Population Monotonicity in Newsvendor Games

Abstract

A newsvendor game studies whether players can collaborate on inventory pooling, where the cost allocation is usually analyzed by the notion of core in cooperative game theory. It is known that the core of the newsvendor game is non-empty and one can use duality theory in stochastic programming to construct an allocation - referred to as the dual-based allocation scheme - belonging to the core. Yet, an allocation that lies in the core does not necessarily guarantee the unhindered formation of a coalition, as some existing members' allocated costs may increase when new members are added in the process. In this work, we use the concept of population monotonic allocation scheme (PMAS), which requires the cost allocated to every member of a coalition to decrease as the coalition grows, to study allocation rules in a growing population. We show that when the demands faced by the newsvendors are independent, log-concavity of their distributions is sufficient to guarantee the existence of PMAS. Specifically, for continuous demands, log-concavity ensures that the game is convex, which in turn implies PMAS exists. We also show that under the same condition the dual-based allocation scheme is a PMAS. For discrete and log-concave demands, however, the game may no longer be convex but we manage to show that the dual-based allocation scheme is a PMAS. When the demands are dependent, the game is in general not convex and we derive a sufficient condition based on the dependence structure, measured by the copula, between each player and the coalition to secure that the dual-based allocation scheme is still a PMAS.