Computing competitive equilibria with mixed manna

Fair division is the problem of allocating a set of items among a set of agents in a fair and efficient way. It arises naturally in a wide range of real-life settings. Competitive equilibrium (CE) is a central solution concept in economics to study markets, and due to its remarkable fairness and efficiency properties (e.g., envy-freeness, proportionality, core stability, Pareto optimality), it is also one of the most preferred mechanisms for fair division even though there is no money involved. The vast majority of work in fair division focuses on the case of disposable goods, which all agents like or can throw away at no cost. In this paper, we consider the case of mixed manna under linear utilities where some items are positive goods liked by all agents, some are bads (chores) that no one likes, and remaining some agents like and others dislike. The recent work of Bogomolnaia et al. [13] initiated the study of CE in mixed manna. They establish that a CE always exists and maintains all the nice properties found in the case of all goods. However, computing a CE of mixed manna is genuinely harder than in the case of all goods due to the non-convex and disconnected nature of the CE set. Our main result is a polynomial-time algorithm for computing a CE of mixed manna when the number of agents or items is constant.