Temporal Fair Division of Indivisible Goods with Scheduling
Kui Wang Choi, Minming LI
TL;DR
This paper advances temporal fair division by analyzing TEF1, TEFX, α-TEFX, and TMMS with and without scheduling. It establishes a nuanced boundary between possibility and impossibility, showing that scheduling can yield TEF1 with a buffer $r\ge n/2$ for identical days, but TEFX and TMMS largely persist as unattainable across broad settings. It also provides polynomial-time algorithms achieving $1/2$-approximate TEFX in several restricted domains (notably two agents with generalized binary valuations) and derives value-dependent α-TEFX bounds. The results illuminate the inherent tension between strict temporal fairness and computational feasibility, and they delineate practical trade-offs for systems that require long-run fairness with online arrivals. The work also sets the stage for future work on tighter scheduling bounds, strategy concerns, and broader domain coverage.
Abstract
We study temporal fair division, where agents receive goods over multiple rounds and cumulative fairness is required. We investigate Temporal Envy-Freeness Up to One Good (TEF1) and Up to Any Good (TEFX), its approximation $α$-TEFX, and Temporal Maximin Share (TMMS). Motivated by known impossibilities in standard settings, we consider the model in various restricted settings and extend it by introducing scheduling. Our main contributions draw the boundary between possibility and impossibility. First, regarding temporal fair division without scheduling, we prove that while constant-factor $α$-TEFX is impossible in general, a $1/2$-approximation is achievable for generalized binary valuations and identical days with two agents. Second, regarding temporal fair division with scheduling, we demonstrate that a scheduling buffer of size at least $n/2$ enables TEF1 for identical days. However, we establish that TEFX and TMMS remain largely impossible even with scheduling or restricted domains. These results highlight the inherent difficulty of strict temporal fairness and quantify the trade-offs required to achieve approximation guarantees.