Fair Division with Indivisible Goods, Chores, and Cake

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.