Piecewise affine parameterized value-function based bilevel non-cooperative games
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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.
KeywordsNoncooperative games Network interdiction Two-level games Linear complementarity Equilibrium solution
Mathematics Subject Classification90C33
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.
- 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.Ehrenmann, A.: Equilibrium problems with equilibrium constraints and their application to electricity markets. Ph.D. thesis, Fitzwilliam College (2004)Google Scholar
- 7.Ferris, M.C., Wets, R.J.B.: MOPEC: multiple optimization problems with equilibrium constraints. http://www.cs.wisc.edu/~ferris/talks/chicago-mar.pdf (2013)
- 10.Nouiehed, M., Pang, J.S., Razaviyayn, M.: On the pervasiveness of difference-convexity in optimization and statistics. Mathem. Program. Ser. B (2017). https://doi.org/10.1007/s10107-018-1286-0
- 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.Sreekumaran, H., Liu, A.: A note on the formulation of max-floow and min-cost-flow network interdiction games (September 2015)Google Scholar
- 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.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.van Stackelberg, H.: The Theory of Market Economy. Oxford University Press, Oxford (1952)Google Scholar