Skip to main content
SpringerLink
Log in

Optimal deterrence of cooperation

  • Original Paper
  • Published:
International Journal of Game Theory Aims and scope Submit manuscript

Abstract

We introduce axiomatically a new solution concept for cooperative games with transferable utility inspired by the core. While core solution concepts have investigated the sustainability of cooperation among players, our solution concept, called contraction core, focuses on the deterrence of cooperation. The main interest of the contraction core is to provide a monetary measure of the robustness of cooperation in the grand coalition. We motivate this concept by providing optimal fine imposed by competition authorities for the dismantling of cartels in oligopolistic markets. We characterize the contraction core on the set of balanced cooperative games with transferable utility by four axioms: the two classic axioms of non-emptiness and individual rationality, a superadditivity principle and a weak version of a new axiom of consistency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The developing theory of oligopoly TU-games comprises many contributions such as Zhao (1999), Norde et al. (2002), Driessen and Meinhardt (2005), Lardon (2012) and Lekeas and Stamatopoulos (2014) among others.

  2. Bejan and Gómez (2012) use a more relaxed feasibility condition based on first best time allocation.

  3. Precisely, the second best time allocation is also a first best one in the non-generical case where \(X_{\Lambda ^*}(N,v)=X_{\Lambda }(N,v)\).

  4. We need to use the set of efficient payoff vectors in the definition of the contraction core in order to deal with the one-player case.

  5. While the contraction core is defined on the set \(\Gamma _c\), the core, the aspiration core and the weak least core are defined on the set \(\Gamma \).

  6. Shapley and Shubik (1966) also define another generalization of the core called the strong \(\varepsilon \)-core. In this case, every coalition faces to the same cost \(\varepsilon \) regardless of its cardinality.

  7. Observe that t \((N,v)=0\) for the one-player case since no cooperation occurs.

  8. We refer to Bejan and Gómez (2009) for a detailed discussion on tax rules.

  9. This is a consequence of the symmetric cost assumption.

  10. Our proof is inspired from that in Peleg and Sudhölter (2003) in the case where \(n\ge 3\). Nevertheless, the main difference is that we do not need to distinguish cases \(n=2\) and \(n\ge 3\). Furthermore, while the weaker version of their axiom of consistency is established for \(s\in \lbrace {1,2}\rbrace \), our weaker version only requires that \(s=1\).

References

  • Albers W (1979) Core and kernel variants based on imputations and demand profiles. In: Moeschlin O, Pallaschke D (eds) Game theory and related fields. North-Holland, Amsterdam

    Google Scholar 

  • Bejan C, Gómez JC (2009) Core extensions for non-balanced TU-games. Int J Game Theory 38:3–16

    Article  Google Scholar 

  • Bejan C, Gómez JC (2012) Axiomatizing core extensions. Int J Game Theory 41:885–898

    Article  Google Scholar 

  • Bennett E (1983) The aspiration approach to predicting coalition formation and payoff distribution in side-payment games. Int J Game Theory 12(1):1–28

    Article  Google Scholar 

  • Bondareva ON (1963) Some applications of linear programming methods to the theory of cooperative games. Problemi Kibernetiki 10:119–139

    Google Scholar 

  • Calleja P, Rafels C, Tijs S (2009) The aggregate-monotonic core. Games Econ Behav 66(2):742–748

    Article  Google Scholar 

  • Chander P, Tulkens H (1997) The core of an economy with multilateral environmental externalities. Int J Game Theory 26:379–401

    Article  Google Scholar 

  • Cross JG (1967) Some theoretic characteristics of economic and political coalitions. J Confl Resolut 11:184–195

    Article  Google Scholar 

  • D’Aspremont C, Jacquemin A (1988) Cooperative and noncooperative R&D in duopoly with spillovers. Am Econ Rev 78(5):1133–1137

    Google Scholar 

  • Davis M, Maschler M (1965) The kernel of a cooperative game. Nav Res Logist Q 12(3):223–259

    Article  Google Scholar 

  • Driessen TS, Meinhardt HI (2005) Convexity of oligopoly games without transferable technologies. Math Soc Sci 50:102–126

    Article  Google Scholar 

  • Gillies DB (1953) Some theorems on n-person games. Princeton University Press, Princeton

    Google Scholar 

  • Gonzalez S, Grabisch M (2015a) Autonomous coalitions. Ann Oper Res 235(1):301–317

    Article  Google Scholar 

  • Gonzalez S, Grabisch M (2015b) Preserving coalitional rationality for non-balanced games. Int J Game Theory 44(3):733–760

    Article  Google Scholar 

  • Gonzalez S, Grabisch M (2016) Multicoalitional solutions. J Math Econ 64:1–10

    Article  Google Scholar 

  • Hart S, Kurz M (1983) Endogeneous formation of coalitions. Econometrica 51(4):1047–1064

    Article  Google Scholar 

  • Lardon A (2012) The \(\gamma \)-core in Cournot oligopoly TU-games with capacity constraints. Theory Decis 72:387–411

    Article  Google Scholar 

  • Lekeas PV, Stamatopoulos G (2014) Cooperative oligopoly games with boundedly rational firms. Ann Oper Res 223(1):255–272

    Article  Google Scholar 

  • Maschler M, Peleg B, Shapley LS (1979) Geometric properties of the kernel, nucleolus, and related solution concepts. Math Oper Res 4(4):303–338

    Article  Google Scholar 

  • Moldovanu B, Winter E (1994) Consistent demands for coalition formation. Essays in game theory: in honor of Michael Maschler. Springer, Berlin

    Google Scholar 

  • Moulin H (2014) Cooperative Microeconomics—a game theoretic introduction. Princeton University Press, Princeton

    Google Scholar 

  • Norde H, Pham Do KH, Tijs S (2002) Oligopoly games with and without transferable technologies. Math Soc Sci 43:187–207

    Article  Google Scholar 

  • Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15(3):187–200

    Article  Google Scholar 

  • Peleg B, Sudhölter P (2003) Introduction to the theory of cooperative games. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  • Shapley LS (1967) On balanced sets and cores. Nav Res Logist Q 14:453–460

    Article  Google Scholar 

  • Shapley LS, Shubik M (1966) Quasi-cores in a monetary economy with non-convex preferences. Econometrica 34(4):805–827

    Article  Google Scholar 

  • Trockel W (2005) Core-equivalence for the Nash bargaining solution. Econ Theory 25(1):255–263

    Article  Google Scholar 

  • Young HP, Okada N, Hashimoto T (1982) Cost allocation in water resources development. Water Resour Res 18(3):463–475

    Article  Google Scholar 

  • Zhao J (1999) A \(\beta \)-core existence result and its application to oligopoly markets. Games Econ Behav 27:153–168

    Article  Google Scholar 

  • Zhao J (2001) The relative interior of the base polyhedron and the core. Econ Theory 18(3):635–648

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Univ Lyon, UJM Saint-Etienne, CNRS, GATE L-SE UMR 5824, 42023, Saint-Étienne, France

    Stéphane Gonzalez

  2. Université Côte d’Azur, GREDEG, CNRS, 250 rue Albert Einstein, 06560, Valbonne, France

    Aymeric Lardon

Authors
  1. Stéphane Gonzalez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aymeric Lardon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aymeric Lardon.

Additional information

This work was supported by the Programme Avenir Lyon Saint-Etienne of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007).

We wish to thank two anonymous referees for providing insightful comments and suggestions.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gonzalez, S., Lardon, A. Optimal deterrence of cooperation. Int J Game Theory 47, 207–227 (2018). https://doi.org/10.1007/s00182-017-0584-8

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00182-017-0584-8

Keywords

JEL Classification

Search

Navigation