Statistical mechanics of budget-constrained auctions

TL;DR

This paper applies statistical mechanics to analyze and solve the budget-constrained auction problem, introducing a message passing algorithm and exploring phase transitions related to long-range correlations.

Contribution

It develops a novel message passing algorithm based on the cavity method for efficiently solving random instances of the auction problem and characterizes its phase diagram.

Findings

The algorithm efficiently solves random instances from a natural distribution.

Identifies two phase transitions related to the average budget parameter.

Maps the phase diagram showing regions with long-range correlations.

Abstract

Finding the optimal assignment in budget-constrained auctions is a combinatorial optimization problem with many important applications, a notable example being the sale of advertisement space by search engines (in this context the problem is often referred to as the off-line AdWords problem). Based on the cavity method of statistical mechanics, we introduce a message passing algorithm that is capable of solving efficiently random instances of the problem extracted from a natural distribution, and we derive from its properties the phase diagram of the problem. As the control parameter (average value of the budgets) is varied, we find two phase transitions delimiting a region in which long-range correlations arise.