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长且未知：每轮新请求到达时，必须立即做出不可撤销选择，系统需持续满足公平性要求。例如食品银行分配实时到货的捐赠物品、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（例如优先处理哪些提案或案件）。此类场景的关键挑战在于公平性要求通常自然具有\emph{尺度依赖性}。例如在公平物品分配中，通常要求不公平性不超过当前所见物品最高价值。因此公平尺度随时间变化。我们提出基于\emph{亏缺量}的通用在线公平方法，该方法衡量每个要求相对于时变基准的当前短缺程度。在该框架内，我们分析了一个简单的完全在线规则：每轮选择最能改善下一轮亏缺分布的行动。我们证明任意时点（前缀层面）的保证：每轮之后，所有追踪要求均满足，松弛量仅以$\sqrt{t}$量级增长（对数因子内），并证明该增长在一般情况下不可避免。我们将该框架应用于不可分割物品的在线分配（得到比例性和无嫉妒性的自然松弛）以及在线公共决策制定。与以往在线公平分配研究不同，我们的规则无需知晓时间范围（总轮数）或任何未来信息（如最大物品价值）。此外，我们的保证在每个独立时间步上永久成立。