Generalized Nash Equilibrium Seeking for Noncooperative Game With Different Monotonicities by Adaptive Neurodynamic...

Area of Science:

• Game Theory
• Optimization Algorithms
• Computational Economics

Background:

• Noncooperative games with constraints are common in economics and engineering.
• Finding generalized Nash equilibria (GNE) is computationally challenging.
• Existing algorithms may lack guaranteed convergence or efficiency.

Purpose of the Study:

• To propose a novel adaptive neurodynamic algorithm (ANA) for seeking GNE in constrained noncooperative games.
• To analyze the convergence properties of the proposed ANA under various monotonicity conditions.
• To introduce a new ANA variant for approximating GNE using Tikhonov regularization.

Main Methods:

• Development of an adaptive neurodynamic algorithm (ANA) with trajectory-dependent penalty parameters.
• Mathematical analysis of finite-time convergence to the action set.
• Proof of exponential or polynomial convergence to GNE based on monotonicity conditions.
• Application of Tikhonov regularization for approximating $\varepsilon $-GNE.

Main Results:

• The ANA ensures finite-time entry into the action set due to adaptive penalty terms.
• Exponential convergence to GNE is proven for strongly monotone games.
• Polynomial convergence to GNE is established for "generalized" strongly monotone games, a novel result.
• Exponential convergence to $\varepsilon $-GNE is demonstrated for generally monotone games.

Conclusions:

• The proposed ANA is effective for finding GNE in constrained noncooperative games.
• The algorithm offers improved convergence properties, including polynomial convergence for the first time.
• The Tikhonov-regularized ANA provides a method for approximating GNE when exact solutions are difficult.
• The algorithm's effectiveness is validated through examples like pollution and base station location games.