In this paper, we study the optional preference model for the facility location game with two heterogeneous facilities on a line interval [0, 1], by further enforcing the requirement of a minimum distance \(0\le d\le 1\) between the two facilities. Each agent has one of three favorable preferences towards the two facilities, i.e., facility 1, facility 2, or optional preference. Here, we consider two variants of the optional preference model: Min (caring for the closer one) and Max (caring for the further one). In both variants, each agent wishes to get close to his preferred facilities, and thus his cost is his distance to his preferred facility. In this game, we consider agents’ locations as public information and agents’ preferences as private information which needs to be reported by agents. The objective is to design a mechanism for the two facilities’ locations such as to minimize the maximum cost of agents (MinMax) and achieve truthful report of agents’ preferences. Given any value of d, for both variants, we propose a strategyproof mechanism with an approximation ratio of 2. We also establish lower bounds of any deterministic strategyproof mechanism for both variants and show that the gaps between the lower bounds and the upper bounds are relatively small.

在本文中，我们研究了直线间隔 [0, 1] 上两个异构设施的设施选址博弈的可选偏好模型，进一步强制要求之间的最小距离 \(0\le d\le 1 \ )两个设施。每个代理对这两个设施具有三个有利偏好之一，即设施1、设施2或可选偏好。在这里，我们考虑可选偏好模型的两种变体：Min（关心较近的一个）和Max（关心较远的一个）。在这两种变体中，每个代理都希望接近他的首选设施，因此他的成本就是他到他的首选设施的距离。在这个游戏中，我们将代理的位置视为公共信息，将代理的偏好视为私人信息，需要代理报告。目标是为两个设施的位置设计一种机制，例如最小化代理商的最大成本（MinMax）并实现代理商偏好的真实报告。给定任何d值，对于这两种变体，我们提出了一种近似比为 2 的策略证明机制。我们还为这两种变体建立了任何确定性策略证明机制的下界，并表明下界和上限之间的差距是相对较大的。小的。