Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Truthful Interval Covering

Argyrios Deligkas, Aris Filos-Ratsikas, Alexandros A. Voudouris

TL;DR

Truthful Interval Covering (TIC) studies mechanism design without money for agents each owning a unit interval on a line, where a single covering interval must be positioned to minimize social or max cost while preventing misreporting. In the fundamental equal-length setting, the authors prove a tight deterministic social-cost bound of $2-\frac{2}{n}$ via the Median mechanism and present a universal-truthful randomized $\frac{5}{3}$-approximation, with max-cost results showing deterministic optimality at $2$ and no improvement from randomization. They also analyze two natural extensions: unknown interval lengths yield impossibility results for both objectives, while known but unequal lengths allow a truthfully deployed Max-Length mechanism with linear social-cost approximation and a $2$-approximation for the max-cost. The work outlines rich future directions, including multiple intervals, alternative cost structures, welfare considerations, obnoxious/hybrid models, and higher-dimensional settings, highlighting TIC as a fertile direction in mechanism design without money with practical public-good motivations.

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.

Truthful Interval Covering

TL;DR

Truthful Interval Covering (TIC) studies mechanism design without money for agents each owning a unit interval on a line, where a single covering interval must be positioned to minimize social or max cost while preventing misreporting. In the fundamental equal-length setting, the authors prove a tight deterministic social-cost bound of via the Median mechanism and present a universal-truthful randomized -approximation, with max-cost results showing deterministic optimality at and no improvement from randomization. They also analyze two natural extensions: unknown interval lengths yield impossibility results for both objectives, while known but unequal lengths allow a truthfully deployed Max-Length mechanism with linear social-cost approximation and a -approximation for the max-cost. The work outlines rich future directions, including multiple intervals, alternative cost structures, welfare considerations, obnoxious/hybrid models, and higher-dimensional settings, highlighting TIC as a fertile direction in mechanism design without money with practical public-good motivations.

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.
Paper Structure (16 sections, 16 theorems, 28 equations, 1 figure)

This paper contains 16 sections, 16 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work and Discussion
  4. The Setting
  5. Social Cost: Deterministic Mechanisms
  6. Social Cost: Randomized Mechanisms
  7. Max Cost
  8. Two Natural Extensions
  9. Unknown interval lengths
  10. Known but unequal interval lengths
  11. Other Open Problems and Directions
  12. Multiple intervals.
  13. Different cost functions and objectives.
  14. Utilities and social welfare.
  15. Obnoxious and hybrid models.
  16. ...and 1 more sections

Key Result

Theorem 2.2

For any $k \in [n]$, the $k$-ordered statistic mechanism is truthful. Furthermore, any convex combination over $k$-ordered statistic mechanisms is universally truthful.

Figures (1)

  • Figure 1: The two worst-case instances for the Uniform-Statistic mechanism. In each figure there are some singleton agents and a group of agents whose intervals overal (these are depicted as a shaded rectangle).

Theorems & Definitions (34)

  • Definition 2.1: $k$-ordered statistic
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 24 more