# Nash equilibrium points and their finding for nonsmooth case

## TL;DR

This paper introduces a numerical method for locating Nash equilibrium points in nonsmooth convex models across various fields, utilizing Steklov averages to handle nonsmoothness and proving convergence of the method.

## Contribution

The paper develops a novel numerical approach for finding Nash equilibria in nonsmooth convex models, extending existing methods to more complex nonsmooth cases.

## Key findings

- The method converges to equilibrium points under certain conditions.
- Limit points of the method are proven to be equilibrium points.
- Convergence rate estimates are provided using the Kantorovich theorem.

## Abstract

The purpose of this paper is to develop a numerical method for finding an equilibrium point in a model, in which the loss function of each object (subject) is described by a convex function with respect to one of its variables. Such models are found in medicine, economics, game theory, and biology. For the more complex case, with nonsmooth functions describing the state of each element of the system as damage, loss, or gain, the Steklov average integrals are used that turn nonsmooth functions into smooth ones. Numerical methods for finding equilibrium points in the more general non-smooth case are constructed. In the process of optimization, the diameters of the sets, over which the averaging takes place, are decreased in accordance with the optimization steps. All limit points are proved to be equilibrium points. Under some conditions, the convergence rate can be estimated using the Kantorovich theorem. The necessity to develop new methods for finding Nash equilibrium points in the nonsmooth case is concluded.