研究集生产、运输和销售为一体的多个制造商在随机市场环境下的两阶段随机非合作博弈问题.首先，建立了该两阶段随机非合作博弈问题的模型，然后将其转化为两阶段随机变分不等式（Stochastic VariationalInequality，简称SVI）.在温和的假设条件下，证明了该问题存在均衡解，并通过Progressive Hedging Method（简称PHM）进行求解.最后，通过改变模型中随机变量的分布和成本参数，分析与研究厂商的市场行为.
In this paper, we discuss the two-stage stochastic non-cooperative game of manufacturers with production, transportation and sales under stochastic market environment. Firstly, we establish a model of the two-stage stochastic non-cooperative game, and then transform it into a two-stage stochastic variational inequality (SVI). Under mild assumptions, it is proved that there exists an equilibrium solution to the game problem, and it is solved by Progressive Hedging Method (PHM). Finally, the market behavior of manufacturers is analyzed and studied by changing the distribution of random variables and cost parameters in the model.
[1] Morgenstern O, Von Neumann J. Theory of Games and Economic Behavior[M]. Princeton:Princeton University Press, 1953.
[2] Nash J. Non-cooperative games[J]. Annals of Mathematics, 1951, 54(2):286-295.
[3] Gibbons R. Game Theory for Applied Economists[M]. Princeton:Princeton University Press, 1992.
[4] Facchinei F, Pang J S. Finite-Dimensional Variational Inequalities and Complementarity Problems[M]. New York:Springer-Verlag, 2003.
[5] Harker P T, Pang J S. Finite-dimensional variational inequality and nonlinear complementarity problems:a survey of theory, algorithms and applications[J]. Mathematical Programming, 1990, 48:161-220.
[6] 韩继业, 修乃华, 戚厚铎. 非线性互补理论与算法[M]. 上海:上海科学技术出版社, 2006.
[7] Jofré A, Rockafellar R T, Wets J B. Variationalinequalities and economic equilibrium[J]. Mathematics ofOperations Research, 2007, 32(1):32-50.
[8] Gürkan G, Pang J S. Approximations of Nash Equilibria[M]. Mathematical Programming, 2009, 117:223-253.
[9] Ravat U, Shanbhag U V. On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games[J]. SIAM Journal of Optimization, 2011, 21:1168-1199.
[10] Ralph D, Smeers Y. Pricing risk under risk measures:an introduction to stochastic-endogenous equilibria[J]. SSRN Electronic Journal, 2011.
[11] Shanbhag U V, Pang J S, Sen S. Inexact best-response schemes for stochastic Nash games:linear convergence and Iteration complexity analysis[C]//Decision & Control IEEE, 2016.
[12] Chen X, Fukushima M. Expected residual minimization method forstochastic linear complementarity problems[J]. Mathematics ofOperations Research. 2005, 30:1022-1038.
[13] Hobbs B F, Pang J S. Nash-cournot equilibria in electric powermarkets with piecewise linear demand functions and joint constraints[J]. Operations Research, 2007, 55(1):113-127.
[14] Jiang H, Xu H. Stochastic approximation approaches to the stochastic variational inequality problem[J]. IEEE Transactions on Automatic Control, 2008, 53(6):1462-1475.
[15] Luo M J, Lin G H. Expected residual minimization method forstochastic variational inequality problems[J]. Journal ofOptimization Theory and Applications, 2009, 140(1):103-116.
[16] Zhang C, Chen X. Smoothing projected gradient method and itsapplication to stochastic linear complementarity problems[J]. SIAM Journal of Optimization, 2009, 20(2):627-649.
[17] Lin G H, Fukushima M. Stochastic equilibrium problems and stochastic mathematical programs with equilibriumconstraints:A survey[J]. Pacific Journal of Optimization,2010, 6(3):455-482.
[18] Shanbhag U V. Stochastic variational inequality problems:applications, analysis, and algorithms[J]. In INFORMS Tutorialsin Operations Research, 2013.
[19] Philpott A, Ferris M, Wets R. Equilibrium, uncertainty and risk inhydro-thermal electricity systems[J]. MathematicalProgramming, 2016, 157(2):483-513.
[20] Pang J S, Sen S, Shanbhag U V. Two-stage non-cooperative games with risk-averse players[J]. Mathematical Programming, 2017, 165:235-290.
[21] Kannan A, Shanbhag U V, Kim H M. Strategic behavior in power markets under uncertainty[J]. Energy Systems, 2011, 2(2):115-141.
[22] Yao J, Adler I, Oren S S. Modeling and computing two-settlement oligopolistic equilibrium in a congested electricity network[J]. Operations Research, 2008, 56(1):34-47.
[23] Khazaei J, Zakeri G, Oren S S. Single and multi-settlement approaches to market clearing mechanisms under demand uncertainty[J]. Operations Research, 2017, 65:1147-1164.
[24] Kannan A, Shanbhag U V, Kim H M. Addressing supply-side risk in uncertain power markets:stochastic Nash models, scalable algorithms and error analysis[J]. Optimization Methods and Software, 2013, 28(5):1095-1138.
[25] Genc T S, Reynolds S S, Sen S. Dynamic oligopolistic games under uncertainty:a stochastic programming approach[J]. Journal of Economic Dynamics & Control, 2007, 31(1):55-80.
[26] Chen X, Pong T K, Wets J B. Two-stage stochastic variational inequalities:an ERM-solution procedure[J]. Mathematical Programming, 2017, 165(1):71-111.
[27] Rockafellar R T, Wets J B. Stochastic variational inequalities:single-stage to multistage[J]. Mathematical Programming, 2017, 165(1):331-360.
[28] Rockafellar R T, Sun J. Solving Lagrangian variational inequalities with applications to stochastic programming[J]. submitted to Mathematical Programming in 2018.
[29] Rockafellar R T, Sun J. Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging[J]. Mathematical Programming, 2019, 174:453-471.
[30] Chen X, Sun H, Xu H. Discrete Approximation of two-stage stochastic and distributionally robust linear complementarity problems[J]. Mathematical Programming, 2017, 117:255-289
[31] Chen X, Shapiro A, Sun H. Convergence analysis of sample average approximation of two-state stochastic generalized equations[J]. SIAM Journal of Optimization, 2018, 29(1):135-161.
[32] Jiang J, Shi Y, Wang X Z, et al. Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty[J]. Journal of Computational Mathematics, 2019.
[33] Ralph D, Xu H. Convergence of stationary points of sample average two-stage stochastic programs:a generalized equation approach[J]. Mathematics of Operations Research, 2011, 36(3):568-592.
