Improved EFX Approximation Guarantees under Ordinal-based Assumptions

Our work studies the fair allocation of indivisible items to a set of agents, and falls within the scope of establishing improved approximation guarantees. It is well known by now that the classic solution concepts in fair division, such as envy-freeness and proportionality, fail to exist in the presence of indivisible items. Unfortunately, the lack of existence remains unresolved even for some relaxations of envy-freeness, and most notably for the notion of EFX, which has attracted significant attention in the relevant literature. This in turn has motivated the quest for approximation algorithms, resulting in the currently best known (φ-1)-approximation guarantee by Amanatidis et al. (2020), where φ equals the golden ratio. So far, it has been notoriously hard to obtain any further advancements beyond this factor. Our main contribution is that we achieve better approximations, for certain special cases, where the agents agree on their perception of some items in terms of their worth. In particular, we first provide an algorithm with a 2/3-approximation, when the agents agree on what are the top n items (but not necessarily on their exact ranking), with n being the number of agents. To do so, we also propose a general framework that can be of independent interest for obtaining further guarantees. Secondly, we establish the existence of exact EFX allocations in a different scenario, where the agents view the items as split into tiers w.r.t. their value, and they agree on which items belong to each tier. Overall, our results provide evidence that improved guarantees can still be possible by exploiting ordinal information of the valuations.