Individual-Based Stability in Hedonic Diversity Games

TL;DR

This paper studies stability concepts in hedonic diversity games, proving computational complexity results and providing efficient algorithms for finding individually stable outcomes, with empirical comparisons of different methods.

Contribution

It extends previous work by establishing NP-completeness for Nash stability, simplifying algorithms for individual stability, and generalizing the model to more than two classes.

Findings

Deciding Nash stability is NP-complete.

All HDGs admit polynomial-time computable IS outcomes.

Empirical analysis compares multiple algorithms and dynamics.

Abstract

In hedonic diversity games (HDGs), recently introduced by Bredereck, Elkind and Igarashi (2019), each agent belongs to one of two classes (men and women, vegetarians and meat-eaters, junior and senior researchers), and agents' preferences over coalitions are determined by the fraction of agents from their class in each coalition. Bredereck et al. show that while an HDG may fail to have a Nash stable (NS) or a core stable (CS) outcome, every HDG in which all agents have single-peaked preferences admits an individually stable (IS) outcome, which can be computed in polynomial time. In this work, we extend and strengthen these results in several ways. First, we establish that the problem of deciding if an HDG has an NS outcome is NP-complete, but admits an XP algorithm with respect to the size of the smaller class. Second, we show that, in fact, all HDGs admit IS outcomes that can be computed in polynomial time; our algorithm for finding such outcomes is considerably simpler than that of Bredereck et al. We also consider two ways of generalizing the model of Bredereck et al. to k > 2 classes. We complement our theoretical results by empirical analysis, comparing the IS outcomes found by our algorithm, the algorithm of Bredereck et al. and a natural better-response dynamics.