We consider competitive facility location as a two-stage multi-agent system with two types of clients. For a given host graph with weighted clients on the vertices, first facility agents strategically select vertices for opening their facilities. Then, the clients strategically select which of the opened facilities in their neighborhood to patronize. Facilities want to attract as much client weight as possible, clients want to minimize congestion on the chosen facility. All recently studied versions of this model assume that clients can split their weight strategically. We consider clients with unsplittable weights but allow mixed strategies. So clients may randomize over which facility to patronize. Besides modeling a natural client behavior, this subtle change yields drastic changes, e.g., for a given facility placement, qualitatively different client equilibria are possible. As our main result, we show that pure subgame perfect equilibria always exist if all client weights are identical. For this, we use a novel potential function argument, employing a hierarchical classification of the clients and sophisticated rounding in each step. In contrast, for non-identical clients, we show that deciding the existence of even approximately stable states is computationally intractable. On the positive side, we give a tight bound of $2$ on the price of anarchy which implies high social welfare of equilibria, if they exist.

翻译：我们考虑将竞争性设施选址建模为一个包含两类客户的两阶段多智能体系统。给定一个在顶点上带有加权客户的宿主图，首先设施智能体策略性地选择顶点以开设其设施。随后，客户策略性地选择其邻域内已开设的设施进行光顾。设施希望吸引尽可能多的客户权重，客户则希望最小化所选设施上的拥堵。该模型所有近期研究版本均假设客户可以策略性地分割其权重。我们考虑具有不可分割权重的客户，但允许混合策略。因此，客户可以随机化选择光顾哪个设施。除了建模一种自然的客户行为外，这一细微变化带来了显著改变，例如，对于给定的设施布局，可能出现性质不同的客户均衡。作为主要结果，我们证明若所有客户权重相同，则纯子博弈完美均衡总是存在。为此，我们采用一种新颖的势函数论证，通过对客户进行分层分类并在每一步进行精细舍入来实现。相比之下，对于非相同客户，我们证明即使判定近似稳定状态的存在性在计算上也是难解的。在积极方面，我们给出了无政府价格的一个紧界$2$，这意味着若均衡存在，则其具有较高的社会福利。