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.

翻译：许多决策过程运行时间漫长且未知：在每一轮中，新请求抵达，必须立即做出不可撤销的选择，且系统需依据持续的公平性要求进行评判。例如，食品银行按抵达顺序分配捐赠物品、计算系统反复在用户间调度稀缺资源，以及机构在保持长期公平的同时做出重复性公共决策（如优先处理哪些提案或案例）。此类场景的关键挑战在于，公平性要求通常具有天然的“尺度依赖性”。例如，在公平物品分配中，通常要求不公平性受限于迄今所见物品的最高价值。因此，公平性的尺度随时间变化。我们提出一种基于“赤字”（衡量各要求相对于时变基准的当前缺口）的在线公平通用方法。在此框架内，我们分析了一个简单的完全在线规则：每轮选择能最大程度改善下一轮赤字配置的行动。我们证明了任意时刻（前缀级）保证：在每一轮后，所有被追踪的要求均能在松弛范围内得到满足，该松弛仅以 $\sqrt{t}$ 量级增长（含对数因子），并证明此增长在一般情况下不可避免。我们为该框架具体应用于不可分割物品的在线分配（产生比例性和无嫉妒性的自然松弛）及在线公共决策。与以往在线公平分配工作不同，我们的规则无需知晓时间跨度（总轮数）或任何未来信息（如最大物品价值）。此外，我们的保证永久成立，适用于每个独立时间步。