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

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Abstract

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.

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Acknowledgements

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.

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Correspondence to Jong-Shi Pang.

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The work was based on research partially supported by the U.S. National Science Foundation Grants CMMI-1402052 and 1538605.

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Hao, T., Pang, JS. Piecewise affine parameterized value-function based bilevel non-cooperative games. Math. Program. 180, 33–73 (2020). https://doi.org/10.1007/s10107-018-1344-7

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