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A Differentiable Path-Following Method with a Compact Formulation to Compute Proper Equilibria

Author

Listed:
  • Yiyin Cao

    (School of Management, Xi’an Jiaotong University, Xi’an 710049, China; Department of Systems Engineering, City University of Hong Kong, Kowloon 999077, Hong Kong)

  • Yin Chen

    (College of Big Data and Internet, Shenzhen Technology University, Shenzhen, Guangdong 518118, China)

  • Chuangyin Dang

    (Department of Systems Engineering, City University of Hong Kong, Kowloon 999077, Hong Kong)

Abstract

The concept of proper equilibrium was established as a strict refinement of perfect equilibrium. This establishment has significantly advanced the development of game theory and its applications. Nonetheless, it remains a challenging problem to compute such an equilibrium. This paper develops a differentiable path-following method with a compact formulation to compute a proper equilibrium. The method incorporates square-root-barrier terms into payoff functions with an extra variable and constitutes a square-root-barrier game. As a result of this barrier game, we acquire a smooth path to a proper equilibrium. To further reduce the computational burden, we present a compact formulation of an ε -proper equilibrium with a polynomial number of variables and equations. Numerical results show that the differentiable path-following method is numerically stable and efficient. Moreover, by relaxing the requirements of proper equilibrium and imposing Selten’s perfection, we come up with the notion of perfect d -proper equilibrium, which approximates a proper equilibrium and is less costly to compute. Numerical examples demonstrate that even when d is rather large, a perfect d -proper equilibrium remains to be a proper equilibrium.

Suggested Citation

  • Yiyin Cao & Yin Chen & Chuangyin Dang, 2024. "A Differentiable Path-Following Method with a Compact Formulation to Compute Proper Equilibria," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 377-396, March.
    • Handle: RePEc:inm:orijoc:v:36:y:2024:i:2:p:377-396
    DOI: 10.1287/ijoc.2022.0148
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