Fair Division Beyond Monotone Valuations

This paper studies fair division of divisible and indivisible items among agents whose cardinal preferences are not necessarily monotone. We establish the existence of fair divisions and develop approximation algorithms to compute them. We address two complementary valuation classes, subadditive and nonnegative, which go beyond monotone functions. Considering both the division of cake (divisible resources) and allocation of indivisible items, we obtain fairness guarantees in terms of (approximate) envy-freeness (EF) and equability (EQ) In the context of envy-freeness, we prove that an EF division of a cake always exists under cake valuations that are subadditive and globally nonnegative. This result complements the nonexistence of EF allocations for burnt cakes known for more general valuations. In the indivisible-items setting, we establish the existence of EF3 allocations for subadditive and globally nonnegative valuations. In addition, we obtain universal existence of EF3 allocations under nonnegative valuations. We study equitability under nonnegative valuations. Here, we prove that EQ3 allocations always exist when the agents' valuations are nonnegative. Also, in the indivisible-items setting, we develop an approximation algorithm that, for given nonnegative valuations, finds allocations that are equitable within additive margins. Our results have combinatorial implications. For instance, the developed results imply the universal existence of proximately equitable graph cuts: Given any graph $G=(V,E)$ and integer $k$ (at most $|V|$), there always exists a partition $V_1, V_2, \ldots, V_k \neq \emptyset$ of the vertex set such that the cut function values of the parts, $V_i$, are additively within $5 \Delta +1$ of each other; here, $\Delta$ is the maximum degree of $G$. Further, such a partition can be computed efficiently.