Social Shaping of Competitive Equilibriums for Resilient Multi-Agent Systems

In this paper, we study entirely self-sustained multi-agent systems with decentralized resource allocation. Agents make local resource decisions, and sometimes, trading decisions to maximize their individual payoffs accruing from the utility of consumption and the income or expenditure from trading. A competitive equilibrium is achieved if all agents maximize their individual payoffs; a social welfare equilibrium is achieved if the total agent utilities are maximized. First, we consider multi-agent systems with static local allocation, and prove from duality theory that under general convexity assumptions, the competitive equilibrium and the social welfare equilibrium exist and agree. Next, we define a social shaping problem for a competitive equilibrium under which the optimal resource price is socially acceptable, and show that agent utility functions can be prescribed in a family of socially admissible quadratic functions, under which the pricing at the competitive equilibrium is always below a threshold. Finally, we extend the study to dynamical multi-agent systems where agents are associated with dynamical states from linear processes, and prove that the dynamic competitive equilibrium and social welfare equilibrium continue to exist and coincide with each other.