Competitive Equilibrium Relaxations in General Auctions

TL;DR

This paper introduces two relaxation models for general auctions to approximate competitive equilibrium, providing algorithms for their solution and ensuring participants are not worse off, addressing the non-existence of exact equilibria.

Contribution

It proposes two novel relaxation models for general auctions and offers algorithms to solve them, advancing the understanding of approximate competitive equilibria.

Findings

The first model guarantees participants are either satisfied or their bids are rejected.

The second model ensures no participant incurs a loss, representing a Pareto-efficient relaxation.

Algorithms for solving both models are developed and analyzed.

Abstract

The goal of an auction is to determine commodity prices such that all participants are perfectly happy. Such a solution is called a competitive equilibrium and does not exist in general. For this reason we are interested in solutions which are similar to a competitive equilibrium. The article introduces two relaxations of a competitive equilibrium for general auctions. Both relaxations determine one price per commodity by solving a difficult non-convex optimization problem. The first model is a mathematical program with equilibrium constraints (MPEC), which ensures that each participant is either perfectly happy or his bid is rejected. An exact algorithm and a heuristic are provided for this model. The second model is a relaxation of the first one and only ensures that no participant incurs a loss. In an optimal solution to the second model, no participant can be made better off without making another one worse off.