We study fair and efficient allocation of divisible goods, in an online manner, among $n$ agents. The goods arrive online in a sequence of $T$ time periods. The agents' values for a good are revealed only after its arrival, and the online algorithm needs to fractionally allocate the good, immediately and irrevocably, among the agents. Towards a unifying treatment of fairness and economic efficiency objectives, we develop an algorithmic framework for finding online allocations to maximize the generalized mean of the values received by the agents. In particular, working with the assumption that each agent's value for the grand bundle of goods is appropriately scaled, we address online maximization of $p$-mean welfare. Parameterized by an exponent term $p \in (-\infty, 1]$, these means encapsulate a range of welfare functions, including social welfare ($p=1$), egalitarian welfare ($p \to -\infty$), and Nash social welfare ($p \to 0$). We present a simple algorithmic template that takes a threshold as input and, with judicious choices for this threshold, leads to both universal and tailored competitive guarantees. First, we show that one can compute online a single allocation that $O (\sqrt{n} \log n)$-approximates the optimal $p$-mean welfare for all $p\le 1$. The existence of such a universal allocation is interesting in and of itself. Moreover, this universal guarantee achieves essentially tight competitive ratios for specific values of $p$. Next, we obtain improved competitive ratios for different ranges of $p$ by executing our algorithm with $p$-specific thresholds, e.g., we provide $O(\log ^3 n)$-competitive ratio for all $p\in (\frac{-1}{\log 2n},1)$. We complement our positive results by establishing lower bounds to show that our guarantees are essentially tight for a wide range of the exponent parameter.

中文翻译：

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广义平均福利的通用且严格的在线算法

我们以在线方式在 $n$ 代理之间研究可分割商品的公平和有效分配。货物按 $T$ 时间段的顺序在线到达。代理对商品的价值仅在商品到达后才会显示，在线算法需要立即且不可撤销地在代理之间对商品进行部分分配。为了统一处理公平性和经济效率目标，我们开发了一个算法框架，用于查找在线分配以最大化代理接收到的值的广义平均值。特别是，假设每个代理对大量商品的价值得到适当调整，我们解决了 $p$-mean 福利的在线最大化问题。由指数项 $p \in (-\infty, 1]$ 参数化，这些意味着封装了一系列福利函数，包括社会福利（$p=1$）、平等主义福利（$p \to -\infty$）和纳什社会福利（$p \to 0$）。我们提出了一个简单的算法模板，该模板将阈值作为输入，并明智地选择该阈值，从而实现通用和量身定制的竞争保证。首先，我们表明可以在线计算单个分配，即 $O (\sqrt{n} \log n)$-近似于所有 $p\le 1$ 的最优 $p$-mean 福利。这种普遍分配的存在本身就很有趣。此外，这种普遍保证对于 $p$ 的特定价值实现了本质上严格的竞争比率。接下来，我们通过使用特定于 $p$ 的阈值执行我们的算法来获得不同 $p$ 范围的改进竞争比率，例如，我们为所有 $p\in ( \frac{-1}{\log 2n},1)$。