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Two-stage distributionally robust noncooperative games: Existence of Nash equilibrium and its application to Cournot–Nash competition

  • *Corresponding author: Atsushi Hori

    *Corresponding author: Atsushi Hori 

This work was supported by JST SPRING, Grant Number JPMJSP2110

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  • Two-stage distributionally robust stochastic noncooperative games with continuous decision variables are studied. In such games, each player solves a two-stage distributionally robust optimization problem depending on the decisions of the other players. Existing studies in this area have been limited with strict assumptions, such as linear decision rules, and supposed that each player solves a two-stage linear distributionally robust optimization with a specifically structured ambiguity set. This limitation motivated us to generalize and analyze the game in a nonlinear case. The contributions of this study are (ⅰ) demonstrating the conditions for the existence of two-stage Nash equilibria under convexity and compactness assumptions, and (ⅱ) consideration of a two-stage distributionally robust Cournot–Nash competition as an application, as well as an investigation into the conditions for the existence of market equilibria in an economic sense. We also report some results of numerical experiments to illustrate how distributional robustness affects the decision of each player in the Cournot–Nash competition.

    Mathematics Subject Classification: 91A15, 90C33.

    Citation:

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  • Figure 1.  Results of numerical experiments in the Cournot–Nash competition ($ N = 2 $)

    Table 1.  Recent works on distributionally robust games

    Finite strategies (mixed strategies)1 Continuous pure strategies2
    One-stage Qu and Goh [35]
    Loizou [29,30]
    Peng et al. [34]
    Sun and Xu [50]
    Liu et al. [28]
    Two-stage ———— Li et al. [25] (linear case)
    Chen et al. [7] (Cournot competition)
    this paper (nonlinear case)
     | Show Table
    DownLoad: CSV

    Table 2.  Boxed pigs

    (piglet) (piglet)
    Pull the lever Wait
    (big pig) Pull the lever $ (p_d(\xi)-6,\xi-p_d(\xi)-2) $ $ (\xi-p_s(\xi)-6,p_s(\xi)) $
    (big pig) Wait $ (p_d(\xi),\xi-p_d(\xi)-2) $ $ (0,0) $
     | Show Table
    DownLoad: CSV

    Table 3.  Stochastic Nash equilibrium under $ P_1 $ (bold fonts)

    (piglet) (piglet)
    Pull the lever Wait
    (big pig) Pull the lever $ (3.0938,1.1562) $ $ \mathit{\boldsymbol{(0.4516,5.7984)}} $
    (big pig) Wait $ \mathit{\boldsymbol{(9.0938,1.1562)}} $ $ (0,0) $
     | Show Table
    DownLoad: CSV

    Table 4.  Distributionally robust Nash equilibrium (bold fonts)

    (piglet) (piglet)
    Pull the lever Wait
    (big pig) Pull the lever $ (-0.3021,-0.9479) $ $ (-3.8495,4.5995) $
    (big pig) Wait $ (5.6979,-0.9479) $ $ \mathit{\boldsymbol{(0,0)}} $
     | Show Table
    DownLoad: CSV
  • [1] M. Aghassi and D. Bertsimas, Robust game theory, Math. Program., 107 (2006), 231-273.  doi: 10.1007/s10107-005-0686-0.
    [2] A. Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a N personnes, SIAM J. Control Optim., 12 (1974), 460-499.  doi: 10.1137/0312037.
    [3] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer, New York, 2011. doi: 10.1007/978-1-4614-0237-4.
    [4] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9.
    [5] X. ChenT. K. Pong and R. J. Wets, Two-stage stochastic variational inequalities: An ERM solution procedure, Math. Program., 165 (2017), 71-111.  doi: 10.1007/s10107-017-1132-9.
    [6] X. ChenA. Shapiro and H. Sun, Convergence analysis of sample average approximation of two-stage stochastic generalized equations, SIAM J. Optim., 29 (2019), 135-161.  doi: 10.1137/17M1162822.
    [7] X. ChenH. Sun and H. Xu, Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems, Math. Program., 177 (2019), 255-289.  doi: 10.1007/s10107-018-1266-4.
    [8] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
    [9] V. DeMiguel and H. Xu, A stochastic multiple-leader Stackelberg model: Analysis, computation, and application, Oper. Res., 57 (2009), 1220-1235.  doi: 10.1287/opre.1080.0686.
    [10] G. Dhaene and J. Bouckaert, Sequential reciprocity in two-player, two-stage games: An experimental analysis, Games Econ. Behav., 70 (2010), 289-303.  doi: 10.1016/j.geb.2010.02.009.
    [11] S. Dirkse and M. Ferris, The path solver: A nommonotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5 (1995), 123-156.  doi: 10.1080/10556789508805606.
    [12] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.
    [13] R. S. GibbonsGame Theory for Applied Economists, Princeton University Press, New Jersey, 1992. 
    [14] A. HaurieG. Zaccour and Y. Smeers, Stochastic equilibrium programming for dynamic oligopolistic markets, J. Optim. Theory Appl., 66 (1990), 243-253.  doi: 10.1007/BF00939537.
    [15] S. HayashiN. Yamashita and M. Fukushima, Robust Nash equilibria and second-order cone complementarity problems, J. Nonlinear Conv. Anal., 6 (2005), 283-296. 
    [16] J. P. HespanhaNoncooperative Game Theory: An Introduction for Engineers and Computer Scientists, Princeton University Press, New Jersey, 2017. 
    [17] M. Hu and M. Fukushima, Variational inequality formulation of a class of multi-leader-follower games, J. Optim. Theory Appl., 151 (2011), 455-473.  doi: 10.1007/s10957-011-9901-8.
    [18] Z. Hu and J. L. Hong, Kullback-Leibler divergence constrained distributionally robust optimization, preprint Optimization Online, 2012, https://optimization-online.org/2012/11/3677/.
    [19] J. JiangY. ShiX. Wang and X. Chen, Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty, J. Compt. Math., 37 (2019), 813-842.  doi: 10.4208/jcm.1906-m2019-0025.
    [20] J. JiangH. Sun and B. Zhou, Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage, Numer. Algorithms, 89 (2022), 167-194.  doi: 10.1007/s11075-021-01110-z.
    [21] R. Jiang and Y. Guan, Data-driven chance constrained stochastic program, Math. Program., 158 (2016), 291-327.  doi: 10.1007/s10107-015-0929-7.
    [22] A. JofréR. T. Rockafellar and R. J. Wets, Variational inequalities and economic equilibrium, Math. Oper. Res., 32 (2007), 32-50.  doi: 10.1287/moor.1060.0233.
    [23] J. Lei and U. V. Shanbhag, Stochastic Nash equilibrium problems: Models, analysis, and algorithms, IEEE Control Syst., 42 (2022), 103-124.  doi: 10.1109/MCS.2022.3171481.
    [24] J. LeiU. V. ShanbhagJ. S. Pang and S. Sen, On synchronous, asynchronous, and randomized best-response schemes for stochastic Nash games, Math. Oper. Res., 45 (2020), 157-190.  doi: 10.1287/moor.2018.0986.
    [25] B. LiJ. SunH. Xu and M. Zhang, A class of two-stage distributionally robust games, J. Ind. Manag. Optim., 15 (2019), 387-400.  doi: 10.3934/jimo.2018048.
    [26] X. Liang and Y. Xiao, Game theory for network security, IEEE Communications Surveys & Tutorials, 15 (2013), 472-486.  doi: 10.1109/SURV.2012.062612.00056.
    [27] Y. Liu and H. Xu, Entropic approximation for mathematical programs with robust equilibrium constraints, SIAM J. Optim., 24 (2014), 933-958.  doi: 10.1137/130931011.
    [28] Y. LiuH. XuS. J. S. Yang and J. Zhang, Distributionally robust equilibrium for continuous games: Nash and Stackelberg models, Eur. J. Oper. Res., 265 (2018), 631-643.  doi: 10.1016/j.ejor.2017.07.050.
    [29] N. Loizou, Distributionally Robust Game Theory, Master's thesis, Imperial College London, London, 2015.
    [30] N. Loizou, Distributionally robust games with risk-averse players, Proceedings of the 5th International Conference on Operations Research and Enterprise Systems (ICORES), Rome, Italy, 2016,186-196. doi: 10.5220/0005753301860196.
    [31] D. Monderer and L. S. Shapley, Potential games, Games Econ. Behav., 14 (1996), 124-143.  doi: 10.1006/game.1996.0044.
    [32] R. NishimuraS. Hayashi and M. Fukushima, Robust Nash equilibria in $N$-person noncooperative games: Uniqueness and reformulation, Pacific J. Optim., 5 (2009), 237-259. 
    [33] J. S. PangS. Sen and U. V. Shanbhag, Two-stage non-cooperative games with risk-averse players, Math. Program., 165 (2017), 235-290.  doi: 10.1007/s10107-017-1148-1.
    [34] G. Peng, T. Zhang and Q. Zhu, A data-driven distributionally robust game using Wasserstein distance, In Decision and Game Theory for Security, Springer, 2020,405-421.
    [35] S. Qu and M. Goh, Distributionally Robust Games with an Application to Supply Chain, Harbin Institute of Technology, Technical report, 2012.
    [36] H. Rahimian and S. Mehrotra, Frameworks and results in distributionally robust optimization, Open J. Math. Optim., 3 (2022), Art. No. 4, 85 pp. doi: 10.5802/ojmo.15.
    [37] D. Ralph and H. Xu, Convergence of stationary points of sample average two-stage stochastic programs: A generalized equation approach, Math. Oper. Res., 36 (2011), 568-592.  doi: 10.1287/moor.1110.0506.
    [38] U. Ravat and U. V. Shanbhag, On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games, SIAM J. Optim., 21 (2011), 1168-1199.  doi: 10.1137/100792644.
    [39] U. Ravat and U. V. Shanbhag, On the existence of solutions to stochastic quasi-variational inequality and complementarity problems, Math. Program., 165 (2017), 291-330.  doi: 10.1007/s10107-017-1179-7.
    [40] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.
    [41] R. T. Rockafellar and J. Sun, Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging, Math. Program., 174 (2019), 453-471.  doi: 10.1007/s10107-018-1251-y.
    [42] R. T. Rockafellar and R. J. Wets, Stochastic variational inequalities: Single-stage to multistage, Math. Program., 165 (2017), 331-360.  doi: 10.1007/s10107-016-0995-5.
    [43] R. T. Rockafellar and R. J. Wets, Scenarios and policy aggregation in optimization under uncertainty, Math. Oper. Res., 16 (1991), 119-147.  doi: 10.1287/moor.16.1.119.
    [44] J. B. Rosen, Existence and uniqueness of equilibrium points for concave $N$-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.
    [45] A. Shapiro and A. Nemirovski, On complexity of stochastic programming problems, Continuous optimization, Appl. Optim., Springer, Boston, MA, 99 (2005), 111-146. doi: 10.1007/0-387-26771-9_4.
    [46] H. D. SheraliA. L. Soyster and F. H. Murphy, Stackelberg-Nash-Cournot equilibria: Characterizations and computations, Oper. Res., 31 (1983), 253-276.  doi: 10.1287/opre.31.2.253.
    [47] V. V. Singh and A. Lisser, Variational inequality formulation for the games with random payoffs, J. Glob. Optim., 72 (2018), 743-760.  doi: 10.1007/s10898-018-0664-8.
    [48] H. Sun and X. Chen, Two-stage stochastic variational inequalities: Theory, algorithms and applications, J. Oper. Res. Soc. China, 9 (2021), 1-32.  doi: 10.1007/s40305-019-00267-8.
    [49] H. Sun, A. Shapiro and X. Chen, Distributionally robust stochastic variational inequalities, Math. Program., published online, 2022. doi: 10.1007/s10107-022-01889-2.
    [50] H. Sun and H. Xu, Convergence analysis for distributionally robust optimization and equilibrium problems, Math. Oper. Res., 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.
    [51] D. M. Topkis, Equilibrium points in nonzero-sum $n$-person submodular games, SIAM J. Control Optim., 17 (1979), 773-787.  doi: 10.1137/0317054.
    [52] W. WiesemannD. Kuhn and M. Sim, Distributionally robust convex optimization, Oper. Res., 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.
    [53] H. Xu, An MPCC approach for stochastic Stackelberg-Nash-Cournot equilibrium, Optimization, 54 (2005), 27-57.  doi: 10.1080/02331930412331323863.
    [54] D. ZhangH. Xu and Y. Wu, A two stage stochastic equilibrium model for electricity markets with two way contracts, Math. Methods Oper. Res., 71 (2010), 1-45.  doi: 10.1007/s00186-009-0283-8.
    [55] M. ZhangJ. Sun and H. Xu, Two-stage quadratic games under uncertainty and their solution by progressive hedging algorithms, SIAM J. Optim., 29 (2019), 1799-1818.  doi: 10.1137/17M1151067.
    [56] B. Zhou and H. Sun, Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty, Numer. Algebra, Control. Optim., 10 (2020), 521-535.  doi: 10.3934/naco.2020049.
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