Equitable Allocations of Mixtures of Goods and Chores

Equitable allocation of indivisible items involves partitioning the items among agents such that everyone derives (almost) equal utility. We consider the approximate notion of \textit{equitability up to one item} (EQ1) and focus on the settings containing mixtures of items (goods and chores), where an agent may derive positive, negative, or zero utility from an item. We first show that -- in stark contrast to the goods-only and chores-only settings -- an EQ1 allocation may not exist even for additive $\{-1,1\}$ bivalued instances, and its corresponding decision problem is computationally intractable. We focus on a natural domain of normalized valuations where the value of the entire set of items is constant for all agents. On the algorithmic side, we show that an EQ1 allocation can be computed efficiently for (i) $\{-1, 0, 1\}$ normalized valuations, (ii) objective but non-normalized valuations, (iii) two agents with type-normalized valuations. Previously, EQX allocations were known to exist only for 2 agents and objective valuations, while the case of subjective valuations remained computationally intractable even with two agents. We make progress by presenting an efficient algorithm that outputs an EQX allocation for $\{-1,1\}$ normalized subjective valuations for any number of agents. We complement our study by providing a comprehensive picture of achieving EQ1 allocations in conjunction with economic efficiency notions such as Pareto optimality and social welfare.