We prove that the classic problem of finding a competitive equilibrium in an exchange economy with indivisible goods, money, and unit-demand agents is PPAD-complete. In this "housing market", agents have preferences over the house and amount of money they end up with, but can experience income effects. Our results contrast with the existence of polynomial-time algorithms for related problems: Top Trading Cycles for the "housing exchange" problem in which there are no transfers and the Hungarian algorithm for the "housing assignment" problem in which agents' utilities are linear in money. Along the way, we prove that the Rainbow-KKM problem, a total search problem based on a generalization by Gale of the Knaster-Kuratowski-Mazurkiewicz lemma, is PPAD-complete. Our reductions also imply bounds on the query complexity of finding competitive equilibrium.

翻译：我们证明了在具有不可分割商品、货币和单位需求代理的交换经济中，寻找竞争均衡的经典问题属于PPAD完全问题。在这个"住房市场"中，代理人对最终获得的房屋和货币金额有偏好，但可能受到收入效应的影响。我们的结果与相关问题的多项式时间算法形成对比：用于"住房交换"问题（无转移支付）的顶级交易循环算法，以及用于"住房分配"问题（代理人效用与货币呈线性关系）的匈牙利算法。在此过程中，我们证明了Rainbow-KKM问题——基于Gale对Knaster-Kuratowski-Mazurkiewicz引理的推广的一个总搜索问题——属于PPAD完全问题。我们的归约还隐含了关于寻找竞争均衡查询复杂度的界限。