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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)

Abstract

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.

Keywords

Nash Equilibrium Complete Characterization Game Form Explicit Characterization Joint Generation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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