  In this paper, we study multi-agent systems with decentralized resource
allocations. Agents have local demand and resource supply, and are
interconnected through a network designed to support sharing of the local
resource; and the network has no external resource supply. It is known from
classical welfare economics theory that by pricing the flow of resource,
balance between the demand and supply is possible. Agents decide on the
consumed resource, and perhaps further the traded resource as well, to maximize
their payoffs considering both the utility of the consumption, and the income
from the trading. When the network supply and demand are balanced, a
competitive equilibrium is achieved if all agents maximize their individual
payoffs, and a social welfare equilibrium is achieved if the total agent
utilities are maximized. First, we consider multi-agent systems with static
local allocations, and prove from duality theory that under general convexity
assumptions, the competitive equilibrium and the social welfare equilibrium
exist and agree. Compared to similar results in the literature based on KKT
arguments, duality theory provides a direct way for connecting the two notions
and for a more general (e.g. nonsmooth) class of utility functions. Next, we
show that the agent utility functions can be prescribed in a family of socially
admissible functions, under which the resource price at the competitive
equilibrium is kept below a threshold. Finally, we extend the study to
dynamical multi-agent systems where agents are associated with dynamical states
from linear processes, and we prove that the dynamic the competitive
equilibrium and social welfare equilibrium continue to exist and coincide with
each other.
