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Asynchronous Proportional Response Dynamics in Markets with Adversarial Scheduling

Yoav Kolumbus, Menahem Levy, Noam Nisan

TL;DR

The paper addresses convergence of asynchronous Proportional Response Dynamics in linear Fisher markets with adversarial activation by introducing an associated game whose exact potential aligns Nash equilibria with market equilibria. The authors prove that any PRD step increasing the potential drives the system to market equilibria, establishing convergence under a liveness schedule and showing unique equilibrium bids for generic markets. They connect PRD to best-response dynamics and no-swap regret, and provide simulations indicating favorable convergence properties relative to best-response dynamics. The work extends the understanding of distributed market dynamics under asynchrony and offers actionable insights for designing robust, decentralized bidding protocols with provable convergence guarantees.

Abstract

We study Proportional Response Dynamics (PRD) in linear Fisher markets where participants act asynchronously. We model this scenario as a sequential process in which in every step, an adversary selects a subset of the players that will update their bids, subject to liveness constraints. We show that if every bidder individually uses the PRD update rule whenever they are included in the group of bidders selected by the adversary, then (in the generic case) the entire dynamic converges to a competitive equilibrium of the market. Our proof technique uncovers further properties of linear Fisher markets, such as the uniqueness of the equilibrium for generic parameters and the convergence of associated best-response dynamics and no-swap regret dynamics under certain conditions.

Asynchronous Proportional Response Dynamics in Markets with Adversarial Scheduling

TL;DR

The paper addresses convergence of asynchronous Proportional Response Dynamics in linear Fisher markets with adversarial activation by introducing an associated game whose exact potential aligns Nash equilibria with market equilibria. The authors prove that any PRD step increasing the potential drives the system to market equilibria, establishing convergence under a liveness schedule and showing unique equilibrium bids for generic markets. They connect PRD to best-response dynamics and no-swap regret, and provide simulations indicating favorable convergence properties relative to best-response dynamics. The work extends the understanding of distributed market dynamics under asynchrony and offers actionable insights for designing robust, decentralized bidding protocols with provable convergence guarantees.

Abstract

We study Proportional Response Dynamics (PRD) in linear Fisher markets where participants act asynchronously. We model this scenario as a sequential process in which in every step, an adversary selects a subset of the players that will update their bids, subject to liveness constraints. We show that if every bidder individually uses the PRD update rule whenever they are included in the group of bidders selected by the adversary, then (in the generic case) the entire dynamic converges to a competitive equilibrium of the market. Our proof technique uncovers further properties of linear Fisher markets, such as the uniqueness of the equilibrium for generic parameters and the convergence of associated best-response dynamics and no-swap regret dynamics under certain conditions.
Paper Structure (15 sections, 26 theorems, 27 equations, 1 figure, 1 algorithm)

This paper contains 15 sections, 26 theorems, 27 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Further Related Work
  3. Model and Preliminaries
  4. The Associated Game
  5. Best Response Dynamics
  6. Simultaneous Play by Subsets of Agents
  7. Convergence of Asynchronous Proportional Response Dynamics
  8. Generic Markets
  9. Simulations
  10. Conclusion
  11. The Associated Game
  12. Best Response Dynamics
  13. Simultaneous Play by Subsets of Agents
  14. Convergence of Asynchronous Proportional Response Dynamics
  15. Generic Markets

Key Result

Theorem 1

For linear Fisher markets, proportional response dynamics with adversarial activation asynchrony, where each player is activated at least once every $T$ steps, approach the set of market equilibrium bid profiles. The prices in the dynamics converge to the unique equilibrium prices.

Figures (1)

  • Figure 1: Proportional response and best response dynamics. The main figures show the progression of the average metrics over time and the insets show a collection of individual dynamics over a longer time period.

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem
  • Theorem 4
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • ...and 45 more