Dynamic Coordination Games with Activation Costs

Motivated by inventory control problems with set-up costs, we consider a coordination game where each player’s dynamics is an inventory model characterized by a controlled input and an uncontrolled output. An activation cost is shared among active players, namely players who control their dynamics at a given time. At each time, each player decides to be active or not depending on its inventory level. The main contribution of this paper is to show that strategies at a Nash equilibrium have a threshold structure on the number of active players. Furthermore, we provide an explicit expression for the lower and upper threshold is given both in the deterministic case, namely when the exogenous signal is known, and in the single-stage game. The relevance of the above results is discussed in the context of inventory control where Nash equilibrium reordering strategies imply that a single retailer reorders only if jointly with a number of other retailers and will reorder to restore a pre-assigned inventory level.