Accelerated Price Adjustment for Fisher Markets with Exact Recovery of Competitive Equilibrium

TL;DR

This work addresses the challenge of efficiently computing competitive equilibrium prices in (linear and quasi-linear) Fisher markets, where traditional tâtonnement methods fail to converge to exact CE and exhibit high iteration costs. By casting CE computation as a box-constrained strongly convex nonsmooth minimization and applying entropy-based smoothing, the authors develop an accelerated price-adjustment method (APM) that achieves an $\epsilon$-CE in $\tilde{O}(1/\sqrt{\epsilon})$ iterations. They further construct a recovery oracle that maps nearby approximate CE prices to exact CE at cost $O((m+n)^2)$, enabling an adaptive APM that converges to CE in finite steps. Theoretical guarantees are complemented by numerical experiments showing APM's fast convergence and the effectiveness of CE recovery, significantly advancing lightweight, decentralized CE computation in large-scale markets.

Abstract

The canonical price-adjustment process, tâtonnement, typically fails to converge to the exact competitive equilibrium (CE) and requires a high iteration complexity of $\tilde{\mathcal{O}}(1/ε)$ to compute $ε$-CE prices in widely studied linear and quasi-linear Fisher markets. This paper proposes refined price-adjustment processes to overcome these limitations. By formulating the task of finding CE of a (quasi-)linear Fisher market as a strongly convex nonsmooth minimization problem, we develop a novel accelerated price-adjustment method (APM) that finds an $ε$-CE price in $\tilde{\mathcal{O}}(1/\sqrtε)$ lightweight iterations, which significantly improves upon the iteration complexities of tâtonnement methods. Furthermore, through our new formulation, we construct a recovery oracle that maps approximate CE prices to exact CE prices at a low computational cost. By coupling this recovery oracle with APM, we obtain an adaptive price-adjustment method whose iterates converge to CE prices in finite steps. To the best of our knowledge, this is the first convergence guarantee to exact CE for price-adjustment methods in linear and quasi-linear Fisher markets. Our developments pave the way for efficient lightweight computation of CE prices. We also present numerical results to demonstrate the fast convergence of the proposed methods and the efficient recovery of CE prices.

Figures (6)

• Figure 1: Comparison of different algorithms for computing $\epsilon$-CE prices in linear Fisher markets with uniform data, where the precision $\epsilon$ is $10^{-4}$.
• Figure 2: Comparison of different algorithms for computing $\epsilon$-CE prices in linear Fisher markets with exponential data, where the precision $\epsilon$ is $10^{-4}$.
• Figure 3: Comparison of different algorithms for computing $\epsilon$-CE prices in quasi-linear Fisher markets with uniform data, where the precision $\epsilon$ is $10^{-4}$.
• Figure 4: Comparison of different algorithms for computing $\epsilon$-CE prices in quasi-linear Fisher markets with exponential data, where the precision $\epsilon$ is $10^{-4}$.
• Figure 5: Comparison of different algorithms for computing $\epsilon$-CE prices in linear (left) and quasi-linear (right) Fisher markets with movie rating dataset, where the precision $\epsilon$ is $10^{-4}$.

Theorems & Definitions (28)

• definition thmcounterdefinition: Competitive Equilibrium in Fisher Market
• lemma thmcounterlemma
• definition thmcounterdefinition: Approximate CE Prices
• theorem thmcountertheorem
• proposition thmcounterproposition
• theorem thmcountertheorem: Recovery of Exact CE
• remark thmcounterremark
• definition thmcounterdefinition: Connection Class of Index Sets
• lemma thmcounterlemma
• lemma thmcounterlemma