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：我们考虑竞争性设施选址问题，将其建模为包含两类客户的两阶段多智能体系统。给定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开设设施的顶点，随后客户策略性地选择其邻域内已开设的设施进行光顾。设施方希望吸引尽可能多的客户权重，客户方则希望最小化所选设施的拥塞程度。近期所有该模型的研究版本均假设客户可策略性分割自身权重。本文考虑具有不可分割权重的客户，但允许混合策略。即客户可随机选择光顾的设施。除建模自然客户行为外，这一细微变化带来了根本性差异：例如在给定设施布局下，可能产生性质迥异的客户均衡状态。作为核心结论，我们证明当所有客户权重相同时，纯子博弈完美均衡始终存在。为此我们提出新颖的势函数论证方法，采用客户的分层分类机制并结合每步的精细舍入。对比而言，当客户权重不同时，证明判定近似稳定态的存在性在计算上不可解。在积极方面，我们给出社会无政府代价的紧界2，这表明均衡存在时具有高社会福利。