Individual-based stability in hedonic games depending on the best or worst players

TL;DR

This paper studies stability in hedonic coalition games based on best or worst players, showing existence, computational methods, and complexity results for different stability concepts.

Contribution

It introduces polynomial-time algorithms for stable partitions in B-hedonic games and analyzes the computational complexity of stability existence in these models.

Findings

Individually stable partitions exist and can be computed efficiently in B-hedonic games.

Existence of Nash stable partitions can be decided in polynomial time for strict preferences.

Checking stability existence is NP-complete in certain cases.

Abstract

We consider coalition formation games in which each player has preferences over the other players and his preferences over coalitions are based on the best player ($\mathcal{B}$-/B-hedonic games) or the worst player ($\mathcal{W}$/W-hedonic games) in the coalition. We show that for $\mathcal{B}$-hedonic games, an individually stable partition is guaranteed to exist and can be computed efficiently. Similarly, there exists a polynomial-time algorithm which returns a Nash stable partition (if one exists) for $\mathcal{B}$-hedonic games with strict preferences. Both $\mathcal{W}$- and W-hedonic games are equivalent if individual rationality is assumed. It is also shown that for B- or $\mathcal{W}$-hedonic games, checking whether a Nash stable partition or an individually stable partition exists is NP-complete even in some cases for strict preferences. We identify a key source of intractability in compact coalition formation games in which preferences over players are extended to preferences over coalitions.