Computing Walrasian equilibria: fast algorithms and structural properties

Abstract

We present the first polynomial time algorithm for computing Walrasian equilibrium in an economy with indivisible goods and general buyer valuations having only access to an aggregate demand oracle, i.e., an oracle that given prices on all goods, returns the aggregated demand over the entire population of buyers. For the important special case of gross substitute valuations, our algorithm queries the aggregate demand oracle O(n) times and takes O(n 3 ) time, where n is the number of goods and the O(·) notation denotes an asymptotic bound up to polylogarithmic factors. Both algorithms are randomized. At the heart of our solution is a method for exactly minimizing certain convex functions which cannot be evaluated but for which the subgradients can be computed. We also give the fastest known algorithm for computing Walrasian equilibrium for gross substitute valuations in the value oracle model. Our algorithm has running time O((mn + n 3 )T V ) where T V is the cost of querying the value oracle.A key technical ingredient is to regularize a convex programming formulation of the problem in a way that subgradients are cheap to compute. En route, we give necessary and sufficient conditions for the existence of robust Walrasian prices, i.e., prices for which each agent has a unique demanded bundle and the demanded bundles clear the market. When such prices exist, the market can be perfectly coordinated by solely using prices. Mathematics Subject Classification 90C25 (Convex programming) · 91B26 (Market models)A shorter conference version of this paper appeared in the ACM-SIAM Symposium on Discrete Algorithms (SODA17) and has been posted on arXiv (https://arxiv.org/abs/1511.04032). The conference version is attached to this submission for the records.

A Two-Step Approach to Optimal Dynamic Pricing in Multi-Demand Combinatorial Markets

“…The concept of Walrasian equilibrium is closely related to optimal allocations and bundles in demand. In particular, Paes Leme and Wong proved the Second Welfare Theorem, which implies that in any optimal allocation the bundle received by each player is a bundle in demand with respect to Walrasian prices [10]. Definition 3.4.…”

A Constant-Factor Approximation for Generalized Malleable Scheduling under $M^\natural$-Concave Processing Speeds

“…Moreover, (GS-W) corresponds to an LP relaxation of the so-called Welfare Maximization problem. Because the functions g * j are M ♮ -concave, this LP is totally dual integral, and integer optimal solutions can be found in polynomial time [24]. Thus, let z be such an integer optimal solution.…”

Solving Strong-Substitutes Product-Mix Auctions

“…We note that to the best of our knowledge, the cutting plane method has not yet been implemented and may be computationally expensive in practice; moreover, solving the convex optimisation problem is not guaranteed to find component-wise minimal prices. While the steepest-descent methods described in the literature run in pseudo-polynomial time in the valuation and demand oracle settings, as compared to the fully polynomial algorithm of Paes Leme and Wong [2017], our steepest descent algorithm uses long steps and exploits the bid representation of bidder demand to close this gap, yielding a competitive fully polynomial algorithm in the bidding-language setting.…”

“…One such work is Murota [2003], which presents an algorithm that works in the multiunit case by reducing the allocation problem to a network flow problem and relies on oracle access to the valuation function of each bidder. Paes Leme and Wong [2017] provides a different algorithm, also in the valuation oracle setting, but this is only applicable in the case in which there is only one unit of each good. We elaborate on this in Section 2.4.…”

“…In Section 3, we consider algorithms for finding component-wise minimal market-clearing prices that have practical running time bounds (ruling out the ellipsoid method, which can in principle be applied [Paes Leme and Wong, 2017]). We adopt a discrete steepest descent method from the discrete optimisation literature [Murota, 2003, Shioura, 2017 that employs submodular minimisation to find the steepest descent directions and present two techniques that reduce the number of iterations required by taking long steps in the steepest descent direction.…”