Equilibrium plans in constrained environments

Abstract

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 classical approach, Kakutani''s Fixed Point theorem, does not work. In particular, best replies against a given strategy profile may not be feasible. We extend Kakutani''s Fixed Point theorem 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 the planning is made in a decentralized manner. A well-known coordination scheme for this setting is the upstream approach where the planning of the buyers is used to decide the request to the suppliers. We show the existence of equilibria for two versions of this coordination model. However, we illustrate with an example that the centralized solution is not, in general, an equilibrium, suggesting that regulation may be needed.We also apply our Fixed Point theorem to a production economy, where both supply and demand are upper bounded.