  Achieving a balance of supply and demand in a multi-agent system with many
individual self-interested and rational agents that act as suppliers and
consumers is a natural problem in a variety of real-life domains---smart power
grids, data centers, and others. In this paper, we address the
profit-maximization problem for a group of distributed supplier and consumer
agents, with no inter-agent communication. We simulate a scenario of a market
with $S$ suppliers and $C$ consumers such that at every instant, each supplier
agent supplies a certain quantity and simultaneously, each consumer agent
consumes a certain quantity. The information about the total amount supplied
and consumed is only kept with the center. The proposed algorithm is a
combination of the classical additive-increase multiplicative-decrease (AIMD)
algorithm in conjunction with a probabilistic rule for the agents to respond to
a capacity signal. This leads to a nonhomogeneous Markov chain and we show
almost sure convergence of this chain to the social optimum, for our market of
distributed supplier and consumer agents. Employing this AIMD-type algorithm,
the center sends a feedback message to the agents in the supplier side if there
is a scenario of excess supply, or to the consumer agents if there is excess
consumption. Each agent has a concave utility function whose derivative tends
to 0 when an optimum quantity is supplied/consumed. Hence when social
convergence is reached, each agent supplies or consumes a quantity which leads
to its individual maximum profit, without the need of any communication. So
eventually, each agent supplies or consumes a quantity which leads to its
individual maximum profit, without communicating with any other agents. Our
simulations show the efficacy of this approach.
