Perpetual Fully Online Horizon-Free Approximate Fairness

2026-05-19

AI Summary

This work addresses the challenge of sustaining fairness guarantees in long-horizon online decision-making under unknown time horizons. The authors propose a general “deficit”-based online fairness framework that, at each round, selects the action optimizing the improvement of the deficit state in the next round. This approach requires no prior knowledge of the total number of rounds or future information—such as the maximum item value—and achieves perpetual, round-by-round fairness guarantees. By integrating a deficit-based metric with a fully online decision rule and employing a prefix-anytime analysis technique, the framework applies to settings including indivisible item allocation and public decision-making. Theoretically, all fairness constraints are satisfied after every round, and the cumulative error grows at a rate of √t (up to logarithmic factors), which is provably unimprovable in general settings.

Abstract

Many decision processes run for a long and unknown duration: in each round new requests arrive, an irrevocable choice must be made immediately, and the system is judged by ongoing fairness requirements. Examples include food banks allocating donated items as they arrive, computing systems repeatedly scheduling scarce resources across users, and institutions making repeated public decisions (e.g., which proposals or cases to prioritize) while remaining fair over time. A key challenge in such settings is that fairness requirements are often naturally \emph{scale-dependent}. For example, in fair item allocation, it is common to require that the unfairness is bounded by the highest values of items seen so far. Thus, the scale of fairness changes over time. We propose a general approach to online fairness based on \emph{deficits}, which measure each requirement's current shortfall relative to a time-varying benchmark. Within this framework, we analyze a simple fully online rule that, in each round, chooses the action that best improves the next-round deficit profile. We prove anytime (prefix-wise) guarantees: after every round, all tracked requirements remain satisfied up to a slack that grows only on the order of $\sqrt{t}$ (up to logarithmic factors), and we show this growth is unavoidable in general. We instantiate the framework for online allocation of indivisible goods (yielding natural relaxations of proportionality and envy-freeness) and for online public decision-making. In contrast to previous works on online fair allocation, our rule does not need to know the horizon (the total number of rounds), nor any other information on the future (e.g. the maximum item value). Moreover, our guarantees hold perpetually, at each individual time step.

Problem

• online fairness
• scale-dependent fairness
• horizon-free
• perpetual fairness
• indivisible goods allocation

Innovation

• online fairness
• horizon-free
• deficit-based framework
• anytime guarantees
• scale-dependent fairness