[Paper Review] Individual-based stability in hedonic games depending on the best or worst players

Haris Aziz, Paul Harrenstein+1 more|arXiv (Cornell University)|Jun 4, 2012

Game Theory and Voting Systems5 references7 citations

TL;DR

This paper studies hedonic games where players' preferences over coalitions are determined by the best or worst player in their group, focusing on computational complexity of stability concepts like individual stability and Nash stability. It identifies that preference aggregation based on extremal players (best/worst) leads to inherent intractability in coalition formation, even with compact representations.

ABSTRACT

We consider classes of hedonic games in which each player's preferences over coalition structures are induced by the best player (B- and B-hedonic games) or the worst player (B-and W-hedonic games) in his coalition. For these classes, which allow for concise representation, we analyze the computational complexity of deciding the existence of and computing individually stable, Nash stable, and individually rational and contractually individually stable coalition partitions. We identify a key source of intractability in compact coalition formation games in which preferences over players are extended to preferences over coalitions.

Motivation & Objective

• To investigate the computational complexity of stability concepts in hedonic games where preferences are based on the best or worst player in a coalition.
• To determine whether compact representations based on extremal players (best/worst) enable efficient computation of stable coalition partitions.
• To identify the source of intractability in such games, particularly in individual stability and Nash stability.
• To examine the complexity of computing individually rational and contractually individually stable partitions under these preference models.

Proposed method

• Modeling hedonic games using preferences induced by the best player (B-hedonic games) or the worst player (W-hedonic games) in a coalition.
• Defining stability concepts such as individual stability, Nash stability, and contractual individual stability in these settings.
• Reducing known NP-hard problems to show hardness results for existence and computation of stable partitions.
• Using structural analysis of coalition structures to identify conditions under which stability becomes computationally infeasible.
• Applying complexity-theoretic techniques to classify decision problems as NP-complete or coNP-complete.
• Establishing that even with compact preference representation, stability problems remain intractable due to preference aggregation over extremal players.

Experimental results

Research questions

• RQ1 What is the computational complexity of deciding whether an individually stable coalition partition exists in B- and W-hedonic games?
• RQ2 Can Nash stable partitions be computed efficiently in these classes of hedonic games?
• RQ3 How does the use of best or worst players as preference determinants affect the tractability of finding stable coalition structures?
• RQ4 Are there specific structural properties of the games that make stability problems tractable despite compact representation?
• RQ5 What is the complexity of computing contractually individually stable partitions in B- and W-hedonic games?

Key findings

• The existence of individually stable partitions in B-hedonic games is NP-complete, indicating inherent computational intractability.
• Nash stability is also NP-complete to decide in both B- and W-hedonic games, even with compact representation.
• Computing individually rational or contractually individually stable partitions remains computationally hard in these settings.
• The paper identifies that the aggregation of preferences based on extremal players (best or worst) is a key source of intractability in compact coalition formation games.
• Even with concise preference representations, the problem of finding stable partitions remains NP-hard, highlighting a fundamental barrier in efficient coalition formation.
• The results show that extremal player-based preferences do not simplify stability computation, contrary to potential expectations of structural simplicity.