Power Diagram Detection with Applications to Information Elicitation

TL;DR

This paper presents a linear programming method to detect whether a given partition of space is a power diagram, with applications in information elicitation and mechanism design, enabling efficient verification of such structures.

Contribution

It introduces a simple linear program for power diagram detection and explores its applications in information elicitation, peer prediction, and mechanism design.

Findings

Efficient linear program for power diagram detection.

Applicable to decompositions of  and restricted domains.

Facilitates design of incentive-compatible mechanisms.

Abstract

Power diagrams, a type of weighted Voronoi diagrams, have many applications throughout operations research. We study the problem of power diagram detection: determining whether a given finite partition of $\mathbb{R}^d$ takes the form of a power diagram. This detection problem is particularly prevalent in the field of information elicitation, where one wishes to design contracts to incentivize self-minded agents to provide honest information. We devise a simple linear program to decide whether a polyhedral cell decomposition can be described as a power diagram. Further, we discuss applications to property elicitation, peer prediction, and mechanism design, where this question arises. Our model is able to efficiently decide the question for decompositions of $\mathbb{R}^d$ or of a restricted domain in $\mathbb{R}^d$. The approach is based on the use of an alternative representation of power diagrams, and invariance of a power diagram under uniform scaling of the parameters in this representation.