Deviation Dynamics in Cardinal Hedonic Games

TL;DR

This paper explores the complexity of convergence in dynamic models of cardinal hedonic games, establishing hardness results and proposing dynamics for finding stable partitions, with implications for various stability notions.

Contribution

It provides meta theorems demonstrating the computational hardness of convergence decisions in dynamic hedonic games and introduces dynamics to find stable partitions.

Findings

Hardness of deciding convergence based on No-instances

CIS dynamics can converge in linear steps from singleton

Worst-case exponential deviations needed for convergence

Abstract

Computing stable partitions in hedonic games is a challenging task because there exist games in which stable outcomes do not exist. Even more, these No-instances can often be leveraged to prove computational hardness results. We make this impression rigorous in a dynamic model of cardinal hedonic games by providing meta theorems. These imply hardness of deciding about the possible or necessary convergence of deviation dynamics based on the mere existence of No-instances. Our results hold for additively separable, fractional, and modified fractional hedonic games (ASHGs, FHGs, and MFHGs). Moreover, they encompass essentially all reasonable stability notions based on single-agent deviations. In addition, we propose dynamics as a method to find individually rational and contractually individual stable (CIS) partitions in ASHGs. In particular, we find that CIS dynamics from the singleton partition possibly converge after a linear number of deviations but may require an exponential number of deviations in the worst case.