Finite Convergence Criteria for Normalized Nash Equilibrium through Weak Sharpness and Linear Conditioning

TL;DR

This paper establishes finite convergence criteria for normalized Nash equilibria in convex generalized games using weak sharpness and linear conditioning, providing theoretical insights and an iterative algorithm with step estimates.

Contribution

It introduces the weak sharpness property for normalized Nash equilibria and links it with linear conditioning, ensuring finite convergence of algorithms in convex generalized Nash games.

Findings

Characterization of weak sharpness via regularized gap function

Equivalence conditions between linear conditioning and weak sharpness

Finite termination of iterative algorithms under these conditions

Abstract

The generalized Nash equilibrium problems play a significant role in modeling and analyzing several complex economics problems. In this work, we consider jointly convex generalized Nash games which were introduced by Rosen. We study two important aspects related to these games, which include the weak sharpness property for the set of normalized Nash equilibria and the linear conditioning technique for regularized Nikaido-Isoda function. Firstly, we define the weak sharpness property for the set of normalized Nash equilibria and then, we provide its characterization in terms of the regularized gap function. Furthermore, we provide the sufficient conditions under which the linear conditioning criteria for regularized Nikaido-Isoda function becomes equivalent to the weak sharpness property for the set of normalized Nash equilibria. We show that an iterative algorithm used for determining a normalized Nash equilibrium for jointly convex generalized Nash games terminates finitely under the weak sharpness and linear conditioning assumptions. Finally, we estimate the number of steps needed to determine a solution for the specific type of jointly convex generalized Nash game as an application.