arXiv 2109.008742021-09-02cs.GT

广义平均福利的泛紧在线算法

Universal and Tight Online Algorithms for Generalized-Mean Welfare

Siddharth Barman, Arindam Khan, Arnab Maiti

备注：: 32 pages

摘要：

我们研究公平和有效的分配可分割的货物，在网上的方式，在$n$代理。货物按$T$时间段的顺序在线到达。代理商对商品的价值只有在商品到达后才能显示，在线算法需要在代理商之间立即和可追溯地分配商品。为了统一处理公平和经济效率的目标，我们开发了一个算法框架，以找到在线分配，最大限度地提高代理商收到的值的广义平均值。特别是，工作的假设，每个代理商的价值的大捆绑的商品是适当的规模，我们解决在线最大化的$p$ -平均福利。通过指数项$p \in (-\infty, 1]$进行参数化，这些平均值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，需要一个阈值作为输入，明智的选择这个阈值，导致普遍和量身定制的竞争保证。首先，我们可以在线计算一个单一的分配，$O (\sqrt{n} \log n)$ -近似的最佳$p$ -平均福利的所有$p\le 1$。这种普遍分配的存在本身就令人感兴趣。此外，这种普遍的保证实现了$p$的特定值的基本上严格的竞争比。接下来，我们通过使用$p$特定阈值执行我们的算法来获得不同范围的$p$的改进的竞争比，例如，我们提供了所有$p\in (\frac{-1}{\log 2n},1)$的$O(\log ^3 n)$ -竞争比。我们补充我们的积极成果，建立下界表明，我们的保证基本上是紧的指数参数的范围很广。

英文摘要：

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.

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摘要：

英文摘要：

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