Competitive Equilibrium for Dynamic Multiagent Systems: Social Shaping and Price Trajectories

In this article, we consider dynamic multiagent systems (MAS) for decentralized resource allocation. The MAS operate at a competitive equilibrium to ensure a balanced supply and demand for available resources. Example resources include energy in a microgrid and carbon permits in a carbon trading system. First, we investigate the MAS over a finite horizon, where the utility functions of agents are parameterized to incorporate individual preferences. We shape individual preferences through a set of utility functions to guarantee the resource price at a competitive equilibrium remains socially acceptable, i.e., the price is upper-bounded by an affordability threshold. We show this problem is solvable implicitly. Next, we consider quadratic MAS and formulate the social shaping problem as a multiagent linear quadratic regulator (LQR) problem. We propose explicit utility sets using quadratic programming and dynamic programming. Moreover, we propose a numerical algorithm for calculating a tight range of the preference function parameters that guarantee a socially accepted price. We then investigate the properties of a competitive equilibrium over an infinite horizon, and show cases where any competitive equilibrium maximizes the social welfare. We show that for initial conditions sufficiently close to the origin, the social welfare maximization solution constitutes a competitive equilibrium with zero price. We also show that for all feasible initial conditions, there exists a time instant after which the optimal price, corresponding to a competitive equilibrium, becomes zero.