Distance Preservation Games

TL;DR

This paper introduces distance preservation games where agents position themselves to maintain ideal distances, analyzing stability, optimality, computational complexity, and efficiency bounds of such configurations.

Contribution

It formally defines DPGs, proves existence and computational complexity results, and provides algorithms and bounds for stable and welfare-optimized solutions.

Findings

Some DPGs lack jump stable profiles.

Finding welfare optimal profiles is NP-complete.

Price of anarchy is at most 2.

Abstract

We introduce and analyze distance preservation games (DPGs). In DPGs, agents express ideal distances to other agents and need to choose locations in the unit interval while preserving their ideal distances as closely as possible. We analyze the existence and computation of location profiles that are jump stable (i.e., no agent can benefit by moving to another location) or welfare optimal for DPGs, respectively. Specifically, we prove that there are DPGs without jump stable location profiles and identify important cases where such outcomes always exist and can be computed efficiently. Similarly, we show that finding welfare optimal location profiles is NP-complete and present approximation algorithms for finding solutions with social welfare close to optimal. Finally, we prove that DPGs have a price of anarchy of at most $2$.