Tâtonnement Dynamics for Fisher Markets with Chores

TL;DR

The paper studies tâtonnement dynamics in Fisher markets with chores, showing that naive price adjustments fail due to non-monotone excess demand, while a relative tâtonnement approach provably converges to competitive equilibria under broad CCH disutilities. For convex CES disutilities with $\rho\in(1,\infty)$, the authors establish a polynomial-time $\tilde{O}(1/\varepsilon^2)$ rate to reach an $\varepsilon$-CE by analyzing the gauge dual and constructing a Lyapunov function that links CE to KKT conditions. The work also provides a detailed stability characterization of CE, including local stability criteria and the Nash welfare justification, and highlights that global stability may fail due to multiple equilibria. These results advance understanding of dynamic market behavior in chores settings and supply algorithmic tools for efficient approximate CE computation under nonconvex, nonsmooth objectives. The framework leverages nonsmooth analysis, gauge-dual representations, and a generalized Eisenberg–Gale program to connect dynamic processes with equilibrium concepts in a challenging but practically relevant domain.

Abstract

In this paper, we initiate the study of tâtonnement dynamics in markets with chores. Tâtonnement is a fundamental market dynamics, capturing how prices evolve when they are adjusted in proportion of their excess demand. While its convergence to a competitive equilibrium (CE) is well understood in goods markets for broad classes of utilities, no analogous results are known for chore markets. Analyzing tâtonnement in the chores market presents new challenges. Several elegant structural properties that facilitate convergence in goods markets-such as convexity of the equilibrium price set and monotonicity of excess demand under the tâtonnement price updates-fail to hold in the chore setting. Consistent with these difficulties, we first show that naïve tâtonnement diverges. To overcome this, we propose a modified process called relative tâtonnement, where prices are updated according to normalized excess demand. We prove its convergence to a CE under suitable step-size choices for a broad class of disutility functions, namely continuous, convex, and 1-homogeneous (CCH) disutilities. This class includes many standard forms such as linear and convex CES disutilities. Our proof proceeds by introducing a nonsmooth, nonconvex yet regular objective function-a generalization of the objective in the Eisenberg-Gale-type dual program introduced by [CKMN24]. For convex CES disutilities, where disutility is the weighted $p$-norm of the individual chore disutilities for $p \in (1, \infty)$, we show that relative tâtonnement converges to an $\varepsilon$-CE in $O(1/\varepsilon^2)$ iterations. This quadratic convergence rate is established by leveraging the polar gauge (or gauge dual) of the disutility function. Finally, following the framework of [AH58], we analyze the stability of CE and provide a complete characterization of local stability.

Figures (5)

• Figure 1: Stability and associated Nash welfare of chores competitive equilibrium visualized as ternary plots (also known as simplex plots). The details on the instances can be found in \ref{['sec:intro']}. We make the plots in the price simplex consisting of nonnegative prices that sum to the sum of budgets. We use blue color to denote the level sets of the objective values of our potential function (see \ref{['eq:potential']}). The darker the color is, the lower the objective value is. We use red and yellow dots to denote "locally stable" and "unstable" chores CE, respectively. We use bold texts to denote the values of "Nash welfare" associated with each CE, and the maximum Nash welfare CE is denoted in light green text. The gray arrows (flows) denote the relative excess demand vector. If there are multiple relative excess demands, we pick the one with the minimal Euclidean norm.
• Figure 2: The left is a function with "downward facing cusps" and hence is not regular, while the right is a "regular" function. Both of them are nonsmooth nonconvex. The left function is not compatible with first-order dynamics because of the descent directions (denoted by red arrows) at the unique stationary point. In contrast, a "regular" function is nice because all first-order generalized gradients pointing to non-descent directions at any stationary point.
• Figure 3: CES disutility functions: $1$-level sets and their smoothness.
• Figure 4: Iso-disutility curves for convex CES disutility functions.
• Figure 5: A trajectory of naïve tâtonnement starting from a neighborhood of a CE. In the right plot, we zoom into the neighborhood. This shows naïve tâtonnement is unstable in the linear chores market.

Theorems & Definitions (63)

• Example 1
• Theorem : Informal
• Theorem
• Theorem
• Lemma
• Theorem : Informal
• Definition 1: Local stability of chores CE
• Definition 2: Stability of dynamical system
• Lemma 1
• proof