This note discusses the possibility of fair gain sharing in cooperative situations where players optimally partition themselves across a number of alternative channels. An example is group purchasing among a set of buyers facing with a range of suppliers. We introduce channel selection games as a new class of cooperative games and give a representation of their cores. With two channels (suppliers), the game has a non-empty core if the gain functions across every individual channel is supermodular. © ۲۰۱۸ Elsevier B.V. All rights reserved.

Introduction

In some collaborative situations, the participants (players) organize themselves across a set of alternative channels to maximize their gain. The alternative channels may represent different suppliers, logistics service providers, or adopted technologies. As a special instance, consider buyers in a group purchasing organization who collaboratively select their suppliers for different products that they procure. An example of such a situation was recently studied in [8] where buyers optimize their purchases across two alternative channels: an intermediary and an original equipment manufacturer. The supplier selection problem has been the subject of extensive study in the literature (see for example [5,16,9] and references therein). The advantages of collaboration in this context can be intuitive—combining bargaining powers or deliveries results in savings. Yet, the mere existence of economies of scale does not necessarily grant the formation and sustenance of a collaborative organization. From a cooperative game theory point of view, the possibility of sharing the obtained savings among the players in a ‘‘fair’’ manner is a crucially important requirement. A widely adopted notion of fairness in the literature requires that each subgroup of players receives at least as much as they could accrue on their own. Accordingly, an allocation of gains among the players is called stable if it satisfies the latter condition. The core of a cooperative game [12] contains all stable allocations which divide total savings among the players. It is worth mentioning that although the literature often associates the definition of the core to Gillies [6], it was Shapley [12] who first defined the core in its current form [17]. In this note, we introduce channel selection games where coalitions of players optimally partition themselves across a set of given channels. The underlying optimization problems in these games are the same as the winner-determination problem in combinatorial auctions (see for example [10]) where a set of products are distributed among a set of bidders with different valuation functions to maximize the sum of all bidders’ valuations. However, to the best of our knowledge, previous literature does not study cooperative games with the same structure as ours. Unlike partition games (e.g. [3]), players in channel selection games choose among a fixed number of distinct options with non-homogeneous costs, and dissimilar to assignment games (e.g. [15]), here multiple players can be assigned to a single channel. We give a representation of the cores of channel selection games in terms of intersections of extended contra-polymatroids associated with gain functions across all channels (e.g. savings obtained by joint purchasing from each supplier), and provide two main observations regarding the existence of allocations in the core of channel selection games. First, if the number of channels is two and the gain function across every channel is supermodular, then the core of the associated game is always non-empty.We prove this result using the generalization of Edmond’s matroid intersection theorem [4]. This closes an open problem in [8] regarding the nonemptiness of the core of games associated with collaborative replenishment situations in the presence of intermediaries. Second, if the number of channels is more than two, then the core of a channel selection game may be empty, even under supermodularity conditions.