Skip to main content
SpringerLink
Log in

Variational Inequality for n-Player Strategic Chance-Constrained Games

  • Original Research
  • Published:
SN Computer Science Aims and scope Submit manuscript

Abstract

In this paper, we focus on n-player strategic chance-constrained games where the payoff of each player follows either Cauchy or normal distributions. We prove the equivalence between a Nash equilibrium problem and a variational inequality problem. We reformulate the latter as a nonlinear complementarity problem (NCP) through the Karush–Kuhn–Tucker (KKT) conditions. Then, we prove the existence of the Nash equilibrium via Brouwer’s fixed-point theorem. To show the efficiency of our approach, we perform numerical experiments on a set of randomly generated instances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blackmore L, Li H, Williams B. A probabilistic approach to optimal robust path planning with obstacles. In: 2006 American control conference. IEEE; 2006. p. 7–14.

  2. Blau RA. Random-payoff two-person zero-sum games. Oper Res. 1974;22(6):1243–51.

    Article  MathSciNet  MATH  Google Scholar 

  3. Cassidy R, Field C, Kirby M. Solution of a satisficing model for random payoff games. Manag Sci. 1972;19(3):266–71.

    Article  MathSciNet  MATH  Google Scholar 

  4. Cavazzuti E, Pappalardo M, Passacantando M, et al. Nash equilibria, variational inequalities, and dynamical systems. J Optimiz Theory Appl. 2002;114(3):491–506.

    Article  MathSciNet  MATH  Google Scholar 

  5. Charnes A, Cooper WW. Deterministic equivalents for optimizing and satisficing under chance constraints. Oper Res. 1963;11(1):18–39.

    Article  MathSciNet  MATH  Google Scholar 

  6. Charnes A, Kirby MJ, Raike WM. Zero-zero chance-constrained games. Theory Probab Appl. 1968;13(4):628–46.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng J, Leung J, Lisser A. Random-payoff two-person zero-sum game with joint chance constraints. Eur J Oper Res. 2016;252(1):213–9.

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheng J, Lisser A. A second-order cone programming approach for linear programs with joint probabilistic constraints. Oper Res Lett. 2012;40(5):325–8.

    Article  MathSciNet  MATH  Google Scholar 

  9. Couchman P, Kouvaritakis B, Cannon M, Prashad F. Gaming strategy for electric power with random demand. IEEE Trans Power Syst. 2005;20(3):1283–92.

    Article  Google Scholar 

  10. De Wolf D, Smeers Y. A stochastic version of a stackelberg-nash-cournot equilibrium model. Manag Sci. 1997;43(2):190–7.

    Article  MATH  Google Scholar 

  11. DeMiguel V, Xu H. A stochastic multiple-leader stackelberg model: analysis, computation, and application. Oper Res. 2009;57(5):1220–35.

    Article  MathSciNet  MATH  Google Scholar 

  12. Dirkse SP, Ferris MC. The path solver: a nommonotone stabilization scheme for mixed complementarity problems. Optimiz Methods Softw. 1995;5(2):123–56.

    Article  Google Scholar 

  13. Facchinei F, Pang J-S. Finite-dimensional variational inequalities and complementarity problems. Berlin: Springer; 2003.

    MATH  Google Scholar 

  14. Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problems. Berlin: Springer Science & Business Media; 2007.

    MATH  Google Scholar 

  15. Jadamba B, Raciti F. Variational inequality approach to stochastic nash equilibrium problems with an application to cournot oligopoly. J Optimiz Theory Appl. 2015;165(3):1050–70.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. SIAM. 2000;

  17. Lee K-H, Baldick R. Solving three-player games by the matrix approach with application to an electric power market. IEEE Trans Power Syst. 2003;18(4):1573–80.

    Article  Google Scholar 

  18. Lepore JJ. Cournot outcomes under bertrand-edgeworth competition with demand uncertainty. J Math Econ. 2012;48(3):177–86.

    Article  MathSciNet  MATH  Google Scholar 

  19. Mazadi M, Rosehart WD, Zareipour H, Malik OP, Oloomi M. Impact of wind integration on electricity markets: a chance-constrained Nash Cournot model. Int Trans Electr Energy Syst. 2013;23(1):83–96.

    Article  Google Scholar 

  20. Nash J. Non-cooperative games. Ann Math 1951;286–95.

  21. Nash JF, et al. Equilibrium points in n-person games. Proc Natl Acad Sci. 1950;36(1):48–9.

    Article  MathSciNet  MATH  Google Scholar 

  22. v. Neumann J. Zur theorie der gesellschaftsspiele. Mathematische annalen. 1928;100(1):295–320.

    Article  MathSciNet  MATH  Google Scholar 

  23. Pang J-S, Fukushima M. Quasi-variational inequalities, generalized nash equilibria, and multi-leader-follower games. Comput Manag Sci. 2005;2(1):21–56.

    Article  MathSciNet  MATH  Google Scholar 

  24. Peng S, Singh VV, Lisser A. General sum games with joint chance constraints. Oper Res Lett. 2018;46(5):482–6.

    Article  MathSciNet  MATH  Google Scholar 

  25. Prékopa A. Stochastic programming, vol. 324. Berlin: Springer Science & Business Media; 2013.

    MATH  Google Scholar 

  26. Ravat U, Shanbhag UV. On the characterization of solution sets of smooth and nonsmooth convex stochastic nash games. SIAM J Optimiz. 2011;21(3):1168–99.

    Article  MathSciNet  MATH  Google Scholar 

  27. Singh VV, Jouini O, Lisser A. Existence of nash equilibrium for chance-constrained games. Oper Res Lett. 2016;44(5):640–4.

    Article  MathSciNet  MATH  Google Scholar 

  28. Singh VV, Jouini O, Lisser A. Solving chance-constrained games using complementarity problems. In: International conference on operations research and enterprise systems. Springer; 2016. p. 52–67.

  29. Singh VV, Jouini O, Lisser A. Distributionally robust chance-constrained games: existence and characterization of nash equilibrium. Optimiz Lett. 2017;11(7):1385–405.

    Article  MathSciNet  MATH  Google Scholar 

  30. Singh VV, Lisser A. Variational inequality formulation for the games with random payoffs. J Global Optimiz. 2018;72(4):743–60.

    Article  MathSciNet  MATH  Google Scholar 

  31. Singh VV, Lisser A. A second-order cone programming formulation for two player zero-sum games with chance constraints. Eur J Oper Res. 2019;275(3):839–45.

    Article  MathSciNet  MATH  Google Scholar 

  32. Son YS, Baldick R, Lee K-H, Siddiqi S. Short-term electricity market auction game analysis: uniform and pay-as-bid pricing. IEEE Trans Power Syst. 2004;19(4):1990–8.

    Article  Google Scholar 

  33. Song T. On random payoff matrix games. In: Systems and management science by extremal methods. Springer; 1992. p. 291–308.

  34. Valenzuela J, Mazumdar M. Cournot prices considering generator availability and demand uncertainty. IEEE Trans Power syst. 2007;22(1):116–25.

    Article  Google Scholar 

  35. Wambach A. Bertrand competition under cost uncertainty. Int J Ind Organiz. 1999;17(7):941–51.

    Article  Google Scholar 

  36. Xu H, Zhang D. Stochastic nash equilibrium problems: sample average approximation and applications. Comput Optimiz Appl. 2013;55(3):597–645.

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang S, Hadji M, Lisser A, Mezali Y. Nonlinear complementarity problems for n-player strategic chance-constrained games. In: International conference on operations research and enterprise systems (ICORES). 2022.

Download references

Funding

IRT-SystemX (Award number: ISX-364).

Author information

Authors and Affiliations

  1. Institut de Recherche Technologique SystemX, 8 Avenue de la Vauve, 91120, Palaiseau, France

    Shangyuan Zhang, Makhlouf Hadji & Yacine Mezali

  2. CentraleSupelec, L2S, Université Paris Saclay, 3 Rue Curie Joliot, 91190, Gif-sur-Yvette, France

    Shangyuan Zhang & Abdel Lisser

Authors
  1. Shangyuan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Makhlouf Hadji
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdel Lisser
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yacine Mezali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shangyuan Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the topical collection “Advances on Operations Research and Enterprise Systems” guest edited by Marc Demange, Federico Liberatore and Greg H. Parlier.

Appendix A

Appendix A

This appendix contains the randomly generated (\(3 \times 3 \times 3\)) games presented in “(\(3\times 3\times 3\)) Random Games with Cauchy Distribution” and “\((3\times 3\times 3)\) Random Games with Normal Distribution” sections. \(\mu _i\) and \(\sigma _i\) are the mean and variance of player i, respectively. As for each game, there are three players and each player has three actions to choose from, \(\mu _i\) and \(\sigma _i\) should be tensors of size (\(3 \times 3 \times 3\)), displayed as three (\(3 \times 3\)) matrices.

The following are three randomly generated games with Cauchy distribution in “(\(3\times 3\times 3\)) Random Games with Cauchy Distribution” section:

  1. 1.

    The first random game with Cauchy distribution:

    $$\begin{aligned} \mu _1(:,:,1)&= \begin{pmatrix} 1&{}1&{}2 \\ 2&{}3&{}1 \\ 1&{}3&{}1 \end{pmatrix} ,\ \mu _1(:,:,2) = \begin{pmatrix} 2&{}1&{}3 \\ 3&{}1&{}2 \\ 1&{}2&{}2 \end{pmatrix} ,\\ \mu _1(:,:,3)&= \begin{pmatrix} 2&{}1&{}2 \\ 1&{}3&{}2 \\ 3&{}1&{}3 \end{pmatrix} \\ \sigma _1(:,:,1)&= \begin{pmatrix} 1&{}2&{}1 \\ 3&{}2&{}2 \\ 3&{}2&{}2 \end{pmatrix} ,\ \sigma _1(:,:,2) = \begin{pmatrix} 3&{}2&{}2 \\ 3&{}3&{}3 \\ 2&{}2&{}3 \end{pmatrix} ,\\ \sigma _1(:,:,3)&= \begin{pmatrix} 2&{}1&{}1 \\ 2&{}1&{}3 \\ 2&{}2&{}1 \end{pmatrix} \\ \mu _2(:,:,1)&= \begin{pmatrix} 1&{}2&{}2 \\ 1&{}1&{}1 \\ 1&{}3&{}3 \end{pmatrix} ,\ \mu _2(:,:,2) = \begin{pmatrix} 3&{}1&{}1 \\ 2&{}2&{}2 \\ 1&{}2&{}3 \end{pmatrix} ,\\ \mu _2(:,:,3)&= \begin{pmatrix} 1&{}1&{}1 \\ 1&{}2&{}1 \\ 1&{}2&{}3 \end{pmatrix} \\ \sigma _2(:,:,1)&= \begin{pmatrix} 1&{}2&{}2 \\ 3&{}2&{}3 \\ 3&{}1&{}2 \end{pmatrix} ,\ \sigma _2(:,:,2) = \begin{pmatrix} 2&{}2&{}2 \\ 1&{}3&{}3 \\ 2&{}2&{}1 \end{pmatrix} ,\\ \sigma _2(:,:,3)&= \begin{pmatrix} 3&{}1&{}3 \\ 3&{}1&{}1 \\ 3&{}2&{}3 \end{pmatrix} \\ \mu _3(:,:,1)&= \begin{pmatrix} 1&{}3&{}3 \\ 2&{}3&{}2 \\ 2&{}3&{}3 \end{pmatrix} ,\ \mu _3(:,:,2) = \begin{pmatrix} 1&{}2&{}2 \\ 1&{}2&{}2 \\ 3&{}3&{}3 \end{pmatrix} ,\\ \mu _3(:,:,3)&= \begin{pmatrix} 3&{}1&{}2 \\ 2&{}3&{}1 \\ 1&{}1&{}3 \end{pmatrix} \\ \sigma _3(:,:,1)&= \begin{pmatrix} 3&{}1&{}1 \\ 2&{}3&{}2 \\ 2&{}1&{}1 \end{pmatrix} ,\ \sigma _3(:,:,2) = \begin{pmatrix} 3&{}1&{}3 \\ 3&{}2&{}2 \\ 3&{}2&{}3 \end{pmatrix} ,\\ \sigma _3(:,:,3)&= \begin{pmatrix} 2&{}1&{}2 \\ 2&{}1&{}3 \\ 3&{}1&{}3 \end{pmatrix} \end{aligned}$$
  2. 2.

    The second random game with Cauchy distribution:

    $$\begin{aligned} \mu _1(:,:,1)&= \begin{pmatrix} 1&{}1&{}1 \\ 1&{}1&{}2 \\ 1&{}2&{}3 \end{pmatrix} ,\ \mu _1(:,:,2) = \begin{pmatrix} 2&{}1&{}3 \\ 2&{}2&{}1 \\ 2&{}3&{}1 \end{pmatrix} ,\\ \mu _1(:,:,3)&= \begin{pmatrix} 2&{}1&{}1 \\ 2&{}3&{}1 \\ 2&{}1&{}1 \end{pmatrix} \\ \sigma _1(:,:,1)&= \begin{pmatrix} 2&{}3&{}1 \\ 2&{}3&{}3 \\ 1&{}3&{}3 \end{pmatrix} ,\ \sigma _1(:,:,2) = \begin{pmatrix} 3&{}2&{}1 \\ 3&{}1&{}1 \\ 3&{}2&{}3 \end{pmatrix} ,\\ \sigma _1(:,:,3)&= \begin{pmatrix} 1&{}2&{}3 \\ 1&{}2&{}2 \\ 1&{}1&{}3 \end{pmatrix} \\ \mu _2(:,:,1)&= \begin{pmatrix} 2&{}1&{}3 \\ 2&{}2&{}1 \\ 1&{}1&{}3 \end{pmatrix} ,\ \mu _2(:,:,2) = \begin{pmatrix} 2&{}1&{}3 \\ 3&{}2&{}3 \\ 3&{}2&{}1 \end{pmatrix} ,\\ \mu _2(:,:,3)&= \begin{pmatrix} 1&{}2&{}1 \\ 2&{}2&{}3 \\ 1&{}1&{}3 \end{pmatrix} \\ \sigma _2(:,:,1)&= \begin{pmatrix} 2&{}3&{}1 \\ 3&{}3&{}2 \\ 1&{}2&{}1 \end{pmatrix} ,\ \sigma _2(:,:,2) = \begin{pmatrix} 1&{}1&{}2 \\ 1&{}1&{}1 \\ 3&{}1&{}3 \end{pmatrix} ,\\ \sigma _2(:,:,3)&= \begin{pmatrix} 1&{}2&{}1 \\ 1&{}1&{}1 \\ 1&{}3&{}1 \end{pmatrix} \\ \mu _3(:,:,1)&= \begin{pmatrix} 1&{}3&{}1 \\ 1&{}2&{}3 \\ 2&{}1&{}3 \end{pmatrix} ,\ \mu _3(:,:,2) = \begin{pmatrix} 2&{}2&{}1 \\ 1&{}1&{}1 \\ 3&{}3&{}3 \end{pmatrix} ,\\ \mu _3(:,:,3)&= \begin{pmatrix} 2&{}3&{}3 \\ 2&{}3&{}2 \\ 3&{}1&{}2 \end{pmatrix} \\ \sigma _3(:,:,1)&= \begin{pmatrix} 1&{}2&{}1 \\ 3&{}3&{}2 \\ 2&{}1&{}2 \end{pmatrix} ,\ \sigma _3(:,:,2) = \begin{pmatrix} 3&{}1&{}3 \\ 3&{}1&{}3 \\ 3&{}1&{}3 \end{pmatrix} ,\\ \sigma _3(:,:,3)&= \begin{pmatrix} 1&{}1&{}2 \\ 3&{}2&{}3 \\ 1&{}1&{}2 \end{pmatrix} \end{aligned}$$
  3. 3.

    The third random game with Cauchy distribution:

    $$\begin{aligned} \mu _1(:,:,1)&= \begin{pmatrix} 1&{}2&{}3 \\ 1&{}2&{}2 \\ 2&{}2&{}2 \end{pmatrix} ,\ \mu _1(:,:,2) = \begin{pmatrix} 2&{}1&{}2 \\ 3&{}2&{}3 \\ 3&{}2&{}1 \end{pmatrix} ,\\ \mu _1(:,:,3)&= \begin{pmatrix} 1&{}2&{}2 \\ 1&{}2&{}2 \\ 3&{}2&{}2 \end{pmatrix} \\ \sigma _1(:,:,1)&= \begin{pmatrix} 3&{}1&{}1 \\ 2&{}1&{}2 \\ 3&{}1&{}2 \end{pmatrix} ,\ \sigma _1(:,:,2) = \begin{pmatrix} 2&{}3&{}1 \\ 3&{}3&{}1 \\ 2&{}3&{}3 \end{pmatrix} ,\\ \sigma _1(:,:,3)&= \begin{pmatrix} 1&{}3&{}3 \\ 2&{}2&{}3 \\ 2&{}2&{}2 \end{pmatrix} \\ \mu _2(:,:,1)&= \begin{pmatrix} 2&{}1&{}1 \\ 1&{}1&{}2 \\ 2&{}3&{}3 \end{pmatrix} ,\ \mu _2(:,:,2) = \begin{pmatrix} 2&{}3&{}1 \\ 3&{}3&{}1 \\ 2&{}2&{}2 \end{pmatrix} ,\\ \mu _2(:,:,3)&= \begin{pmatrix} 1&{}2&{}3 \\ 1&{}3&{}2 \\ 3&{}2&{}3 \end{pmatrix} \\ \sigma _2(:,:,1)&= \begin{pmatrix} 3&{}3&{}3 \\ 2&{}3&{}2 \\ 1&{}1&{}3 \end{pmatrix} ,\ \sigma _2(:,:,2) = \begin{pmatrix} 3&{}3&{}2 \\ 1&{}2&{}1 \\ 3&{}3&{}1 \end{pmatrix} ,\\ \sigma _2(:,:,3)&= \begin{pmatrix} 2&{}3&{}1 \\ 3&{}1&{}1 \\ 3&{}2&{}1 \end{pmatrix} \\ \mu _3(:,:,1)&= \begin{pmatrix} 3&{}2&{}3 \\ 2&{}1&{}2 \\ 1&{}1&{}3 \end{pmatrix} ,\ \mu _3(:,:,2) = \begin{pmatrix} 3&{}3&{}3 \\ 2&{}3&{}2 \\ 3&{}1&{}3 \end{pmatrix} ,\\ \mu _3(:,:,3)&= \begin{pmatrix} 2&{}1&{}2 \\ 2&{}2&{}3 \\ 3&{}3&{}3 \end{pmatrix} \\ \sigma _3(:,:,1)&= \begin{pmatrix} 2&{}1&{}3 \\ 1&{}2&{}3 \\ 2&{}2&{}3 \end{pmatrix} ,\ \sigma _3(:,:,2) = \begin{pmatrix} 3&{}2&{}3 \\ 2&{}1&{}2 \\ 3&{}1&{}3 \end{pmatrix} ,\\ \sigma _3(:,:,3)&= \begin{pmatrix} 1&{}1&{}1 \\ 2&{}2&{}2 \\ 2&{}2&{}1 \end{pmatrix} \end{aligned}$$

The following are three randomly generated games with normal distribution in 4.2:

  1. 1.

    The first random game with normal distribution:

    $$\begin{aligned} \mu _1(:,:,1)= & {} \begin{pmatrix} 3&{}1&{}2 \\ 1&{}1&{}2 \\ 3&{}2&{}3 \end{pmatrix} ,\ \mu _1(:,:,2) = \begin{pmatrix} 1&{}3&{}1 \\ 3&{}2&{}2 \\ 3&{}2&{}3 \end{pmatrix} ,\\ \mu _1(:,:,3)= & {} \begin{pmatrix} 1&{}3&{}2 \\ 1&{}3&{}3 \\ 1&{}3&{}3 \end{pmatrix} \\ \sigma _1^2(:,:,1)= & {} \begin{pmatrix} 9&{}1&{}1 \\ 4&{}9&{}1 \\ 4&{}1&{}4 \end{pmatrix} ,\ \sigma _1^2(:,:,2) = \begin{pmatrix} 4&{}1&{}1 \\ 9&{}9&{}1 \\ 9&{}1&{}1 \end{pmatrix} ,\\ \sigma _1^2(:,:,3)= & {} \begin{pmatrix} 4&{}4&{}4 \\ 4&{}4&{}4 \\ 1&{}9&{}1 \end{pmatrix} \\ \mu _2(:,:,1)= & {} \begin{pmatrix} 2&{}1&{}3 \\ 2&{}1&{}2 \\ 1&{}1&{}3 \end{pmatrix} ,\ \mu _2(:,:,2) = \begin{pmatrix} 2&{}2&{}3 \\ 2&{}3&{}1 \\ 1&{}2&{}2 \end{pmatrix} ,\\ \mu _2(:,:,3)= & {} \begin{pmatrix} 2&{}3&{}1 \\ 3&{}1&{}1 \\ 3&{}2&{}2 \end{pmatrix} \\ \sigma _2^2(:,:,1)= & {} \begin{pmatrix} 4&{}4&{}1 \\ 1&{}9&{}4 \\ 1&{}4&{}9 \end{pmatrix} ,\ \sigma _2^2(:,:,2) = \begin{pmatrix} 9&{}4&{}4 \\ 1&{}4&{}9 \\ 4&{}1&{}1 \end{pmatrix} ,\\ \sigma _2^2(:,:,3)= & {} \begin{pmatrix} 4&{}4&{}4 \\ 4&{}4&{}1 \\ 9&{}1&{}9 \end{pmatrix} \\ \mu _3(:,:,1)= & {} \begin{pmatrix} 3&{}3&{}2 \\ 1&{}1&{}1 \\ 3&{}3&{}2 \end{pmatrix} ,\ \mu _3(:,:,2) = \begin{pmatrix} 3&{}3&{}2 \\ 1&{}3&{}2 \\ 2&{}2&{}3 \end{pmatrix} ,\\ \mu _3(:,:,3)= & {} \begin{pmatrix} 3&{}3&{}2 \\ 2&{}3&{}1 \\ 2&{}3&{}1 \end{pmatrix} \\ \sigma _3^2(:,:,1)= & {} \begin{pmatrix} 1&{}9&{}9 \\ 9&{}9&{}1 \\ 9&{}4&{}1 \end{pmatrix} ,\ \sigma _3^2(:,:,2) = \begin{pmatrix} 4&{}9&{}1 \\ 1&{}4&{}1 \\ 1&{}9&{}9 \end{pmatrix} ,\\ \sigma _3^2(:,:,3)= & {} \begin{pmatrix} 9&{}1&{}9 \\ 1&{}4&{}9 \\ 4&{}4&{}9 \end{pmatrix} \\ \mu _1(:,:,1)= & {} \begin{pmatrix} 1&{}2&{}3 \\ 1&{}3&{}3 \\ 3&{}2&{}3 \end{pmatrix} ,\ \mu _1(:,:,2) = \begin{pmatrix} 3&{}3&{}1 \\ 1&{}3&{}2 \\ 1&{}1&{}3 \end{pmatrix} ,\\ \mu _1(:,:,3)= & {} \begin{pmatrix} 2&{}2&{}2 \\ 1&{}2&{}1 \\ 3&{}2&{}1 \end{pmatrix} \\ \sigma _1^2(:,:,1)= & {} \begin{pmatrix} 4&{}9&{}4 \\ 1&{}1&{}4 \\ 9&{}9&{}9 \end{pmatrix} ,\ \sigma _1^2(:,:,2) = \begin{pmatrix} 9&{}1&{}9 \\ 4&{}9&{}9 \\ 1&{}9&{}9 \end{pmatrix} ,\\ \sigma _1^2(:,:,3)= & {} \begin{pmatrix} 4&{}4&{}9 \\ 4&{}1&{}4 \\ 1&{}4&{}4 \end{pmatrix} \\ \mu _2(:,:,1)= & {} \begin{pmatrix} 2&{}1&{}2 \\ 3&{}3&{}1 \\ 1&{}3&{}1 \end{pmatrix} ,\ \mu _2(:,:,2) = \begin{pmatrix} 1&{}1&{}1 \\ 2&{}2&{}3 \\ 3&{}1&{}1 \end{pmatrix} ,\\ \mu _2(:,:,3)= & {} \begin{pmatrix} 1&{}2&{}1 \\ 2&{}1&{}2 \\ 1&{}3&{}1 \end{pmatrix} \\ \sigma _2^2(:,:,1)= & {} \begin{pmatrix} 4&{}9&{}4 \\ 9&{}4&{}4 \\ 4&{}4&{}9 \end{pmatrix} ,\ \sigma _2^2(:,:,2) = \begin{pmatrix} 4&{}4&{}9 \\ 1&{}4&{}9 \\ 9&{}9&{}1 \end{pmatrix} ,\\ \sigma _2^2(:,:,3)= & {} \begin{pmatrix} 4&{}9&{}4 \\ 4&{}1&{}1 \\ 9&{}9&{}4 \end{pmatrix} \\ \mu _3(:,:,1)= & {} \begin{pmatrix} 1&{}2&{}1 \\ 3&{}3&{}1 \\ 3&{}3&{}2 \end{pmatrix} ,\ \mu _3(:,:,2) = \begin{pmatrix} 3&{}3&{}3 \\ 3&{}3&{}1 \\ 1&{}1&{}2 \end{pmatrix} ,\\ \mu _3(:,:,3)= & {} \begin{pmatrix} 2&{}1&{}3 \\ 2&{}1&{}3 \\ 1&{}2&{}1 \end{pmatrix} \\ \sigma _3^2(:,:,1)= & {} \begin{pmatrix} 1&{}9&{}4 \\ 4&{}1&{}1 \\ 1&{}1&{}1 \end{pmatrix} ,\ \sigma _3^2(:,:,2) = \begin{pmatrix} 9&{}9&{}9 \\ 9&{}4&{}9 \\ 9&{}1&{}4 \end{pmatrix} ,\\ \sigma _3^2(:,:,3)= & {} \begin{pmatrix} 1&{}4&{}1 \\ 1&{}9&{}9 \\ 9&{}4&{}1 \end{pmatrix} \end{aligned}$$
  2. 2.

    The third random game with normal distribution:

    $$\begin{aligned} \mu _1(:,:,1)= & {} \begin{pmatrix} 2&{}3&{}1 \\ 2&{}3&{}3 \\ 2&{}1&{}2 \end{pmatrix} ,\ \mu _1(:,:,2) = \begin{pmatrix} 2&{}2&{}2 \\ 2&{}3&{}1 \\ 2&{}2&{}2 \end{pmatrix} ,\\ \mu _1(:,:,3)= & {} \begin{pmatrix} 2&{}1&{}1 \\ 1&{}1&{}2 \\ 3&{}3&{}1 \end{pmatrix} \\ \sigma _1^2(:,:,1)= & {} \begin{pmatrix} 1&{}4&{}1 \\ 9&{}1&{}1 \\ 1&{}1&{}4 \end{pmatrix} ,\ \sigma _1^2(:,:,2) = \begin{pmatrix} 4&{}9&{}4 \\ 4&{}4&{}1 \\ 9&{}4&{}4 \end{pmatrix} , \\ \sigma _1^2(:,:,3)= & {} \begin{pmatrix} 9&{}9&{}9 \\ 9&{}1&{}4 \\ 1&{}1&{}1 \end{pmatrix} \\ \mu _2(:,:,1)= & {} \begin{pmatrix} 3&{}2&{}2 \\ 3&{}3&{}1 \\ 3&{}1&{}2 \end{pmatrix} ,\ \mu _2(:,:,2) = \begin{pmatrix} 1&{}1&{}1 \\ 3&{}1&{}1 \\ 1&{}1&{}1 \end{pmatrix} ,\\ \mu _2(:,:,3)= & {} \begin{pmatrix} 1&{}2&{}1 \\ 1&{}1&{}3 \\ 1&{}2&{}1 \end{pmatrix} \\ \sigma _1^2(:,:,1)= & {} \begin{pmatrix} 1&{}4&{}1 \\ 9&{}1&{}1 \\ 1&{}1&{}4 \end{pmatrix} ,\ \sigma _1^2(:,:,2) = \begin{pmatrix} 4&{}9&{}4 \\ 4&{}4&{}1 \\ 9&{}4&{}4 \end{pmatrix} ,\\ \sigma _1^2(:,:,3)= & {} \begin{pmatrix} 9&{}9&{}9 \\ 9&{}1&{}4 \\ 1&{}1&{}1 \end{pmatrix} \\ \mu _3(:,:,1)= & {} \begin{pmatrix} 3&{}3&{}2 \\ 1&{}3&{}3 \\ 1&{}1&{}2 \end{pmatrix} ,\ \mu _3(:,:,2) = \begin{pmatrix} 2&{}3&{}3 \\ 3&{}3&{}1 \\ 2&{}1&{}3 \end{pmatrix} ,\\ \mu _3(:,:,3)= & {} \begin{pmatrix} 2&{}3&{}3 \\ 2&{}2&{}3 \\ 3&{}3&{}2 \end{pmatrix} \\ \sigma _3^2(:,:,1)= & {} \begin{pmatrix} 1&{}1&{}9 \\ 4&{}4&{}9 \\ 9&{}1&{}1 \end{pmatrix} ,\ \sigma _3^2(:,:,2) = \begin{pmatrix} 9&{}9&{}1 \\ 1&{}4&{}1 \\ 1&{}9&{}1 \end{pmatrix} ,\\ \sigma _3^2(:,:,3)= & {} \begin{pmatrix} 4&{}4&{}9 \\ 9&{}4&{}4 \\ 4&{}1&{}1 \end{pmatrix} \end{aligned}$$

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, S., Hadji, M., Lisser, A. et al. Variational Inequality for n-Player Strategic Chance-Constrained Games. SN COMPUT. SCI. 4, 82 (2023). https://doi.org/10.1007/s42979-022-01488-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42979-022-01488-0

Keywords

Search

Navigation