Population Monotonicity in Newsvendor Games

A newsvendor game allows the players to collaborate on inventory pooling and share the resulting total cost. There are several possible ways to allocate the cost. Previous studies have focused on the core of the game. It is known that the core of the newsvendor game is nonempty, and one can use duality theory in stochastic programming to construct an allocation-referred to as the dual-based allocation-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 schemes 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 a PMAS. Specifically, for continuous demands, log-concavity ensures that the game is convex, which in turn implies a 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, even so, the dual-based allocation scheme is a PMAS. When the demands are dependent, the game is in general not convex. We derive a sufficient condition based on the dependence structure, measured by the copula, to ensure that the dual-based allocation scheme is still a PMAS. We also include an example of a game with no PMAS.