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Coalition Formation and Stability

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Abstract

This paper aims to develop, for any cooperative game, a solution notion that enjoys stability and consists of a coalition structure and an associated payoff vector derived from the Shapley value. To this end, two concepts are combined: those of strong Nash equilibrium and Aumann–Drèze coalitional value. In particular, we are interested in conditions ensuring that the grand coalition is the best preference for all players. Monotonicity, convexity, cohesiveness and other conditions are used to provide several theoretical results that we apply to numerical examples including real-world economic situations.

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Notes

  1. A suggestion for which we are grateful to a reviewer.

  2. A survey that shows the impact of the Shapley value in several scientific disciplines is due to Moretti and Patrone (2008).

  3. In a previous work Hart and Kurz (1983), they defined two more notions, \(\alpha \)-stability and \(\beta \)-stability, for NTU cooperative games, but we will restrict our study to TU games.

  4. Here \(u_T\) will not mean the unanimity game associated to coalition T.

  5. The difference between the \(\gamma \)-model and the \(\delta \)-model lies here. In the \(\delta \)-model, Hart and Kurz assume that, if only some members of a coalition choose it, the subcoalition consisting of these members forms, while the others become singletons. Of course, if a coalition forms in the \(\gamma \)-model it also forms in the \(\delta \)-model, but the converse is not true. Here we prefer using the \(\gamma \)-model solely because the subcoalition might have a utility no longer interesting to its members.

  6. The lack of solution appears here and in other examples below. Following Segal (2003), we could call collusion proof to any game where this occurs. In such a game, players are reduced to form, in principle, the trivial structure \(\mathcal {B}^n\), that we have called the “disagreement point”.

  7. In Segal (2003), the expression \([u(T\cup \{i,j\})-u(T\cup \{i\})-u(T\cup \{j\})+u(T)]\) is denoted as \(\Delta _{ij}^2[u](T)\), and \(\Delta _{ij}^2\) is called the “second-order difference operator”.

  8. This inequality is, in fact, equivalent to the convexity of the game (Ichiishi 1993, Theorem 2.1.3).

  9. As we do not impose \(u(N)=1\), we accept as simple the null game \(u=0\), because the restriction of a simple game may well be a null subgame.

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Acknowledgements

The authors wish to thank Prof. Gregory Kersten, Editor-in-Chief, and two anonymous reviewers for their helpful comments. Research partially supported by Grant MTM2015-66818-P of the Economy and Competitiveness Spanish Ministry and the European Regional Development Fund.

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  1. Departament de Matemàtiques, Universitat Politècnica de Catalunya (UPC), ESEIAAT, C. Colom 11, 08222, Terrassa, Barcelona, Spain

    Antonio Magaña & Francesc Carreras

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  1. Antonio Magaña
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  2. Francesc Carreras
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Correspondence to Antonio Magaña.

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Magaña, A., Carreras, F. Coalition Formation and Stability. Group Decis Negot 27, 467–502 (2018). https://doi.org/10.1007/s10726-018-9570-1

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