Fair Division with Indivisible Goods, Chores, and Cake
Haris Aziz, Xinhang Lu, Simon Mackenzie, Mashbat Suzuki
TL;DR
This work addresses fair division with a mix of indivisible items and a heterogeneous divisible resource (cake) under additive utilities. It establishes the existence of envy-freeness for mixed resources ($EFM$) in the presence of both cake and indivisible items and, with indivisible items and chores, shows the existence of an EF1 allocation that is envy-freeable, enabling EFM via a cake-to-money reduction. The main technical strategy reduces the problem to finding an EF1 and envy-freeable allocation of the indivisible items by bundling items into meta-goods and applying an iterative maximum-weight matching framework, combined with a cake-to-money reduction to lift the discrete solution to the mixed-resource setting. The results connect envy-freeness, EF1, envy-freeability, and EFM, generalize several core fair-division theorems, and open avenues for extending to broader valuation classes and subsidy-based mechanisms.
Abstract
We study the problem of fairly allocating indivisible items and a desirable heterogeneous divisible good (i.e., cake) to agents with additive utilities. In our paper, each indivisible item can be a good that yields non-negative utilities to some agents and a chore that yields negative utilities to the other agents. Given a fixed set of divisible and indivisible resources, we investigate almost envy-free allocations, captured by the natural fairness concept of envy-freeness for mixed resources (EFM). It requires that an agent $i$ does not envy another agent $j$ if agent $j$'s bundle contains any piece of cake yielding positive utility to agent $i$ (i.e., envy-freeness), and agent $i$ is envy-free up to one item (EF1) towards agent $j$ otherwise. We prove that with indivisible items and a cake, an EFM allocation always exists for any number of agents with additive utilities.