Perpetual Fully-Online Approximate Fairness
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 donations, computing systems repeatedly scheduling scarce resources across users, and institutions making repeated decisions while remaining fair over time. We propose a general approach based on \emph{deficits}, which measure how far the current outcome is from satisfying each fairness requirement. The goal is to keep all deficits small at each time step, without knowing the horizon or future agent valuations. This viewpoint also highlights a natural modeling question for long-running systems: how much of the past should be counted when fairness is evaluated? We first study the full-history model, where all past rounds count equally. We propose an efficient fully-online rule. For $n$ agents, we prove anytime guarantees: after any $t$ rounds, all requirements remain satisfied up to a slack of order $\tilde O(\sqrt{t/n})$. We instantiate the rule for online allocation of indivisible goods, yielding natural relaxations of proportionality and envy-free, and for online public decision-making. We show that this slack is tight even for weak proportionality. For unrestricted classical $\mathrm{EF}c$, the exact worst-case parameter at horizon $T$ is $\lceil T/n\rceil$. We then study discounted-memory fairness, where older deficits carry smaller weight. The same fully-online rule applies to these discounted deficits, and the resulting threshold is controlled by the discount function. In particular, the time dependence is never worse than the full-history $\sqrt t$ dependence. Overall, our results show that memory is a central part of perpetual fairness. The question is not only which requirement to impose, but also how the system should count past unfairness.