Mixed Fair Division: A Survey
Shengxin Liu, Xinhang Lu, Mashbat Suzuki, Toby Walsh
TL;DR
This survey maps the landscape of fair division when resources are mixed across three core domains: indivisible goods and chores, combinations of divisible and indivisible goods (mixed goods), and indivisible goods coupled with subsidies. It unifies cake-cutting foundations with discrete fair division, introducing and systematizing solution concepts such as $EF$, $EF1$, $EFX$, $PROP$, and $MMS$ across these mixed models. It surveys existence results and algorithms for envy-freeness relaxations, MMS guarantees, and proportionality, highlighting both positive results (e.g., EF1 and MMS in several mixed settings, $EFM$ and $PROP-\alpha$ in mixed resources) and open questions (notably EF1 and PO coexistence in multi-agent mixed domains). It further discusses subsidies and transfer payments as mechanisms to achieve envy-freeness or proportionality, analyzes their bounds and computational aspects, and points to future directions including randomized mechanisms, externalities, and broader applicability to public-resource allocation. Overall, the work synthesizes state-of-the-art theory and identifies key gaps and promising avenues for algorithm design in realistic mixed-resource environments.
Abstract
Fair division considers the allocation of scarce resources among agents in such a way that every agent gets a fair share. It is a fundamental problem in society and has received significant attention and rapid developments from the game theory and artificial intelligence communities in recent years. The majority of the fair division literature can be divided along at least two orthogonal directions: goods versus chores, and divisible versus indivisible resources. In this survey, besides describing the state of the art, we outline a number of interesting open questions and future directions in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (mixed goods), and (iii) indivisible goods with subsidy which can be viewed like a divisible good.