Product-Mix Auctions and Tropical Geometry

• 成果类型: Article
• 署名作者: Ngoc Mai Tran; Yu, Josephine
• 署名单位: University of Texas System; University of Texas Austin; University of Bonn; University System of Georgia; Georgia Institute of Technology
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2018.0975
• 发表日期: 2019
• 页码: 1396-1411
• 关键词: discrete convexity; polytopes

摘要

In a recent and ongoing work, Baldwin and Klemperer explore a connection between tropical geometry and economics. They give a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the unimodularity theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis. We give a new proof of the unimodularity theorem via the classical unimodularity theorem in integer programming. We give a unified treatment of these results via tropical geometry and formulate a new sufficient condition for competitive equilibrium when there are only two types of products. Generalizations of our theorem in higher dimensions are equivalent to various forms of the Oda conjecture in algebraic geometry.