Bullwhip and Reverse Bullwhip Effects under the Rationing Game

Abstract:     
When an unreliable supplier serves multiple retailers, the retailers may compete with each other by inflating their order quantities in order to obtain their desired allocation from the supplier, a behavior known as the rationing game. In this paper, we study the retailers' ordering behavior and find that no Nash Equilibrium (NE) of the retailers' order quantities exists for a certain class of supply processes when the supplier applies the well known proportional allocation rule. We also prove that, if the retailers make a certain reasonable assumption about their competitors' behavior, the bullwhip effect (BWE) always occurs downstream under the rationing game when the mean demand changes over time. Moreover, we prove that the reverse bullwhip effect (RBWE) occurs upstream in the same setting, a consequence of the disruption caused by the supplier. We then introduce an additional model in which the retailers pay a reservation payment for each unit ordered. We prove that an NE exists in this setting, regardless of how small the reservation payment is, and we characterize the conditions under which the BWE and RBWE occur. Finally, we find that capacity information sharing does not necessarily mitigate the [R]BWE and that, although information sharing benefits the supplier, it may reduce the profitability of the retailers and of the supply chain as a whole.

Keywords: rationing game, bullwhip effect, reverse bullwhip effect, supply uncertainty, order variance

Working Paper Series