Solving Four Open Problems about Core Stability in Altruistic Hedonic Games
Jörg Rothe, Ildikó Schlotter
TL;DR
This work resolves four open questions on the computational complexity of verifying core stability in altruistic hedonic games (AHGs) across average- and minimum-based, EQ and AL preference models. It develops clique-based reductions from Clique, employing sophisticated gadgets (circulant and dome structures) to ensure that any blocking coalition corresponds to a clique, establishing coNP-completeness for all four variants. The results extend the hardness landscape from SF-based models to the more altruistic EQ and AL frameworks within both AHGs and broader CFGs. The findings have implications for understanding stability verification in friend-centered coalition formation and guide future inquiries into related stability notions and algorithmic feasibility.
Abstract
Hedonic games -- at the interface of cooperative game theory and computational social choice -- are coalition formation games in which the players have preferences over the coalitions they can join. Kerkmann et al. [13] introduced altruistic hedonic games where the players' utilities depend not only on their own but also on their friends' valuations of coalitions. The complexity of the verification problem for core stability has remained open in four variants of altruistic hedonic games: namely, for the variants with average- and minimum-based "equal-treatment" and "altruistic-treatment" preferences. We solve these four open questions by proving the corresponding problems coNP-complete; our reductions rely on rather intricate gadgets in the related networks of friends.