# Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty

## TL;DR

This paper formulates a two-stage stochastic variational inequality model for Cournot-Nash equilibrium under uncertainty, providing solution existence conditions, a regularized approximation method, and demonstrating its application to oil market data.

## Contribution

It introduces a regularized sample average approximation method for two-stage stochastic variational inequalities and proves its convergence, applied to oligopolistic market modeling.

## Key findings

- Proven convergence of the proposed method as regularization and sample size increase.
- Effective modeling of oil market shares using the two-stage SVI approach.
- Numerical validation with historical crude oil data.

## Abstract

A convex two-stage non-cooperative multi-agent game under uncertainty is formulated as a two-stage stochastic variational inequality (SVI). Under standard assumptions, we provide sufficient conditions for the existence of solutions of the two-stage SVI and propose a regularized sample average approximation method for solving it. We prove the convergence of the method as the regularization parameter tends to zero and the sample size tends to infinity. Moreover, our approach is applied to a two-stage stochastic production and supply planning problem with homogeneous commodity in an oligopolistic market. Numerical results based on historical data in crude oil market are presented to demonstrate the effectiveness of the two-stage SVI in describing the market share of oil producing agents.