Robust Dynamic Cooperative Games

Classical cooperative game theory is no longer a suitable tool for those situations where the values of coalitions are not known with certainty. Recent works address situations where the values of coalitions are modelled by random variables. In this work we still consider the values of coalitions as uncertain, but model them as unknown but bounded disturbances. We do not focus on solving a specific game, but rather consider a family of games described by a polyhedron: each point in the polyhedron is a vector of coalitions’ values and corresponds to a specific game. We consider a dynamic context where while we know with certainty the average value of each coalition on the long run, at each time such a value is unknown and fluctuates within the bounded polyhedron. Then, it makes sense to define “robust” allocation rules, i.e., allocation rules that bound, within a pre- defined threshold, a so-called complaint vector while guaranteeing a certain average (over time) allocation vector. We also present as motivating example a joint replenishment application

Distribution center consolidation games

We study the location-inventory model as introduced by Teo et al. (2001) to analyze the impact of consolidation of distribution centers on facility and inventory costs. We extend their result on profitability of consolidation. We associate a cooperative game with each location-inventory situation and prove that this game has a non-empty core for identical and independent demand processes. This illustrates that consolidation does not only lower joint costs (which was shown by Teo et al. (2001)), but it allows for a stable division of the minimal costs as well

A General Framework for Cooperation under Uncertainty

In this paper, we introduce a general framework for situations with decision making under uncertainty and cooperation possibilities. This framework is based upon a two stage stochastic programming approach. We show that under relatively mild assumptions the cooperative games associated with these situations are totally balanced and, hence, have non-empty cores. Finally, we consider several example situations, which can be studied using this general framework.Two-stage stochastic programming;cooperative game theory;core

Inventory Games

AMS classifications: 90D12, 90B05.inventory management;information;cooperative games;proportional division

Supply chain collaboration

In the past, research in operations management focused on single-firm analysis. Its goal was to provide managers in practice with suitable tools to improve the performance of their firm by calculating optimal inventory quantities, among others. Nowadays, business decisions are dominated by the globalization of markets and increased competition among firms. Further, more and more products reach the customer through supply chains that are composed of independent firms. Following these trends, research in operations management has shifted its focus from single-firm analysis to multi-firm analysis, in particular to improving the efficiency and performance of supply chains under decentralized control. The main characteristics of such chains are that the firms in the chain are independent actors who try to optimize their individual objectives, and that the decisions taken by a firm do also affect the performance of the other parties in the supply chain. These interactions among firms’ decisions ask for alignment and coordination of actions. Therefore, game theory, the study of situations of cooperation or conflict among heterogenous actors, is very well suited to deal with these interactions. This has been recognized by researchers in the field, since there are an ever increasing number of papers that applies tools, methods and models from game theory to supply chain problems

Cooperation between Multiple Newsvendors with Warehouses

This study considers a supply chain that consists of n retailers, each of them facing a newsvendor problem, m warehouses and a supplier.The retailers are supplied with a single product via some warehouses.In these warehouses, the ordered amounts of goods of these retailers become available after some lead time.At the time that the goods arrive at the warehouses, demand realizations are known by the retailers.The retailers can increase their expected joint profits by coordinating their orders and making allocations after demand realization.For this setting, we consider an associated cooperative game between the retailers.We show that this associated cooperative game has a nonempty core. Finally, we study a variant of this game, where the retailers are allowed to leave unsold goods at the warehouses.supply chain management;newsvendor;warehouse;game theory;balancedness

Sharing Supermodular Costs

We study cooperative games with supermodular costs. We show that supermodular costs arise in a variety of situations; in particular, we show that the problem of minimizing a linear function over a supermodular polyhedron—a problem that often arises in combinatorial optimization—has supermodular optimal costs. In addition, we examine the computational complexity of the least core and least core value of supermodular cost cooperative games. We show that the problem of computing the least core value of these games is strongly NP-hard and, in fact, is inapproximable within a factor strictly less than 17/16 unless P = NP. For a particular class of supermodular cost cooperative games that arises from a scheduling problem, we show that the Shapley value—which, in this case, is computable in polynomial time—is in the least core, while computing the least core value is NP-hard.National Science Foundation (U.S.) (DMI-0426686

Allocating Pooled Inventory According to Contributions and Entitlements

Inventory pooling, whether by centralization of stock or by mutual assistance, is known to be beneficial when demands are uncertain. But when the retailers are independent, the question is how to divide the benefits of pooling.We consider a decentralized inventory pooling scheme where retailers' entitlements to allocation in case of shortage depend on their contributions to the pool. We derive the Nash equilibrium, and specialize it to symmetric cases

Allocating Pooled Inventory According to Contributions and Entitlements

Inventory pooling, whether by centralization of stock or by mutual assistance, is known to be beneficial when demands are uncertain. But when the retailers are independent, the question is how to divide the benefits of pooling.We consider a decentralized inventory pooling scheme where retailers' entitlements to allocation in case of shortage depend on their contributions to the pool. We derive the Nash equilibrium, and specialize it to symmetric cases

Large Newsvendor Games

We consider a game, called newsvendor game, where several retailers, who face a random demand, can pool their resources and build a centralized inventory that stocks a single item on their behalf. Profits have to be allocated in a way that is advantageous to all the retailers. A game in characteristic form is obtained by assigning to each coalition its optimal expected profit. A similar game (modeled in terms of costs) was considered by Muller et al. (2002), who proved that this game is balanced for every possible joint distribution of the random demands. In this paper we consider newsvendor games with possibly an infinite number of newsvendors. We prove in great generality results about balancedness of the game, and we show that in a game with a continuum of players, under a nonatomic condition on the demand, the core is a singleton. For a particular class of demands we show how the core shrinks to a singleton when the number of players increases.newsvendor games, nonatomic games, core, balanced games.