# American Institute of Mathematical Sciences

December  2020, 10(4): 521-535. doi: 10.3934/naco.2020049

## Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty

 Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

* Corresponding author: Hailin Sun

Received  April 2020 Revised  September 2020 Published  September 2020

Fund Project: The work is supported by NSFC grant 11871276

A convex two-stage non-cooperative game with risk-averse players under uncertainty is formulated as a two-stage stochastic variational inequality (SVI) for point-to-set operators. Due to the indifferentiability of function $(\cdot)_+$ and the discontinuity of solution mapping of the second-stage problem, under standard assumptions, we propose a smoothing and regularization method to approximate it as a two-stage SVI in point-to-point case with continuous second stage solution functions. The corresponding convergence analysis is also given.

Citation: Bin Zhou, Hailin Sun. Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049
##### References:

show all references

##### References:
 [1] Zhimin Liu, Shaojian Qu, Hassan Raza, Zhong Wu, Deqiang Qu, Jianhui Du. Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2783-2804. doi: 10.3934/jimo.2020094 [2] Zhiping Chen, Youpan Han. Continuity and stability of two-stage stochastic programs with quadratic continuous recourse. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 197-209. doi: 10.3934/naco.2015.5.197 [3] René Henrion, Christian Küchler, Werner Römisch. Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 363-384. doi: 10.3934/jimo.2008.4.363 [4] Rüdiger Schultz. Two-stage stochastic programs: Integer variables, dominance relations and PDE constraints. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 713-738. doi: 10.3934/naco.2012.2.713 [5] Yuwei Shen, Jinxing Xie, Tingting Li. The risk-averse newsvendor game with competition on demand. Journal of Industrial & Management Optimization, 2016, 12 (3) : 931-947. doi: 10.3934/jimo.2016.12.931 [6] Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025 [7] Bin Li, Jie Sun, Honglei Xu, Min Zhang. A class of two-stage distributionally robust games. Journal of Industrial & Management Optimization, 2019, 15 (1) : 387-400. doi: 10.3934/jimo.2018048 [8] Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645 [9] Jie Jiang, Zhiping Chen, He Hu. Stability of a class of risk-averse multistage stochastic programs and their distributionally robust counterparts. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2415-2440. doi: 10.3934/jimo.2020075 [10] Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35 [11] Jianxun Liu, Shengjie Li, Yingrang Xu. Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021083 [12] Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547 [13] Urszula Foryś, Beata Zduniak. Two-stage model of carcinogenic mutations with the influence of delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2501-2519. doi: 10.3934/dcdsb.2014.19.2501 [14] Tugba Sarac, Aydin Sipahioglu, Emine Akyol Ozer. A two-stage solution approach for plastic injection machines scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1289-1314. doi: 10.3934/jimo.2020022 [15] Burcu Özçam, Hao Cheng. A discretization based smoothing method for solving semi-infinite variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 219-233. doi: 10.3934/jimo.2005.1.219 [16] G. Bellettini, G. Fusco, G. F. Gronchi. Regularization of the two-body problem via smoothing the potential. Communications on Pure & Applied Analysis, 2003, 2 (3) : 323-353. doi: 10.3934/cpaa.2003.2.323 [17] Chien Hsun Tseng. Applications of a nonlinear optimization solver and two-stage comprehensive Denoising techniques for optimum underwater wideband sonar echolocation system. Journal of Industrial & Management Optimization, 2013, 9 (1) : 205-225. doi: 10.3934/jimo.2013.9.205 [18] Qingqing Ye. Algorithmic computation of MAP/PH/1 queue with finite system capacity and two-stage vacations. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2459-2477. doi: 10.3934/jimo.2019063 [19] Chao Mi, Jun Wang, Weijian Mi, Youfang Huang, Zhiwei Zhang, Yongsheng Yang, Jun Jiang, Postolache Octavian. Research on regional clustering and two-stage SVM method for container truck recognition. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1117-1133. doi: 10.3934/dcdss.2019077 [20] Dan Liu, Shigui Ruan, Deming Zhu. Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. Mathematical Biosciences & Engineering, 2012, 9 (2) : 347-368. doi: 10.3934/mbe.2012.9.347

Impact Factor: