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We design a computational approach to find equilibria in a class of Nash games possessing a hierarchical structure. By using tools from mixed-integer programming and the characterization of variational equilibria in terms of the Karush--Kuhn--Tucker conditions, we propose a mixed-integer game formulation for solving such a challenging instance. Besides providing an equivalent reformulation, we build upon our previous work and design a proximal Gauss--Seidel method with global convergence guarantees for the case in which the game enjoys a potential structure. In addition to proving the convergence of the resulting scheme, we show its performance on a numerical instance of a ride-hail market problem.
Fabiani et al. (Wed,) studied this question.