Fair division with convertible items

Abstract

This work addresses the allocation of goods or chores among agents with heterogeneous valuations. While traditional models focus on either divisible or indivisible items, real-world scenarios often involve both. We introduce a model that allows converting some indivisible items into divisible ones, while preserving agents' heterogeneous utilities. We define Feasible Allocation and introduce new efficiency notions based on discrete-Pareto Optimality (dPO) and fractional-Pareto Optimality (fPO). Our key results show that a bounded number of converted items ensures an EF allocation. We can always find an instance satisfying EF + Restricted-fPO (R-fPO), but EF + Global-PO (G-PO) may not always exist for both goods and chores settings. Futhermore, we also proved that decreasing the number of converted items by one may violate the EF guarantees. Our model bridges the gap between divisible and indivisible item allocation and provides new insights for fair division in mixed-resource settings.