Fair and Truthful Allocations Under Leveled Valuations
George Christodoulou, Vasilis Christoforidis
TL;DR
This work analyzes fair allocation of indivisible goods under leveled valuations, focusing on MMS and EFX across submodular, XOS, and subadditive subclasses. It delivers constructive MMS guarantees, proving a 2/3-MMS allocation exists for leveled submodular valuations and that two submodular agents also admit a 2/3-MMS allocation, while establishing tight 1/2-MMS bounds for subadditive leveled valuations; it also proves the existence of exact EFX allocations via simple, ordinal-friendly protocols and presents truthful mechanisms achieving constant MMS approximations. The results illuminate the trade-offs between fairness and truthfulness in leveled settings, providing both positive existence results and tight impossibility bounds, and offer mechanism designs (SDQ/Serial Quota Dictatorship) that achieve EFX, MMS guarantees, and strategic robustness. Overall, the paper advances understanding of fair and truthful allocations when agents exhibit leveled preferences, with implications for mechanism design and fair division under restricted valuation structures.
Abstract
We study the problem of fairly allocating indivisible goods among agents which are equipped with {\em leveled} valuation functions. Such preferences, that have been studied before in economics and fair division literature, capture a simple and intuitive economic behavior; larger bundles are always preferred to smaller ones. We provide a fine-grained analysis for various subclasses of leveled valuations focusing on two extensively studied notions of fairness, (approximate) MMS and EFX. In particular, we present a general positive result, showing the existence of $2/3$-MMS allocations under valuations that are both leveled and submodular. We also show how some of our ideas can be used beyond the class of leveled valuations; for the case of two submodular (not necessarily leveled) agents we show that there always exists a $2/3$-MMS allocation, complementing a recent impossibility result. Then, we switch to the case of subadditive and fractionally subadditive leveled agents, where we are able to show tight (lower and upper) bounds of $1/2$ on the approximation factor of MMS. Moreover, we show the existence of exact EFX allocations under general leveled valuations via a simple protocol that in addition satisfies several natural economic properties. Finally, we take a mechanism design approach and we propose protocols that are both truthful and approximately fair under leveled valuations.