A differentiable homotopy method to compute perfect equilibria

  • Yin Chen
  • Chuangyin DangEmail author
Full Length Paper Series A


The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large.


Noncooperative game Nash equilibrium Perfect equilibrium Differentiable homotopy method Simplicial homotopy method 

Mathematics Subject Classification

90C33 91A10 90-08 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Department of Systems Engineering and Engineering ManagementCity University of Hong KongKowloonHong Kong

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