Abstract
In this note, we investigate stochastic Nash equilibrium problems by means of monotone variational inequalities in probabilistic Lebesgue spaces. We apply our approach to a class of oligopolistic market equilibrium problems, where the data are known through their probability distributions.
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References
Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)
Gabay, D., Moulin, H.: On the uniqueness and stability of Nash Equilibria in non cooperative games. In: Bensoussan, A., Kleindorfer, P., Tapiero, C.S. (eds.) Applied Stochastic Control in Econometrics and Management Sciences, pp. 271–294. North Holland, Amsterdam (1980)
Barbagallo, A., Maugeri, A.: Duality theory for the dynamic oligopolistic market equilibrium problem. Optimization 60(1–2), 29–52 (2011)
Barbagallo, A., Di Vincenzo, R.: Lipschitz continuity and duality for dynamic oligopolistic market equilibrium problem with memory term. J. Math. Anal. Appl. 382, 231–247 (2011)
Barbagallo, A., Mauro, P.: Evolutionary variational formulation for oligopolistic market equilibrium problems with production excesses. J. Math. Anal. Appl. 155, 288–314 (2012)
Gwinner, J., Raciti, F.: On a class of random variational inequalities on random sets. Numer. Funct. Anal. Optim. 27(5–6), 619–636 (2006)
Gwinner, J., Raciti, F.: Some equilibrium problems under uncertainty and random variational inequalities. Ann. Oper. Res. 200, 299–319 (2012). doi:10.1007/s10479-012-1109-2
Gwinner, J., Raciti, F.: Random equilibrium problems on networks. Math. Comput. Model. 43, 880–891 (2006)
Gwinner, J., Raciti, F.: On monotone variational inequalities with random data. J. Math. Inequal. 3(3), 443–453 (2009)
Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)
Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Progr. B 117, 51–80 (2009)
De Miguel, V., Xu, H.: A stochastic multiple-leader Stackelberg model: analysis, computation, and application. Oper. Res. 57, 1220–1235 (2009)
Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Progr. 84, 313–333 (1999)
Lu, S., Budhiraja, A.: Confidence regions for stochastic variational inequalities. Math. Oper. Res. 38 (3) (2013). doi:10.1287/moor.1120.0579
Patriche, M.: Equilibrium of Bayesian fuzzy economies and quasi-variational inequalities with random fuzzy mappings. J. Inequal. Appl. 2013, 374 (2013). doi:10.1186/1029-242X-2013-374
Patriksson, M.: On the applicability and solution of bilevel optimization models in transportation science: a study on the existence, stability and computation of optimal solutions to stochastic mathematical programs with equilibrium constraints. Trans. Res. B 42, 843–860 (2008)
Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation. Optimization 57, 395–418 (2008)
Ravat, U., Shanbhag, U. V.: On the characterization of solution sets of smooth and nonsmooth stochastic Nash games. Proceedings of the American Control Conference (ACC), Baltimore (2010)
Ravat, U., Shanbhag, U.V.: On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games. SIAM J. Optim. 21(3), 1168–1199 (2011)
Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia-Pac. J. Oper. Res. 27, 103–119 (2010)
Ravat, U., Shanbhag, U. V.: On the existence of solutions to stochastic variational inequality and complementarity problems. arXiv:1306.0586v1 [math.OC] (2013)
Jadamba, B., Khan, A.A., Raciti, F.: Regularization of stochastic variational inequalities and a comparison of an \(L_p\) and a sample-path approach. Nonlinear Anal. A 94, 65–83 (2014). doi: 10.1016/j.na.2013.08.009
Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)
Cournot, A.A.: Researches into the Mathematical Principles of the Theory of Wealth. 1838, English Translation. MacMillan, London (1897)
Chen, Y., Hobbs, B.F., Leyffer, S., Munson, T.S.: Leader-follower equilibria for electric power and NO\(_x\) allowences markets. Comput. Manag. Sci. 3(4), 307–330 (2006)
Chen, Y., Sijm, J., Hobbs, B.F., Lise, W.: Implications of CO\(_2\) emissions trading for short-run electricity market outcomes in northwest Europe. J. Regul. Econ. 34, 23–44 (2008)
Bonenti, F., Oggioni, G., Allevi, E., Marangoni, G.: Evaluating the EU ETS impacts on profits, investments and prices of the Italian electricity market. Energy Policy 59, 242–256 (2013)
Maugeri, A., Raciti, F.: On existence theorems for monotone and nonmonotone variational inequalities. J. Convex Anal. 16(3&4), 899–911 (2009)
Murphy, F.H., Sheraly, H., Soyster, A.L.: A mathematical programming approach for determining oligopolistic market equilibrium. Math. Progr. 24, 92–106 (1982)
Nagurney, A.: Network Economics: A Variational Inequality Approach, Second and Revised Edition. Kluwer Academic Publishers, Dordrecht (1999)
Dentcheva, D., Ruszczyński, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14, 548–566 (2003)
Jadamba, B., Raciti, F.: On the modelling of some environmental games with uncertain data. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0389-2
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Communicated by Roland Glowinski.
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Jadamba, B., Raciti, F. Variational Inequality Approach to Stochastic Nash Equilibrium Problems with an Application to Cournot Oligopoly. J Optim Theory Appl 165, 1050–1070 (2015). https://doi.org/10.1007/s10957-014-0673-9
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DOI: https://doi.org/10.1007/s10957-014-0673-9