Competitive Equilibrium for Dynamic Multi-Agent Systems: Social Shaping and Price Trajectories

TL;DR

This paper analyzes dynamic multi-agent systems operating at competitive equilibrium, introducing social shaping of preferences to ensure socially acceptable prices and exploring equilibrium properties over finite and infinite horizons.

Contribution

It develops a framework for social shaping in dynamic MAS, formulates the problem as a multi-agent LQR, and provides explicit algorithms and results for price bounds and equilibrium properties.

Findings

Socially acceptable price bounds can be computed via quadratic programming.

Under certain conditions, the equilibrium price converges to zero over time.

Maximizing social welfare aligns with competitive equilibrium under feasibility assumptions.

Abstract

In this paper, we consider dynamic multi-agent systems (MAS) for decentralized resource allocation. The MAS operates at a competitive equilibrium to ensure supply and demand are balanced. First, we investigate the MAS over a finite horizon. 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 at the conceptual level. Next, we consider quadratic MAS and formulate the associated social shaping problem as a multi-agent linear quadratic regulator (LQR) problem which enables us to propose explicit utility sets using quadratic programming and dynamic programming. Then, a numerical algorithm is presented for calculating a tight range of the preference function parameters which guarantees a socially accepted price. We investigate the properties of a competitive equilibrium over an infinite horizon. Considering general utility functions, we show that under feasibility assumptions, any competitive equilibrium maximizes the social welfare. Then, we prove that for sufficiently small initial conditions, the social welfare maximization solution constitutes a competitive equilibrium with zero price. We also prove for general feasible initial conditions, there exists a time instant after which the optimal price, corresponding to a competitive equilibrium, becomes zero. Finally, we specifically focus on quadratic MAS and propose explicit results.