资源不足情形下带容量设施选址的机制设计
Mechanism Design for Locating Facilities with Capacities with Insufficient Resources
AI总结：
本文研究资源不足的带容量设施选址问题，提出结合绝对真实性与均衡稳定性的机制，证明百分位机制满足该性质并给出近似比上界，同时扩展至高维场景。
摘要：
本文探讨了总设施容量小于代理数量的 $m$-Capacitated Facility Location Problem 的机制设计问题。遵循 Aziz 等人提出的框架，设施选址的社会福利由先到先得（FCFS）博弈决定，代理们在设施位置确定后展开竞争。当设施数量 $m > 1$ 时，FCFS 博弈的纳什均衡（NE）不唯一，导致代理的效用和真实性的概念变得不明确。为解决这些问题，我们考虑了绝对真实的机制，即无论在 FCFS 博弈中采取何种策略，都能防止代理误报的机制。我们将这一更严格的真实性要求与均衡稳定（ES）机制的概念相结合，这类机制的社会福利不依赖于 FCFS 博弈的 NE。我们证明了百分位机制类是绝对真实的，并确定了其满足 ES 的条件。我们还表明每个 ES 百分位机制的近似比是有界的，并计算了其具体值。值得注意的是，当所有设施容量相同且代理数量足够大时，可以实现小于 $1+\ rac{1}{2m-1}$ 的近似比。最后，我们将研究扩展到高维问题。在此框架下，我们证明 ES 百分位机制类受到的限制更为严格，并对同时满足 ES 和绝对真实性的机制进行了刻画。我们进一步通过实验评估了代理作为分布样本时机制的性能，以此支持我们的理论发现。
英文摘要：
This paper explores the Mechanism Design aspects of the $m$-Capacitated Facility Location Problem where the total facility capacity is less than the number of agents. Following the framework outlined by Aziz et al., the Social Welfare of the facility location is determined through a First-Come-First-Served (FCFS) game, in which agents compete once the facility positions are established. When the number of facilities is $m > 1$, the Nash Equilibrium (NE) of the FCFS game is not unique, making the utility of the agents and the concept of truthfulness unclear. To tackle these issues, we consider absolutely truthful mechanisms, i.e. mechanisms that prevent agents from misreporting regardless of the strategies used during the FCFS game. We combine this stricter truthfulness requirement with the notion of Equilibrium Stable (ES) mechanisms, which are mechanisms whose Social Welfare does not depend on the NE of the FCFS game. We demonstrate that the class of percentile mechanisms is absolutely truthful and identify the conditions under which they are ES. We also show that the approximation ratio of each ES percentile mechanism is bounded and determine its value. Notably, when all the facilities have the same capacity and the number of agents is sufficiently large, it is possible to achieve an approximation ratio smaller than $1+\frac{1}{2m-1}$. Finally, we extend our study to encompass higher-dimensional problems. Within this framework, we demonstrate that the class of ES percentile mechanisms is even more restricted and characterize the mechanisms that are both ES and absolutely truthful. We further support our findings by empirically evaluating the performance of the mechanisms when the agents are the samples of a distribution.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
