Decision-making by hierarchies of discordant agents

We study the following decision-making scenario: A linear program is solved by a set of agents arranged hierarchically in a tree, where each agent decides the level of certain variables, and has a distinct objective function, known to all agents. Authority is reflected in two ways: Agents higher in the tree set their variables first; and agents that are siblings in the tree resolve their game by focusing on the Nash equilibrium that is optimum for the agent above them. We give a necessary and sufficient condition for such a hierarchy to be efficient (i.e., to have perfect coordination, to ultimately optimize the objective of the firm). We study problems related to designing a hierarchy (assigning decision makers to positions in the tree) in order to achieve efficiency or otherwise optimize coordination.