Truthful Interval Covering

TL;DR

This paper introduces the Truthful Interval Covering problem in mechanism design without money, focusing on designing truthful mechanisms for placing a covering interval to minimize social or egalitarian costs, with tight bounds and improved randomized solutions.

Contribution

It formalizes the TIC problem, provides tight bounds for deterministic mechanisms, and develops a randomized truthful mechanism that surpasses deterministic approaches.

Findings

Tight bounds on approximation ratios for deterministic mechanisms.

A new randomized truthful mechanism outperforming deterministic ones.

Analysis of extensions and limitations of the TIC model.

Abstract

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.