Piecewise affine parameterized value-function based bilevel non-cooperative games

  • Tianyu Hao
  • Jong-Shi PangEmail author
Full Length Paper Series A


Generalizing certain network interdiction games communicated to us by Andrew Liu and his collaborators, this paper studies a bilevel, non-cooperative game wherein the objective function of each player’s optimization problem contains a value function of a second-level linear program parameterized by the first-level variables in a non-convex manner. In the applied network interdiction games, this parameterization is through a piecewise linear function that upper bounds the second-level decision variable. In order to give a unified treatment to the overall two-level game where the second-level problems may be minimization or maximization, we formulate it as a one-level game of a particular kind. Namely, each player’s objective function is the sum of a first-level objective function ± a value function of a second-level maximization problem whose objective function involves a difference-of-convex (dc), specifically piecewise affine, parameterization by the first-level variables. This non-convex parameterization is a major difference from the family of games with min-max objectives discussed in Facchinei et al. (Comput Optim Appl 59(1):85–112, 2014) wherein the convexity of the overall games is preserved. In contrast, the piecewise affine (dc) parameterization of the second-level objective functions to be maximized renders the players’ combined first-level objective functions non-convex and non-differentiable. We investigate the existence of a first-order stationary solution of such a game, which we call a quasi-Nash equilibrium, and study the computation of such a solution in the linear-quadratic case by Lemke’s method using a linear complementarity formulation.


Noncooperative games Network interdiction Two-level games Linear complementarity Equilibrium solution 

Mathematics Subject Classification




The authors learned about the network interdiction games when Dr. Andrew Liu (Purdue University) was invited to give a seminar in the Daniel J. Epstein Department of Industrial and Systems Engineering at the University of Southern California in Fall 2015. Our research is a significant extension of these applied games and includes for instance a two-stage stochastic game with finite scenarios. The authors are grateful to two referees for their constructive comments that have helped improved the presentation of the paper.


  1. 1.
    Chen, Y., Hobbs, B.F., Leyffer, S., Munson, T.S.: Leader-follower equilibria for electric power and NOx allowances markets. Comput. Manag. Sci. 3(4), 307–330 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem, SIAM Classics in Applied Mathematics, vol. 60, Philadelphia (2009) [Originally published by Academic Press, Boston (1992)]Google Scholar
  3. 3.
    Ehrenmann, A.: Equilibrium problems with equilibrium constraints and their application to electricity markets. Ph.D. thesis, Fitzwilliam College (2004)Google Scholar
  4. 4.
    Facchinei, F., Pang, J.S.: Nash equilibria: the variational approach. In: Eldar, Y., Palomar, D. (eds.) Convex Optimization in Signal Processing and Communications, pp. 443–493. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  5. 5.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)zbMATHGoogle Scholar
  6. 6.
    Facchinei, F., Pang, J.S., Scutari, G.: Non-cooperative games with minmax objectives. Comput. Optim. Appl. 59(1), 85–112 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ferris, M.C., Wets, R.J.B.: MOPEC: multiple optimization problems with equilibrium constraints. (2013)
  8. 8.
    Leyffer, S., Munson, T.S.: Solving multi-leader-common-follower games. Optim. Methods Softw. 25(4), 601–623 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs With Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  10. 10.
    Nouiehed, M., Pang, J.S., Razaviyayn, M.: On the pervasiveness of difference-convexity in optimization and statistics. Mathem. Program. Ser. B (2017).
  11. 11.
    Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2(1), 21–56 (2005). with erratumMathSciNetCrossRefGoogle Scholar
  12. 12.
    Pang, J.S., Razaviyayn, M., Alvarado, A.: Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. (2016). MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pang, J.S., Scutari, G.: Nonconvex games with side constraints. SIAM J. Optim. 21(4), 1491–1522 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pang, J.S., Sen, S., Shanbhag, U.: Two-stage non-cooperative games with risk-averse players. Math. Prgram. Ser. B 165(1), 235–290 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Philpott, A.B., Ferris, M.C., Wets, R.J.B.: Equilibrium, uncertainty and risk in hydro-thermal electricity systems. Math. Program. B 157(2), 483–513 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sreekumaran, H.: Decentralized algorithms for Nash equilibrium problems-applications to multi-agent network interdiction games and beyond. Ph.D. thesis, Purdue University (September 2015)Google Scholar
  17. 17.
    Sreekumaran, H., Liu, A.: A note on the formulation of max-floow and min-cost-flow network interdiction games (September 2015)Google Scholar
  18. 18.
    Sreekumaran, H., Hota, A.R., Liu, A.L., Uhan, N.A., Sundaram, S.: Multi-agent decentralized network interdiction games (July 2015). arXiv:1503.01100v2
  19. 19.
    Su, C.L.: Equilibrium problems with equilibrium constraints: stationarities, algorithms, and applications. Ph.D. thesis, Department of Management Science and Engineering, Stanford University (2005)Google Scholar
  20. 20.
    van Stackelberg, H.: The Theory of Market Economy. Oxford University Press, Oxford (1952)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.The Daniel J. Epstein Department of Industrial and Systems EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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