Skip to main content
Log in

The boundary of the core of a balanced game: face games

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


This paper extends the concept of face games, introduced by González-Díaz and Sánchez-Rodríguez (Games Econ Behav 62:100–105, 2008) for convex games, to the general class of balanced games. Each face of the core is the core of a face game and contains the best stable allocations for a coalition provided that the members of the complement coalition get their miminum worth inside the core. Since face games are exact we investigate several properties of the exact envelope of a balanced game that allow us to characterize exactness, convexity and decomposability of a game in terms of its face games. The close connection between extreme points of the core and extreme points of the face games is analyzed. In particular, we show that the marginal vectors that belong to the core and the lexinal vectors must be marginal vectors and lexinal vectors, respectively, of the single player face games. Finally, we present several subclasses of games where face games could provide some insight on the core structure.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others


  1. A game such that all of its subgames are exact is called totally exact by Biswas et al. (1999). Obviously if a game is convex then it is totally exact. Conversely, if v is totally exact, then given \(S, T \in 2N\) there exists \(x\in C(v_{| (S\cup T)})\) such that \(x(S\cap T)= v(S\cap T)\) and \(x(S\cup T)=v(S\cup T)\). Then, \(v(S)+v(T)\le x(S)+x(T)=x(S\cup T)+x(S\cap T)=v(S\cup T)+v(S\cap T)\), so v is convex.

  2. Shapley (1971) defines \(F_T^v\) as \(C(v)\cap H_{T}^v\). Although Shapley’s definition might seem more natural, ours follows the one given in González-Díaz and Sánchez-Rodríguez (2008).

  3. This result was initially stated for convex games but, as it is remarked in Shapley (1971), the proof does not use convexity.


  • Biswas AK, Parthasarathy T, Potters JAM, Voorneveld M (1999) Large cores and exactness. Games Econ Behav 28:1–12

    Article  Google Scholar 

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

    Google Scholar 

  • Csóka P, Herings PJ, Kóczy L (2011) Balancedness conditions for exact games. Math Methods Oper Res 74:41–52

    Article  Google Scholar 

  • Estévez-Fernández A, Fiestras-Janeiro MG, Mosquera MA, Sánchez-Rodríguez E (2012) A bankruptcy approach to the core cover. Math Methods Oper Res 76:343–359

    Article  Google Scholar 

  • Faigle U, Kern W (1992) The shapley value for cooperation games under precedence constraints. Int J Game Theory 21:249–266

    Article  Google Scholar 

  • Fiestras-Janeiro MG, Sánchez-Rodríguez E, Schuster M (2015) A precedence constraint value revisited. TOP 24:156–179

    Article  Google Scholar 

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

  • González-Díaz J, Mirás Calvo MA, Quinteiro Sandomingo C, Sánchez Rodríguez E (2016) Airport games: the core and its center. Math Soc Sci 82:105–115

    Article  Google Scholar 

  • González-Díaz J, Sánchez-Rodríguez E (2008) Cores of convex and strictly convex games. Games Econ Behav 62:100–105

    Article  Google Scholar 

  • Hamers H, Klijn F, Solymosi T, Tijs S, Pere Villar J (2002) Assigment games satisfy the CoMa-property. Games Econ Behav 38:231–239

    Article  Google Scholar 

  • Ichiischi T (1981) Super-modularity: applications to convex games and to the greedy algorithm for lp. J Econ Theory 25(2):283–286

    Article  Google Scholar 

  • Núñez M, Rafels C (1998) On extreme points of the core and reduced games. Ann Oper Res 84:121–133

    Article  Google Scholar 

  • Núñez M, Rafels C (2003) Characterization of the extreme core allocations of the assignment game. Games Econ Behav 44:311–331

    Article  Google Scholar 

  • Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15(120–148):187–200

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Shapley LS (1971) Cores of convex games. Int J Game Theory 1(1):11–26

    Article  Google Scholar 

  • Tijs S, Borm P, Lohmann E, Quant M (2011) An average lexicographic value for cooperative games. Eur J Oper Res 213:210–220

    Article  Google Scholar 

Download references


This work has been supported by the European Regional Development Fund (ERDF) and Ministerio de Economía, Industria y Competitividad through Grant MTM2017-87197-C3-2-P and by the Xunta de Galicia through the European Regional Development Fund (Grupos de Referencia Competitiva ED431C-2016-040). We also benefited from Grant ECO2016-75712-P (AEI/FEDER,UE) of Ministerio de Economía, Industria y Competitividad and Grant RGEAF-ECOBAS: ED431B 2019/35 of Xunta de Galicia.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Carmen Quinteiro Sandomingo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirás Calvo, M.Á., Quinteiro Sandomingo, C. & Sánchez Rodríguez, E. The boundary of the core of a balanced game: face games. Int J Game Theory 49, 579–599 (2020).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: