deviating from this coalition structure, and may be divided among the players in any way they wish. We define the cost of stability as the minimum external payment that stabilizes the game. We provide tight bounds on the cost of stability, both for games where the coalitional values are nonnegative (profit-sharing games) and for games where the coalitional values are nonpositive (cost-sharing games), under natural assumptions on the characteristic function, such as superadditivity, anonymity, or both. We also investigate the relationship between the cost of stability and several variants of the least core. Finally, we study the computational complexity of problems related to the cost of stability, with a focus on weighted voting games.

COALITIONAL VALUES AND COST ALLOCATION PROBLEMS

In this paper, we explain why the field of cost allocation problems with a coalition structure is a wide and promising unexplored research direction. In particular, we pose some open questions for airport games with a coalition structure.