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Sufficient conditions for the existence of Nash equilibria in bimatrix games in terms of forbidden $$2 \times 2$$ 2 × 2 subgames

Author

Listed:
  • Endre Boros

    (Rutgers University)

  • Khaled Elbassioni

    (Masdar Institute of Science and Technology)

  • Vladimir Gurvich

    (Rutgers University)

  • Kazuhisa Makino

    (Kyoto University)

  • Vladimir Oudalov

    (Salient Management Company)

Abstract

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its $$2 \times 2$$ 2 × 2 submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every $$2 \times 2$$ 2 × 2 subgame has one. Nevertheless, Shapley’s claim can be extended to bimatrix games as follows. We partition all $$2 \times 2$$ 2 × 2 bimatrix games into fifteen classes $$C = \{c_1, \ldots , c_{15}\}$$ C = { c 1 , … , c 15 } depending only on the preferences of two players. A subset $$t \subseteq C$$ t ⊆ C is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a method to construct 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) strongest NE-theorems. Let us remark that the suggested approach, which may be called “math-pattern recognition”, is very general: it allows to characterize completely an arbitrary “target” in terms of arbitrary “attributes”.

Suggested Citation

  • Endre Boros & Khaled Elbassioni & Vladimir Gurvich & Kazuhisa Makino & Vladimir Oudalov, 2016. "Sufficient conditions for the existence of Nash equilibria in bimatrix games in terms of forbidden $$2 \times 2$$ 2 × 2 subgames," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 1111-1131, November.
  • Handle: RePEc:spr:jogath:v:45:y:2016:i:4:d:10.1007_s00182-015-0513-7
    DOI: 10.1007/s00182-015-0513-7
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    References listed on IDEAS

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    More about this item

    Keywords

    Matrix and bimatrix games; Saddle point; Nash equilibrium; Nash-solvability; Dual hypergraphs; Transversal hypergraphs; Dualization;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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