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

Abstract

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.

Figures (6)

• Figure 1: An instance of a Fisher market with $2$ agents and $1$ chore. The gray region (polyhedral) is the feasible region of \ref{['chores EG dual']}. The (only) KKT point / CE ($(\beta_1, \beta_2, p_1) = (1, 2, 2)$) is marked by the yellow point. The blue arrow denotes the direction of gradient at the KKT point. In the left, two green trajectories are iterates generated by the gradient ascending algorithm, starting from two feasible points. In the right, the blue region (2-dimensional polyhedral) is the feasible region of \ref{['chores dual redundant']}.
• Figure 2: The trajectory of GFW until termination on a $2$-agent-$8$-chore instance, starting from a feasible interior point. The gray region is the feasible set projected onto the $(\beta_1, \beta_2)$ subspace. The blue lines with arrows denote the trajectory of GFW. The orange arrows and dotted lines denote gradient ascending directions of $f$ and corresponding planes perpendicular to these directions, respectively.
• Figure 3: Numerical results on iid randomly generated disutility matrix instances. Each row corresponds to a different disutility distribution (noted on the left). Left: Average number of iterations to solve a given instance size. Middle: Average wall clock running time to solve a given instance size (using Gurobi for both algorithms). Right: Fraction of instances solved for each algorithm. GFW solves every instance for every distribution.
• Figure 4: Numerical results on AAMAS PC member bidding data. The top row uses the "original" valuations. The bottom row adds Gaussian noise. Left: Average number of iterations to solve a given instance size. Middle: Average wall clock running time to solve a given instance size. Right: Fraction of instances solved.
• Figure 5: Numerical results for 100 buyers and up to 1000 chores. GFW solves every instance for every distribution.

Theorems & Definitions (32)

• definition 1
• definition 2: $\epsilon$-approximate CE boodaghians2022polynomial
• definition 3
• theorem 1
• proof
• theorem 2
• proof
• lemma 1
• theorem 3
• proof