A Complete Characterization of Nash-Solvability of Bimatrix Games in Terms of the Exclusion of Certain 2×2 Subgames

  • Endre Boros
  • Khaled Elbassioni
  • Vladimir Gurvich
  • Kazuhisa Makino
  • Vladimir Oudalov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)


In 1964 Shapley observed that a matrix has a saddle point whenever every 2 ×2 submatrix of it has one. In contrast, a bimatrix game may have no Nash equilibrium (NE) even when every 2 ×2 subgame of it has one. Nevertheless, Shapley’s claim can be generalized for bimatrix games in many ways as follows. We partition all 2 ×2 bimatrix games into fifteen classes C = {c 1, ..., c 15} depending on the preference pre-orders of the two players. A subset t ⊆ C is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a general method for getting all minimal (that is, strongest) NE-theorems based on the procedure of joint generation of transversal hypergraphs given by a special oracle. By this method we obtain all (six) minimal NE-theorems.


Nash Equilibrium Complete Characterization Game Form Explicit Characterization Joint Generation 
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  1. 1.
    Boros, E., Gurvich, V., Makino, K.: Minimal and locally minimal games and game forms, Rutcor Research Report 28-2007, Rutgers UniversityGoogle Scholar
  2. 2.
    Chvatal, V., Lenhart, W.J., Sbihi, N.: Two-colourings that decompose perfect graphs. J. Combin. Theory Ser. B 49, 1–9 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fredman, M., Khachiyan, L.: On the Complexity of Dualization of Monotone Disjunctive Normal Forms. J. Algorithms 21(3), 618–628 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gurvich, V.A., Gvishiani, A.D.: Dual set systems and their applications. Izvestiya Akad. Nauk SSSR, ser. Tekhnicheskaya Kibernetika (in Russian) 4, 31–39 (1983); English translation in: Soviet J. of Computer and System Science (formerly Engineering Cybernetics) MathSciNetGoogle Scholar
  5. 5.
    Gurvich, V., Khachiyan, L.: On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions. Discrete Applied Mathematics 96-97, 363–373 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gurvich, V., Libkin, L.: Absolutely determined matrices. Mathematical Social Sciences 20, 1–18 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kukushkin, N.S.: Shapley’s 2 ×2 theorem for game forms. Economics Bullletin,; see also Technical report, Department of Mathematical Methods for Economic Decision Analysis, Computing Center of Russian Academy of Sciences,
  8. 8.
    Shapley, L.S.: Some topics in two-person games. In: Drescher, M., Shapley, L.S., Tucker, A.W. (eds.) Advances in Game Theory. Annals of Mathematical Studies, AM52, pp. 1–28. Princeton University Press, Princeton (1964)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Endre Boros
    • 1
  • Khaled Elbassioni
    • 2
  • Vladimir Gurvich
    • 1
  • Kazuhisa Makino
    • 3
  • Vladimir Oudalov
    • 1
  1. 1.RUTCORRutgers UniversityPiscataway
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany
  3. 3.Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan

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