Solving Strong-Substitutes Product-Mix Auctions

Abstract

This paper develops algorithms to solve strong-substitutes product-mix auctions. That is, it finds competitive equilibrium prices and quantities for agents who use this auction's bidding language to truthfully express their strong-substitutes preferences over an arbitrary number of goods, each of which is available in multiple discrete units. (Strong substitutes preferences are also known, in other literatures, as M -concave, matroidal and well-layered maps, and valuated matroids). Our use of the bidding language, and the information it provides, contrasts with existing algorithms that rely on access to a valuation or demand oracle to find equilibrium.We compute market-clearing prices using algorithms that apply existing submodular minimisation methods. Allocating the supply among the bidders at these prices then requires solving a novel constrained matching problem. Our algorithm iteratively simplifies the allocation problem, perturbing bids and prices in a way that resolves tie-breaking choices created by bids that can be accepted on more than one good. We provide practical running time bounds on both price-finding and allocation, and illustrate experimentally that our allocation mechanism is practical.

Strong substitutes: structural properties, and a new algorithm for competitive equilibrium prices

“…The SSPMA is neither based on a value nor a demand oracle. 6 A collection of bids specifies a large number of package values, which mitigates the missing bids problem. This type of preference elicitation permits efficient ways to compute Walrasian prices, and allows us to uncover new properties of strong substitutes valuations.…”

“…More precisely, we show that the Toland-Singer dual [33,47] of L(p) is the minimum difference between the positive and negative linear programs. [6] have recently shown that a standard steepest-descent algorithm based on the Lyapunov function (following [38]) can solve the SSPMA pricing problem, but their method takes only limited advantage of the special features of the geometric representation. 14 By taking fuller advantage of the structure of strong substitutes analyzed in this paper, we find an alternative to steepest descent on the Lyapunov function.…”

“…We say that the set B is valid when the indirect utility u B is concave. (See Theorem 1 of [6]; further discussion of this notion is given below after Definition 2.) To define the aggregate demand set with positive and negative bids, first define the demand D b (p) associated with an individual negative bid b as…”

“…If B also contains negative bids, the problem of efficiently computing equilibrium prices is less obvious. One route, taken by [6], is to minimize the Lyapunov function L : R n → R [4], defined for target t as…”

“…( 3). The set of minimizers of L coincides with the set of equilibrium prices, and structural properties of L allow for polynomial-time steepest descent algorithms to find these minima [6,36,42]. However, this approach works by invoking a rather generic submodular function minimization algorithm, under the assumption that a demand oracle is available.…”