Beyond Proportional Individual Guarantees for Binary Perpetual Voting
Yotam Gafni, Ben Golan
TL;DR
The paper investigates fair perpetual voting in binary decision settings by introducing an adaptive maxi-min-share notion ($MMS^{adapt}$) that captures an individual’s guaranteed number of favorable decisions under adversarial bundle assignments. It develops online and offline algorithms (notably graceful sequences and per-type round-robin variants) and establishes precise existence and nonexistence results across small and large numbers of agents ($n$), showing $n=3$ is achievable online, $n=4$ offline, and $n\ge 7$ impossible online; it also demonstrates that MNW can significantly underperform $MMS^{adapt}$ (no better than $O(1/n)$ in the worst case) and relates MMS to a Random Dictator baseline. The work clarifies the landscape of fair perpetual decision-making, highlighting the trade-offs between adaptivity, computational feasibility, and fairness guarantees in contexts like DAOs. It also contrasts adaptive MMS with egalitarian variants, motivating further study of existence frontiers and practical mechanisms for scalable, transparent governance.
Abstract
Perpetual voting studies fair collective decision-making in settings where many decisions are to be made, and is a natural framework for settings such as parliaments and the running of blockchain Decentralized Autonomous Organizations (DAOs). We focus our attention on the binary case (YES/NO decisions) and \textit{individual} guarantees for each of the participating agents. We introduce a novel notion, inspired by the popular maxi-min-share (MMS) for fair allocation. The agent expects to get as many decisions as if they were to optimally partition the decisions among the agents, with an adversary deciding which of the agents decides on what bundle. We show an online algorithm that guarantees the MMS notion for $n=3$ agents, an offline algorithm for $n=4$ agents, and show that no online algorithm can guarantee the $MMS^{adapt}$ for $n\geq 7$ agents. We also show that the Maximum Nash Welfare (MNW) outcome can only guarantee $O(\frac{1}{n})$ of the MMS notion in the worst case.