Solving Four Open Problems about Core Stability in Altruistic Hedonic Games

TL;DR

This paper addresses four open problems in altruistic hedonic games by proving their core stability verification problems are coNP-complete, using complex gadget-based reductions in social network models.

Contribution

It resolves four previously open complexity questions about core stability verification in altruistic hedonic games, establishing coNP-completeness for each variant.

Findings

Core stability verification is coNP-complete for four variants of altruistic hedonic games.

The proofs involve intricate gadget constructions in social network graphs.

Results close open questions in the computational complexity of coalition stability.

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.