PhD Defense

(XOS) valuations.

In addition, we consider ordinal approximations of MMS and prove the existence of 1-out-of-d4n/3e MMS allocations in the additive setting. The second part of the dissertation focuses on envy-based fairness notions. For indivisible items, the most prominent envy-based criterion is envy-freeness up to any item (EFX). Although the existence of EFX
allocations remains open in many general settings, we contribute to this area by proving the existence of EFX allocations for three agents under minimal constraints on their valuations. Furthermore, we establish the existence of relaxed forms of EFX, including epistemic EFX, and approximate EFX with charity.

In the third and final part of this thesis, we move beyond single fairness criteria—whether share-based or envy-based—to establish the existence of allocations that satisfy multiple fairness guarantees. Specifically, we prove the existence of (partial) allocations that are 2/3-MMS and EFX simultaneously. This line of research, while less
established, represents a promising direction for future research.