Traditionally, the problem of apportioning the seats of a legislative body has been viewed as a one-shot process with no dynamic considerations. While this approach is reasonable for some settings, dynamic aspects play an important role in many others. We initiate the study of apportionment problems in an online setting. Specifically, we introduce a framework for proportional apportionment with no information about the future. In this model, time is discrete and there are $n$ parties that receive a certain share of the votes at each time step. An online algorithm needs to irrevocably assign a prescribed number of seats at each time, ensuring that each party receives its fractional share rounded up or down, and that the cumulative number of seats allocated to each party remains close to its cumulative share up to that time. We study deterministic and randomized online apportionment methods. For deterministic methods, we construct a family of adversarial instances that yield a lower bound, linear in $n$, on the worst-case deviation between the seats allocated to a party and its cumulative share. We show that this bound is best possible and is matched by a natural greedy method. As a consequence, a method guaranteeing that the cumulative number of seats assigned to each party up to any step equals its cumulative share rounded up or down (global quota) exists if and only if $n\leq 3$. Then, we turn to randomized allocations and show that, for $n\leq 3$, we can randomize over methods satisfying global quota with the additional guarantee that each party receives, in expectation, its proportional share in every step. Our proof is constructive: Any method satisfying these properties can be obtained from a flow on a recursively constructed network. We showcase the applicability of our results to obtain approximate solutions in the context of online dependent rounding procedures.

翻译：传统上，立法机构席位分配问题被视为一个不考虑动态因素的单一过程。尽管这种方法在某些情境下是合理的，但在许多其他情境中，动态方面起着重要作用。我们开创性地研究了在线环境下的席位分配问题。具体而言，我们引入了一个无未来信息的比例分配框架。在该模型中，时间是离散的，存在 $n$ 个政党，每个时间步获得一定比例的选票。在线算法需要在每个时间步不可撤销地分配规定数量的席位，确保每个政党获得其分数份额的向上或向下取整值，并且分配给每个政党的累计席位数量始终接近其截至该时刻的累计份额。我们研究了确定性和随机性的在线分配方法。对于确定性方法，我们构建了一系列对抗性实例，这些实例产生了一个关于政党分配席位与其累计份额之间最坏情况偏差的下界，该下界与 $n$ 呈线性关系。我们证明这个下界是最优的，并且可以通过一种自然的贪心方法达到。因此，存在一种方法能保证截至任何步骤分配给每个政党的累计席位数量等于其累计份额的向上或向下取整值（全局配额），当且仅当 $n\leq 3$。接着，我们转向随机分配，并证明对于 $n\leq 3$，我们可以对满足全局配额的方法进行随机化，同时额外保证每个政党在每一步中期望获得其比例份额。我们的证明是构造性的：任何满足这些性质的方法都可以通过递归构造的网络上的流来获得。我们展示了我们的结果在在线依赖舍入过程中获得近似解的应用潜力。