Individual Rationality in Topological Distance Games is Surprisingly Hard

TL;DR

This paper investigates the computational difficulty of ensuring individually rational outcomes in topological distance games, revealing that even simple cases are often intractable and that restrictions are needed for tractability.

Contribution

It provides a comprehensive complexity analysis of the problem and identifies conditions under which finding rational solutions becomes computationally feasible.

Findings

Deciding existence of rational outcomes is intractable in basic cases.

Combining restrictions like number of agents and topology can lead to tractability.

Even minimal stability requirements are computationally hard to achieve.

Abstract

In the recently introduced topological distance games, strategic agents need to be assigned to a subset of vertices of a topology. In the assignment, the utility of an agent depends on both the agent's inherent utilities for other agents and its distance from them on the topology. We study the computational complexity of finding individually rational outcomes; this notion is widely assumed to be the very minimal stability requirement and requires that the utility of every agent in a solution is non-negative. We perform a comprehensive study of the problem's complexity, and we prove that even in very basic cases, deciding whether an individually rational solution exists is intractable. To reach at least some tractability, one needs to combine multiple restrictions of the input instance, including the number of agents and the topology and the influence of distant agents on the utility.