Exploring Relations among Fairness Notions in Discrete Fair Division

Fair allocation of indivisible items among agents is a fundamental and extensively studied problem. However, fairness does not have a single universally accepted definition, leading to many competing fairness notions. Some of these notions are considered stronger or more desirable, but they are also more difficult to guarantee. In this work, we examine 22 different fairness notions and organize them into a hierarchy. Formally, we say that a notion $F_1$ implies another notion $F_2$ if every $F_1$-fair allocation is also $F_2$-fair. We give a near-complete picture of implications among fairness notions: for almost every pair of notions, we either prove an implication or give a counterexample demonstrating that the implication does not hold. Although some of these results are already known, many are new. We examine multiple settings, including the allocation of goods, chores, and mixed manna, and different valuation classes like additive, submodular, and subadditive. We believe this work clarifies the relative strengths and applicability of these notions, providing a foundation for future research in fair division. Moreover, we develop an inference engine to automate part of our work. It is available as a user-friendly web application and may have broader applications beyond fair division.