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Characterizations of the core of TU and NTU games with communication structures

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Abstract

It is well-known that the core on several domains of cooperative transferable utility (TU) and nontransferable utility (NTU) games is characterized by various combinations of axioms containing some versions of the reduced game property, of its converse, or of the reconfirmation property with respect to the Davis–Maschler reduced game. We show that these characterizations are still valid for games with communication structures à la Myerson when using the notion of the reduced communication structure that establishes a new link between two inside players if they can communicate via outside players. Thus, it is shown that, if communication structures are present, the core may still be characterized on balanced TU games, on totally balanced TU games, on NTU games with a nonempty core, on the domains of all TU or NTU games, and on several other interesting domains of TU and NTU games. As a byproduct we construct, for any NTU game with communication structure, a certain classical NTU game with the same core that may be regarded as its Myerson restricted NTU game.

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Notes

  1. Note that \(\nu (N,v,g)\) may not coincide with \(\nu (N,v/g)\) in general already for \(|N|=3\) as shown by an example Khmelnitskaya and Sudhölter (2013, p. 297), but using Kohlberg ’s (1971) characterization of the (pre)nucleolus by balanced collections of coalitions it may be deduced that \(\nu (N,v,g)=\nu (N,v/g)\) for all balanced (Nvg).

  2. When applied to sets, the “+” denotes the “Minkowski sum”, i.e., \(x+\mathbb R^S_+=\{x+y\mid y\in \mathbb R^S_+\}\).

References

  • Albizuri MJ, Zarzuelo JM (2009) Conference structures and consistency. Discrete Math 309:4969–4976

    Article  Google Scholar 

  • Bilbao JM (1999) The core of games on convex geometries. Eur J Oper Res 119:365–372

    Article  Google Scholar 

  • Billera LJ, Bixby RE (1973) A characterization of polyhedral market games. Int J Game Theory 2:253–261

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • Faigle U (1989) Cores of games with restricted cooperation. Zeitschrift für Operations Research-Mathematical Methods of Operations Research 33:405–422

    Article  Google Scholar 

  • Grabisch M, Sudhölter P (2012) The bounded core for games with precedence constraints. Ann Oper Res 201:251–264

    Article  Google Scholar 

  • Herings P, van der Laan G, Talman A, Yang Z (2010) The average tree solution for cooperative games with communication structure. Games Econ Behav 68:626–633

    Article  Google Scholar 

  • Hwang Y-A, Sudhölter P (2001) Axiomatizations of the core on the universal domain and other natural domains. Int J Game Theory 29:597–623

    Article  Google Scholar 

  • Keiding H, Thorlund-Petersen L (1987) The core of a cooperative game without side payments. J Optim Theory Appl 54:273–288

    Article  Google Scholar 

  • Khmelnitskaya A, Sudhölter P (2013) The prenucleolus and the prekernel for games with communication structures. Math Methods Oper Res 78:285–299

    Article  Google Scholar 

  • Kohlberg E (1971) On the nucleolus of a characteristic function game. SIAM J Appl Math 20:62–66

    Article  Google Scholar 

  • Llerena F (2007) An axiomatization of the core of games with restricted cooperation. Econ Lett 95:80–84

    Article  Google Scholar 

  • Maschler M, Peleg B, Shapley LS (1972) The kernel and bargaining set for convex games. Int J Game Theory 1:73–93

    Article  Google Scholar 

  • Myerson RB (1977) Graphs and cooperation in games. Math Oper Res 2:225–229

    Article  Google Scholar 

  • Myerson RB (1980) Conference structures and fair allocation rules. Int J Game Theory 9:169–182

    Article  Google Scholar 

  • Peleg B (1985) An axiomatization of the core of cooperative games without side payments. J Math Econ 14:203–214

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Peleg B (1989) An axiomatization of the core of market games. Math Oper Res 14:448–456

    Article  Google Scholar 

  • Predtetchinski A, Herings PJ-J (2004) A necessary and sufficient condition for nonemptiness of the core of a non-transferable utility game. J Econ Theory 116:84–92

    Article  Google Scholar 

  • Pulido MA, Sánchez-Soriano J (2006) Characterization of the core in games with restricted cooperation. Eur J Oper Res 175:860–869

    Article  Google Scholar 

  • Pulido MA, Sánchez-Soriano J (2009) On the core, the Weber set and convexity in games with a priori unions. Eur J Oper Res 193:468–475

    Article  Google Scholar 

  • Scarf HE (1967) The core of an \(n\)-person game. Econometrica 35:50–69

    Article  Google Scholar 

  • Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17:1163–1170

    Article  Google Scholar 

  • Serrano R, Volij O (1998) Axiomatizations of neoclassical concepts for economies. J Math Econ 30:87–108

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Sobolev AI (1975) The characterization of optimality principles in cooperative games by functional equations. In: Vorobiev NN (ed) Mathematical methods in the social sciences, Vilnius, vol 6. Academy of Sciences of the Lithuanian SSR, pp 95–151 (in Russian)

  • Sudhölter P, Peleg B (2002) A note on an axiomatization of the core of market games. Math Oper Res 27:441–444

    Article  Google Scholar 

  • Tadenuma K (1992) Reduced games, consistency, and the core. Int J Game Theory 20:325–334

    Article  Google Scholar 

Download references

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Correspondence to Peter Sudhölter.

Additional information

The authors are grateful to an anonymous referee and an associate editor of this journal for their detailed comments that helped to improve the writing of this paper. This research was supported by the Spanish Ministerio de Ciencia e Innovación under project ECO2012-33618, co-funded by the ERDF. Moreover, the first author was supported by the Basque Country University (UFI11/51 and GIU13/31), and the second author was supported by The Danish Council for Independent Research|Social Sciences under the FINQ project (Grant ID: DFF-1327-00097).

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Albizuri, M.J., Sudhölter, P. Characterizations of the core of TU and NTU games with communication structures. Soc Choice Welf 46, 451–475 (2016). https://doi.org/10.1007/s00355-015-0924-1

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