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 and Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049
References:
 [1] X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0. [2] X. Chen, T. Pong and R. Wets, Two-stage stochastic variational inequalities: an ERM-solution procedure, Mathematical Programming, 165 (2017), 71-111.  doi: 10.1007/s10107-017-1132-9. [3] X. Chen, A. Shapiro and H. Sun, Convergence analysis of sample average approximation of two-stage stochastic generalized equations, SIAM Journal on Optimization, 29 (2019), 135-161.  doi: 10.1137/17M1162822. [4] X. Chen, H. Sun and H. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Mathematical Programming, 177 (2019), 255-289. doi: 10.1007/s10107-018-1266-4. [5] X. Chen, Y. Shi and X. Wang, Equilibrium oil market share under the COVID-19 pandemic, arXiv: 2007.15265, preprint, 2020. [6] F. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309. [7] A. Ehrenmann and Y. Smeers, Generation capacity expansion in a risky environment: a stochastic equilibrium analysis, Operations Research, 59 (2011), 1332-1346.  doi: 10.1287/opre.1110.0992. [8] F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003. [9] T. S. Genc, S. S. Reynolds and S. Sen, Dynamic oligopolistic games under uncertainty: a stochastic programming approach, Journal of Economic Dynamics and Control, 31 (2007), 55-80.  doi: 10.1016/j.jedc.2005.09.011. [10] G. Gürkan, A. Y. Özge and S. M. Robinson, Sample-path solution of stochastic variational inequalities, Mathematical Programming, 84 (1999), 313-333.  doi: 10.1007/s101070050024. [11] G. Gürkan and J. S. Pang, Approximations of Nash equilibria, Mathematical Programming, 117 (2009), 223-253.  doi: 10.1007/s10107-007-0156-y. [12] T. Hao and J. S. Pang, Piecewise affine parameterized value-function based bilevel non-cooperative games, Mathematical Programming, 180 (2020) 33–73. doi: 10.1007/s10107-018-1344-7. [13] B. Hobbs and J. Pang, Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints, Operations Research, 55 (2007), 113-127.  doi: 10.1287/opre.1060.0342. [14] J. Jiang, Y. Shi, X. Wang and X. Chen, Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty, Journal of Computational Mathematics, 37 (2019), 813-842.  doi: 10.4208/jcm.1906-m2019-0025. [15] A. Jofré, R. Rockafellar and R. Wets, Variational inequalities and economic equilibrium, Mathematics of Operations Research, 32 (2007), 32-50.  doi: 10.1287/moor.1060.0233. [16] G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: a survey, Pacific Journal of Optimization, 6 (2010), 455-482. [17] J. Pang, S. Sen and U. V. Shanbhag, Two-stage non-cooperative games with risk-averse players, Mathematical Programming, 165 (2017), 235-290.  doi: 10.1007/s10107-017-1148-1. [18] D. Ralph and Y. Smeers, Pricing risk under risk measures: an introduction to stochastic-endogenous equilibria, Available at SSRN 1903897, preprint, 2011. [19] N. Rashedi and H. Kebriaei, Cooperative and non-cooperative Nash solution for linear supply function equilibrium game, Applied Mathematics and Computation, 244 (2014), 794-808.  doi: 10.1016/j.amc.2014.07.041. [20] U. Ravat and U. Shanbha, On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games, SIAM Journal on Optimization, 21 (2011), 1168-1199.  doi: 10.1137/100792644. [21] R. Rockafellar and J. Sun, Solving Lagrangian variational inequalities with applications to stochastic programming, Mathematical Programming, 155 (2020), 1-17.  doi: 10.1007/s10107-019-01458-0. [22] R. Rockafellar and R. Wets, Stochastic variational inequalities: single-stage to multistage, Mathematical Programming, 165 (2017), 331-360.  doi: 10.1007/s10107-016-0995-5. [23] D. A. Schiro, B. F. Hobbs and J. S. Pang, Perfectly competitive capacity expansion games with risk-averse participants, Computational Optimization and Applications, 65 (2015), 511-539.  doi: 10.1007/s10589-015-9798-5. [24] A. Shapiro, On a time consistency concept in risk averse multistage stochastic programming, Operations Research Letters, 37 (2009), 143-147.  doi: 10.1016/j.orl.2009.02.005. [25] A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2014. [26] U. V. Shanbhag, Stochastic variational inequality problems: applications, analysis, and algorithms, in INFORMS Tutorials in Operations Research, (2013), 70–107. [27] H. Sun and X. Chen, Two-stage stochastic variational inequalities: theory, algorithms and applications, Journal of the Operations Research Society of China, publish online, 2019. [28] M. Zhang, J. Sun and H. Xu, Two-stage quadratic games under uncertainty and their solution by progressive hedging algorithms, SIAM Journal on Optimization, 29 (2019), 1799-1818.  doi: 10.1137/17M1151067.

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References:
 [1] X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0. [2] X. Chen, T. Pong and R. Wets, Two-stage stochastic variational inequalities: an ERM-solution procedure, Mathematical Programming, 165 (2017), 71-111.  doi: 10.1007/s10107-017-1132-9. [3] X. Chen, A. Shapiro and H. Sun, Convergence analysis of sample average approximation of two-stage stochastic generalized equations, SIAM Journal on Optimization, 29 (2019), 135-161.  doi: 10.1137/17M1162822. [4] X. Chen, H. Sun and H. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Mathematical Programming, 177 (2019), 255-289. doi: 10.1007/s10107-018-1266-4. [5] X. Chen, Y. Shi and X. Wang, Equilibrium oil market share under the COVID-19 pandemic, arXiv: 2007.15265, preprint, 2020. [6] F. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309. [7] A. Ehrenmann and Y. Smeers, Generation capacity expansion in a risky environment: a stochastic equilibrium analysis, Operations Research, 59 (2011), 1332-1346.  doi: 10.1287/opre.1110.0992. [8] F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003. [9] T. S. Genc, S. S. Reynolds and S. Sen, Dynamic oligopolistic games under uncertainty: a stochastic programming approach, Journal of Economic Dynamics and Control, 31 (2007), 55-80.  doi: 10.1016/j.jedc.2005.09.011. [10] G. Gürkan, A. Y. Özge and S. M. Robinson, Sample-path solution of stochastic variational inequalities, Mathematical Programming, 84 (1999), 313-333.  doi: 10.1007/s101070050024. [11] G. Gürkan and J. S. Pang, Approximations of Nash equilibria, Mathematical Programming, 117 (2009), 223-253.  doi: 10.1007/s10107-007-0156-y. [12] T. Hao and J. S. Pang, Piecewise affine parameterized value-function based bilevel non-cooperative games, Mathematical Programming, 180 (2020) 33–73. doi: 10.1007/s10107-018-1344-7. [13] B. Hobbs and J. Pang, Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints, Operations Research, 55 (2007), 113-127.  doi: 10.1287/opre.1060.0342. [14] J. Jiang, Y. Shi, X. Wang and X. Chen, Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty, Journal of Computational Mathematics, 37 (2019), 813-842.  doi: 10.4208/jcm.1906-m2019-0025. [15] A. Jofré, R. Rockafellar and R. Wets, Variational inequalities and economic equilibrium, Mathematics of Operations Research, 32 (2007), 32-50.  doi: 10.1287/moor.1060.0233. [16] G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: a survey, Pacific Journal of Optimization, 6 (2010), 455-482. [17] J. Pang, S. Sen and U. V. Shanbhag, Two-stage non-cooperative games with risk-averse players, Mathematical Programming, 165 (2017), 235-290.  doi: 10.1007/s10107-017-1148-1. [18] D. Ralph and Y. Smeers, Pricing risk under risk measures: an introduction to stochastic-endogenous equilibria, Available at SSRN 1903897, preprint, 2011. [19] N. Rashedi and H. Kebriaei, Cooperative and non-cooperative Nash solution for linear supply function equilibrium game, Applied Mathematics and Computation, 244 (2014), 794-808.  doi: 10.1016/j.amc.2014.07.041. [20] U. Ravat and U. Shanbha, On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games, SIAM Journal on Optimization, 21 (2011), 1168-1199.  doi: 10.1137/100792644. [21] R. Rockafellar and J. Sun, Solving Lagrangian variational inequalities with applications to stochastic programming, Mathematical Programming, 155 (2020), 1-17.  doi: 10.1007/s10107-019-01458-0. [22] R. Rockafellar and R. Wets, Stochastic variational inequalities: single-stage to multistage, Mathematical Programming, 165 (2017), 331-360.  doi: 10.1007/s10107-016-0995-5. [23] D. A. Schiro, B. F. Hobbs and J. S. Pang, Perfectly competitive capacity expansion games with risk-averse participants, Computational Optimization and Applications, 65 (2015), 511-539.  doi: 10.1007/s10589-015-9798-5. [24] A. Shapiro, On a time consistency concept in risk averse multistage stochastic programming, Operations Research Letters, 37 (2009), 143-147.  doi: 10.1016/j.orl.2009.02.005. [25] A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2014. [26] U. V. Shanbhag, Stochastic variational inequality problems: applications, analysis, and algorithms, in INFORMS Tutorials in Operations Research, (2013), 70–107. [27] H. Sun and X. Chen, Two-stage stochastic variational inequalities: theory, algorithms and applications, Journal of the Operations Research Society of China, publish online, 2019. [28] M. Zhang, J. Sun and H. Xu, Two-stage quadratic games under uncertainty and their solution by progressive hedging algorithms, SIAM Journal on Optimization, 29 (2019), 1799-1818.  doi: 10.1137/17M1151067.
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