Methods for finding leader--follower equilibria with multiple followers
This addresses a computationally challenging problem in game theory with applications in real-world scenarios, but it is incremental as it builds on existing work for single followers.
The paper tackles the problem of finding leader-follower equilibria with multiple followers, which is computationally hard, and proposes nonconvex mathematical programming formulations and global optimization methods to find exact and approximate equilibria, with thorough computational evaluation.
The concept of leader--follower (or Stackelberg) equilibrium plays a central role in a number of real--world applications of game theory. While the case with a single follower has been thoroughly investigated, results with multiple followers are only sporadic and the problem of designing and evaluating computationally tractable equilibrium-finding algorithms is still largely open. In this work, we focus on the fundamental case where multiple followers play a Nash equilibrium once the leader has committed to a strategy---as we illustrate, the corresponding equilibrium finding problem can be easily shown to be $\mathcal{FNP}$--hard and not in Poly--$\mathcal{APX}$ unless $\mathcal{P} = \mathcal{NP}$ and therefore it is one among the hardest problems to solve and approximate. We propose nonconvex mathematical programming formulations and global optimization methods to find both exact and approximate equilibria, as well as a heuristic black box algorithm. All the methods and formulations that we introduce are thoroughly evaluated computationally.