Tracing Equilibrium in Dynamic Markets via Distributed Adaptation

In real-world decentralized systems, agents' actions are often coupled with changes in the environment which are out of the agents' control. Yet, in many important domains, the existing analyses presume static environments. The theme of our work is to bridge such a gap between existing work and reality, with a focus on markets. Competitive (market) equilibrium is a central concept in economics with numerous applications beyond markets, such as scheduling, fair allocation of goods, or bandwidth distribution in networks. Natural and decentralized processes like tatonnement and proportional response dynamics (PRD) are known to converge quickly towards equilibrium in large classes of static Fisher markets. In contrast, many large real-world markets are subject to frequent and dynamic changes. We provide the first provable performance guarantees of discrete-time tatonnement and PRD in dynamic markets. We analyze the prominent class of CES (Constant Elasticity of Substitution) Fisher markets and quantify the impact of changes in supplies of goods, budgets of agents, and utility functions of agents on the convergences of the processes to equilibrium. Since the equilibrium becomes a dynamic object and will rarely be reached, we provide bounds expressing the distance to equilibrium that will be maintained. Our results indicate that in many cases, the processes trace the equilibrium rather closely and quickly recover conditions of approximate market clearing. Our analyses proceed by quantifying the impact of variation in market parameters on several potential functions which guarantee convergences in static settings. This approach is captured in two general yet handy frameworks for Lyapunov dynamical systems. They are of independent interest, which we demonstrate with the analysis of load balancing in dynamic environment setting.