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: |
[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.![]() ![]() ![]() |