Solving the Arctic Product-Mix Auction

2026-09-28

AI Summary

This study addresses the computational complexity of computing competitive equilibria in Arctic product-mix auctions under budget constraints and separable convex piecewise-linear costs. To overcome this challenge, it proposes the first strongly polynomial-time algorithm by reducing the original problem to a cost-free auction and an associated linear market model, thereby efficiently computing equilibrium prices and allocations. Furthermore, it establishes polynomial-time reductions to both Fisher and Arrow–Debreu markets. The primary contribution lies in achieving the first strongly polynomial-time solution for settings involving costs, substantially improving computational efficiency. Additionally, this work proves that goods with positive sales volumes maintain consistent prices across different equilibria.

Abstract

Arctic product-mix auctions allow budget-constrained bidders to express preferences over multiple substitute goods. We give the first strongly polynomial algorithm to find the competitive equilibria when the auctioneer has separable convex piecewise-linear costs. It combines a strongly polynomial reduction to the costless Arctic auction with an existing strongly polynomial algorithm for costless Arctic auctions. We also give a polynomial-time Turing reduction from costless Arctic auctions to linear Fisher markets, and a strongly polynomial many-to-one reduction to linear Arrow-Debreu exchange markets. If a good is sold in positive quantity in two competitive equilibria, its price is the same in both.

Problem

• Arctic product-mix auction
• competitive equilibrium
• budget-constrained bidders
• separable convex piecewise-linear costs

Innovation

• Arctic product-mix auction
• strongly polynomial algorithm
• competitive equilibrium
• polynomial-time reduction
• linear Fisher markets