In this paper, we initiate the study of tâtonnement dynamics in markets with chores. Tâtonnement is a fundamental market dynamics, that captures 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 utility functions, no analogous results are known for chore markets.

Tâtonnement is a simple, intuitive market process where prices are iteratively adjusted based on the difference between demand and supply. Many variants under different market assumptions have been studied and shown to converge to a market equilibrium, in some cases at a fast rate. However, the classical case of linear Fisher markets have long eluded the analyses, and it remains unclear whether tâtonnement converges in this case. We show that, for a sufficiently small stepsize, the prices given by the tâtonnement process are guaranteed to converge to equilibrium prices, up to a small approximation radius that depends on the stepsize. To achieve this, we consider the dual Eisenberg-Gale convex program in the price space, view tâtonnement as subgradient descent on this convex program, and utilize novel last-iterate convergence results for subgradient descent under error bound conditions. In doing so, we show that the convex program satisfies a particular error bound condition, the quadratic growth condition, and that the price sequence generated by tâtonnement is bounded above and away from zero. We also show that a similar convergence result holds for tâtonnement in quasi-linear Fisher markets. Numerical experiments are conducted to demonstrate that the theoretical linear convergence aligns with empirical observations.

We study the computation of competitive equilibrium for Fisher markets with n agents and m divisible chores. Prior work showed that competitive equilibria correspond to the nonzero KKT points of the Nash welfare minimization program, which is a non-convex analogue of the Eisenberg-Gale convex program. We introduce an analogue of the Eisenberg-Gale dual for chores: we show that all KKT points of this dual correspond to competitive equilibria, and while it is not a dual of the non-convex primal program in a formal sense, the objectives touch at all KKT points. Similar to the primal program, the dual has problems from an optimization perspective: there are many feasible directions where the objective tends to positive infinity, and these attract iterative optimization methods. We then derive a new constraint for the dual, which restricts optimization to a hyperplane that avoids all these directions. We show that restriction to this hyperplane retains all KKT points, and surprisingly, does not introduce any new ones. This allows, for the first time ever, application of iterative optimization methods over a convex region for computing competitive equilibria for chores.

We consider the problem of large-scale Fisher market equilibrium computation through scalable first-order optimization methods. It is well-known that market equilibria can be captured using structured convex programs such as the Eisenberg-Gale and Shmyrev convex programs. Highly performant deterministic full-gradient first-order methods have been developed for these programs. In this paper, we develop new block-coordinate first-order methods for computing Fisher market equilibria, and show that these methods have interpretations as tâtonnement-style or proportional response-style dynamics where either buyers or items show up one at a time. We reformulate these convex programs and solve them using proximal block coordinate descent methods, a class of methods that update only a small number of coordinates of the decision variable in each iteration. Leveraging recent advances in the convergence analysis of these methods and structures of the equilibrium-capturing convex programs, we establish fast convergence rates of these methods.