Existence of equilibria in a decentralized two-level supply chain

In this paper, we analyze equilibria in competitive environments under constraints across players’ strategies. This means that the action taken by one player limits the possible choices of the other players. In this context, the usual approach to show existence of equilibrium, Kakutani’s fixed point theorem, cannot be applied directly. In particular, best replies against a given strategy profile may not be feasible. We devise a new fixed point correspondence to deal with the feasibility issue. Our main motivation to study this problem of co-dependency comes from the field of supply chain planning. A set of buyers is faced with external demand over a planning horizon, and to satisfy this demand they request inputs from a set of suppliers. Both suppliers and buyers face production capacities and they plan their own production in a decentralized manner. A well-known coordination scheme for this setting is the upstream approach where the plan of the buyers is used to decide the request to the suppliers. We show the existence of equilibria for a (shared) inventory cost minimization version of this coordination scheme in which a distribution center manages the inventory of the inputs. However, we illustrate with an example that the centralized solution is not, in general, an equilibrium, suggesting that regulation may be needed.