Fair and Efficient Allocations Without Obvious Manipulations

TL;DR

This paper studies fair and efficient allocations of indivisible goods among strategic agents with additive valuations under a relaxed incentive model called non-obvious manipulability ($NOM$). It shows that EF1 allocations can be achieved by deterministic $NOM$ mechanisms (e.g., Round-Robin) and that a utilitarian SW maximizer can be $NOM$ for $n\ge3$ (but not for $n=2$), while egalitarian and Nash SW maximizers are $NOM$-impossible for any $n$. A central contribution is a black-box reduction from $EF1$+$NOM$ mechanism design to $EF1$ algorithm design that preserves Pareto efficiency, enabling construction of NOM mechanisms from EF1 algorithms and connecting fairness guarantees with $NOM$ properties. The work also derives best-of-both-worlds impossibilities, showing that certain ex-ante and ex-post guarantees cannot be achieved together even when $NOM$ is allowed, and positions these results with respect to existing literature on fairness, efficiency, and practical mechanism design without monetary transfers. Overall, the paper builds a cohesive framework linking $EF1$, $NOM$, and various welfare objectives, yielding both positive algorithmic results and fundamental limitations for robust fair allocation without money.

Abstract

We consider the fundamental problem of allocating a set of indivisible goods among strategic agents with additive valuation functions. It is well known that, in the absence of monetary transfers, Pareto efficient and truthful rules are dictatorial, while there is no deterministic truthful mechanism that allocates all items and achieves envy-freeness up to one item (EF1), even for the case of two agents. In this paper, we investigate the interplay of fairness and efficiency under a relaxation of truthfulness called non-obvious manipulability (NOM), recently proposed by Troyan and Morrill. We show that this relaxation allows us to bypass the aforementioned negative results in a very strong sense. Specifically, we prove that there are deterministic and EF1 algorithms that are not obviously manipulable, and the algorithm that maximizes utilitarian social welfare (the sum of agents' utilities), which is Pareto efficient but not dictatorial, is not obviously manipulable for $n \geq 3$ agents (but obviously manipulable for $n=2$ agents). At the same time, maximizing the egalitarian social welfare (the minimum of agents' utilities) or the Nash social welfare (the product of agents' utilities) is obviously manipulable for any number of agents and items. Our main result is an approximation preserving black-box reduction from the problem of designing EF1 and NOM mechanisms to the problem of designing EF1 algorithms. En route, we prove an interesting structural result about EF1 allocations, as well as new "best-of-both-worlds" results (for the problem without incentives), that might be of independent interest.

Theorems & Definitions (34)

• Definition 1: Not Obviously Manipulable troyan2020obvious
• Theorem 1
• proof
• Corollary 1
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof