Beyond Proportional Individual Guarantees for Binary Perpetual Voting

TL;DR

This paper explores fair decision-making in perpetual binary voting, introducing a new MMS-inspired guarantee, and provides algorithms and impossibility results for different numbers of agents.

Contribution

It introduces a novel MMS-inspired fairness notion for binary perpetual voting and presents algorithms and impossibility results for various agent counts.

Findings

Guaranteed MMS for 3 agents with an online algorithm.

Offline algorithm guarantees MMS for 4 agents.

No online algorithm can guarantee MMS^{adapt} for 7 or more agents.

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.