We initiate the study of a novel problem in mechanism design without money, which we term Truthful Interval Covering (TIC). An instance of TIC consists of a set of agents each associated with an individual interval on a line, and the objective is to decide where to place a covering interval to minimize the total social or egalitarian cost of the agents, which is determined by the intersection of this interval with their individual ones. This fundamental problem can model situations of provisioning a public good, such as the use of power generators to prevent or mitigate load shedding in developing countries. In the strategic version of the problem, the agents wish to minimize their individual costs, and might misreport the position and/or length of their intervals to achieve that. Our goal is to design truthful mechanisms to prevent such strategic misreports and achieve good approximations to the best possible social or egalitarian cost. We consider the fundamental setting of known intervals with equal lengths and provide tight bounds on the approximation ratios achieved by truthful deterministic mechanisms. For the social cost, we also design a randomized truthful mechanism that outperforms all possible deterministic ones. Finally, we highlight a plethora of natural extensions of our model for future work, as well as some natural limitations of those settings.

翻译：我们首次研究了无货币机制设计中的一个新颖问题，称为“真实区间覆盖”（Truthful Interval Covering, TIC）。TIC问题包含一组智能体，每个智能体在直线上关联一个独立区间，目标是通过选择一个覆盖区间来最小化所有智能体的社会总成本或平均主义成本，该成本由所选区间与各智能体区间的交集决定。这一基础问题可建模公共物品供给场景，例如在发展中国家使用发电机预防或缓解负荷削减。在策略性版本中，智能体旨在最小化自身成本，可能虚报其区间的中心和/或长度。我们的目标是设计真实机制以遏制此类策略性虚报，并实现对社会最优总成本或平均主义成本的近似优化。我们考虑已知等长区间的基础设定，并给出了真实确定性机制所能达到的近似比紧界。针对社会成本，我们还设计了一种优于所有确定性机制的随机化真实机制。最后，我们强调了模型中可供未来研究的多种自然扩展方向，以及这些设定中的固有局限性。