Fully Distributed T\^atonnement for Chores Markets

Abstract

We study price-adjustment dynamics for computing competitive equilibria (CE) in Fisher markets with chores. Unlike in classical goods markets, prices in chores markets are payments for taking on undesirable tasks, and natural excess-demand dynamics can fail; even the na\"ive analogue of Walrasian t\^atonnement may diverge. Recent work of Chaudhury et al. [2025] overcomes this obstacle via relative t\^atonnement, which subtracts the average excess-demand signal from the excess demand vector. This recovers convergence, but at the cost of coupling the price updates across all chores. This leaves open whether such global coupling is inherent, or whether convergent t\^atonnement can be recovered through a genuinely local update in which each chore reacts only to its own excess demand. We answer this question affirmatively through multiplicative t\^atonnement, a fully distributed dynamics in which each chore price is updated using only its current price and its own excess-demand signal. Although the update contains no explicit normalization term, Walras' law and the multiplicative form of the update implicitly preserve the relevant aggregate price geometry. We prove that multiplicative t\^atonnement converges to a CE in any chores Fisher market with continuous, convex, and $1$-homogeneous (CCH) disutilities. For convex CES disutilities, we further prove an approximate-CE convergence rate with the same $O(1/\varepsilon^2)$ dependence as relative t\^atonnement, but with improved dependence on problem constants. Experiments on real-world and simulated instances show that multiplicative t\^atonnement is substantially faster in practice, often by an order of magnitude.