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

Yiyin Cao (), Yin Chen () and Chuangyin Dang ()
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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

INFORMS Journal on Computing, 2024, vol. 36, issue 2, 377-396

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.

Keywords: noncooperative game; Nash equilibrium; proper equilibrium; perfect d -proper equilibrium; differentiable path-following method (search for similar items in EconPapers)
Date: 2024
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