  In this work, we approach the problem of resource allocation in a team of
agents through the framework of Centroidal Voronoi Tessellations. CVTs provide
a natural way to embed a desired global trend in the team through probability
distributions, and in one-dimensional spaces, CVTs offer an inherent line
structure allowing for a simple communication graph and scalability. We first
consider the amount of resource to be allocated to be a constant and provide an
analytical solution to such static resource allocation problem by embedding the
allocation constraint within the distribution through a system of nonlinear
equations. Using the solution of such a constrained CVT minimization problem as
an initialization step, we propose a decentralized dynamic resource allocation
solution that employs a one-step update when the desired distribution is
Gaussian. We introduce a "civility model" for negotiations between the agents
to allow for flexibility in local preferences and maintaining robustness
against local disturbances. We demonstrate the effectiveness of the proposed
method by considering the application of demand-response in smart grids through
the problem of power allocation in a group of building thermal loads.
