Substitute Valuations: Generation and Structure

TL;DR

This paper introduces a Monte Carlo algorithm for generating substitute valuations in combinatorial auctions and explores the geometric structure of the set of all such valuations, revealing its complex polyhedral composition.

Contribution

It provides a novel Monte Carlo method for generating substitute valuations and characterizes the geometric structure of their set for any number of goods.

Findings

Set of substitute valuations forms a union of polyhedra.

Maximal polyhedra grow exponentially with number of goods.

Algorithmic complexity increases nearly as 2^K with goods.

Abstract

Substitute valuations (in some contexts called gross substitute valuations) are prominent in combinatorial auction theory. An algorithm is given in this paper for generating a substitute valuation through Monte Carlo simulation. In addition, the geometry of the set of all substitute valuations for a fixed number of goods K is investigated. The set consists of a union of polyhedrons, and the maximal polyhedrons are identified for K=4. It is shown that the maximum dimension of the maximal polyhedrons increases with K nearly as fast as two to the power K. Consequently, under broad conditions, if a combinatorial algorithm can present an arbitrary substitute valuation given a list of input numbers, the list must grow nearly as fast as two to the power K.