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Complementarity Enhanced Nash’s Mappings and Differentiable Homotopy Methods to Select Perfect Equilibria

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Abstract

To extend the concept of subgame perfect equilibrium to an extensive-form game with imperfect information but perfect recall, Selten (Int J Game Theory 4:25–55, 1975) formulated the notion of perfect equilibrium and attained its existence through the agent normal-form representation of the extensive-form game. As a strict refinement of Nash equilibrium, a perfect equilibrium always yields a sequential equilibrium. The selection of a perfect equilibrium thus plays an essential role in the applications of game theory. Moreover, a different procedure may select a different perfect equilibrium. The existence of Nash equilibrium was proved by Nash (Ann Math 54:289–295, 1951) through the construction of an elegant continuous mapping and an application of Brouwer’s fixed point theorem. This paper intends to enhance Nash’s mapping to select a perfect equilibrium. By incorporating the complementarity condition into the equilibrium system with Nash’s mapping through an artificial game, we successfully eliminate the nonnegativity constraints on a mixed strategy profile imposed by Nash’s mapping. In the artificial game, each player solves against a given mixed strategy profile a strictly convex quadratic optimization problem. This enhancement enables us to derive differentiable homotopy systems from Nash’s mapping and establish the existence of smooth paths for selecting a perfect equilibrium. The homotopy methods start from an arbitrary totally mixed strategy profile and numerically trace the smooth paths to a perfect equilibrium. Numerical results show that the methods are numerically stable and computationally efficient in search of a perfect equilibrium and outperform the existing differentiable homotopy method.

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Acknowledgements

We are very grateful to the editor and two anonymous reviewers for their valuable comments and suggestions, which have significantly enhanced the quality of the paper. This work was partially supported by NSFC: 61976184.

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Authors and Affiliations

  1. City University of Hong Kong, Kowloon, Hong Kong

    Yiyin Cao & Chuangyin Dang

  2. Xi’an Jiaotong University, Xi’an, China

    Yiyin Cao

  3. Shanxi University, Taiyuan, China

    Yabin Sun

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  1. Yiyin Cao
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  2. Chuangyin Dang
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  3. Yabin Sun
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Correspondence to Yiyin Cao.

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Communicated by Xinmin Yang.

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Cao, Y., Dang, C. & Sun, Y. Complementarity Enhanced Nash’s Mappings and Differentiable Homotopy Methods to Select Perfect Equilibria. J Optim Theory Appl 192, 533–563 (2022). https://doi.org/10.1007/s10957-021-01977-x

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  • DOI: https://doi.org/10.1007/s10957-021-01977-x

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