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.

中文翻译：

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基于分段仿射参数化价值函数的双层非合作博弈

本文概括了 Andrew Liu 和他的合作者传达给我们的某些网络拦截博弈，研究了一个双层非合作博弈，其中每个玩家优化问题的目标函数包含一个由一级参数化的二级线性程序的值函数 -非凸方式的水平变量。在应用的网络拦截游戏中，这种参数化是通过分段线性函数实现的，该函数为二级决策变量设置了上限。为了统一处理二级问题可能是最小化或最大化的整体二级博弈，我们将其表述为特定类型的一级博弈。即，每个玩家的目标函数是一级目标函数的总和 ± 二级最大化问题的值函数，其目标函数涉及凸差 (dc)，特别是分段仿射，由一级变量参数化. 这种非凸参数化是与 Facchinei 等人讨论的具有最小-最大目标的游戏系列的主要区别。(Comput Optim Appl 59 (1): 85-112, 2014)，其中保留了整体游戏的凸性。相比之下，要最大化的二级目标函数的分段仿射 (dc) 参数化使玩家组合的一级目标函数非凸且不可微。我们研究了这种博弈的一阶平稳解的存在，我们称之为准纳什均衡，