Control in Hedonic Games
Jiehua Chen, Jakob Guttmann, Merisa Mustajbašić, Sofia Simola
TL;DR
The paper introduces and analyzes control in hedonic games, where an external actor can add or delete agents to enforce NA, PA, or GR under four stability notions (IR, IS, NS, CS) across FriHG and AddHG. It provides a near-complete complexity classification: polynomial-time algorithms exist for several NA/PA objectives in the friend-oriented setting via minimum-weight paths/cycles and Steiner-network reductions; many NS/IS/CS scenarios remain NP-hard or harder, with symmetry or DAG restrictions offering some tractability. For grand coalition control, the AddHG setting exhibits W[2]-hardness with respect to the budget k (even under symmetry), while certain XP algorithms exist for fixed k; deletions often yield easier GR results than additions in friHG. Across both preference models, the paper identifies immune and never cases, giving a nuanced view of how external control interacts with stability concepts, and highlights a rich landscape of positive results and hardness barriers that inform practical design of interventions in coalition formation contexts.
Abstract
We initiate the study of control in hedonic games, where an external actor influences coalition formation by adding or deleting agents. We consider three basic control goals (1) enforcing that an agent is not alone (NA); (2) enforcing that a pair of agents is in the same coalition (PA); (3) enforcing that all agents are in the same grand coalition (GR), combined with two control actions: adding agents (AddAg) or deleting agents (DelAg). We analyze these problems for friend-oriented and additive preferences under individual rationality, individual stability, Nash stability, and core stability. We provide a complete computational complexity classification for control in hedonic games.