Competitive Equilibrium for Chores: from Dual Eisenberg-Gale to a Fast, Greedy, LP-based Algorithm

We study the computation of competitive equilibrium for Fisher markets with $n$ agents and $m$ divisible chores. Competitive equilibria for chores are known to correspond to the nonzero KKT points of a program that minimizes the product of agent disutilities, which is a non-convex program whose zero points foil iterative optimization methods. We introduce a dual-like analogue of this program, and show that a simple modification to our "dual" program avoids such zero points, while retaining the correspondence between KKT points and competitive equilibria. This allows, for the first time ever, application of iterative optimization methods over a convex region for computing competitive equilibria for chores. We next introduce a greedy Frank-Wolfe algorithm for optimization over our program and show a new state-of-the-art convergence rate to competitive equilibrium. Moreover, our method is significantly simpler than prior methods: each iteration of our method only requires solving a simple linear program. We show through numerical experiments that our method is extremely practical: it easily solves every instance we tried, including instances with hundreds of agents and up to 1000 chores, usually in 10-30 iterations, is simple to implement, and has no numerical issues.