We study deterministic mechanism design without money for k-facility location games with envy ratio on a real line segment, where a set of strategic agents report their locations and a social planner locates k facilities for minimizing the envy ratio. The objective of envy ratio, which is defined as the maximum over the ratios between any two agents' utilities, is derived from fair division to measure the fairness with respect to a certain facility location profile.

The problem is studied in two settings. In the homogeneous k-facility location game where k facilities serve the same purpose, we propose a $\frac{2k}{2k-1}$-approximate deterministic group strategyproof mechanism which is also the best deterministic strategyproof mechanism. In the heterogeneous k-facility location game where each facility serves a different purpose, when k is even, we devise an optimal and group strategyproof mechanism; when k is odd, we provide a $\frac{k+1}{k-1}$-approximate deterministic group strategyproof mechanism.

我们研究了在实线细分市场上具有嫉妒比率的k设施定位游戏的不花钱的确定性机制设计，在该策略中一组战略代理人报告了他们的位置，而社会计划者则定位了k设施以最小化嫉妒率。嫉妒比率的目标是定义为任意两个代理商效用之间的比率中的最大值，它是根据公平划分得出的，以衡量相对于特定设施位置状况的公平性。

在两种情况下研究该问题。在k个设施服务于相同目的的齐次k设施定位博弈中，我们提出了一个$\frac{\mathrm{2个}ķ}{\mathrm{2个}ķ-\mathrm{1个}}$-近似确定性组策略证明机制，也是最佳的确定性策略证明机制。在每个设施的服务目的不同的异构k设施位置博弈中，当k为偶数时，我们设计了一种最优的群体策略证明机制。当k为奇数时，我们提供一个$\frac{ķ+\mathrm{1个}}{ķ-\mathrm{1个}}$-近似确定性群体策略证明机制。