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

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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.


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