Abstract
In coalition formation games where agents have preferences over coalitions to which they belong, the set of fixed points of an operator and the core of coalition formation games coincide. An acyclicity condition on preference profiles guarantees the existence of a unique core. An algorithm using that operator finds all core partitions whenever there exists one.
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Notes
These games are also called hedonic games.
In economics literature, there are only a few papers on roommate problem compared to the computer science literature. See Gusfield and Irving (1989) for a survey on roommate problem in computer science literature.
Tan (1991) gives conditions (see Chung (2000), p. 211 for a clear explanation of these conditions) necessary and sufficient for the existence of core matchings in roommate markets when preferences are strict. Chung (2000) provides a condition on weak preferences sufficient for the existence of core matchings.
When preferences are strict, labor markets (many-to-one matchings) without peer effects are not special cases of coalition formation games as in those markets workers are indifferent about their co-workers.
A preference profile satisfies the top-coalition property if for each group of agents, there exists a subgroup of agents preferred by each of its members to all other subgroups of the group.
No \(n\)-cycle is a weaker condition than the \(k\)-acyclicity under strict preferences.
A collection of permissible coalitions satisfy the single-lapping property if for each pair of coalitions there exists at most one agent in common, and for each cycle of overlapping coalitions there is one agent common in all of these coalitions in the cycle.
A firm’s preference relation satisfies the substitutes condition if, roughly speaking, adding a worker to the set of workers in consideration does not make the firm include another worker in his choice when this worker was not chosen before.
To see that \(\fancyscript{V^{**}}\setminus \fancyscript{V^{*}}\ne \emptyset \), consider the roommates problem with all pairs permitted and agents’ preferences over pairs as \(\{1,2\}{\succ _1} \{1,3\}{\succ _1}\{1\}, \{2,3\}{\succ _2} \{1,2\}{\succ _2}\{2\}\) and \(\{1,3\}{\succ _3} \{2,3\}{\succ _3}\{3\}\). The core is empty, and the pre-partition (not a partition) \(v(1,2,3)=(\{1,2\},\{2,3\},\{1,3\})\in \fancyscript{V^{**}}\setminus \fancyscript{V^{*}}\).
A coalition \(B\in \fancyscript{K}\) is singleton if \(|B|=1\).
A binary relation \(\trianglerighteq \subseteq X\times X\) is a partial order if it is reflexive, antisymmetric, and transitive.
\(\overline{x}_Y\in X\) is the least upper bound of subset \(Y\subseteq X\) if (i) for each \(y\in Y\), \(\overline{x}_Y\trianglerighteq y\), and (ii) for each \(x\in X\) \(x\trianglerighteq \overline{x}_Y\) where \(x\trianglerighteq y\) for all \(y\in Y\).
\(\underline{x}_Y\in X\) is the greatest lower bound of subset \(Y\subseteq X\) if (i) for each \(y\in Y\), \(y\trianglerighteq \underline{x}_Y\), and (ii) for each \(x\in X\) \(\underline{x}_Y\trianglerighteq x\) where \(y\trianglerighteq x\) for all \(y\in Y\).
I would like to thank to an anonymous referee for correcting the original proof and pointing out this simpler proof.
A matching has a simple extension if unmatched agents can be paired to form another simple matching.
I would like to thank to an anonymous referee for bringing this to my attention.
A coalition is a proper coalition if it is a strict subset of the grand coalition.
Given a non-empty coalition \(D\subseteq N\), a non-empty sub-coalition \(C\subseteq D\) is a top coalition of \(D\) if for each \(i\in C\) and for each \(C'\subseteq D\) with \(i\in C'\), \(C\succeq _i C'\).
In this section, the set of agents \(N\) is assumed to be the set containing natural numbers \(1\) through \(|N|\).
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Acknowledgments
I am grateful to Utku Ünver and Jan Werner for their comments, advice and support. I would like to thank Atila Abdülkadiroğlu, Hüseyin Yıldırım, and the participants of the Society for Economic Design 2011 and Games 2012 conferences for their valuable comments. I also would like to thank Sumru Altuğ for her invaluable support during my visit to Economic Research Forum at Koç University. I also thank the Co-editor Bhaskar Dutta, an associate editor, and two anonymous reviewers for their excellent comments, which significantly improved the paper. All errors are mine.
