[Paper Review] Tracing Equilibrium in Dynamic Markets via Distributed Adaptation

Yun Kuen Cheung, Martin Hoefer+1 more|arXiv (Cornell University)|Apr 21, 2018

Economic theories and models5 references3 citations

TL;DR

This paper provides the first provable performance guarantees for discrete-time tatonnement and proportional response dynamics (PRD) in dynamic Fisher markets with time-varying parameters. It establishes bounds on the distance to equilibrium under perturbations in supplies, budgets, and utility functions, showing that both mechanisms track the moving equilibrium closely and maintain approximate market clearing despite dynamic changes.

ABSTRACT

Competitive equilibrium is a central concept in economics with numerous applications beyond markets, such as scheduling, fair allocation of goods, or bandwidth distribution in networks. Computation of competitive equilibria has received a significant amount of interest in algorithmic game theory, mainly for the prominent case of Fisher markets. Natural and decentralized processes like tatonnement and proportional response dynamics (PRD) converge quickly towards equilibrium in large classes of Fisher markets. Almost all of the literature assumes that the market is a static environment and that the parameters of agents and goods do not change over time. In contrast, many large real-world markets are subject to frequent and dynamic changes. In this paper, we provide the first provable performance guarantees of discrete-time tatonnement and PRD in markets that are subject to perturbation over time. We analyze the prominent class of Fisher markets with CES utilities and quantify the impact of changes in supplies of goods, budgets of agents, and utility functions of agents on the convergence of tatonnement to market equilibrium. Since the equilibrium becomes a dynamic object and will rarely be reached, our analysis provides bounds expressing the distance to equilibrium that will be maintained via tatonnement and PRD updates. Our results indicate that in many cases, tatonnement and PRD follow the equilibrium rather closely and quickly recover conditions of approximate market clearing. Our approach can be generalized to analyzing a general class of Lyapunov dynamical systems with changing system parameters, which might be of independent interest.

Motivation & Objective

• To analyze the performance of decentralized price and allocation dynamics—tatonnement and PRD—in dynamic markets where agent budgets, good supplies, and utility functions change over time.
• To quantify how closely tatonnement and PRD can track the moving equilibrium in response to adversarial perturbations in market parameters.
• To extend convergence analysis from static to dynamic Lyapunov systems by introducing a perturbation-aware potential function framework.
• To provide formal bounds on the distance to equilibrium that are maintained during adaptation, ensuring approximate market clearing.
• To generalize the analysis to a broad class of Lyapunov dynamical systems with time-varying parameters.

Proposed method

• Uses a Lyapunov potential function framework to analyze convergence in dynamic markets, adapting static convergence guarantees to time-varying settings.
• Introduces a perturbation-aware condition (Theorem D.1) that bounds the change in Bregman divergence between current prices and the time-varying equilibrium.
• Applies mirror descent updates with Bregman divergence to model tatonnement and PRD dynamics, ensuring convergence under parameter shifts.
• Employs a telescoping argument on the potential function to derive recursive bounds on the distance to equilibrium over time.
• Quantifies the impact of perturbations via a term Δt that captures changes in system parameters between consecutive rounds.
• Analyzes CES utility functions in Fisher markets, focusing on constant elasticity of substitution to model diverse agent preferences.

Experimental results

Research questions

• RQ1 How well can tatonnement and PRD track the equilibrium in a dynamic market where supplies, budgets, and utility functions change over time?
• RQ2 What is the maximum distance to equilibrium that tatonnement and PRD can maintain under time-varying perturbations?
• RQ3 Can the convergence guarantees of static market dynamics be extended to dynamic environments with changing system parameters?
• RQ4 How do perturbations in market parameters affect the stability and performance of decentralized adaptation processes?
• RQ5 What general framework can be used to analyze Lyapunov systems with time-varying fixed points and dynamic parameters?

Key findings

• The distance between current prices and the time-varying equilibrium is bounded by a geometric decay term and a perturbation sum, ensuring stable tracking.
• For tatonnement and PRD, the potential function Φ(s^t, p^t) decays at a rate determined by the ratio q1/q2, with convergence speed dependent on the strong convexity and smoothness parameters.
• When perturbations Δt are bounded, the system maintains a constant distance to equilibrium that scales with the magnitude of Δt and the convergence rate.
• The analysis shows that both tatonnement and PRD maintain approximate market clearing conditions even when the equilibrium is not exactly reached.
• The framework generalizes to any Lyapunov dynamical system with time-varying parameters, provided the perturbation condition (15) holds.
• For mirror descent updates with Bregman divergence, the parameters satisfy q1 = L − σ and q2 = L, yielding a convergence rate of (q1/q2)^T.